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TUTORIALS in

CONTEMPORARY

NONLINEAR METHODS

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EDITED BY MICHAEL A. RILEY & GUY C. VAN ORDEN
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A DIGITAL PUBLICATION AVAILABLE AT:

WWW.NSF.GOV/SBE/BCS/PAC/NMBS/NMBS.JSP

EDITORS

MICHAEL A. RILEY

UNIVERSITY OF CINCINNATI

DEPARTMENT OF PSYCHOLOGY

&

GUY C. VAN ORDEN

ARIZONA STATE UNIVERSITY

DEPARTMENT OF PSYCHOLOGY

&

NATIONAL SCIENCE FOUNDATION

PROGRAM IN PERCEPTION, ACTION, & COGNITION

The volume is intended to be freely distributed and reproduced for

educational purposes. The volume may be posted, in its entirety, at

world wide web or FTP locations other than the one listed above,

provided that the Editors are notified in advance. Each chapter in this

volume © 2005 the respective author(s).
HOW TO CITE THIS WORK

Cite chapters from this digital volume just as you would cite

chapters from a printed volume, with the exception that the typical

information about the publisher is substituted with “Retrieved [date]

from [URL of web site from which you downloaded the book].” For

example,

Tuller, B. (2005). Categorization and learning in speech perception as

dynamical processes. In M. A. Riley & G. C. Van Orden (Eds.), Tutorials in

contemporary nonlinear methods for the behavioral sciences (pp. 353-400).

Retrieved March 1, 2005, from http://www.nsf.gov/sbe/bcs/pac/nmbs/nmbs.jsp

Cite the entire book in a similar way:

Riley, M. A., & Van Orden, G. C. (2005). Tutorials in contemporary nonlinear

methods for the behavioral sciences. Retrieved March 1, 2005, from

http://www.nsf.gov/sbe/bcs/pac/nmbs/nmbs.jsp
TABLE OF CONTENTS

Preface M. A. Riley & G. C. Van Orden iv

Chapter 1. Why nonlinear methods? C. Carello & M. Moreno 1

Chapter 2. Recurrence quantification analysis of

nonlinear dynamical systems

C. L. Webber, Jr., & J. P. Zbilut 26

Chapter 3. Application of recurrence quantification

analysis: Influence of cognitive activity on postural

fluctuations

G. L. Pellecchia & K. Shockley 95

Chapter 4. Cross recurrence quantification of

interpersonal postural activity

K. Shockley 142

Chapter 5. Introduction to fractals L. S. Liebovitch &. L. A. Shehadeh 178

Chapter 6. Gauging the fractal dimension of

cognitive performance

J. G. Holden 267

Chapter 7. 1/f dynamic in complex visual search:

Evidence for self-organized criticality in human

perception

D. J. Aks 319

Chapter 8. Categorization and learning in speech

perception as dynamical processes

B. Tuller 353
PREFACE

Chapters 2-8 in this volume were contributed by speakers at an

October, 2003, workshop, “Nonlinear Methods in Psychology,” held at

George Mason University in Fairfax, VA, and sponsored by the National

Science Foundation. The workshop served two specific purposes. The

first was to provide the attendees (mostly graduate students) with

tutorial lectures describing concepts and methods from nonlinear

dynamics that have already shed light on problems in the behavioral

sciences. The emphasis was on recurrence quantification analysis

(Chapters 2, 3, 4) and fractals (Chapters 5, 6, 7), but the workshop

included other, more established topics, such as applications of

nonlinear dynamics to speech perception (Chapter 8), social group

coordination, and human development. In addition to the authors of the

chapters, we thank Stephen Guastello and Paul van Geert for their

presentations at the workshop. The second purpose of the workshop

was to provide the present authors with critical feedback. The chapters

are intended to present these topics to a target audience of graduate

students and other researchers who might lack specialized

backgrounds often required to grasp the concepts and use the methods

with real data. The students who attended the workshop also served as

ad hoc reviewers of the chapters in this volume. They were instructed

specifically to evaluate the chapters in terms of accessibility and

usefulness for beginners. The reviewers and authors were thus

charged with very difficult tasks that they enthusiastically embraced.

By design, this volume does not represent the full breadth of

nonlinear methods that have been or are currently being used by

researchers in the behavioral sciences. Instead, we focus on a narrow
subset of contemporary methods that are less well represented in other

methods texts. The narrow focus allowed us to present formal, yet

accessible, descriptions of methods and concepts along with detailed

applications in the behavioral sciences. We hope to provide the reader

with an in-depth, tutorial-style treatment—a detailed guide for the

particular methods and a launching point for further studies. Thanks to

the very hard work of the authors and the reviewers, we believe this

volume achieves these goals.

This volume is somewhat unique also in the publication format

and venue. Our aim was to balance availability and access with the

utmost standards of academic rigor. We chose to publish the volume

online (initially on the National Science Foundation’s web site, linked

from the web page of the Program in Perception, Action, and

Cognition) so that it may be widely available and free of charge. By

making the volume freely and easily available we may ensure that the

methods and concepts described herein have a maximal impact. We

chose a traditional format for the text (rather than a web-page format

with hyperlinks) to maintain the feel of a typical, edited academic book.

However, the electronic format also allowed us to post supplements to

several of the chapters. Liebovitch and Shehadeh (Chapter 5)

requested that their PowerPoint file be made available along with the

chapter, so we have posted that file in addition to the PDF-formatted

chapter. Pellecchia and Shockley (Chapter 3) made available the data

set that they discuss in detail in their chapter. Aks (Chapter 7) plans to

release a Flash demo of her eye movement model that we will also

make available with this volume. We also provide links to pages from

which free data analysis software can be obtained.
We are grateful to the authors and reviewers for their

contributions to this project. We also acknowledge the support of the

National Science Foundation, Social, Behavioral, & Economic Sciences

Directorate, Division of Behavioral & Cognitive Sciences. Philip Rubin

and Peg Barrat, Directors of the Division of Behavioral & Cognitive

Sciences, supported this project from its inception to its completion.

We especially thank Christy Contreras and Philip Johnson at NSF and

Chris Kello, Debbie Kranz, and Daragh Sibley at George Mason

University. Chris Kello arranged that we could meet at George Mason.

Christy, Debbie, and Daragh handled the logistics of the workshop.

Philip Johnson set the book up on the NSF website.

The views expressed in this volume are the authors’ and do not

necessarily represent those of the National Science Foundation.

Michael A. Riley Guy C. Van Orden

University of Cincinnati National Science Foundation

Cincinnati, OH Arlington, VA

michael.riley@uc.edu

Arizona State University

Tempe, AZ

guy.van.orden@asu.edu
CHAPTER 1

WHY NONLINEAR METHODS?

Claudia Carello

Center for the Ecological Study of Perception and Action

University of Connecticut

406 Babbidge Road

Storrs, CT 06269-1020

U. S. A.

E-mail: claudia.carello@uconn.edu

Miguel A. Moreno

Haskins Laboratories

300 George Street

New Haven, CT 06511-6695

U. S. A.

E-mail: miguel.moreno@uconn.edu
Carello and Moreno

2

As behavioral scientists, we’re in the business of understanding

pieces of behavior. Everyone has his or her favorite types of behaviors,

oftentimes things that simply derive from personal interest—we are

fascinated by language or sport or animals and somehow contrive to

make those into experimental topics. Apart from our idiosyncratic

preferences we also bring our intellectual preferences, our

assumptions about the kinds of explanations we expect to work. And

although our theoretical positions are reasonably explicit, we also have

metatheoretical positions that are typically somewhat hidden.

Nonetheless, they sit implicitly behind what we do. Whether we are

connectionists or computationalists or direct realists, the inherent

philosophies of those positions dictate the kinds of problems we study

and the kinds of variables by which we choose to define them. Our

metatheories tell us what we think ought to be important.

But that’s not the end of how we frame problems. In designing

our studies, we still have a number of choices to make. Some of those

choices are dictated by the requirements of the analyses that we’ll

use—repeated measures or factorial designs, randomized or blocked

trials. Unlike our theoretical and metatheoretical positions, we tend to

think of analyses as objective and benign with respect to intellectual

assumptions. To be sure, all analyses assume criterial characteristics of

the data that render the analysis in question legitimate. But we tend to

think of those assumptions as mathematical. An important lesson of the

chapters in this volume, however, is that our statistical analyses buy

into intellectual assumptions as well. As you’ll see, what we analyze and

how we analyze it entails assumptions about the kinds of things that

exist and assumptions about how those things can fit together. The
Why nonlinear methods?

3

chapters show us how we might begin to change the way we

understand the very nature of those pieces of behavior that interest us.

At the very least, these chapters show that analyses that acknowledge

the dynamical nature of certain behaviors reveal a good deal of rich

structure that cannot be extracted with more familiar analyses.

Consider Figure 1.1, which illustrates a few of the many different

kinds of patterns of data that behavioral scientists encounter. Panel A

shows distributions of the kind that we assume are typical of our

experiments. Some manipulation increased the likelihood of larger

responses, although, in this case, with greater variability among the

responses. Consequently, the means of the two distributions are

numerically different but the variances are such that there is

considerable overlap between the two distributions. Conventional

analyses allow us to assess the extent to which the variability seems to

be systematic (i.e., due to the manipulation) or random (e.g., due to the

vagaries of individual differences among people) in order to determine

whether those means are different enough to be reliable.

The remaining panels show data of the kind considered in these

chapters. Whether they fit this conventional characterization is an issue.

Panel B, for example, shows two distributions that appear to be of the

same general sort as panel A. Distribution 2 is a little more variable

than Distribution 1, but in this case their peaks are in the same location.

A closer look, however, suggests a subtle difference. The mean of

Distribution 2 is larger than the mean of Distribution 1, and by the same

amount as in panel A. But this time the increase is not due to a

straightforward, overall addition. There appears to be a stretching of

the high end of the distribution so that more large values get included
Carello and Moreno

4

Figure 1.1. (A) Two normal distributions with different means and variances. (B) A normal

distribution (solid line) and a distribution with a stretched “tail” (dashed line). (C) A time

series. (D) Two time series with identical summary means. (E) Two time-ordered velocity ×

position profiles. (F) Categorical responses with different orders of presentation (indicated by

the arrows).

in the calculation of the mean (Moreno, 2001). Is it appropriate to say

that the means of these two distributions differ? The central tendency is

at the same value. Eliminating values of the dependent variable that are
Why nonlinear methods?

5

larger than a certain cut-off might eliminate this tail but would that be

an accurate depiction of the consequences of this particular

experimental manipulation? Panels C-F show data for which the mean

may be an even less appropriate measure. They are all time-series of

one kind or another, depictions of individual responses being tracked

over time or, at least, over order of presentation. In panel C you can

see that the data are very noisy trial by trial. But there also appears to

be some kind of large-scale wavy pattern overlaid on this noise.

Summary statistics such as the mean and variance would be hardpressed to capture this (see Chapter 6 by Holden). Panel D plots two

time series together. The average position of the two series can be said

to be the same if we simply add up the values and divide by the

number of observations. But it is quite apparent that this single value is

not an appropriate characterization of either time series or of the

differences between them. Indeed, there is not really a single mean for

either series; the mean of each changes over time, making it “illegal” to

conduct conventional analyses. Panel E suggests that tracking the

coincident changes in two variables might be informative. There is

more to these data than a correlation could reveal. Whether small

values of X1 go along with large or small values of X2 depends on when

the observations occurred in the series. Finally, in panel F, two parallel

functions are displaced from one another, not as a function of the X

variable but as a function of whether that variable was encountered in

an ascending series or a descending series (indicated by the arrows).

There might be a temptation to average over the two presentation

orders so as to identify “the” transition point, or to have ignored order

altogether in a randomized presentation (but see Chapter 8 by Tuller).
Carello and Moreno

6

In this opening chapter, we have two goals. First, we’ll take one

measure that is common in cognitive psychology research and use it to

illustrate the kinds of intellectual assumptions that standard data

analyses embrace. Second, we’ll provide an overview of the issues that

are treated in detail in the individual chapters.

A BRIEF HISTORY OF DECOMPOSING PERFORMANCE INTO

COMPONENTS

Reaction (or response) time is the workhorse for exploring the

nature of cognitive systems. Traditional approaches have tried to

understand responses as the sum of component effects. While such

approaches allow that the intrinsic dynamics of components may be

complex, they severely restrict the kinds of interactions that can occur

between components. In particular, it is common to assume that

interactions between components must be linear. Traditional

approaches have gambled that the effect of each cognitive component

combines additively with the effects of other components, which

together define the shape of response time distributions. This brief

history tracks the payoff, so far, of this gamble.

Linear interactions mean that the effect of an unobservable

component can be recovered in an overall measure like response time

because each component effect spans a sub-interval of response time.

The overall finishing time of the same component doing the same job

will vary from occasion to occasion, however. Thus, the overall time

course of all components would appear, to an experimenter, as a

distribution of finishing times (like one of those shown in Figure 1.1A).
Why nonlinear methods?

7

The enterprise of decomposing response time performance into

component processes is an old one. In 1868, Donders proposed a

subtractive method for identifying stages of information processing. The

subtractive method was based on the idea that a stage could be

inserted into (or deleted from) a sequence of stages. Donders

hypothesized that a new stage would be added to accompany specific

modifications to an experimental task. Comparing response times from

two tasks could estimate the duration of the added stage.

The subtractive method was the preferred procedure for

revealing mental stages for decades (Wundt, 1874; Cattell, 1886;

Jastrow, 1890). It fell out of favor for several reasons. One criticism was

that task modifications are more likely to alter the entire sequence of

stages than to insert or delete individual stages (Külpe, 1895). Devising

an experimental manipulation that unambiguously introduced a new

processing stage proved to be the downfall of the subtractive method.

A more contemporary effort to identify component processes

adopted Donders’ assumption of additive finishing times plus

assumptions geared to the asymmetrical shape of response time

distributions. Empirical response time distributions typically have a

hyperbolic shape with an elongated, slow tail, much like the dashedline distribution in Figure 1.1B. The slow tails of response time

distributions resemble exponential distributions (Christie & Luce, 1956;

McGill, 1963) and the fast tails resemble the left half of Gaussian

distributions. Christie and Luce hypothesized that empirical response

time distributions are the intertwining of an exponential distribution

and a base distribution of an unspecified form.
Carello and Moreno

8

Hohle (1965) suggested that the form of the base distribution was

Gaussian and indeed the convolution of an exponential and a Gaussian

distribution can approximate very closely the shape of empirical

response time distributions (Luce, 1986). Based on this idea, response

time distributions are the sum of numerous component distributions

with similar variances plus an exponentially distributed component

with a much greater variance.

Hohle’s assumptions include Donder’s core assumption of

additivity, that the interactions between components are linear.

Consequently the shape of response time distributions should reduce

to three parameters. Two parameters, μ and Σ, summarize the shape of

an underlying Gaussian distribution. μ describes the location of the

Gaussian distribution along the time axis and Σ describes the extent of

the distribution’s spread. A single parameter, τ, summarizes the

location and spread of the exponential distribution.

Different component processes can be inferred if the parameter

estimates systematically dissociate across experimental manipulations.

Some manipulated factors should selectively influence one distribution

(e.g., the exponential distribution) without affecting the other (e.g., the

Gaussian distribution). This strategy for identifying component

processes avoids one of the pitfalls of Donders’ subtractive method, the

requirement that two different tasks add (or delete) a stage of

processing. Hohle’s method requires that different conditions of the

same task influence the exponential and Gaussian parameters

independently.

As part of a model of response time performance, Hohle

assumed that the exponential distribution is the effect of a response-
Why nonlinear methods?

9

choice process and the Gaussian distribution is the sum of all other

processes (see also, Christie & Luce, 1956). The mapping of parameters

to component processes was merely intuitive, however. For example,

McGill (1963) had opposite intuitions. He assigned response-choice

and other decision processes to the Gaussian sum of processes and

suggested that the exponential distribution represents motor

processes.

Unfortunately these ex-Gaussian strategies (combining

exponential and Gaussian distributions) fared no better than Donder’s

subtractive method (Sternberg, 1969). Different factor manipulations

did not systematically discriminate among different component

parameters across experiments (Hohle, 1967; Gholson & Hohle, 1968a,

1968b). This outcome presents a problem because the ex-Gaussian

hypothesis combines so many assumptions. When results are

inconclusive, it is difficult, or impossible, to decide which assumption is

false. Another problem, pointed out by Sternberg, is that combinations

of many other distributions also approximate response time

distributions (see also Van Zandt & Ratcliff, 1995).

Sternberg (1969) realized that the core assumption that cognitive

systems are composed of successive stages could be isolated from

supplemental assumptions such as specifying the form of component

distributions. Sternberg stripped the assumptions regarding the nature

of cognitive systems down to their core and asked, “How do component

processes interact?”—the recurring question in this brief history.

Sternberg proposed that if the component processes interact linearly

then there must exist some factors that when manipulated will

selectively influence different component distributions. If components
Carello and Moreno

10

interact linearly, then component distributions selectively influenced

by separate factors will combine additively.

Sternberg’s strategy of testing for linear interactions requires

experimental manipulations of two or more factors. If the influence of

one factor on overall performance is completely independent of the

influence of another factor—a statistically additive interaction—then the

two experimental factors relate to two different component processes.

Alternatively, if the influence of one factor is modulated by another

factor—a non-additive interaction—then both factors influence at least

one common component process.

Assessments of additive interactions between component

processes require estimates of component distributions that combine

additively. Appropriate estimates of component finishing times,

according to Sternberg (1969, p. 286), are arithmetic means. The mean

of a sum of component distributions is the sum of component

distribution means. Response time means, therefore, can be treated as

the sum of component means.

Unfortunately, Sternberg’s additive factors method has yet to

identify any component process unequivocally. Additive interactions

are the exception in cognitive studies. This situation could imply that

the right set of factors has yet to be identified. The right set of factors

could provide the necessary context to discover fundamental additive

interactions. However, there is no guarantee that such a set of factors

exists. Moreover, it may not be feasible to prove that such a set of

factors does not exist. Sternberg provided an elegant and scientifically

conservative test of the traditional assumption regarding the nature of
Why nonlinear methods?

11

cognitive systems. Unfortunately, the results have yet to answer

Sternberg’s question, “how do components interact?”

Following Sternberg’s lead, in some sense, the chapters in this

volume focus on the question, “how do components interact?”

Nonlinear dynamical systems provide another way to explore this

question. Nonlinear dynamical systems do not exclude the possibility of

linear interactions; linear interactions are special circumstances within

the range of possibilities of a dynamical system. Thus, modeling

response times, among other things, as a dynamical system is a very

general and conservative approach. There are fewer a priori

assumptions regarding the components of the system. Furthermore,

there are fewer restrictions on how components may interact.

THE CHAPTERS

The workshop focused on two types of analyses—recurrence

quantification and fractals—that seem particularly fruitful for behavioral

research. The general premises of these techniques are summarized in

the chapters by Webber and Zbilut (“Recurrence Quantification

Analysis of Nonlinear Dynamical Systems”) and by Liebovitch and

Shehadeh (“Introduction to Fractals”) and each is followed by

particular experimental implementations. Also included is an

illustration of what can be gained by treating an established

phenomenon dynamically from the start.

Recurrence Quantification Analysis

Much of the behavior of living systems is complex and seemingly

non-predictable. Nonetheless, aspects of this behavior can be counted

on to repeat. The bits that repeat may do so over long stretches,
Carello and Moreno

12

perhaps producing a pattern, or the recurrences can be quite shortlived. Consider an activity like a square dance. Much of a dancer’s time

is spent synchronized with the group in a large and obvious pattern,

say, concentric circles alternately moving clockwise and

counterclockwise. Only occasionally and briefly does one dancer get

back together with his or her original partner. Both levels of

recurrence—the circular patterns of the group and the momentary

contact between partners—can be quantified and tell us something

additional about the activity.

The levels of recurrence in the “RQA Dance”1 as executed by

other kinds of particles may not be as obvious, particularly the rare

recurrences, but they are just as informative. And, it seems, the more

complex the behavior the rarer and less obvious the recurrences, and

the greater the need for ways to discover them. As noted by Webber

and Zbilut, “the degree to which those systems exhibit recurrent

patterns speaks volumes regarding their underlying dynamics.” Even if

we don’t have a recurrent behavior as obvious as dancing partners

holding hands, a system’s underlying dynamics are accessible. Picking

up on the theme that everything is connected to everything else,

Takens (1981) introduced a theorem allowing a behavior space to be

reconstructed from any measured variable. To be sure, a complex

system is ultimately characterized by a number of participating

variables. But these variables are necessarily coupled to one another

and, therefore, each reflects the behavior of the system. In keeping

 

1 The metaphor of RQA as describing a dance of particles was first illustrated by the duo of M.

T. Turvey and Nobuhiro Furyama during a typically staid Turvey lecture at the University of

Tokyo in May, 2004.
Why nonlinear methods?

13

with our intuitions that behavior evolves over time, Takens’

reconstruction is accomplished through time-delayed copies of some

nominated variable. That is, some variable x is chosen as a preliminary

index of the system’s behavior and we track what happens to x over

time. But we also want to know how x behaves relative to itself at later

points in time, say, t plus a delay of δ or t plus a delay of 2δ So the

original variable x becomes a dimension of the system in question and

each time-delayed copy becomes another dimension of the system.

Trajectories are traced through this multi-dimensional space and

recurrences are measured: Do the trajectories come together at a

point, do they travel together for a sequence of points, and so on? Each

of these becomes an objective indication of some aspect of the system’s

dynamics. An advantage here is that the analysis allows you to

characterize the dynamics of the system from the measurement of any

variable, not necessarily a variable that seems like it ought to be right

(what standard analyses refer to as face validity).

This technique is illustrated in Shockley’s chapter “Cross

Recurrence Quantification of Interpersonal Postural Activity.” He

exploits RQA in a line of research aimed at quantifying the

synchronization between two people who are engaged in a

conversation. You can appreciate the challenge this behavior poses.

What do you measure? The history of interpersonal synchrony is to

treat it as a phenomenon of social coordination and to look for overt

signs of that coordination. This means that the problem has been

addressed fairly subjectively. For example, researchers might examine

videotapes and look for signs of synchrony (e.g., similar gestures by a

talker and a listener). As rigorously objective as researchers try to be,
Carello and Moreno

14

they must still interject themselves into the process of identifying an

occasion of synchrony. Shockley and his colleagues have instead

chosen a behavior—the postural sway of the participants—that is not an

overt part of the act of conversing and used RQA to sift through the

trajectories and extract the recurrent patterns. This is unlikely to be a

behavior that people are controlling consciously. It changes “for free,”

pushed around by the fact that our heads are pretty massive sitting up

on top of the relatively skinny sticks that are our bodies. So when we

breathe and talk and gesture, those big heads move around, moving

the body’s center of mass, CM, along with them. The trajectory of the

CM is tracked over time, time-delayed copies of its trajectory can be

generated, and you’re on your way to generating a behavior space.

The subtle measures of RQA allow Shockley and colleagues to

manipulate the constraints on just how coordinated the joint behavior

is. They have people talk to each other or to someone else, at the same

time or in the course of taking turns, using words that differ in their

similarity, and so on—in order to uncover influences on the degree of

coordination. This means that motor behavior, a level of behavior that

some might like to relegate to the bin of basic behaviors that we can

take for granted, can be used as an index of language, something we

take to be one of our fanciest behaviors.

In the chapter on related work by Pellecchia and Shockley

“Application of Recurrence Quantification Analysis: Influence of

Cognitive Activity on Postural Fluctuations,” RQA is applied to a single

postural trajectory. This time, your big head is moved around not by

coordinating with another person but simply by standing while

directing some of your attention to another task. Here the emphasis is
Why nonlinear methods?

15

not so much on the objectivity to be gained—postural sway indexed by

the excursions of the center of pressure (COP) is a common measure in

some domains. But Pellecchia and Shockley point out that more

traditional postural measures are summary measures, for example, of

COP path magnitude and variability. But these summary measures turn

out to be insufficiently sensitive to the varied ways in which cognitive

attentional load can influence certain aspects of postural stability but

not others. They “do not reflect the dynamical properties of postural

control.” In particular, they miss the temporal structure of a COP time

series. One challenge is that posture data are what is called

nonstationary. No single summary measure adequately captures them

because the mean changes over time and the variability changes over

time (see Figure 1.1C). The notion of an average postural location

doesn’t make sense. But this nonstationarity also makes posture data

inappropriate for analyses that assume stationarity. RQA, in contrast,

makes no such assumptions about the way the data are distributed.

Pellecchia and Shockley deal with some of the technical aspects

of carrying out RQA. When you’re evaluating the RQA dance, what

qualifies as an instance of the two original partners having come

together? Do they actually have to touch hands or can they simply slap

hands or wave in the vicinity of each other? When you’re generating

your behavior space through time-delayed copies, what should the

delay be? How many dimensions define your hyper-space?

Pedagogically, our preferences is to illustrate it with three because we

can visualize three-space. But there are no such mundane constraints

on RQA. Pellecchia and Shockley suggest a kind of exploratory strategy

in which the RQA quantities are calculated for a range of parameter
Carello and Moreno

16

values (numbers of dimensions, size of time-delay), and then settle on a

value of the latter from a range that doesn’t have dramatic

consequences for the RQA measures. The upshot in the research they

describe is that RQA promotes a different understanding of what

attention does. In the particular experiment they described, for

example, summary measures of COP were all affected in the same way

by attentional demands, suggesting that attention causes a decrement

in postural control. The RQA measures, in contrast, yielded differences

that suggest a nuanced understanding of the ways in which postural

components (e.g., the front to back vs. side to side movements) might

be modulated by a stander in order to meet attentional demands.

Variability is something to be harnessed by the system to achieve a

goal.

Fractal Analyses

Most of the things that we need to measure tend to be irregular.

This is no less true when the things we measure are behaviors rather

than objects. Geometrically, this means that things are more like

coastlines than rectangles. As Liebovitch and Shehadeh point out in

“Introduction to Fractals,” this not only makes them hard to measure, it

makes the ruler that we use important. Quite surprisingly, the

measured size of the coastline depends on the size of the ruler. Smaller

rulers get into more of the nooks and crannies, thereby including more

stretches of coastline than would be the case with a large ruler that is

forced to bridge those gaps. So the level of resolution that we choose to

achieve in our measurement affects the values we get. Consider one

classic value that is typically used to characterize the behavior of a

system, its mean. The mean is a measure derived from a collection of
Why nonlinear methods?

17

property or performance values that all tend to distribute around this

more or less central value. It is considered a value that typifies the thing

being measured. But with irregular objects like coastlines, there really

is no typical value. The mean depends on the resolution of the

measurement and isn’t all that meaningful.

Subjectivity in measurement is certainly a problem to be

reckoned with. You can appreciate how standard assumptions about

normal distributions (most notably, more samples should lead to more

precision, not more stuff) will be inappropriate in such circumstances.

But so-called fractal objects, which are defined, in part, by this

dependency on measurement resolution, have an additional property

that makes them especially interesting. Fractal objects are self-similar,

that is, structure at the large scale (or the structure of behavior at the

large scale) is duplicated at the small scale: “...the statistics of the small

pieces are similar to the statistics of the large pieces.” This is so

whether we are talking about the structure at different scales of space

or at different scales of time. What happens within a square centimeter

mimics what happens within a square meter; what happens within a one

second window mimics what happens within a one minute window.

Self-similarity can reveal much about the dynamics of a system.

Depending on how the variability relates to the size of the window—a

relationship indexed by what is called the Hurst exponent—it tells you

whether an increase in the value of a measure taken now is likely to be

followed by an increase or a decrease in that measure taken later. In

essence, continuous dynamical processes manifest a kind of memory

without the logical attributions and storage metaphors we gravitate to

in the behavioral sciences.
Carello and Moreno

18

A respect for unfolding dynamics encourages the treatment of

order effects as entities of interest in an experiment rather than as

sources of contamination. This is illustrated by Holden in “Gauging the

Fractal Dimension of Cognitive Performance.” He uses a simple

repetitive time-estimation task to demonstrate the options for analyzing

a time series of response times. A participant attempted to produce a

sequence of equal intervals to mimic a presented target interval. Here

our interest is not so much in accuracy as in the kind of process that

produced the performance that was obtained. The way in which the

variability changes over time is the source of the insight. Holden

provides a nice intuitive metaphor here. If the interval estimates did not

vary, then a graph (of value over time) would yield a one-dimensional

straight line. To the extent that the time series is messy, it more closely

resembles a two-dimensional plane. The fractal dimension of the time

series can be calculated from the time series, with a value of 1.5

indicating randomness and 1.2 indicating pink noise, a time-scale

dependent variability. It is, of course, more complicated than this, since

order matters. Shuffling the data yields white noise because you’ve

destroyed the fractal structure, the pattern of variability at short time

scales that is echoed at ever-longer time scales.

Holden’s chapter also contains several caveats for conducting

these kinds of nonlinear analyses. Caveats are necessary because there

are options and we are sensitive to the fact that options often entail

assumptions. Let’s consider a few that routinely arise due to the

practical finiteness of data collection. Ordinary statistics tell us that

outliers are a problem and should be eliminated. Sympathy for

dynamics burdens us with the knowledge that outliers are not
Why nonlinear methods?

19

necessarily produced by an aberrant procedural hiccup. We have to

assess the extent to which they would dominate the analysis. You also

have to be careful that the analysis is not dominated by spurious trends

(e.g., a stretch of linearity) in the finite data set that might be part of

something else if the data collection had continued. So the longer the

time series the better in order to see the cascading structure of varied

time scales. But since our interest is in emergence over time, we cannot

pretend that data collected over several days are the same as data

collected in one sitting. Finally, with all of these options, you’re bound

to come up with varied characterizations depending on which choices

you made. So you need to conduct more than one analysis type (e.g.,

spectral and dispersion analyses) as converging operations.

In her chapter “1/f Dynamic in Complex Visual Search: Evidence

for Self-Organized Criticality in Human Perception” Aks applies this

perspective to visual search behavior, the eye movements that people

engage in when looking for a specific small detail amidst a clutter of

distracting detail. These movements appear quite haphazard. Our eyes

dart back and forth, up and around, in a mix of short jumps and long,

often landing on the same places time and again. We do not follow a

systematic path, say, from upper left to lower right that would

seemingly guarantee that the target would be encountered. Our

behavior doesn’t show in any obvious way that we remember where we

looked in vain before. Yet the repetitive, jerky movements are

surprisingly effective—we find our friend in the crowd; we select the
Carello and Moreno

20

perfect bolt from a stash of culch.2 Indeed, the ordinary conditions

under which visual search happens don’t really favor the tidy, thorough

search. We don’t have all the time in the world; our targets or our goals

are often on the move.

At issue in this research domain is what guides visual search. If

eye movements truly were random, we would have to conclude that

they were driven by something other than what the system had done

before, that there was no real role for memory. But it is in the

superficially random noisiness that a fractal analysis uncovers subtle

structure. By quantifying how the noise changes over time we gain

insight into the kind of system that has produced that noise. Presenting

subjects with a difficult search task (looking for a target with a

conjunction of features, not just one) reveals that a bout of scanning has

its own internal history. What we do early in the bout does indeed

influence what we do later in that same bout. But this dynamic history is

quite different from more standard characterizations of memory that

entail a certain degree of address-specific tagging. It implicates more

subtle contingencies, indexed by what is called 1/f behavior.

The final chapter by Tuller does not incorporate either

recurrence or fractal analyses. Instead, she illustrates the advantage of

a general dynamical attitude in designing an experiment, with

subsequent opportunities for new interpretations of seemingly wellunderstood results. The generality is especially apparent in that this is

strictly a perception experiment. The relevant series is not of the timing

 

2 In the New England vernacular, culch refers to items that may (or may not) come in handy

someday.
Why nonlinear methods?

21

of the responses but of their content—why should a speech token sound

like one syllable versus another? Tuller tackles the classic phenomenon

of categorical perception. Syllables are synthesized to vary

incrementally on some acoustic property. Even though each syllable

token is defined by a different value taken from a wide range of the

acoustic property, each is heard as either one syllable or the other;

they belong to one syllable category or the other. Categorical

perception has historically been treated statically: You hear a token

and its maps better onto one representation than another. But Tuller

shows that thinking of the process as dynamical—with stable, attractive

states that change abruptly—focuses the experimenter on finding what

encourages those nonlinear shifts from one stability to another.

Tuller’s major caveat is that some standard methodological

choices, most notably, randomizing the order in which stimuli are

presented, obscure the dynamics of a system. In a now-familiar refrain

for this volume, she notes that far from being a nuisance that has to be

controlled, order effects allow a system’s dynamical signature to

emerge. A dynamical perspective requires that the stimuli be

presented in order, for example, alternating increasing and decreasing

levels of the acoustic property. This allows an interpretation of the

acoustic property as a control parameter rather than as a cue. Now the

categorical shift can be mined. Does it happen at the same value of the

control parameter in one order as the other (see Figure 1.1F)? Does the

syllable you first hear persist? Or do you switch to the other syllable at

a low level on the way up but a high level on the way down? Tuller

notes that all three of these patterns have been observed and one or the

other can be encouraged. Most notably, the abrupt change from one
Carello and Moreno

22

syllable to another, from one stable state to another, comes about

during situations of instability. What had been unanimity of responses

becomes somewhat mixed. Such instability links this perceptual

phenomenon to the general phenomenon of self-organized pattern

formation. This linkage allows one to write the differential equations

that “define systems with attractor properties that fit the observed

experimental data.” That is to say, the linkage is far from metaphorical.

A perception system is modeled in the same way, with the same

ontological status, as an action system, clearly a different direction from

the traditional treatment of categorical perception.

CONCLUSIONS

Behavioral scientists study the actions of humans and animals.

Some of these actions make sense at the level of the individual and

some emerge only in a social setting. As we noted at the outset, we

make a number of choices in conducting our studies. The practical

issue of what kind of equipment we have, how many participants are

available, and so on, are supplemented by the kinds of analyses we

know how to do and the kinds of data we collected to put into those

analyses. We have suggested that such choices are not necessarily as

benign and objective as we would like to believe. Do we try to avoid

data like those shown in panels B-F of Figure 1.1 because they are

messy? Or do we try to contrive our manipulations to produce those

kinds of data because they allow the richness of dynamical systems to

be seen? The following chapters are of a mind that we should not be

afraid of variability. It may well be the driving force of nature.
Why nonlinear methods?

23

AUTHOR NOTE

Preparation of this work was supported by grants from the National

Science Foundation (SBR 00-04097) and the National Institutes of Health

(HD-01994).

REFERENCES

Cattell, J. McK. (1886). The time taken up by the cerebral operations.

Mind, 11, 220-242.

Christie, L S., & Luce, R. D. (1956). Decision structure and time relations

in simple choice behavior. Bulletin of Mathematical Biophysics, 18, 89-

112.

Cramer, H. (1946). Mathematical methods of statistics. Princeton:

Princeton University Press.

Donders, F. C. (1868). Over de snelheid van psychische processen.

Onderzoekingen gedaan in het Physiologisch Laboratorium der

Utrechtsche Hoogenschool, 1968-1869, Tweede reeks, II, 92-120.

Transl. By W. G. Koster, Acta Psychologica, 30, 412-431.

Gholson, B., & Hohle, R. H. (1968a). Verbal reaction time in hues and

hue names and forms vs. form names. Perception & Psychophysics, 3,

191-196.

Gholson, B., & Hohle, R. H. (1968b). Choice reaction time to hues

printed in conflicting hue names and nonsense words. Journal of

Experimental Psychology, 76, 413-418.
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24

Hohle, R. H. (1965). Inferred components of reaction time as functions of

foreperiod duration. Journal of Experimental Psychology, 69, 382-386.

Hohle, R. H. (1967). Component process latencies in reaction times of

children and adults. In L.P. Lipsitt & C.C. Spiker (Eds.), Advances in

child development and behavior, Vol III. New York: Academic Press.

Jastrow, J. (1890). The time-relations of mental phenomena. New York:

N.D.C. Hodges.

Külpe, O. (1895). Outlines of psychology. New York: MacMillan, Secs.

69, 70.

Luce, R. D. (1986). Response times: Their role in inferring elementary

mental organization. New York: Oxford University Press.

McGill, W. J. (1963). Stochastic latency mechanisms. In R. D. Luce, R. R.

Bush, & E. Galanter (Eds.), Handbook of mathematical psychology, Vol I.

New York: Wiley.

Moreno, M. A. (2001) A nonlinear dynamical systems perspective on

response time distributions. Unpublished doctoral dissertation, Arizona

State University, Tempe, AZ.

Sternberg, S. (1969). The discovery of processing stages: Extensions of

Donders’ method. Acta Psychologica, 30, 276-315.

Van Zandt, T., & Radcliff, R. (1995). Statistical mimicking of reaction

time data: Single-process models, parameter variability, and mixtures.

Psychonomic Bulletin & Review, 2, 20-54.
Why nonlinear methods?

25

Wundt, W. (1874). Grundzuge der Physiologischen Psychologie. Leipzig:

Engelmann.
CHAPTER 2

Recurrence Quantification Analysis

of Nonlinear Dynamical Systems

Charles L. Webber, Jr.

Department of Physiology

Loyola University Medical Center

Maywood, Illinois 60153-3304

U. S. A.

E-mail: cwebber@lumc.edu

Joseph P. Zbilut

Department of Molecular Biophysics and Physiology

Rush University Medical Center

Chicago, IL 60612-3839

U. S. A.

E-mail: jzbilut@rush.edu
Webber & Zbilut 

27

RECURRENCES IN NATURE

Comprehension of the physical world depends upon

observation, measurement, analysis, and, if possible, prediction of

patterns expressed ubiquitously in nature. Linear or near-linear

systems possess a regularity and simplicity that are readily amenable

to scientific investigations. Yet numerous other systems, especially

those in the animate realm, possess complexities that are at best nonlinear and non-predictable. Common ground between living and nonliving systems resides in their shared property of recurrence. That is,

within the dynamical signals expressed by living and non-living signals

are stretches, short or long, of repeating patterns. Actually, recurrence

properties are taken advantage of in signal compression techniques

(Wyner, Ziv, & Wyner, 1998)—one billion sine waves at a constant

frequency will perfectly collapse to a fraction of one sine cycle with loss

of no information. As signals grow in complexity, however,

recurrences become rarer, and efficient compressibility is resisted.

For so-called random systems such as radioactive decay, recurrences

occur theoretically by chance alone. But the lesson is clear: Insofar as

natural patterns are found in all dynamical systems, the degree to

which those systems exhibit recurrent patterns speaks volumes

regarding their underlying dynamics. And, on reflection, it should be

appreciated that the entire scientific enterprise is based upon the

concept of recurrence: To be accepted as valid, experimental results

must be repeatable in the hands of the primary investigator and

verifiable by independent laboratories.

Patterns of recurrence in nature necessarily have mathematical

underpinnings (Eckmann, Kamphorst, & Ruelle, 1987) which will
Recurrence Quantification Analysis

28

readily become apparent in the sections that follow. Although

recurrence quantification analysis (RQA) is scantly a decade old (Zbilut

& Webber, 1992; Webber & Zbilut, 1994), the concept of recurrence in

mathematics has a much longer history (Feller, 1950; Kac, 1959).

However, any appreciation of nonlinear dynamics and associated terms

the reader brings to the table (fractals, attractors, non-stationarity,

singularities, etc.) will greatly facilitate the comprehension of

recurrence analysis (see Glass & Mackey, 1988; Bassingthwaighte,

Liebovitch, & West, 1994; Kaplan & Glass, 1995). Indeed, our primary

goal in this chapter is to provide clear details regarding the proper

implementation of one nonlinear technique, RQA, in the assessment or

diagnosis of complex dynamical systems. Using carefully selected

examples pertinent to the field of psychology and parallel disciplines,

we will build a case for the fundamentals of RQA, its derivation, and its

utility. Patience is required of the learner, for there are no short cuts to

RQA strategies; proper implementations are application-specific. A

mathematical framework is constructed to give quantitative legitimacy

to RQA, but for those unfamiliar with mathematical notation, an

Appendix is provided with discrete numerical examples (Webber,

2004).

The up and down motions of sea waves originally inspired

Joseph Fourier to characterize signals in terms of their frequencydomain features (e.g., cycles per sec); that technique now bears his

name (Grafakos, 2003). Likewise, for our recurrence purposes,

consider a system of literal waves on the sea as measured from buoy

instrumentation and plotted in Figure 2.1A. During the 9.4 days of 226

hourly measurements the wave heights rise and fall in a nonlinear,
Webber & Zbilut 

29

Figure 2.1. (A) Wave height measurements recorded hourly for 226 hours from buoy

instrumentation located 33 nautical miles south of Islip, LI, NY (National Data Buoy Center

Station 44025). Wave heights are expressed in feet to the nearest 0.1 ft from hour 14, July 10

through hour 23, July 19, 2002. Dots are placed on all waves having identical heights of exactly

0.9 ft that occur at 25 aperiodic points in time (hours: 4, 7-9, 29, 61, 71, 117, 120, 124-126, 143,

145, 147-148, 179-180, 187, 210, 220-221, 223-225). These wave data are owned by the NSW

Department of Land and Water Conservation (DLWC) as collected and provided by the Manly

Hydraulic Laboratory (MHL), Sydney, Australia. Time series data from data file BOUY (n=226

points). (B) Matrix plot of identical wave heights at 0.9 ft. The same time series of 25 aperiodic

time points is aligned along both horizontal and vertical time axes, and intersecting pixels are

darkened to map out recurrence matches at 0.9 ft wave heights only. All other wave heights

are excluded from the matrix. Recurrence data computed from data file BUOY9 using program

RQD.EXE. RQA parameters: P1-Plast = 1-226; RANDSEQ = n; NORM = max norm; DELAY = 1;

EMBED = 1; RESCALE = absolute; RADUS = 0; COLORBND = 1; LINE = 2.
Recurrence Quantification Analysis

30

aperiodic fashion over different time scales (long-periods, large

amplitudes; short-periods, small amplitudes). To capture the

fundamental notion of recurrence, a dot is placed on every wave of the

time series that is exactly 0.9 ft in height. As illustrated, this forms an

imaginary line of horizontal dots cutting through the waves at 25

specific time points. That is, each of these waves is exactly recurrent

(same height) with one another at certain instances in time that are nonperiodic. To graphically represent the recurrent structuring of the data

matrix at 0.9 ft height only, the time-positions of these waves are

plotted against each other in Figure 2.1B. All other wave heights are

excluded.

To get a more accurate picture, however, it is necessary to

include waves of all possible heights. Because the waves range in

height from 0.5 to 1.4 ft with a measurement resolution of 0.1 ft, there

are necessarily 10 different possible wave heights. The recurrence plot

of Figure 2.2A shows the distribution of recurrent points (darkened

pixels) for all waves of exactly the same height, including low-,

medium-, and high-amplitude waves. For the sea wave data it can be

seen that the recurrence plot forms a delicate, lace-like pattern of

recurrent points. By necessity, there is a long diagonal line (wave

heights always self-match), the plot is symmetrical across that diagonal

(if height of wave i matches height of wave j, then the height of wave j

matches height of wave i), and the recurrence plot is 2-dimensional.

What happens if we relax the constraint that wave heights must

be exactly the same before registering them as recurrent? For

example, if we let sea waves within 0.2 ft of each other be considered

recurrent, a 1.0 ft wave would recur with other waves ranging in height
Webber & Zbilut 

31

Figure 2.2. Recurrence plots of wave height data for different radius (RADIUS) settings and

computed recurrence densities (%REC). (A) Exact recurrences (RADIUS = 0.0 ft; %REC =

13.479%). (B) Approximate recurrences (RADIUS = 0.2 ft; %REC = 54.938%). (C) Saturated

recurrences (RADIUS = 0.9 ft, %REC = 100.000%). Like color-contour maps, the recurrence

plot in C is color-coded in 0.2 ft increments of RADIUS: blue (0.0-0.2 ft); cyan (0.2-0.4 ft); green

(0.4-0.6 ft); yellow (0.6-0.8 ft); red (0.8-0.9 ft). Time series of wave height data (WH, feet versus

hours) align along the horizontal and vertical axes of each recurrence plot (time calibration of

hour 1 through hour 226). Recurrence data computed from file BUOY using program RQD.EXE.

RQA parameters: P1-Plast = 1-226; RANDSEQ = n; NORM = max norm; DELAY = 1; EMBED = 1;

RESCALE = absolute; RADIUS = 0.0 (A), 0.2 (B) or 1.0 (C); COLORBND = 1 (A and B) or 0.2 (C);

LINE = 2.
Recurrence Quantification Analysis

32

from 0.8 to 1.2 ft. Therefore, applying this rule to all points in the time

series should result in the recruitment of more recurrent points. Indeed

this is the case, as demonstrated in the recurrence plot of Figure 2.2B.

The density of recurrences is decidedly higher, and the lace-like

quality of the former plot is replaced by broader areas of approximate

recurrences. Carrying this relaxation rule to its limit, what would the

recurrence plot look like if all waves were recurrent with one another?

This is easily done for the present example by defining two time-points

recurrent if they are within 0.9 ft in height. The 0.9 ft limit comes from

the difference between the largest wave (1.4 ft) and the smallest wave

(0.4 ft). In this case, we would correctly expect every pixel in the

recurrence plot to be selected, creating one huge dark square with loss

of all subtle details (not shown). Discriminating details of distances can

be retrieved, however, by color-coding the differences in wave heights

according to a simple coloring scheme as illustrated in Figure 2.2C. All

we need to do is change the assigned color in 0.2 ft increments,

constructing a color-contour map of wave heights. Actually, this type of

coloring is not unlike any contour mapping using different colors to

designate various elevations above or below sea level. It is very

instructive to make mental correlations between the wave height data

(time series along horizontal and vertical axes) and the attendant

recurrence plots.

RECURRENCE PLOTS

With this simple introduction to recurrence plots, it is now

necessary to become more mathematical. Published numerical

examples (Webber, 2004) are reiterated in the Appendix to assist in the
Webber & Zbilut 

33

learning of these important details. As will be carefully shown, the

recurrence plot is nothing more (or less) than the visualization of a

square recurrence matrix (RM) of distance elements within a cutoff

limit. We will work up to the definition of seven recurrence parameters

(user-adjustable constants) which control the distances within the

matrix, their rescaling, and recurrence plot texture. To do this, we will

introduce the straightforward ECG signal recorded from a human

volunteer as plotted in Figure 2.3A. We could easily generate a

recurrence plot of this signal like was done for the sea wave data,

identifying as recurrent those time points with identical or similar

voltages. But to do so would only be the first-dimensional approach to

the problem (represented in a 2-dimensional recurrence plot); as we

describe in the following paragraphs, such a 1-dimensional approach

may not suffice.

Consider the ECG signal in its one-dimensaional representation

of voltage as a function of time (Figure 2.3A). Does not this signal

actually “live” in higher dimensions? Of course it does. Since the ECG

derives from summed cardiac potentials that move simultaneously in

three dimensions (frontal, saggital, and horizontal orthogonal planes),

the 1- and 2-dimensional representations are mere projections of the

signal to lower dimensions. To accurately represent the ECG in three

dimensions it would be necessary to simultaneously record electrical

potentials in three orthogonal planes from the subject. But here is

where things get very interesting. About a quarter century ago Takens

(1981), elaborating upon key conceptual ideas of Ruelle, introduced his

theorem of higher-dimensional reconstruction by the method of time

delays. What this theorem states is that the topological features of any
Recurrence Quantification Analysis

34

Figure 2.3. (A) Time series of a one-dimensional, lead II electrocardiogram recorded from a

normal human volunteer and digitized at a rate of 200 Hz. Displayed are the first three cardiac

cycles with characteristic P waves (atrial depolarization), QRS complexes (ventricular

depolarization) and T waves (ventricular repolarization). (B) Three-dimensional reconstruction

of the ECG signal in phase space by the method of time delays (τ = 10 ms). Each ECG wave (P,

QRS, T) forms its own unique loop (three times), returning to isopotential (0,0,0 amplitude units)

between waves (T-P interval). Time series data from file ECG (A) with 5 ms per point. Phasespace data from file ECG with DELAY = 2 points (10 ms) or 4 points (20 ms).

higher-dimensional system consisting of multiple coupled variables

can be reconstructed from but a single measured variable of that

system. That is, the actual dimension of the system (D) is set by a
Webber & Zbilut 

35

number of participant variables, but we may only need to measure one

of those variables to learn interesting features of the system’s

underlying dynamics. The reconstruction is performed by defining

time-delayed vectors (Vi) of M points (Pi) that are delayed or offset in

time (τ). Thus, each point represents a single amplitude (scalar) at a

specific instance in time.

D-dimensional vector, Vi = Pi + Pi+τ + Pi+2 τ + ... + Pi+(D-1) τ [2.1]

Think of what this means graphically for our single-lead ECG

signal. If the 1-dimensional data (ECG vector) are plotted against itself

twice delayed (τ and 2τ) on a three-axis plot, the signal is promoted

into 3-dimensional space as shown in Figure 2.3B. As seen, the

morphology of this normal electrocardiogram forms three loops in

phase space corresponding to the P, QRS and T waves of the ECG time

series. Topologically, these loops are identical to the simultaneous

plotting of three orthogonal leads, had they been recorded. That is, the

one measured variable (ECG lead II) becomes a surrogate for the two

unmeasured variables (ECG leads I & III).

Careful examination of Equation 2.1 explicitly introduces two of

the seven recurrence parameters mentioned previously. First we need

to know into what dimension the dynamic should be projected, the

embedding dimension (M), where M >= D. And second, for M > 1, we

need to pick an appropriate delay (τ) between sequential time points in

the 1-dimensional signal. It must be appreciated that the selections of

embedding dimension and delay are not trivial, but are based on
Recurrence Quantification Analysis

36

nonlinear dynamical theory. Moreover, the presence of noise

complicates the situation, as will be discussed later.

Embedding dimension (M or EMBED), the first recurrence

parameter, can in principle be estimated by the nearest-neighbor

methodology of Kennel, Brown, and Abarbanel (1992). Parameter M is

increased in integer steps until the recruitment of nearest neighbors of

the dynamic under scrutiny becomes unchanging. At this particular

value of M, the information of the system has been maximized (and,

technically speaking, the attractor has been completely unfolded).

Thus, there is no need to explore higher dimensions since no new

information would be recruited anyway. This methodology works well

on stable and low-noise systems, which are most notably found in

mathematical examples such as the Lorenz attractor (Lorenz, 1963). But

when it comes to real-world data, noise inflates the dimension (D)

(Parker & Chua, 1989), and non-stationarities (transients, drifts) in the

system modulate the critical M. Thus, in practice, M > D. Because of

these practical limitations, we typically use embedding dimensions of

10 to 20 on biological systems, but no higher. It is a curious fact that

when the embedding dimension is set to too high, even

random/stochastic systems (which in theory exhibit recurrence only by

chance) display strong, yet artifactual, patterns of recurrence.

Delay (τ or DELAY), the second recurrence parameter, should be

selected so as to minimize the interaction between points of the

measured time series. This, in effect, opens up the attractor (assuming

one exists), by presenting its largest profile. An analogy would be the

construction of a circle by plotting a sine wave against itself delayed by

90 degrees (whereas delaying by a mere 1 degree would yield a very
Webber & Zbilut 

37

thin, slightly bowed line profile). Two common ways of selecting a

proper delay include finding the first minimum in either the (linear)

autocorrelation function or (nonlinear) mutual information function

(Frazer & Swinney, 1986) of the continuous time series. For

discontinuous signals (maps as opposed to flows), such as R-R intervals

extracted from the continuous ECG signals, the delay is best set to 1 (no

points in the time series are skipped). In special cases, parameter τ

can also be set to 1 for continuous flows if the goal to is to perform

waveform matching (recurrence matching of similar waveforms, pointfor-sequential-point). As a corollary to this discussion, Grassberger,

Schreiber, and Schaffrath (1991) demonstrated that the delay is a noncritical critical parameter, provided the attractor is sufficiently opened

up. When any parameter is non-critical it simply means that the

quantitative features of the system are robust and stable against

changes in the named parameter. This comes as good news for

physiological systems in which we need not over concern ourselves

with finding the optimal delay (which anyway is non-existent for

transient states), since many delays will suffice for the same system.

The third recurrence parameter is the range, defined by the

selected starting point (Pstart) and ending point (Pend) in the time series

to be analyzed. For an embedded time series (M > 1) of N points, it

becomes obvious that M – 1 embedded points must extend beyond

Pend. Thus Pend can extend no further than N – M + 1 points into the

available data. In effect, the range defines a window (W = Pend – Pstart +

1) on the dynamic under investigation. As will be stressed, short

windows focus on small-scale recurrences, whereas long windows

focus on large-scale recurrences (see Figure 2.8).
Recurrence Quantification Analysis

38

The fourth recurrence parameter is the norm, of which there are

three selections possible: Minimum norm, maximum norm, and

Euclidean norm. As implied by its name, the norm function

geometrically defines the size (and shape) of the neighborhood

surrounding each reference point. The recurrence area is largest for

the max norm, smallest for the min norm, and intermediate for the

Euclidean norm (see Marwan, 2003a). (Since a vector must have at

least two points, each norm is unique if and only if M > 1, else all norms

are exactly equivalent for M = 1.) But before we can get explicit about

norms, it is necessary to be absolutely clear as to how vectors are

defined from the 1-dimensional time series. Think of it this way. Take a

vector time series (T) of N scalar points (P) and double label the time

series as follows:

Ti = Tj = P1, P2, P3, P4, ..., PN. [2.2]

The dual subscripts (i ≠ j ) refer to different points (Pi and Pj) in Ti

and Tj when M = 1, but to different vectors (Vi and Vj) in Ti and Tj when

M > 1. Each vector is constructed by starting with an initial point and

taking M – 1 subsequent points offset by τ (see Equations 2.3 and 2.4

and compare with Equation 2.1). The distances between all possible

combinations of i-vectors (Vi) and j-vectors (Vj) are computed

according to the norming function selected. The minimum or maximum

norm at point Pi,Pj is defined, respectively, by the smallest or largest

difference between paired points in vector-i and vector-j (Vi – Vj). The

Euclidean norm is defined by the Euclidean distance between paired

vectors (Equation 2.5).
Webber & Zbilut 

39

Vi = Pi + Pi+τ + Pi+2τ + ... + Pi+(M-1) τ [2.3]

Vj = Pj + Pj+τ + Pj+2τ + ... + Pj+(M-1) τ [2.4]

Euclidean distance = √(Σ(Vi – Vj)

2

) [2.5]

Computed distance values are distributed within a distance

matrix DM[j, i] which aligns vertically with Tj and horizontally with Ti (i

= 1 to W; j = 1 to W; where maximum W = N – M + 1). The distance

matrix has W2 elements with a long central diagonal of W distances all

equal to 0.0. This ubiquitous diagonal feature arises because individual

vectors are always identical matches with themselves (Vi = Vj whenever

i = j). The matrix is also symmetrical across the diagonal (if vector Vi is

close to vector Vj then by necessity vector Vj is close to vector Vi ).

The fifth recurrence parameter is the rescaling option. The

distance matrix can be rescaled by dividing down each element in the

distance matrix (DM) by either the mean distance or maximum distance

of the entire matrix. In general, DM rescaling allows systems operating

on different scales to be statistically compared. Mean distance

rescaling is useful in smoothing out any matrix possessing an outlier

maximum distance. But maximum distance rescaling is the most

commonly used (and recommended) rescaling option, which redefines

the DM over the unit interval (0.0 to 1.0 or 0.0% to 100.0%). It is also

possible to leave the DM unaltered by not rescaling. In such special

cases, DM distances are expressed in absolute units identical to the

amplitude units of the input time series (volts, mm Hg, angstroms,

seconds, etc.).
Recurrence Quantification Analysis

40

The sixth recurrence parameter is the radius (RADIUS), which is

always expressed in units relative to the elements in the distance

matrix, whether or not those elements have been rescaled. In the sea

wave example, the radius was actually set to three different values:

RADIUS = 0.0, 0.2, and 0.9 ft (Figures 2.2A, B, and C, respectively). In

that case we can speak in absolute distances because the distance

matrix was never rescaled. In effect, the radius parameter implements

a cut-off limit (Heavyside function) that transforms the distance matrix

(DM) into the recurrence matrix (RM). That is, all (i, j) elements in DM

with distances at or below the RADIUS cutoff are included in the

recurrence matrix (element value = 1), but all other elements are

excluded from RM (element value = 0). Note carefully that the RM

derives from the DM, but the two matrices are not identical. As seen for

the wave data (Figure 2.2), as the radius increases, the number of

recurrent points increases. Only when RADIUS equals or exceeds the

maximum distance is each cell of the matrix filled with values of 1 (the

recurrence matrix is saturated). The “shotgun plot” of Figure 2.4

provides a conceptual framework for understanding why an increasing

RADIUS captures more and more recurrent points in phase space.

Thus, if one becomes too liberal with RADIUS, points (M = 1) or vectors

(M > 1) that are actually quite distant from one another will nonetheless

be counted as recurrent (the recurrence matrix is too inclusive).

Proper procedures for selecting the optimal RADIUS parameter will be

described below.
Webber & Zbilut 

41

Figure 2.4. Representation of a hypothetical system in higher-dimensional phase space with a

splay of points (closed dots) surrounding a single reference point (open dot). The points falling

within the smallest circle (RADIUS = 1 distance units) are the nearest neighbors of the reference

point. That is, those points are recurrent with the reference point. The second concentric

circle (RADIUS = 2 distance units) includes a few more neighbors, increasing the number of

recurrences from 4 to 14. Increasing the radius further (RADIUS = 3 or 4 distance units)

becomes too inclusive, capturing an additional 20 or 60 distant points as nearest neighbors

when, in fact, they are not.

The seventh and last parameter is termed the line parameter

(LINE). This parameter is important when extracting quantitative

features from recurrence plots (see next section), but exerts no effect

on the recurrence matrix itself. If the length of a recurrence feature is

shorter than the line parameter, that feature is rejected during the

quantitative analyses. Typically, the line parameter is set equal to 2

because it takes a minimum of two points to define any line. But it is

possible to increase the line parameter (in integer steps) and thereby

implement a quantitative filter function on feature extractions, but this is

not necessarily recommended.
Recurrence Quantification Analysis

42

RECURRENCE QUANTIFICATION

Simply put, recurrence plots, especially colored versions

expressing recurrence distances as contour maps, are beautiful to look

at (e.g., Figure 2.2C). With little debate, global recurrence plots of

time series and signals extant in nature captivate one’s attention.

Admittedly, such curious and intriguing graphical displays tend more

to evoke artistic than scientific appreciation, and rightfully so.

Recalling the brief history of recurrence analysis, recurrence plots

were originally posited as qualitative tools to detect hidden rhythms

graphically (Eckmann et al., 1987). From the outset, color was not the

key; rather the specific patterns of multi-dimensional recurrences gave

hints regarding the underlying dynamic. Early on it was understood

how important it was to hold the radius parameter to small values so as

to keep the recurrence matrix sparse. In so doing, emphasis was

placed on local recurrences that formed delicate, lacy patterns. All of

this is well and good, but the next logical step was to promote

recurrence analysis to quantitative status (Zbilut & Webber, 1992;

Webber and Zbilut, 1994). Instead of trusting one’s eye to “see”

recurrence patterns, specific rules had to be devised whereby certain

recurrence features could be automatically extracted from recurrence

plots. In so doing, problems relating to individual biases of multiple

observers and subjective interpretations of recurrence plots were

categorically precluded.

We will highlight the fundamental rules of recurrence

quantification analysis (RQA) by employing the classic strange attractor

of Hénon (1976). This chaotic attractor is a geometrical structure

(system) that derives its form (dynamic) from the nonlinear coupling of
Webber & Zbilut 

43

two variables. Note in Equation 2.6 that the next data point, Xi+1, is a

nonlinear function of the previous Xi and Yi terms (the Xi

2 term provides

the nonlinear interaction), whereas in Equation 2.7 the next Yi+1 is a

linear function of the previous Xi term.

Xi+1 = Yi + 1.0 – (1.4Xi

2

) [2.6]

Yi+1 = 0.3Xi [2.7]

We seeded the coupled Hénon variables with initial zero values

(e.g., X0 = Y0 = 0.0) and iterated the system 2000 times to create a

sample time series. To make sure the dynamic settled down on its

attractor, the first 1000 iterations were rejected as transients. The next

200 iterations of the system are plotted in Figure 2.5 (cycles 1001

through 1200), which shows the complex dynamics of the coupled

variables. Plotting Yi as a function of Xi generates the Hénon strange

attractor (Figure 2.6A). It is called an attractor because dynamical

points are “attracted” to certain positions on the map and “repelled”

from other positions (the white space). Note that the points remain

confined within tight limits (±1.3 for X and ±0.4 for Y) without flying off

to infinity. The dimension of the Hénon attractor is estimated to be

around 1.26 (Girault, 1991), which is a fractal or non-integer dimension.

Fractal dimensions relate more to the mathematical concept of scaling

than real-world dimensions, which must be integers (see Liebovitch &

Shehadeh, Chapter 5). Recalling the method of time delays as

discussed for the ECG example (Figure 2.3), plotting Xi (current value)

as a function of Xi+1 (next value) topologically reproduces the Hénon
Recurrence Quantification Analysis

44

attractor (Figure 2.6B). So does plotting Yi as a function of Yi+1 (Figure

2.6C). These relationships underscore the remarkable power of

surrogate variables to adequately substitute for unmeasured variables

by using the method of time delays (Takens, 1981).

Figure 2.5. Fluctuations of the Xi and Yi variables comprising the Hénon system in its chaotic

mode (Equations 2.6 and 2.7). The first 1000 iterations are rejected before plotting the next 200

iterations (cycles 1001-1200). The dynamic behaviors of both variables are complex, but

bounded to the Hénon attractor (see Figure 2.6). The dynamics are also fully deterministic, yet

noisy, depending upon the round-off routines implemented by the digital computer that

generated the data. Iterated data from files HENCX and HENCY.
Webber & Zbilut 

45

Figure 2.6. Reconstruction of the Hénon strange attractor in phase space by three separate

methods. (A) Plotting Yi versus Xi generates the familiar Hénon strange attractor (dots;

Equations 2.6 and 2.7) or 16-point Hénon periodic attractor (crosshairs; Equations 2.18 and

2.19). (B) Plotting Xi versus Xi+1 reconstructs the topological features of the original Hénon

strange attractor by the method of time delays. (C) Plotting Yi versus Yi+1 also reconstructs the

attractor. Plots generated from data files HENCX, HENCY, HENPX, HENPY (with DELAY = 1 for B

and C).
Recurrence Quantification Analysis

46

Let us select the Hénon X variable (Figure 2.5A) and generate a

recurrence plot of this single variable, as shown in Figure 2.7. The

window size (W) is 200 points (e.g., cycles 1001-1200). This is an autorecurrence plot since the Hénon X variable is being compared with

itself (points Xi = points Xj, 3 points at a time since M = 3). Qualitative

examination of the symmetrical recurrence plot reveals short line

segments parallel to the central diagonal, a cluster of points

correspondent to a brief period-2 structure in the dynamic (cycles

1025-1036), and a few isolated points representing chance recurrences.

We will focus on the diagonal and vertical structures since from those

stem the seven recurrence (dependent) variables or quantifications.

Because the recurrence plot is symmetrical across the central diagonal,

all quantitative feature extractions take place within the upper triangle,

excluding the long diagonal (which provides no unique information)

and lower triangle (which provides only redundant information).

The first recurrence variable is %recurrence (%REC). %REC

quantifies the percentage of recurrent points falling within the specified

radius. This variable can range from 0% (no recurrent points) to 100%

(all points recurrent). For a given window size W, Equation 2.8 holds

true. For the Hénon attractor (Figure 2.7), %REC = 1.201% confirming

that the recurrence matrix is sparse (as is desired).

%REC = 100 (#recurrent points in triangle) / (W(W – 1) / 2) [2.8]

The second recurrence variable is %determinism (%DET).

%DET measures the proportion of recurrent points forming diagonal

line structures. Diagonal line segments must have a minimum length
Webber & Zbilut 

47

Figure 2.7. Recurrence plot of the Hénon chaotic X variable. The same variable is plotted on

the horizontal (chaotic Xi) and vertical axes (chaotic Xj) from iterations of coupled Equations 2.6

and 2.7. When vectors of 3 points (e.g., M = 3) match, recurrent points are plotted at the

corresponding (i, j) intersections. The most striking feature of this plot is the short diagonal line

structures parallel to the main diagonal. Recurrence data computed from file HENCX using

program RQD.EXE. RQA parameters: P1-Plast = 1001-1200; RANDSEQ = n; NORM = Euclid;

DELAY = 1; EMBED = 3; RESCALE = max dist; RADUS = 3; COLORBND = 1; LINE = 2. RQA

variables: %REC = 1.201; %DET = 88.703; LMAX = 12; ENT = 2.557; TND = –4.505; %LAM =

2.510; TT = 2.000.

defined by the line parameter, lest they be excluded. The name

determinism comes from repeating or deterministic patterns in the

dynamic. Periodic signals (e.g. sine waves) will give very long

diagonal lines, chaotic signals (e.g. Hénon attractor) will give very

short diagonal lines, and stochastic signals (e.g. random numbers) will
Recurrence Quantification Analysis

48

give no diagonal lines at all (unless parameter RADIUS is set too high).

For the Hénon attractor (Figure 2.7), %DET = 88.703% showing that

most of the recurrent points present are found in deterministic

structures.

%DET = 100 (#points in diagonal lines)/(#recurrent points) [2.9]

The third recurrence variable is linemax (LMAX), which is simply

the length of the longest diagonal line segment in the plot, excluding

the main diagonal line of identity (i = j). This is a very important

recurrence variable because it inversely scales with the most positive

Lyapunov exponent (Eckmann et al., 1987; Trulla et al., 1996). Positive

Lyapunov exponents gauge the rate at which trajectories diverge, and

are the hallmark for dynamic chaos. Thus, the shorter the linemax, the

more chaotic (less stable) the signal. For the Hénon attractor (Figure

2.7), LMAX = 12 points.

LMAX = length of longest diagonal line in recurrence plot [2.10]

The fourth recurrence variable is entropy (ENT), which is the

Shannon information entropy (Shannon, 1948) of all diagonal line

lengths distributed over integer bins in a histogram. ENT is a measure

of signal complexity and is calibrated in units of bits/bin. Individual

histogram bin probabilities (Pbin) are computed for each non-zero bin

and then summed according to Shannon’s equation. For the Hénon

attractor (Figure 2.7), ENT = 2.557 bits/bin due to a wide distribution of

diagonal line lengths. For simple periodic systems in which all

diagonal lines are of equal length, the entropy would be expected to be
Webber & Zbilut 

49

0.0 bins/bin (but see Figure 2.12). The units of bits per bin come from

taking the base-2 logarithm. For further discussion of ENT see

Pellechia and Shockley (Chapter 3).

ENT = –∑(Pbin)log2(Pbin) [2.11]

The fifth recurrence variable is trend (TND), which quantifies the

degree of system stationarity. If recurrent points are homogeneously

distributed across the recurrence plot, TND values will hover near zero

units. If recurrent points are heterogeneously distributed across the

recurrence plot, TND values will deviate from zero units. TND is

computed as the slope of the least squares regression of %local

recurrence as a function of the orthogonal displacement from the

central diagonal. Multiplying by 1000 increases the gain of the TND

variable. For the Hénon attractor (Figure 2.7), TND = –4.505 units,

which is within the ± 5 units, confirming system stationarity (Webber et

al., 1995) as achieved by rejecting the first 1000 points.

TND = 1000(slope of %local recurrence vs. displacement) [2.12]

The sixth and seventh recurrence variables, %laminarity

(%LAM) and trapping time (TT), were introduced by Marwan, Wessel,

Meyerfeldt, Schirdewan, and Kurths (2002). %LAM is analogous to

%DET except that it measures the percentage of recurrent points

comprising vertical line structures rather than diagonal line structures.

The line parameter still governs the minimum length of vertical lines to

be included. TT, on the other hand, is simply the average length of
Recurrence Quantification Analysis

50

vertical line structures. For the Hénon attractor (Figure 2.7), %LAM =

2.510% and TT = 2.000, showing that vertical line structures are not

common for this system.

%LAM = 100(#points in vertical lines)/(#recurrent points) [2.13]

TT = average length of vertical lines ≥ parameter line [2.14]

Recurrence plots and recurrence quantifications are strongly

dependent on the sequential organization of the time series or data

string. By contrast, standard statistical measures such as mean and

standard deviation are sequence independent. Random shuffling of the

original sequence destroys the small-scale structuring of line segments

(diagonal as well as vertical) and alters the computed recurrence

variables, but does not change the mean and standard deviation. A

good analogy would be that of Morse code. Random shuffling of the

dots and dashes would not change the percentage of dots and dashes in

the code, but it would certainly alter/destroy the encoded message!

This important idea will be expanded upon when we discuss linguistic

texts and protein codes.

RECURRENCE EPOCHS

So far we have demonstrated that time series data (linear vectors

of sequential scalar measurements of length N) can be embedded into

higher dimensional space by the method of time delays (Takens, 1981).

Distances between all possible vectors are computed and registered in

a distance matrix, specific distance values being based on the selected
Webber & Zbilut 

51

norm parameter. A recurrence matrix (RM) is derived from the

distance matrix (DM) by selecting an inclusive radius parameter such

that only a small percentage of points with small distances are counted

as recurrent (yielding a sparse RM). The recurrence plot (RP), of

course, is just the graphical representations of RM elements at or below

the radius threshold. Seven features (recurrence variables) are

extracted from the recurrence plot within each window (W) of

observation on the time series. The question before us now is how can

these recurrence variables be useful in the diagnosis of dynamical

systems?

Whenever any dynamic is sampled, we are taking a “slice of

life,” as it were. The dynamic was “alive” before we sampled it, and

probably remained “alive” after our sampling. Consider, for example,

the EMG signal recorded from the biceps muscle of a normal human

volunteer and its attendant recurrence plot in Figure 2.8 (Webber,

Schmidt, & Walsh, 1995). The focus is on the first 1972 points of the time

series digitized at 1000 Hz (displayed from 37 ms to 1828 ms). But how

might these digitized data be processed in terms of recurrence

analysis? It would certainly be feasible to perform recurrence

quantifications within the entire window (Wlarge = 1972 points) as

represented by the single, large, outer RM square. On the other hand,

the data can be windowed into four smaller segments (Wsmall = 1024

points) as represented by the four smaller and overlapping RM

squares. In the latter case the window offset of 256 points means the

sliding window jogs over 256 points (256 ms) between windows. Two

effects are at play here. First, larger windows focus on global dynamics

(longer time frame) whereas smaller windows focus on local dynamics
Recurrence Quantification Analysis

52

Figure 2.8. Windowed recurrence analysis of resting biceps brachii EMG signal. The large

outer square displays the large scale recurrence plot (W = 1792 = N points). The four small

inner squares (epochs) block off the small scale recurrence plots (W = 1024 < N) with an offset

of 256 points between windows. Recurrence data computed from file EMG using program

RQD.EXE. RQA parameters: P1-Plast = 1-1972 (for large square) or 1-1024, 257-1280, 513-1536,

769-1792 (for small squares); RANDSEQ = n; NORM = Euclid; DELAY = 4; EMBED = 10;

RESCALE = max dist; RADUS = 15; COLORBND = 1; LINE = 2. RQA variables (for large square):

%REC = 0.335; %DET = 60.905; LMAX = 118; ENT = 1.711; TND = –0.129; %LAM = 55.660; TT =

2.501. RQA variables (for small squares, respectively): %REC = 0.471, 0.407, 0.331, 0.311;

%DET = 68.560, 65.603, 66.474, 68.140; LMAX = 118, 118, 118, 59; ENT = 1.753, 1.778, 1.751,

1.572; TND = –0.488, –0.663, –0.644, –0.464; %LAM = 59.635, 54.435, 45.759, 44.506; TT = 2.454,

2.500, 2.396, 2.425.

(shorter time frame). Second, larger window offsets yield lower timeresolution RQA variables, whereas smaller window offsets yield higher

time-resolution variables. Remember, seven RQA variables are

computed (extracted) from each RM (or RP). By implementing a sliding

window design, each of those variables is computed multiple times,
Webber & Zbilut 

53

creating seven new derivative dynamical systems expressed in terms

of %REC, %DET, LMAX, ENT, TND, %LAM, and TT. Alignment of those

variables (outputs) with the original time series (input) (adjusting for

the embedding dimension, M) might reveal details not obvious in the 1-

dimensional input data.

Here are the important details of this the muscle fatigue

experiment performed by Webber et al. (1995) which illustrate these

fundamental rules of sliding windows (termed epochs). While seated

comfortably, normal human volunteers were instrumented for the

recording of surface EMG signals from the biceps brachii as shown in

Figure 2.9A. Subjects were asked to hold their forearm parallel to the

floor and their elbow at 90°. A 1.4 kg weight was placed in the open

palm and a control EMG recording was digitized at 1000 Hz. After 60

seconds of recording, the weight load was increased to 5.1 kg, which

led to total muscle fatigue in 1 to 6 minutes (or 2.8 minutes for the

example subject in Figure 2.9), depending upon the biceps muscle

mass of the subject. The experiment was designed to compare the

performance of nonlinear RQA and linear spectral analysis on identical

EMG signals. Might the two techniques have differential sensitivities?

The recorded time series (N = 227,957 points) was partitioned in

shorter windows (W = 1024) or epochs, each 1.024 seconds long.

Adjacent windows were offset by 256 points (75% overlap), fixing the

time resolution to 256 ms. Spectral features and recurrence

quantifications were then computed for each of the 887 sliding

windows. As shown in Figure 2.9B and 2.9C, respectively, the spectral

center frequency (FC) and recurrence %DET were stable during the 60

second, low-weight control period. For statistical purposes, 95%
Recurrence Quantification Analysis

54

Figure 2.9. Muscle fatigue experiment in a human volunteer, designed to force the system

through a series of state changes until task failure occurs. (A) Bipolar surface EMG recording

from the biceps brachii muscle during light-weight loading (1.4 kg) before time zero, and

heavy-weight loading (5.1 kg) starting at time zero. After 60 sec of control (steady-state

dynamics), the subject was forced onto a trajectory (transient dynamic) ending in task failure

over the next 167 sec. (B) Spectral center frequency (FC) remains constant during the lightweight loading, but slowly decreases during the heavy-weight loading. The first spectral

detection of fatigue occurs at 63.3 sec when the 3rd degree polynomial breaks out of the control

95% confidence limits (dot). (C) Recurrence variable %DET also remains constant during the

light-weight loading, but increases during the heavy-weight loading. Fatigue is detected by

%DET after only 45.6 sec (dot) or 28% sooner than FC. EMG digitized at a rate of 1000 Hz.

Recurrence data computed from data file EMG using program RQE.EXE. RQA parameters: P1-

Plast (first epoch) = 1-1024; SHIFT = 256; #EPOCHS = 887; RANDSEQ = n; NORM = Euclid;

DELAY = 4; EMBED = 10; RESCALE = max dist; RADUS = 15; LINE = 2.
Webber & Zbilut 

55

confidence limits were drawn for each control trace, and a 3rd degree

polynomial was constructed through each data set. Increasing the load,

however, perturbed the steady-state dynamics and set the biceps’

contractile machinery on a transition trajectory toward muscle fatigue

and ultimate task failure. Examination of the output data shows that

during the fatiguing process, FC decreased whereas %DET increased.

Both variables can be interpreted as indicators of larger motor unit

recruitment and synchronization, but that is not the point being made

here. What is significant is the fact that the %DET broke out of its 95%

confidence limit before FC broke out of its 95% confidence limit (e.g. at

45.6 sec versus 63.3 sec after the heavy loading, respectively, or 28%

sooner). It can be concluded that whereas both spectral and

recurrence strategies are responsive to dynamical changes in

physiological systems such as fatiguing muscle, recurrence

quantification is substantially more sensitive. In other words, subtle

dynamical departures from “steady state” occurring in time series data

might be delayed or even missed by spectral tools, but detected

sooner and/or more accurately by recurrence tools.

This EMG example illustrates the power of sliding recurrence

windows, but we glanced over the determination of the several

important recurrence parameters. As mentioned, each window

consisted of 1024 points (range). But we also selected the Euclidean

norm, maximum distance rescaling, and a line of 2 (all typical choices).

But what about the delay, radius, and embedding dimension

parameters—how were they selected? First, to estimate an ideal delay

time, the control EMG data (58 windows of non-overlapping, adjacent

windows, 1.024 sec each) were subjected to autocorrelation analysis.
Recurrence Quantification Analysis

56

The average delay, based on the first zero crossing of the

autocorrelation function (a zero value indicates no correlation), was

found to be 4 digitized points or 4 ms. Second, to estimate the proper

radius threshold, the beginning control EMG data (1 window or epoch

or 1024 points) were subjected to recurrence scaling analysis. In this

case, recurrence variables were recomputed for a family of radius

values from 0% to 100% in steps of 1% (of the maximum distance in the

distance matrix). Figure 2.10 depicts the results with respect to %REC

and %DET. With increasing RADIUS values, %REC increased smoothly

to its maximum, following either a sigmoidal curve with linear scaling

(Figure 2.10A) or a more linear curve with double logarithmic scaling

(Figure 2.10B). On the other hand, %DET exhibited a hitch or shelf with

a first local minimum at a radius of 17%. This oddity is due to the faster

recruitment of isolated recurrent points than points contributing to

diagonal line structures as RADIUS is incremented (Figure 2.10C; see

Equation 2.9).

There are three guidelines for selecting the proper radius (in

order of preference): (1) RADIUS must fall with the linear scaling region

of the double logarithmic plot; (2) %REC must be kept low (e.g., 0.1 to

2.0%); and (3) RADIUS may or may not coincide with the first minimum

hitch in %DET. Weighing all three factors together, a radius of 15%

was selected for the example EMG data (vertical dashed lines in Figure

2.10), which fits all three criteria. Because there are mathematical

scaling rules linking log(%REC) with log(RADIUS), as will be discussed

below, the first guideline for RADIUS selection is preferred. In contrast,

since there are no known rules describing the hitch region in %DET,

this latter method must be applied with caution—user beware.
Webber & Zbilut 

57

Figure 2.10. Methods for selecting the proper radius parameter for recurrence analysis of the

control EMG recording. (A) With step increases in RADIUS, the density of recurrence points

(%REC) increases along a sigmoid curve (M = 10). (B) Double-logarithmic plot of %REC as a

function of RADIUS defines a linear scaling region from RADIUS = 8% to 15%. RADIUS is

selected at 15% where %REC is 0.471% (sparse recurrence matrix). (C) Linear plot of %DET

as a function of RADIUS showing a short plateau and small trough near RADIUS = 15% which

may or may not be coincidental. Recurrence data computed from file EMG using program

RQS.EXE. RQA parameters: P1-Plast = 1-1024; RANDSEQ = n; NORM = Euclid; DATA MIN = DATA

MAX = 4; EMBED MIN = EMBED MAX = 10; RESCALE = max dist; RADUS MIN = 0; RADIUS MAX

= 100; RADIUS STEP = 1; LINE = 2.
Recurrence Quantification Analysis

58

The third parameter requiring explanation is the embedding

dimension. Why did we set M = 10? The mathematics underlying the

distance matrix are equivalent to the correlation integral implemented

by Grassberger and Procaccia (1983) for dimensional analysis of

dynamical systems. Theoretically, the Grassberger-Procaccia

dimension (GPD) is a measure of the number of independent variables

participating in the system at any given instant. Thus, GPD is another

measure of complexity, since the greater the number of participant

variables, the more complex the system. Practically, the G-P

dimension can be estimated by observing how the number of

recurrences (#REC) scales with increases in absolute radius (R), as

shown by Equation 2.15. The exponential form of this equation is easily

converted to its linear form by taking the logarithm of both sides,

yielding Equation 2.16. Then, substitution into this equation of

%recurrence (%REC) for #REC and relative radius (RADIUS) for

absolute radius (R), respectively, gives Equation 2.17.

#REC = RGPD [2.15]

log(#REC) = log(R)GPD [2.16]

log(%REC) = log(RADIUS)GPD [2.17]

The G-P Dimension is easily computed by taking the ratio

(or slope) of ∆log(%REC) to ∆log(RADIUS) over a linear scaling

region, as discussed by Mayer-Kress and Holzfuss (1987). Such a

double-logarithmic plot is illustrated in Figure 2.11A for embedding
Webber & Zbilut 

59

Figure 2.11. Relationship between log(%REC) and log(RADIUS) over a range of embedding

dimensions (M = 1 to 20) for the control EMG recording. (A) Each curve in the family of curves

has a linear scaling region that is used to estimate the Grassberger-Procaccia dimension (GPD)

when the slope becomes unchanging with increases in M. The dashed line refers to the same

line plotted in Figure 2.10B at M = 10. (B) GPD plotted as a function of embedding dimension

saturates at GPD = 4.292 when M = 10, the proper embedding for this dynamic. Recurrence

data computed from file EMG using program RQS.EXE. RQA parameters: P1-Plast = 1-1024;

RANDSEQ = n; NORM = Euclid; DATA MIN = DATA MAX = 4; EMBED MIN = 1; EMBED MAX =

20; RESCALE = max dist; RADUS MIN = 0; RADIUS MAX = 100; RADIUS STEP = 1; LINE = 2.

dimensions ranging from 1 to 20. As shown in Figure 2.11B, as the

embedding dimension (M) increases, the slope increases to a

“plateau” of GPD = 4.292 (a fractal dimension) starting at M = 10.

Further embeddings do not result in any further increases in GPD. This

is the reason why we selected M = 10. As is most often the case for real

data, it should be recognized that M > GPD. The reason for this is that

noise artificially inflates the dimension. Parka and Chua (1989) stated
Recurrence Quantification Analysis

60

that for noisy systems, the embedding dimension maximizes at M = 2 ×

GPD + 1. An M of 10 for our EMG signal is very close to this theoretical

limit (M = 2 × 4.292 + 1 = 9.6). There are numerous practical problems

associated with the estimation of M, including non-stationary dynamics,

high-dimensional attractors, and the presence of noise. For example,

inspection of the log-log plots (Figure 2.11B) shows a worsening

“double hump” of unknown origin in the curves for M > 10. Therefore,

we typically bypass any G-P analysis or nearest-neighbor strategy, and

usually set parameter M to 10, but never higher than 20, for real data.

RECURRENCE INTERVALS

Recurrence intervals (or recurrence times) quantify the

perioidicities expressed by dynamical systems. By way of one

mundane example, it is proper to say that the recurrence interval (or

recurrence time) for the earth’s sun is 24 hours, day in and day out. But

for heuristic purposes, let us revisit the coupled Hénon equations

discussed above (Equations 2.6 and 2.7) in which the constant

multiplier of the nonlinear term Xi

2 is changed from 1.4 to 1.054 as

follows:

Xi+1 = Yi + 1.0 – (1.054Xi

2

), [2.18]

Yi+1 = 0.3Xi. [2.19]

Such minor tweaking of the equation converts the Hénon chaotic

attractor into the Hénon 16-point periodic attractor in phase space (see

16 crosshairs in Figures 2.6A, B, & C). Expressing this periodic
Webber & Zbilut 

61

attractor in recurrence space, however, produces a series of parallel

lines spanning from border to border as shown in Figure 2.12.

Recurrence interval analysis verifies that the vertical spacing between

recurrent points throughout the entire plot (including recurrent points

in the upper and lower triangles, as well as the central diagonal) is a

perfect 16 points, as expected. Since the border effectively truncates

diagonal lines to different lengths, an artifactual ENT > 0.0 bit/bin is

observed.

Figure 2.12. Recurrence plot of the 16-point Hénon periodic X variable. The same variable is

plotted on the horizontal (periodic Xi) and vertical axes (periodic Xj) from iterations of coupled

Equations 2.18 and 2.19. When vectors of 3 points (e.g. M = 3) match, recurrent points are

plotted at the corresponding (i,j) intersections. The most striking feature of this plot is the long

diagonal line structures parallel to the main diagonal and offset by exactly 16 points.

Recurrence data computed from file HENPX using program RQD.EXE. RQA parameters: P1-Plast

= 1001-1200; RANDSEQ = n; NORM = Euclid; DELAY = 1; EMBED = 3; RESCALE = max dist;

RADUS = 0.5; COLORBND = 1; LINE = 2. RQA variables: %REC = 5.789; %DET = 100.000; LMAX

= 184; ENT = 3.585 (line truncation effect); TND = 12.449; %LAM = 0.000; TT = undefined.
Recurrence Quantification Analysis

62

Moving from theoretical mathematics to practical physiology, it

is important to demonstrate the power and utility of recurrence interval

analysis on revealing subtle features in human ectroencephalographic

(EEG) signals. To illustrate this, EEG data from a normal quiescent

subject were obtained from a local clinical electrophysiology

laboratory. The subject was instrumented with 21 active unipolar

electrodes referenced to the nose according to the standard 10-20 EEG

recording system (electrodes spaced at intervals 10%-20% of head

circumference) (Misulis & Head, 2003). The signals were digitized at

500 Hz and bandpass filtered (0.15 Hz - 70.0 Hz). However, instead of

selecting just one of the active electrode sites for analysis, Euclidean

distances (ED) were computed for each instance in time (every 2 ms)

across all 21 electrodes (En) according to Equation 2.20. In effect, an

[N,21] matrix of 21 parallel vectors was collapsed into a single [N]

vector time series with N points.

EDi = √ (E1i

2

 + E2i

2

 + E3i

2

 + ... + E20i

2

 + E21i

2

) [2.20]

Recurrence analysis was conducted on 2000 points (4.0 s) of the

composite EEG signal (Euclidean vector), which is plotted horizontally

and vertically in Figure 2.13. Because the signal was already projected

into higher dimensional space (D=21), it was inappropriate to set the

embedding dimension to anything other than one (M = 1). The

minimum norm was chosen and the delay parameter was set equal to

one (τ = 1), but neither parameter was critical since with an embedding

dimension of one, no points were time delayed. The radius was set to

0.2% of the maximum distance rescaling, insuring that %REC < 1%.
Webber & Zbilut 

63

Figure 2.13. Recurrence interval analysis of a normal human electroencephalogram (EEG)

digitized at 500 Hz. The time series data are derived from computing the Euclid distances

across 21 simultaneously recorded leads of the 10-20 system of electrodes positioned around

the skull. Voltages were updated at each instant in time (2 msec) for 40 sec (2000 data points)

for recurrence analysis. The recurrence plot has both fine-scale and large-scale features that

were captured by measuring the vertical time interval between all recurrent points (see Figure

2.14). Recurrence data computed from file EEG using program RQD.EXE. RQA parameters: P1-

Plast = 1-2000; RANDSEQ = n; NORM = min norm; DELAY = 1; EMBED = 1; RESCALE = max dist;

RADIUS = 0.2; COLORBND = 1; LINE = 2. RQA variables: %REC = 0.804; %DET = 7.318; LMAX

= 5; ENT = 0.445; TND = –0.102; %LAM = 11.207; TT = 2.087. These EEG data were kindly

provided by Lukasz Konopka, Ph.D., Director of Biological Psychiatry, Hines V.A., Hines, IL

60141.

The recurrence plot for this system (Figure 2.13) reveals a much

more complex structuring than that observed with the Hénon 16-point

periodic system (Figure 2.12). In fact, the seven recurrence variables

were quantified for this EEG signal as follows: %REC = 0.804%; %DET

= 7.318%; LMAX = 5; ENT = 0.445; TND = –0.102; %LAM = 11.207%; TT
Recurrence Quantification Analysis

64

= 2.087. The homogeneous distribution of recurrent points is quantified

by the near zero TND value, indicating the stationary state of the EEG

signal in this individual. The %DET value indicates that the signal has

deterministic features arising from repeated (recurrent) EEG waves at

various frequencies. Likewise, the %LAM value reveals significant

laminate structuring of recurring dynamical features in the vertical

plane (strings of multiple j-points recurring with single i-points).

However, it is the vertical spacing between recurrent points

(recurrence intervals) that is of interest to us.

Recurrence quantification interval (RQI) analysis was conducted

on this example EEG signal using the exact same parameter settings as

indicated above. The distribution of recurrence intervals is plotted in a

double logarithmic histogram in Figure 2.14. Of the 32,142 intervals

counted, 2,075 (6.46%) recurrence intervals are located in the first bin

at 2 ms. These points are usually excluded as noise since they come

from adjacent points forming vertical line structures (whence see

%LAM). Recurrence intervals spanning 4 to 398 ms were distributed

rather uniformly, but over the range from 400 through 1,600 msec the

recurrence counts scaled with recurrence time. The remainder of

intervals from 1,602 to 3,398 ms contained mostly single-digit counts

and comprised the noise floor beyond the scaling region. Within the

scaling regions, however, slope (S) defines the scaling relationship

between the number of intervals (#INT) and interval length (L).

#INT = LS [2.21]

log(#INT) = log(L)S [2.22]
Webber & Zbilut 

65

Figure 2.14. Double-logarithmic scaling relation between number of recurrence intervals

(tally counts) and duration of recurrence intervals (ms) computed from a normal EEG signal

(Figure 2.13). A scaling relation spans the range 400 to 1600 ms and has a significant negative

slope (p < 0.0000001) of –1.974 (dashed line) indicative of a 1/INT2 process. Recurrence data

computed from file EEG using program RQI.EXE. RQA parameters: P1-Plast = 1-2000; RANDSEQ

= n; NORM = max norm; DELAY = 1; EMBED = 1; RESCALE = max dist; RADUS = 0.2.

The linear scaling region (Figure 2.14) has a negative slope S =

–1.974, which is consistent with a 1/INT2 process (where INT is the

recurrence interval) and is highly significant (p < 0.000001) for the 380

points defining the scaling region. These data indicate that there are

scaling rules in place for the lowest EEG frequency (delta < 4 Hz), but

not the higher frequencies (beta > 13 Hz; alpha > 8 Hz; theta > 4 Hz)

(Misulis & Head, 2003). This type of RQI analysis is analogous to

uncovering 1/f scaling rules, but in the latter case the lower

frequencies (longer intervals) carry most of the power

(Bassingthwaighte et al., 1994; see also Aks, Chapter 7, and Liebovitch

& Shehadeh, Chapter 5).

Thomasson, Heoppner, Webber, and Zbilut (2001) and

Thomasson, Webber, and Zbilut (2002) performed RQI analysis on
Recurrence Quantification Analysis

66

human EEG signals and were able to demonstrate different scaling

regions for control EEGs versus pre-ictal EEGs. That is, just prior to

seizure activity, the scaling relation became more correlated (steeper

negative slope), consistent with a self-organizing process leading to

synchronized and focal brain activity. These studies provide evidence

that subtle recurrence changes in the electrical dynamic may forecast

seizure states in the brain.

CROSS RECURRENCE

So far we have been speaking of recurrence quantitative analysis

from the single signal perspective. That is, in RQA recurrence patterns

are sought within individual signals in the spirit of autocorrelation.

However, in the spirit of cross correlation, it is possible to detect

recurrence patterns shared by paired signals by cross recurrence

analysis (KRQA) (Zbilut, Giuliani, & Webber, 1998a; Marwan et al.,

2002; Marwan 2003a; see also Shockley, Chapter 4). The mathematics

of cross recurrence, as well as the parameters and variables of cross

recurrence, all are the same as explained for auto-recurrence. There

are two principle differences, however. First, in KRQA distances are

computed between two different signals. It is assumed that the two

signals arose from coupled systems, were sampled simultaneously at

the same digitization rate (which prohibits the drifting of one signal

with respect to the other), and are on the same amplitude scale (which

permits low distances between signals to be computed). The last

requirement can be achieved by rescaling the input data over the unit

interval (minimum:maximum mapped to 0.0:1.0), provided that neither

of the paired input data streams possess strong nonstationaity (upward
Webber & Zbilut 

67

or downward drifts). Second, in KRQA the shared symmetry between

the upper and lower triangles of the auto-recurrence plot is lost. In fact

the entire central line of identity vanishes. Because of these expected

differences from auto-RQA, quantitative cross-recurrence

quantifications must be made across the entire matrix (plot), not just the

upper half (upper triangle).

Shockley, Butwill, Zbilut, and Webber (2002) performed a very

simple experiment on mechanically coupled oscillators. A tray (which

served as a driver), filled with fluids of differing viscosities (oil, syrup,

or honey), was set into reciprocal, sine-wave motion by a strong

sinusoidal motor. A rotor blade (the follower—what was driven by the

driver) was positioned within the viscous fluid and was set into

independent motion by a pulley system and falling weight (gravity

activated). The position of the tray and rotor were monitored by

separate motion-tracking sensors and digitized at 60 Hz. Four

experimental runs were performed at each of three viscosities.

Transients due to acceleration were rejected and 600 points of steady

state data (10 s) were collected for each trial. KRQA was performed

using the driver-tray as the i-signal (i-vector, Vi) and the follower-rotor

as the j-signal (j-vector, Vj). The cross recurrence plot for one of the

trials at a medium coupling viscosity is presented in Figure 2.15 (M = 5;

DELAY = 1; RADIUS = 2.0%). The long diagonal, always present for

auto-recurrence plots (e.g., Figures 2.2, 2.7, 2.8, 2.12, and 2.13), is

noticeably absent. But short diagonals appear in the plot wherever

waveform coupling occurs between the driver (independent and

constant frequency) and rotor (dependent and variable frequency).
Recurrence Quantification Analysis

68

Figure 2.15. Cross recurrence plot of coupled oscillators in a low-viscosity coupling. The

positions of the different oscillators are plotted along the horizontal axis (driver oscillator) or

vertical axis (rotor oscillator). When vectors of 5 points each match (e.g. M = 5), recurrent

points are plotted at the corresponding i,j intersections. The most striking feature of this plot is

the unusual distribution of recurrent points reflective of various phasic couplings between the

two oscillators. Recurrence data computed from files DRIVER and ROTOR using program

KRQD.EXE. RQA parameters: P1-Plast = 1-596; RANDSEQ = n; NORM = Euclid; DELAY = 1;

EMBED = 5; RESCALE = max dist; RADUS = 2; COLORBND = 1; LINE = 2. RQA variables: %REC

= 1.480; %DET = 94.103; LMAX = 24; ENT = 3.041; TND = –0.813; %LAM = 74.834; TT = 3.300.

Cross recurrence interval analysis (KRQI) on the coupled

oscillators results in the frequency spectrum plotted in Figure 2.16. To

accomplish this, recurrence intervals were transformed to the

frequency domain by taking their reciprocals. For spectral

comparison, the Fast Fourier transform (FFT) spectrum is computed on
Webber & Zbilut 

69

Figure 2.16. Frequency characteristics of the low-viscosity coupled driver-rotor system from

linear (FFT) and non-linear (KRQI) perspectives. Spectral analysis (FFT) of the rotor dynamic

shows a dominant (uncoupled) frequency faster than the driver frequency at low resolution.

Cross recurrence analysis (1/KRQI) reveals similar features but with greater details at high

resolution. For example, the high frequency peak is splayed out over a wider frequency band

due to subtle nonlinear interactions not detected by FFT. Recurrence data computed from

coupled files DRIVER and ROTOR using program KRQI.EXE. RQA parameters: P1-Plast = 1-596;

RANDSEQ = n; NORM = Euclid; DELAY = 1; EMBED = 5; RESCALE = max dist; RADUS = 2; LINE

= 2.

the rotor data only. The linear FFT spectrum reports two peaks, a low

amplitude peak near the driver frequency (dashed line), and a high

amplitude peak at a higher frequency (non-harmonic). The nonlinear

KRQI spectrum provides much more details on the partially coupled

system. More spectral power is found at the higher, uncoupled

frequency, but the frequency band is splayed out due to nonlinear

interactions between the rotor and driver. At the lower, coupled

frequency (dashed line), the KRQI resolution is superior to the FFT

resolution. More importantly, subtle changes in relative viscosity from
Recurrence Quantification Analysis

70

0.693 to 0.711 units (1.00 units for high viscosity coupling) gave

significant changes in KRQA variables %REC, %DET, and LMAX (not

shown). This example illustrates the ease with which cross-recurrence

analysis handles non-linearities in short data sets, and at a higher

resolution than the FFT to boot.

There is still much work to be done with respect to crossrecurrence analysis, but it is posited that KRQA may be a formidable

tool in the examination of coupled oscillators. This may be particularly

true for coupled biological oscillators where subtle, short scale

changes in coupling may indicate (forecast) altered dynamics that may

or may not be healthful for the examined system in part or even the

entire organism as a whole. Shockley (Chapter 4) presents a

psychological application of KRQA.

RECURRENCE PATTERNS OF SPEECH

Recurrence analysis, auto-recurrence or cross-recurrence, is

fully amenable to linguistic systems or symbolic dynamics. Actually,

one of the easiest explanations of recurrence can be purchased at the

price of the simple children’s book authored by Dr. Seuss (Geisel,

1960), Green Eggs and Ham. Webber and Zbilut (1996, 1998) have

repeatedly used this example for instructive purposes. The reasoning

goes as follows. Ask a child this riddle, “How can Dr. Seuss write a

book with 812 words if he has a limited vocabulary of only 50 words?”

The obvious answer is that words must be reused—that is, words recur.

While we are at it, why not ask the child, “How can books with

thousands of words be written in English, if there are only 26 alphabet

letters available?” In this case, letters must be reused. So at the word
Webber & Zbilut 

71

level or orthographic (spelling) level, symbols are simply reused in

any combination desired by the author, as long as they correspond to

allowable words in the language of choice. Common experience

informs us that letters in words or words in sentences do not, and need

not, appear at periodic intervals. Rather, actual linguistic sequences

are at once highly nonlinear and highly meaningful as well. In this

context, Orsucci, Walter, Giuliani, Webber, and Zbilut (1999)

implemented RQA to study the linguistic structuring of American

poems, Swedish poems, and Italian translations of the Swedish poems.

They found invariance among the various language exemplars,

suggesting hidden structuring at the orthographic level.

It is intriguing to consider the potential ability of recurrence

strategies in the analysis of written text or spoken words as first

explored by Orsucci et al. (1999) using American and Italian speech

samples. Do different authors or various speakers have specific

recurrence signatures that betray their individual identities? We might

proceed at the orthographic level, rendering any speech text numeric

by arbitrarily substituting integers for letters: A=1; B=2; C=3;...; X=24;

y=25; Z=26; and for numbers: 0=27; 1=28; 2=29 ...; 7=34; 8=35; 9=36.

We can keep things simple by ignoring cases of letters, all punctuation

marks, spaces, carriage returns, and linefeeds. But how should the

recurrence parameters be set? Well, since the encoding scheme is

entirely arbitrary (we could have used: Z=1; Y=2; X=3; ...; etc.), the

most important constraint is that the radius must be set to 0 distance

units. This will insure that only identical letters (unique integers) will

recur with each other. The embedding dimension can be set to one or

higher, but for M > 1 the delay should be set to one so as not to skip any
Recurrence Quantification Analysis

72

letters in the string. The length of the text sets the maximum size of the

window, but smaller windows can parcel the text into a series of

epochs. It makes no difference whether the distance matrix is rescaled

or not, because the only allowable radius is 0 units. The line parameter

should be set to 2, lest one wants to exclude short words longer than

two characters each (but that is not advised). With these preliminaries

complete, what might diagonal line structures in the recurrence plot

signify? If only identical letters count as recurrent points, a string of

diagonal recurrences must indicate that the identical string of

characters appears at different positions in the text. Actually, lines of

varying length must represent words of varying length. Very loosely

speaking, we might (correctly) envisage words as vectors of letters! In

any case, recurrence quantifications can be captured in the seven

recurrence variables we have discussed at length with respect to other

dynamical systems.

But let us leave the letter level, and focus rather on words. We

now face a new problem, one that involves choosing a scheme for

encoding words. Unlike English letters that are limited to 26 characters

and 10 digits, English words can number in the hundreds of thousands.

For example, The Oxford English Dictionary Online (2003) has some

290,000 entries and 616,500 different English word forms. To encode

words, we can assign ascending integer values to each new word, but

whenever a former word recurs, its old integer value must be reused.

To keep it simple, we can treat all punctuation marks as spaces. After

the full text is encoded, the total number of integers in the derived file

must equal the total number of words in the text. The number of
Webber & Zbilut 

73

different words in the text (vocabulary size) will be represented by the

numerical value of the largest integer.

It has long been known that the psychiatric illness of

schizophrenia is characterized by disordered speech (Kasanin & Lewis,

1944). So to provide a practical example of quantitative textual analysis

using RQA strategies, let us examine the speech patterns of a

schizophrenic patient and “normal” academic as quoted by Wróbel

(1990). Each quote consists of exactly 165 words. First quoted is the

schizophrenic (Wróbel, 1990, p. 77), the context of which reveals this

patient’s altered sense of reality. To get the gist of how the 165 words

were encoded, here are the codes for the first 26 words (“pre-started”

counts as 2 words due to replacement of the dash with a space): 1-2-3-

4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-14-15-16-3-19-18-14-20 (note

the repeat pattern of 14-15-16: “before me and”):

In Wroclaw I pre-started to pray, you know, the

psychiatrist Kobohen came before me and he stood before

me and I also stood before him, because he came to the

ward on a visit, you know, he came before and he says:

oh, that's the new one—he says—he arrived today—he

says and he made a sign of the cross, you know, like this

before me. I felt in the presence of that such terrible

desires to pray because of that cross, that I began praying

incredibly, I prostrated myself, I prayed on my knees,

prostrate, I so implored the Lord God as much as possible,

you know, and I felt myself a ruler, you know, I thought I

was the supreme ruler on this earth, that over the whole

world I was the supreme ruler, I began praying so

incredibly with various crosses, yes I prayed so incredibly

with crosses, with perfection and in different ways. He

was dismissed from there, ...
Recurrence Quantification Analysis

74

Second quoted is a normal academic (Wróbel, 1990, p. 99) who

speaks logically and with intent purpose. The first 26 words (of 165

words) were encoded as follows: 1-2-3-4-5-6-7-8-9-10-11-9-10-12-13-14-

15-6-16-17-18-19-20-21-22-23 (note the repeat pattern of 9-10: “of the”).

Newtonian mechanics, for example, imposes a unified

form of the description of the world. Let us imagine a

white surface with irregular black spots on it. We then say

that whatever kind of picture these make, I shall be able to

approximate as closely as I wish to the description of it by

covering the surface with a sufficiently fine square mesh,

and saying of every square whether it is black or white. In

this way I shall have imposed a unified form on the

description of surface. The form is optional, since I could

have achieved the same result by using a net with a

triangular or hexagonal mesh. Possibly the use of a

triangular mesh would have made the description simpler:

that is to say, it might be that we could describe the

surface more accurately with a coarse triangular mesh

than with a fine square mesh (or conversely), and so on.

The different nets correspond to different systems for

describing the world.

Recurrence plots were constructed for the 165 words of text

comprising each subject’s speech as shown in Figure 2.17. Here, M =

1, DELAY = 1, and RADIUS = 0. The “time series” beside each

recurrence plot present as “saw-tooth” patterns due to the reusing

(recurrence) of words. Recurrent points are plotted only when exact

word matches are found. Recurrence quantifications are reported in

Table 2.1 for these two subjects, both at the orthographic and word

levels, before and after random shuffling. The 742-character normal

text was truncated to 670 characters to match the number of characters
Webber & Zbilut 

75

Figure 2.17. Recurrence analysis of human speech patterns at the word level. (A)

Schizophrenic speech. (B) Normal speech. Recurrences occur only for exact word matches.

Recurrence data computed from individual files SCHIZWRD and NORMWRD using program

RQD.EXE. RQA parameters: P1-Plast = 1-165; RANDSEQ = n; NORM = Euclid; DELAY = 1; EMBED

= 1; RESCALE = absolute; RADUS = 0; COLORBND = 1; LINE = 2. RQA variables for

schizophrenic speech (A): %REC = 1.870; %DET = 26.087; LMAX = 5; ENT = 0.675; TND =

–8.361; %LAM = 0.000; TT = undefined. RQA variables for normal speech (B): %REC = 1.567;

%DET = 23.113; LMAX = 3; ENT = 0.773; TND = –1.898; %LAM = 0.000; TT = undefined. See

also Table 1.
Recurrence Quantification Analysis

76

in the schizophrenic text. Cursory review of the data reveal more

differences at the word level than orthographic level. Shuffling

decidedly distorts the deterministic structures. Admittedly, these

examples are only anecdotal exercises without statistical merit. But

what is being emphasized is the potential for nonlinear recurrence

methodology to be employed on different speech patterns from many

subjects, with careful attention being paid to proper experimental

design.

Table 2.1. Recurrence quantification analysis of human speech patterns. Data were

computed from native sequences (and shuffled sequences) of letters (data files

SCHIZLET and NORMLET) and words (data files SCHIZWRD and NORMWRD) using

program RQD.EXE.

RQA

Variable

Schizophrenic

Letters

(N = 670)

Normal

Letters

(N = 670)

Schizophrenic

Words

(N = 165)

Normal

Words

(N = 165)

%REC 6.511%

(6.511%)

6.312%

(6.312%)

1.870%

(1.870%)

1.567%

(1.567%)

%DET 22.257%

(12.019%)

22.280%

(11.600%)

26.087%

(3.953%)

23.113%

(0.943%)

LMAX 19

(4)

16

(4)

5

(2)

3

(2)

ENT 0.995

(0.317)

0.986

(0.359)

0.675

(0.000)

0.773

(0.773)

TND 0.104

(0.029)

0.251

(0.240)

–8.361

(2.108)

–1.898

(-1.898)

%LAM 4.495%

(12.636%)

1.484%

(18.576%)

0.000%

(11.067%)

0.000%

(0.943%)

TT 2.000

(2.185)

2.000

(2.104)

2.000

(undefined)

undefined

(2.000)
Webber & Zbilut 

77

RECURRENCE PATTERNS OF PROTEINS

It is a simple matter to move from recurrence analysis of English

texts to amino acid patterns of proteins. The amino acid alphabet is a

20-letter code of naturally occurring amino acids found in eukaryotic

proteins. Protein function depends upon protein shape in 3-

dimensional space (tertiary and quartenary structures), and protein

shape depends upon the specific 1-dimensional amino acid sequence.

Thus, recurrence analysis of amino acid sequences may yield

interesting differences among the various classes of proteins. One

approach would be to encode amino acids with unique integers from 1

to 20 and perform RQA computations using M = 1, DELAY = 1, and

RADIUS = 0. When this is done, however, it has been shown that amino

acid sequences are only slightly different from shuffled sequences

(Webber & Zbilut, 1998). That is, naturally occurring sequences are

almost random sequences, at least in terms of arbitrary amino acid

names (residues). But even at this lowly name level, cross recurrence

plots may prove useful in identifying stretches of identical residue

patches (e.g., between protein isoforms) which would graphically plot

as long diagonal lines (identical matches) with intervening gaps (mismatches).

A change in perspective is in order. Thinking through the

problem, it becomes clear that what imparts a 3-dimensional shape to

proteins is the physical properties of their 1-dimensional string of

amino acids. Specific sequences of amino acids twist and fold on

themselves, forming protein globules with specific features. Two of the

strongest physical properties that amino possess are their

hydrophobicity and net charge. By merely substituting the name of
Recurrence Quantification Analysis

78

each amino acid by one of its physical properties, it is possible to

generate new sequence amenable for recurrence analysis in higher

dimensions.

To illustrate, consider human hemoglobin, a quartenary protein

with two alpha-chains (141 amino acids each) and two beta-chains (146

amino acids each) of amino acids. Recurrence quantifications were

performed on the alpha chain, first at the orthographic level (M = 1;

DELAY = 1; RADIUS = 0% maximum distance), and second following

hydrophobic substitutions using the hydrophobicity scale of Miyazawa

and Jernigan (1996) (M = 3; DELAY = 1; RADIUS = 5% maximum

distance). Recurrence plots for both representations of the same

system are shown in Figure 2.18. Recurrences seem denser for the

hemoglobin residue names (orthographic level) then for their

hydrophobic substitutions. Conversely, the diagonal line structuring

seems to be more common in hydrophobic space. But we cannot trust

our eyes, which always see things from our own biased perspective.

Nevertheless, these first impressions are confirmed in the data of Table

2.2, which lists the results of RQA performed on native residue

sequences as well as their randomly shuffled versions. Thus the

hydrophobic representation of hemoglobin has a lower %REC but

higher %DET than at the residue name (“spelling”) level. The

hydrophobic signal is also more complex (higher ENT value), a

reflection of its greater non-stationarity (higher absolute TND value).

The bottom line is that by transforming single-letter amino acid

residues into hydrophobic signals, the amino-acid sequence can be
Webber & Zbilut 

79

Figure 2.18. Recurrence analysis of human hemoglobin protein (alpha chain) at two levels.

(A) Residue name level. Recurrences occur only for identical residue matches. Recurrence

data computed from file HEMOALET using program RQD.EXE. RQA parameters: P1-Plast = 1-139;

RANDSEQ = n; NORM = Euclid; DELAY = 1; EMBED = 1; RESCALE = absolute; RADUS = 0;

COLORBND = 1; LINE = 2. RQA variables: %REC = 7.518; %DET = 15.633; LMAX = 3; ENT =

0.219; TND = –2.002; %LAM = 12.129; TT = 2.000. (B) Residue hydrophobicity level.

Recurrences occur for similar hydrophobic patches. Recurrence data computed from file

HEMOAHYD using program RQD.EXE. RQA parameters: P1-Plast = 1-139; RANDSEQ = n; NORM =

Euclid; DELAY = 1; EMBED = 3; RESCALE = max dist; RADUS = 10; COLORBND = 1; LINE = 2.

RQA variables: %REC = 2.669; %DET = 46.875; LMAX = 5; ENT = 1.113; TND = –6.289; %LAM =

0.781; TT = 2.000. See also Table 2.
Recurrence Quantification Analysis

80

projected into higher dimensional space (M > 1) where approximate

recurrences become legitimate (RADIUS > 0). For these and other

reasons RQA is making significant headway into the problem of protein

folding (Zbilut, Sirabella, Giuliani, Manetti, Colosimo, & Webber, 2002;

Giuliani, Benigni, Zbilut, Webber, & Sirabella, 2002).

Table 2.2. Recurrence quantification analysis of human hemoglobin protein. Data were

computed from native sequences (and shuffled sequences) of protein letters (data file

HEMOALET) and protein hydrophobicities (data file HEMOAHYD) using program

RQD.EXE.

RQA

Variable

Hemoglobin-A

Residue Names

(N = 141)

Hemoglobin-A

Hydrophobicity

(N = 141)

%REC 7.518%

(7.518%)

2.669%

(2.627%)

%DET 15.633%

(10.377%)

46.875%

(41.667%)

LMAX 3

(3)

5

(4)

ENT 0.219

(0.176)

1.113

(0.957)

TND -2.002

(-8.333)

-6.289

(-5.814)

%LAM 12.129%

(11.860%)

0.781%

(8.730%)

TT 2.000

(2.000)

2.000

(2.000)
Webber & Zbilut 

81

SUMMARY

The focus on this chapter has been on the proper implementation

strategies of RQA. We have not emphasized the large and growing

recurrence literature amassed over the last decade. Most of this

literature, exceeding 170 references, is accessible from the website of

Marwan (2003b). But it is appropriate here to just list several of the

fields impacted by nonlinear recurrence analysis over the last decade:

Chaos science (Trulla, Giuliani, Zbilut, & Webber, 1996), respiration

(Webber, 1991), heartbeat dynamics (Giuliani, Piccirillo, Marigliano, &

Colosimo, 1998), blood pressure (Mestivier, Dabire, Safar, & Chau,

1998), otoacoustic emissions (Zimatore, Hatzopoulos, Giuliani, Martini,

& Colosimo, 2002), muscle fatigue (Farina, Fattorini, Felici, & Filligoi,

2002), postural control (Riley, Balasubramaniam, & Turvey, 1999),

genomics (Bultrini, Pizzi, Del Giudice, & Frontali, 2003), proteomics

(Zbilut et al., 2002), molecular dynamics (Manetti, Ceruso, Giuliani,

Webber, & Zbilut, 1999), and even finance (Antoniou & Vorlow, 2000).

In fact, any time series (or spatial series, for that matter) is amenable to

RQA. If it wiggles in time (physiology) or space (anatomy) RQA can

quantify the dynamic.

For parting thoughts, the mere fact that ants can lift 50 times their

body weight whereas humans cannot is one lowly illustration of nonlinearity in our shared world with insects. As we have seen, many of

the dynamical, real-world systems common to man are notoriously nonlinear, non-stationary, and noisy. To successfully analyze such systems

it seems best to use non-linear tools that are independent of prescribed

statistical distributions of the data (e.g., Gaussian), can deal with short

data sets (e.g., polypeptides), are not stymied by actual signal
Recurrence Quantification Analysis

82

transients and outliers (e.g., observations beyond a few standard

deviations of the mean), and can project the input signal into higherdimensional space (e.g., embedding capabilities). So far, recurrence

analysis seems to fit the bill in all of these important aspects. But the

recurrence approach is not a stand-alone technique, some kind of endall, be-all of extant nonlinear dynamical tools. For instance, significant

advances are also being made by combining recurrence quantification

analysis (RQA) with principle component analysis (PCA). By this means

the complex relationship among the several recurrence variables can

be objectively compared and processed as an ensemble (Zbilut,

Giuliani, & Webber, 1998b; Giuliani, Colafranceschi, Webber, & Zbilut,

2001).

RQA functions like a microscope, snooping out higherdimensional subtleties in the dynamics that are not obvious its firstdimensional representation. Returning to the animal world, one

analogy would be with farmers who can best forecast weather patterns

by observing altered behaviors of their barnyard livestock. Evidently,

these domesticated animals can sense actual physical phenomena

beyond the human sensory range. Indeed, Marino and Frilot (2003)

used RQA strategies to convincingly demonstrate the sensitivity of

rabbits to magnetic fields. In addition, Mohr, Langbein, and Nürnberg

(2002) applied RQA to assess stress levels in calves and cows. But

whatever the case, whether it be forecasting dynamical events in the

medical field, geophysics, or meteorology, the future of recurrence

analysis looks bright and promising.
Webber & Zbilut 

83

SOFTWARE

Some investigators are writing their own programs to perform

recurrence quantifications. But for those wishing to download

recurrence software from the World Wide Web, at least three sites are

available as shown below: Visual Recurrence Analysis (VRA) from

Eugene Kononov; Cross Recurrence Plots-Toolbox (CRP) from Norbert

Marwan; RQA (Recurrence Quantification Analysis) from Charles

Webber.

VRA 4.6: http://home.netcom.com/~eugenek/download.html

CRP 4.5: http://www.agnld.uni-potsdam.de/~marwan/intro.html

RQA 8.1: http://homepages.luc.edu/~cwebber
Recurrence Quantification Analysis

84

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Recurrence Quantification Analysis

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APPENDIX: Mathematical Construction of the Recurrence Matrix

(Webber, 2004)

RQA employs the method of time delays to embed experimental data into

higher dimensions. Explicit examples of how distance matrices (DM) and

recurrence matrices (RM) are constructed for Euclidean, maximum, and

minimum norms are detailed below for a contrived time-series vector (TS)

with 29 elements.

TS = [3.7, 9.2, 2.1, –5.4, 0.0, –10.9, 9.2, 3.1, 1.7, 1.8, –0.3, –4.9, 2.7, 3.5, 

7.5, –9.9, –9.9, –4.7, 1.3, 2.7, 7.6, 3.9, 7.3, 8.0, 0.3, –1.9, 5.1, 8.8, 8.2]

For DELAY=8, EMBED=4, FIRST=1 and LAST=5, the following 5 time-delay

vectors are constructed:

V1 = [+3.7, +1.7, –9.9, +0.3]

V2 = [+9.2, +1.8, –4.7, –1.9]

V3 = [+2.1, –0.3, +1.3, +5.1]

V4 = [–5.4, –4.9, +2.7, +8.8]

V5 = [+0.0, +2.7, +7.6, +8.2].

Comparison of the 5 vectors constructs a single 5 × 5 recurrence matrix of

distances for each of the 3 norm types. For example, the Euclidean distance

between vectors V4 and V5 is calculated as follows:

DM(Euclid norm at i=4, j=5) = SQRT (SQR(–5.4 – 0.0)+(SQR(–4.9 – 2.7)+

SQR(2.7 – 7.6) + SQR(8.8 – 8.2)) = 10.549.

To compute the maximum and minimum distances between vectors V4 and

V5, the vectors are compared element by element.

ABS(–5.4 – 0.0) = 5.4

ABS(–4.9 – 2.7) = 7.6 (largest difference)

ABS(2.7 – 7.6) = 4.9

ABS(8.8 – 8.2) = 0.6 (smallest difference)

By definition, maximum and minimum distances are simply the maximum and

minimum differences, respectfully.

DM(max norm at i=4, j=5) = 7.6

DM(min norm at i=4, j=5) = 0.6
Webber & Zbilut 

93

This procedure is repeated for each cell, giving the following results for

DM(i,j). Only the distances in the upper triangle are shown since the lower

half is perfectly symmetrical. Note that the central diagonal is designated by

0.00 distances (vector identity matches).

DM(Euclid norm) =

[1,5]=19.58; [2,5]=18.41; [3,5]=7.92; [4,5]=10.55; [5,5]=0.00

[1,4]=18.90; [2,4]=20.67; [3,4]=9.65; [4,4]=0.00

[1,3]=12.45; [2,3]=11.83; [3,3]=0.00

[1,2]=7.88; [2,2]=0.00

[1,1]=0.00

DM(max norm) =

[1,5]=17.50; [2,5]=12.30; [3,5]=6.30; [4,5]=7.60; [5,5]=0.00

[1,4]=12.60; [2,4]=14.60; [3,4]=7.50; [4,4]=0.00

[1,3]=11.20; [2,3]=7.10; [3,3]=0.00

[1,2]=5.50; [2,2]=0.00

[1,1]=0.00

DM(min norm) =

[1,5]=1.00; [2,5]=0.90; [3,5]=2.10; [4,5]=0.60; [5,5]=0.00

[1,4]=6.60; [2,4]=6.70; [3,4]=1.40; [4,4]=0.00

[1,3]=1.60; [2,3]=2.10; [3,3]=0.00

[1,2]=0.10; [2,2]=0.00

[1,1]=0.00

These distances are all expressed in absolute units and can be retained as

such by selecting rescaling option 1.

By selecting rescaling option 2, RM distances are normalized to the mean

distance of the RM by dividing each cell by the absolute mean distance and

multiplying by 100. All central diagonal values of zero are excluded from the

calculation of mean distance.

DM mean distance (Euclid norm) = 13.783 = 100.0%

DM mean distance (max norm) = 10.220 = 100.0%

DM mean distance (min norm) = 2.310 = 100.0%
Recurrence Quantification Analysis

94

By selecting rescaling option 3, DM distances are normalized to the maximum

distance of the DM by dividing each cell by the absolute maximum distance

and multiplying by 100. Be careful not to confuse max distance and max

norm, which are different.

DM max distance (Euclid norm) = 20.671 = 100.0%

DM max distance (max norm) = 17.500 = 100.0%

DM max distance (min norm) = 6.700 = 100.0%

Recurrence matrices are derived from distance matrices by setting a RADIUS

threshold. As shown below, the Heavyside function assigns values of 0 or 1 to

array elements. The RADIUS parameter is always relative to the reported

MAXDIST, whether it be expressed in absolute units or relative units. Only

those distances in RM[I,J] equal to or less than the RADIUS are defined as

recurrent points at coordinates I, J.

RM (Euclid norm with RADIUS of 8.0 ) =

 [1,5]=0; [2,5]=0; [3,5]=1; [4,5]=0; [5,5]=1

 [1,4]=0; [2,4]=0; [3,4]=0; [4,4]=1

 [1,3]=0; [2,3]=0; [3,3]=1

 [1,2]=1; [2,2]=1

 [1,1]=1

 RM (max norm with RADIUS of 12.3) =

 [1,5]=0; [2,5]=1; [3,5]=1; [4,5]=1; [5,5]=1

 [1,4]=0; [2,4]=0; [3,4]=1; [4,4]=1

 [1,3]=1; [2,3]=1; [3,3]=1

 [1,2]=1; [2,2]=1

 [1,1]=1

RM (min norm with RADIUS of 1.2) =

 [1,5]=1; [2,5]=1; [3,5]=0; [4,5]=1; [5,5]=1

 [1,4]=0; [2,4]=0; [3,4]=0; [4,4]=1

 [1,3]=0; [2,3]=0; [3,3]=1

 [1,2]=1; [2,2]=1

 [1,1]=1

RQA looks for patterns among these recurrent points, and this need not/must

not be done manually (too subjective). Objective pattern recognition

algorithms are written into the many recurrence programs to define the RQA

variables %REC, %DET, ENT, LMAX, TND, %LAM, and TT.
CHAPTER 3

Application of Recurrence

Quantification Analysis: Influence

of Cognitive Activity on Postural

Fluctuations

Geraldine L. Pellecchia

Department of Physical Therapy

School of Allied Health

University of Connecticut

358 Mansfield Rd., Unit 2101

Storrs, CT 06269-2101

U. S. A.

E-mail: geraldine.pellecchia@uconn.edu

Kevin Shockley

Department of Psychology

University of Cincinnati

ML 0376, 429 Dyer Hall

Cincinnati, OH 45221-0376

U. S. A.

E-mail: kevin.shockley@.edu
RQA of Postural Fluctuations

96

Control of a stable standing posture is requisite to many

everyday actions. During upright standing, the body undergoes

continuous, low-amplitude sway. These spontaneous postural

fluctuations are often indexed by the center of pressure (COP). The

COP is the location of the net vertical ground reaction force and is

calculable from the forces and moments measured by a device called a

force platform. During upright standing with equal weight bearing on

each foot, the COP is located midway between the feet. The path

traversed by the COP over time reflects the dynamic nature of postural

control. Figure 3.1 depicts a typical COP path during 30 s of upright

standing on a compliant surface. The ease with which COP measures

can be obtained with a force platform provides a means to explore

factors that may influence postural control.

Figure 3.1. A sample 30 s center of pressure (COP) profile. COP path is shown for an

individual standing upright with feet together on a compliant surface.
Pellechia & Shockley

97

An important aspect of many routine activities of daily living is

the ability to concurrently maintain an upright posture and perform an

unrelated cognitive task. For example, we often walk while talking.

Despite that such scenarios are commonplace, executing cognitive and

postural tasks concurrently is not without consequence. Numerous

studies have demonstrated changes in performance on either the

cognitive, postural, or both tasks when carried out simultaneously

compared to when the same tasks are performed separately (e.g.,

Dault, Geurts, Mulder, & Duysens, 2001; Kerr, Condon, & McDonald,

1985; Lajoie, Teasdale, Bard, & Fleury, 1993; Maylor, Allison, & Wing,

2001; Maylor & Wing, 1996; Stelmach, Zelaznik, & Lowe, 1990).

In a recent paper, Pellecchia (2003) examined the effect of

attentional demand on postural sway. COP was recorded as

participants stood on a force platform and performed cognitive tasks

that varied in attentional requirements. Attentional demand was

quantified using information reduction tasks modeled after those

described by Posner (1964; Posner & Rossman, 1965). An information

reduction task is one in which the required cognitive operation results

in a reduction of information from stimulus to response. The size of that

transformation is quantified by the difference in the amount of

information contained in the stimulus and the response. Posner

asserted that the attentional requirements of a cognitive task could be

manipulated by varying the processing demands of the task. For a set

of numeric tasks, Posner demonstrated a direct relation between task

difficulty and the magnitude of information reduced in carrying out a

task. Information reduction tasks have been an effective means of
RQA of Postural Fluctuations

98

manipulating the attentional requirements of cognitive activity (for an

example, see Pellecchia & Turvey, 2001).

To examine the influence of attentional demand on postural

sway, Pellecchia (2003) adopted a conventional approach to data

analysis. Specifically, the mean magnitude and variability of the COP

path during upright standing were compared across a set of

information reduction tasks. This traditional method has been used

extensively in previous studies examining effects of various

experimental manipulations on postural control (for examples, see

Derave, De Clercq, Bouckaert, & Pannier, 1998; Gravelle et al., 2002;

Guerraz, Thilo, Bronstein, & Gresty, 2001; Polonyova & Hlavacka, 2001;

Vuillerme, Forestier, & Nougier, 2002; Vuillerme, Nougier, & Teasdale,

2000). Although well established, this approach to the analysis of COP

data is of limited usefulness. Summary measures of COP path

magnitude and variability do not reflect the dynamical properties of

postural control (Newell, 1998). In contrast, recurrence quantification

analysis (RQA), a relatively new analytical method, examines the time

evolution of data series (see Webber & Zbilut, Chapter 2). In recent

years, investigators have begun using RQA to explore the dynamics of

postural control. For example, Riley and Clark (2003) employed RQA to

investigate how changes in sensory information influenced the

temporal structure of spontaneous postural sway, and thereby to gain

insight into the adaptive nature of postural control. RQA is a useful tool

for identifying structure that is inherent in postural fluctuations but not

evident when using conventional methods of analyzing COP data. The

purpose of this chapter is to demonstrate the application of RQA to the

study of postural fluctuations during standing as a function of varying
Pellechia & Shockley

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attentional demands of an unrelated, concurrent, cognitive task.

Following a description of the experimental method, we review the

results of the traditional analysis of COP data as previously reported by

Pellecchia (2003). Next, we employ the methods of RQA to examine the

dynamical properties of COP time series.

COGNITIVE ACTIVITY & POSTURAL CONTROL: THE EXPERIMENT

An AMTI Accusway System for Balance and Postural Sway

Measurement (Advanced Mechanical Technology, Inc., Watertown,

Massachusetts) was used to collect data. The Accusway System consists

of a portable force platform and SWAYWIN software for data acquisition

and analysis. The force platform produces six signals—three force

measures, Fx, Fy, and Fz, and three moment measures, Mx, My, and Mz,

where the subscripts x, y, and z denote medio-lateral (ML; side-toside), anterior-posterior (AP; front-to-back), and vertical directions,

respectively. SWAYWIN software uses the forces and moments to

calculate x and y coordinates of the position of the COP. The Accusway

System samples at a rate of 50 Hz. Therefore, a 30 s trial period yielded

1500 data points for the ML COP (position of the center of pressure in

the ML direction) time series and 1500 data points for the AP COP

(position of the center of pressure in the AP direction) time series.

The postural task consisted of standing on a 10 cm thick foam

pad that had been placed on top of the force platform, as shown in

Figure 3.2. The foam pad created a compliant surface, thereby altering

the somatosensory information available for postural control and

making the postural task more challenging than simply standing on a

firm, flat surface.
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Figure 3.2. The experimental set-up used in the present experiment. Participants stood with

feet together, arms by side, and looking straight ahead at a blank wall. A foam pad was placed

on the force platform to create a compliant surface.

Three information reduction tasks—digit reversal, digit

classification, and counting backward by 3s—were used to vary the

attentional demands of the concurrent cognitive task. The amount of

information reduced in performing each task was determined using the

method described by Posner (1964; Posner & Rossman, 1965; see also

Note 1 in Pellecchia & Turvey, 2001). In digit reversal, the task was to

reverse the order of a pair of digits. For example, on hearing the
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stimulus 4, 7, a correct response would be 7, 4. The input contained 6.5

bits of information, and the output contained 6.5 bits of information.

Therefore, digit reversal was a 0-bit reduction task. In digit

classification, the task was to combine a pair of single digits into a

double-digit number and to classify that number as high (if > 50) or low

(if < 50), and odd or even. For example, the single digits 4, 7, combine

to form the double-digit number 47; correct classification would be low,

odd. The input contained 6.5 bits of information; the output contained 2

bits of information. Therefore, digit classification required 4.5 bits of

information reduction. In the counting back by 3s task, participants

were given a 3 digit number from which to start counting. Participants

were instructed to first recite the starting number, and then count

backward by 3s from that number. Correct responses to the stimulus

365 would be 365, 362, 359, 356, and so on. We determined that

counting back by 3s from a randomly chosen three-digit number

required approximately 5.9 bits of information reduction.

A pre-recorded audiotape provided stimuli for the digit reversal

and digit classification tasks. The audiotape consisted of pairs of

random single digits presented at a rate of 2 digits/s with a 2 s pause

between pairs. For the counting backward by 3s task, a different

starting number was selected for each trial. Starting numbers ranging

between 200 and 999 were chosen from a random number table. Prior

to data collection, participants practiced the three information

reduction tasks for a minimum of 15 s each while seated in a chair.

During the experiment, participants stood in stocking feet on the

foam pad that rested on the force platform. The force platform was

positioned approximately 2 m from a blank wall. Participants were
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instructed to stand with the feet together, the arms by the sides, and

with the eyes open and looking straight ahead. COP data were

collected under four cognitive task conditions: Quiet standing (i.e.,

performing no cognitive task), standing combined with digit reversal,

standing combined with digit classification, and standing combined

with counting backward by 3s. Participants performed two 30 s trials of

each condition, for a total of eight trials. The order of the four

experimental conditions was randomized. Data collection for each

participant’s first trial began 30-60 s after the participant assumed the

proper position on the force platform. For those trials in which standing

was combined with a cognitive task, force platform data collection

began after the participant voiced their first response. There was a 30-

60 s break between trials, during which time the participant remained

standing on the platform. Participants’ verbal responses to the

cognitive tasks were audiotape-recorded for subsequent analysis.

Practice and data collection together lasted approximately 30 min.

TRADITIONAL APPROACH TO ANALYSIS OF COP DATA

Data Analysis

SWAYWIN software was used to calculate five dependent

measures: Total COP path length (LCOP), anterio-posterior (AP) and

medio-lateral (ML) COP range, and AP and ML COP variability. LCOP is

the total distance traveled by the COP over the 30 s trial period (see

Figure 3.1 for a visual display of LCOP). AP COP range and ML COP

range are the differences between the two extreme position values in

the respective directions. AP and ML COP variability are the standard

deviations of the COP in the respective directions. Means of those
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quantities were calculated for the two trials in each experimental

condition, and the means were used in all subsequent analyses.

Separate repeated-measures analyses of variance (ANOVA) were used

to determine the effect of cognitive task condition on each COP

measure. Pearson product-moment correlations examined the relation

between bits of information reduced and each dependent variable.

To examine performance on the cognitive tasks, the number of

errors was determined for each trial by listening to the audiotape

recording of participants’ responses. Error rate was calculated as the

number of errors divided by the total number of responses for each 30

s trial. Error rates were averaged for each participant’s two trials in

each experimental condition. The mean error scores were used in

subsequent repeated measures ANOVAs to examine cognitive task

performance. Post-hoc analyses were conducted using least significant

difference pair-wise multiple comparison tests.

Results of Traditional Analyses

The effects of attentional requirements on postural sway and

cognitive task performance were previously reported (Pellecchia,

2003) and are summarized in Figure 3.3. Repeated-measures ANOVAs

revealed a main effect of cognitive task condition on LCOP, F(3, 57) =

8.09, p < .001, as shown in Figure 3.3a. Figures 3.3b-e depict similar

results for the other four COP measures. Separate ANOVAs revealed a

main effect of cognitive task condition on AP range F(3, 57) = 9.84, p <

.001, ML range F(3, 57) = 3.03, p < .05, and AP variability, F(3, 57) =

5.70, p < .01. Post-hoc tests showed sway measures of LCOP, AP range,

ML range, and AP variability were greater for the counting back by 3s
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1.8

2.2

2.6

3

3.4

Quiet Standing Reversal Classification Back by 3s

Cognitive Task

Range AP COP (cm)

b

1.9

2

2.1

2.2

2.3

2.4

Quiet Standing Reversal Classification Back by 3s

Cognitive Task

Range ML COP (cm)

c

1.8

2.2

2.6

3

3.4

Quiet Standing Reversal Classification Back by 3s

Cognitive Task

SD AP COP (cm)

d

0.36

0.38

0.4

0.42

0.44

Quiet Standing Reversal Classification Back by 3s

Cognitive Task

SD ML COP (cm)

e

44

48

52

56

60

Quiet Standing Reversal Classification Back by 3s

Cognitive Task

LCOP (cm) .

a

Figure 3.3. Five summary measures of the center of pressure (COP) as a function of cognitive

task condition. (a) Total length of the path of the center of pressure (LCOP); (b) Range of COP

motion in the anterior-posterior (AP) direction; (c) Range of COP motion in the medio-lateral

(ML) direction; (d) standard deviation (SD) of AP COP motion; and (e) SD of ML COP motion.

task than for the other three cognitive task conditions. In addition, AP

sway range was greater for digit classification than for quiet standing.

For ML variability, the main effect of cognitive task condition

approached statistical significance, F(3, 57) = 2.35, p = .08. Pair-wise
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comparisons suggested greater ML variability for the counting back by

3s task than for digit reversal (p < .05) and quiet standing (p = .06).

As noted previously, digit reversal, digit classification, and

counting back by 3s required 0, 4.5, and 5.9 bits of information

reduction, respectively. Inspection of Figures 3.3a-e suggests that

postural sway was directly related to the magnitude of information

reduced for the cognitive tasks. Separate correlation analyses

confirmed this relation between bits of information reduced and each

COP measure. Specifically, Pearson correlation coefficients for LCOP,

AP range, ML range, AP variability, and ML variability were .79, .89,

.82, .89, and .97, respectively.

In the evaluation of cognitive task performance, repeated

measures ANOVA revealed a main effect of cognitive task on error rate,

F(2, 36) = 7.58, p < .01. The error rate for counting back by 3s (M =

0.113) was greater than error rates for digit classification (M = 0.026)

and digit reversal (M = 0.003).

To summarize, the traditional approach to analysis of COP data

revealed greater magnitude and variability of COP motion with higher

attentional demands of a concurrent, unrelated, cognitive task. Upon

further inspection of Figures 3.3a-e, the various COP measures appear

to provide redundant information. In fact, these measures are highly

correlated. Pearson correlation coefficients examining the relation

among mean values of the five COP measures ranged from .96 to .99.

Discussion of Results of Traditional Analysis

Considering the results of the traditional analysis of COP data,

one might assume that only a single measure of the COP need be

considered. That is, each summary measure of the COP suggested a
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similar conclusion about the influence of attentional requirements on

postural sway. More specifically, performing a concurrent cognitive

task was associated with increases in all five measures. Furthermore,

there were no apparent differential effects of attentional demands on

AP or ML COP motion.

A conventional viewpoint holds that the degree of postural sway

reflects performance of the postural control system. In individuals free

of neuromuscular or balance disorders, small amplitude and variability

of COP excursions is considered to indicate good balance, whereas

large amplitude and variability of COP motion is considered to indicate

poor balance. From this perspective, the observed increases in sway

magnitude and variability in the present experiment suggest that

carrying out a concurrent cognitive task compromises postural

stability. Consistent with this view and the traditional notion of attention

as limited capacity or limited processing resources, one might

conclude that counting backward by 3s while standing upright exceeds

an individual’s attentional capacity, and brings about a decline in

performance of the postural control system (Woollacott & ShumwayCook, 2002). This interpretation is intuitively appealing and broadly

held. On second blush, however, the notion that concurrent

performance of a fundamental motor task such as maintaining an

upright posture and a relatively simple arithmetic task could exceed

human attentional resources is somewhat suspect.

The traditional approach to the analysis of COP data provides

limited information about the postural control system’s response to

concurrent performance of an unrelated cognitive task. In particular,

summary measures of COP magnitude and variability do not inform
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about the temporal structure of the COP time series. Consider Figure

3.4, which depicts the AP COP and ML COP time series for the COP

profile shown in Figure 3.1. Using the methods of RQA, we can explore

the temporal structure of these postural fluctuations, and perhaps gain

further insight into changes brought about in the postural control

system by varying the attentional requirements of cognitive activity.

-2.00

-1.60

-1.20

-0.80

-0.40

0.00

Time

AP COP displacement (cm) z

-1.00

-0.60

-0.20

0.20

0.60

1.00

Time

ML COP displacement (cm) z

Figure 3.4. AP COP (top) and ML COP (bottom) times series for the COP profile shown in

Figure 3.1.
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RECURRENCE QUANTIFICATION ANALYSIS

RQA, a relatively new tool for the analysis of nonlinear dynamical

systems, can be used to identify subtle patterns of recurrence in a data

series (Webber & Zbilut, 1994, 1996; Zbilut & Webber, 1992; for a

detailed tutorial, see Webber & Zbilut, Chapter 2). Time-delayed

copies of a single scalar time series are used as surrogate variables to

reconstruct a higher dimensional phase space. Through examination of

the reconstructed phase space, RQA is able to detect system dynamics

that are intrinsic to, though not readily apparent in, the onedimensional time series. In the reconstructed phase space, the

distances between all possible vectors are determined and used to

create a distance matrix. Next, a recurrence matrix of recurrent points

is generated from the distance matrix. Recurrent points are those points

in the distance matrix that fall within a specified distance of one

another.

A recurrence plot is simply a graphical depiction of the

recurrence matrix. The recurrence plot is an autocorrelation plot of x(t)

with x(i) along the abscissa and x(j) along the ordinate. Only those

points that satisfy x(i) = x(j), defined as values of i and j that fall within a

specified radius or distance of one another, are plotted. Visual

inspection of recurrence plots may reveal patterns in the data not

evident from examination of the time series. RQA uses pattern

recognition algorithms (see discussion of quantification of qualitative

features below) to quantify the recurrence features depicted in

recurrence plots, and, therefore, is more objective than visual

inspection of recurrence plots.
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In the present chapter, we apply RQA to the COP data generated

by the experiment described above to examine the effects of cognitive

activity on postural fluctuations. An important characteristic of RQA is

that, unlike other analytic techniques, it does not assume data

stationarity. This is of particular relevance in the analysis of COP time

series, which have been shown to be nonstationary—drift in the first

(mean) and second (standard deviation) moments over time (Newell,

1998; Newell, Slobounov, Slobounova, & Molenaar, 1997; Schumann,

Redfern, Furman, El-Jaroudi, & Chaparro, 1995). In addition, RQA

requires no assumptions about data set size or distribution of the data.

RQA of the COP data from the present experiment was

performed using recurrence software available free of charge from

http://homepages.luc.edu/~cwebber/. Recurrence Quantification

Analysis version 6.2 was used to conduct the present analyses, but a

more recent version of the software is now available. The software

includes 20 programs for examining recurrence in a single time series

and cross-recurrence in two time series. All programs run in MSDOS,

requiring the user to have a basic knowledge of how to work in a DOS

environment. The README.TXT file is a valuable resource and should

be read by all first-time users of the software. Toward the beginning of

that file, programs are listed by purpose for which they are used. This

provides a useful guide for selecting an appropriate program. For

example, when the aim is to generate a recurrence plot, the user can

look under the heading “2 programs display recurrence plots” and

select from RQD.EXE, which is used to generate recurrence

quantification plots for a single time series, and KRQD.EXE, which is

used to generate cross-recurrence plots from two different files. Later
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in the README.TXT file descriptions of each program detail program

usage, input parameters that must be defined, and output that will be

generated. The section of the README.TXT file titled “Mathematical

Construction of the Recurrence Matrix” (see also Webber & Zbilut,

Chapter 2) is a particularly helpful tool for understanding the process of

RQA. In addition, near the end of the file, the creators of the software

discuss several important points to consider in conducting RQA.

We used program RQD.EXE (Recurrence Quantification Display)

to create recurrence plots. Figure 3.5 depicts recurrence plots for the

AP COP and ML COP time series shown in Figure 3.4. The data used to

generate these recurrence plots are available for download for the

reader who wishes to reproduce these plots. As noted above, points

plotted in the recurrence plot are those points determined to be

“neighbors” in the reconstructed phase space, that is, COP values that

are within a specified distance of one another. The basic features of

recurrence plots and our choices of parameter values used to generate

the plots with program RQD.EXE are explained below in the

subsections entitled Quantification of Qualitative Features of Recurrence

Plots and Parameter Selection. Additional information about the

qualitative features of recurrence plots can be found in Riley,

Balasubramaniam, and Turvey (1999).

Our plan was to use program RQE.EXE (Recurrence

Quantification Epochs) to examine effects of cognitive task condition on

five recurrence variables: %recurrence (%REC), %determinism

(%DET), maxline (MAXL), entropy (ENT), and trend (TND).

In contrast to the program RQD.EXE, which we used to generate

the recurrence plots, the output of RQE.EXE is entirely quantitative.
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Figure 3.5. Recurrence plots for the AP COP (top) and ML COP (bottom) time series shown in

Figure 3.4. The time series are plotted at the bottom of the figure. Recurrence parameters and

recurrence output are listed to the left of the plot. Distribution of line lengths is graphed at the

bottom left.
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Recurrence variables are calculated from the upper triangular area of

the recurrence plot, excluding the central diagonal, because the plot is

symmetrical about the main diagonal. %REC is the percentage of data

points that are recurrent, defined as those points falling within a

distance specified by a selected radius value (see below). %DET, an

index of degree of determinism, is the percentage of recurrent points

that form diagonal lines in a recurrence plot (parallel to the central

diagonal). In other words, %DET refers to the percentage of

consecutive recurring points. The number of consecutive points

needed to constitute a line is determined by the value selected for the

line length parameter. MAXL is the length of the longest diagonal line,

excluding the main diagonal. MAXL is inversely proportional to the

largest positive Lyapunov exponent, and thereby provides a measure

of the dynamical stability of the system. According to Webber and

Zbilut (Chapter 2), “the shorter the [MAXL], the more chaotic (less

stable) the signal.” ENT is calculated as the Shannon information

entropy of a histogram of diagonal line lengths, and is an index of the

complexity of the deterministic structure of the time series. TND

provides a measure of the degree of system stationarity, with values of

TND at or near zero reflecting stationarity and values deviating from

zero indicating drift in the system.

Quantification of Qualitative Features of Recurrence Plots

Visual inspection of recurrence plots may be useful for a

qualitative understanding of the quantitative recurrence measures

described above. To this end, we have generated several recurrence

plots for time series with known structure. In particular, we show

recurrence plots for a simple sinusoid (Figure 3.6), the same sinusoid
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Figure 3.6. Recurrence plot for a simple sinusoid.

with superimposed white noise (Figure 3.7), the same sinusoid with a

linear drift (Figure 3.8), a sample time series from a known complex

mathematical system—the Lorenz attractor (Figure 3.9), and time series

from two regimes of another mathematical system, the Hénon attractor

(Figure 3.10). Comparison of the recurrence plots will help to illustrate

what the quantitative recurrence measures actually mean.

%REC & %DET. Consider the simple sinusoid, which is an

entirely deterministic signal, depicted in the bottom of Figure 3.6. By

entirely deterministic, we mean that each value in the time series

recurs and is part of a string of consecutive recurring values. This

aspect of the time series is illustrated by every illuminated pixel in the

recurrence plot corresponding to part of a diagonal line. This means
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Figure 3.7. Recurrence plot for a sinusoid with superimposed white noise.

Figure 3.8. Recurrence plot for a sinusoid with linear drift.
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Figure 3.9. Recurrence plot for a time series from the Lorenz attractor.

that the proportion of recurring points that are part of a diagonal line is

100% (i.e., %DET = 100%). Note that just because every value in the

time series recurs does not mean that every possible point in the

recurrence plot is recurrent. In this particular example, of all of the

possible locations that could be recurrent in a time series of a length of

950 data points (950 × 950 / 2 = 451,250), 26,853 were recurrent (~6%)

(the total number of possible recurrent points [950 × 950] is divided by

2 because only one of the triangular regions is used to calculate

recurrence, since the plot is symmetrical about the main diagonal).

For the time series depicted in Figure 3.7, we no longer have an

entirely deterministic signal, given that we have added a random

component (white noise). Each value in the time series no longer

recurs and each value that does recur is no longer necessarily part of a
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Figure 3.10. Recurrence plot for a times series from a periodic regime (top) and a chaotic

regime (bottom) of the Hénon attractor.
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diagonal line. The fact that each value in the time series no longer

recurs is illustrated by fewer illuminated pixels in the plot (4,889 as

compared to 26,853 in Figure 3.6) and a lower proportion of values

recurring (~1%). The fact that the noisy signal in Figure 3.7 is no longer

entirely deterministic is illustrated by the fact that fewer of the

illuminated pixels form diagonal lines, which results in a lower value of

%DET (~ 12%) than the signal in Figure 3.6.

TND. The time series depicted in Figure 3.8 is nonstationary—

the mean state drifts (becomes lower in this case) over time. This was

achieved simply by adding a monotonic decrease (a negatively sloped

straight line) to the time series depicted in Figure 3.6. Note that we only

have a modest change in %REC (~5%), and no change in %DET (100%)

as compared to the time series in Figure 3.6 (~6% and 100%,

respectively). However, one can see a qualitative difference between

the recurrence plots depicted in Figures 3.6 and 3.8.

Recall that the central diagonal corresponds to sameness in time.

This location in the recurrence plot is ubiquitously recurrent because it

represents comparison of a value to itself. But note that as one moves

perpendicularly away from the central diagonal, this represents

deviation in time. For example, as one moves upward and left away

from the diagonal, this means that one is comparing a point early in the

time series (indicated by a value on the x-axis near the origin) to a

point later in the time series (indicated by a value on the y-axis near the

extreme). As one moves perpendicularly away from the central

diagonal in Figure 3.8, the pixel density decreases. This occurs

because over time there is a drift in the mean state of the time series.

The average value of the first 100 points in the time series is
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approximately 0.09, while the average of the last 100 points is

approximately –0.09. The color density does not change, however, in

the recurrence plot depicted in Figure 3.6. This qualitative aspect of the

recurrence plot is quantified by the measure of trend (TND)—the slope

of %REC as a function of distance away from the diagonal. Note that

TND for Figure 3.8 is considerably different than zero (~7) while the

value of TRD for Figure 3.6 is approximately equal to zero.

ENT. The time series depicted in Figure 3.9 is a sample of data

generated from the Lorenz model. The Lorenz system is a nonlinear,

chaotic system that would be considered a complex system by most.

The equations representing the Lorenz system and the true and

reconstructed phase spaces of the system may be seen in Shockley

(Chapter 4). We have selected sample data from this system to

illustrate how recurrence analysis may be used to quantify the

complexity of a time series. Note that in the time series depicted in the

bottom of Figure 3.9 the system appears to be somewhat periodic, as

indicated by the peaks and valleys occurring at similar periods.

However, the amplitude of the signal changes over time and abrupt

shifts in the value of the system occur at irregular intervals (compare

the first part of the time series to the later parts). Note that most of the

illuminated pixels form diagonal lines (as indicated by %DET = 99%)

and that we see a similar proportion of recurrent points as in Figure 3.6

(~7%). However, the recurrence plot in Figure 3.9 looks different than

the recurrence plot in Figure 3.6. This distinction between the plots can

be captured most readily by the frequency distribution of line lengths

shown in the lower left of each figure. The distribution of line lengths

for the Lorenz system (Figure 3.9) has a richer variety than that for the
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119

simple sinusoid (Figure 3.6). This variety of structure is what is meant

by complexity in recurrence analysis. This aspect of the time series is

quantified by the Shannon entropy (ENT; the negative sum of the

normalized log2 probabilities [P] of lines corresponding to given line

lengths) of the line length distributions in question (see Equation 2.11 in

Webber & Zbilut, Chapter 2). Note that the entropy for the Lorenz

system (ENT = ~5) is greater than the entropy for a simple sinusoid

(ENT = ~2), indicating that the Lorenz system is more complex than a

simple sinusoid.

MAXL. To illustrate the meaning of the recurrence measure

maxline (MAXL), we have selected data sets generated from the Hénon

system. The Hénon system is a model of the dynamics of stars moving

within galaxies. It is governed by the following two equations of motion:

2

x& = y +1− ax [3.1]

y& = bx [3.2]

where x and y correspond to the dimensions of change, x and y with

overdots correspond to rate of change along those dimensions, and a

and b are parameters.

One of the interesting features of the Hénon system is that

depending on the values of the parameters (a and b) the Hénon system

may exhibit behavior that is highly predictable (periodic) or chaotic

behavior that is only predictable in the very short term. Figure 3.10

shows recurrence plots of a periodic regime (oscillation among 16

values;e.g., a = 1.055, b = 0.3) and a chaotic regime of the Hénon
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system (e.g., a = 1.4, b = 0.3) (the data sets used in the present example

are provided with the RQA software a t

http://homepages.luc.edu/~cwebber/). By definition, the chaotic

regime is less stable than the periodic regime. By stability we mean

that two trajectories that are initially nearby one another stay nearby

one another longer in a more stable system than in a less stable system.

MAXL has been shown to be sensitive to the stability of the system in

question (Eckmann, Kamphorst, & Ruelle, 1987).1 The larger MAXL, the

more stable the system—nearby trajectories diverge less quickly than

for a system with a smaller MAXL. For the chaotic regime of the Hénon

attractor the longest diagonal line is quite short (MAXL = 7) as

compared to the longest diagonal line for the periodic regime (MAXL =

942). While it is the case that MAXL will change considerably

depending on the system under scrutiny (as can be seen by

comparison of MAXL values for Figures 3.6-3.10), what is of interest is

how MAXL changes within the same system (in this case the Hénon

system) under different conditions.

Parameter Selection

Prerequisite to generating plots and calculating recurrence

variables is the selection of appropriate settings for seven parameters:

 1 Lyapunov exponents quantify the exponential rate of divergence of nearby trajectories along

a given dimension in the system. A negative Lyapunov exponent quantifies the average rate of

convergence of trajectories over time, while positive Lyapunov exponents characterize the

average rate of divergence over time. For a highly stable system (e.g., periodic systems), two

trajectories that are initially nearby one another will continue to be nearby one another at any

given later point in time. This means that the Lyapunov exponent would be at or near zero (i.e.,

no divergence over time). One hallmark of chaotic systems, however, is that they have at least

one positive Lyapunov exponent (meaning that along at least one dimension, two trajectories

that are initially nearby one another will diverge exponentially over time). Chaotic systems that

exhibit bounded regions in which trajectories unfold (e.g., the Lorenz attractor or the Henon

attractor for certain parameter ranges) also have at least one negative Lyapunov exponent.

MAXL has been shown to be inversely proportional to the largest positive Lypunov exponent

(larger MAXL smaller value of Lyapunov exponent; see Eckmann, Kamphorst, & Ruelle, 1987).
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Embedding dimension, delay, range, norm, rescaling, radius, and line

length (see Webber & Zbilut, Chapter 2). Selection of these parameter

values is challenging. Although some guidelines are available, there

are as of yet no absolute standards for identifying the most appropriate

parameter values. A summary of the decision making that was involved

in our choice of parameters follows.

Selection of some parameters is more difficult than others.

Choosing a proper embedding dimension, delay, and radius are

among the most challenging decisions that must be made. We followed

the approach described by Zbilut and Webber (1992) and used by

Riley et al. (1999) to select values for those three parameters. The

general strategy is to calculate RQA measures for a range of parameter

values, and select a value from a range in which small changes in

parameter settings result in small, continuous changes in the RQA

measures. To follow that strategy, we enlisted program RQS.EXE

(Recurrence Quantification Scale), which “...scales recurrence

quantifications for a single epoch of data by incrementing parameter

values over specified ranges” (Webber, 2004, p. 6). In the following

paragraphs, we describe first the decision making involved in selecting

a range of parameters for embedding dimension, delay, and radius for

use in RQS.EXE, and next the choice of a single setting for each

parameter.

Embedding Dimension. Embedding dimension specifies the ndimensions of the reconstructed phase space, that is, the dimension

into which the dynamic of the system under study will be projected

(see discussion of delay below). Selecting an embedding dimension

that is too high can amplify the effects of noise. Choosing an
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embedding dimension that is too low will result in underdetermination,

that is, the dynamics of the system will not be fully revealed. Webber

(2004) suggested, for physiological data, starting with embedding

dimensions between 10 and 20 and working downward. Investigators

applying RQA to the study of COP data have reported embedding

dimension 8 (Schmit et al., submitted), 9 (Riley & Clark, 2003) and 10

(Balasubramaniam, Riley, & Turvey, 2000; Riley et al., 1999). Based on

Webber’s suggestion and previous papers, we decided to examine

RQA output for embedding dimensions 7 through 10.

Delay. As mentioned previously, time-delayed copies of the data

series are used as surrogate variables to project the data into higherdimensional space. The delay parameter specifies the time lag to use in

reconstructing that phase space. For example, imagine a time series for

which we selected a delay (τ) of 10 and embedding dimension of 3. Our

first embedding dimension [x(t)] in the reconstructed phase space

would start at data point 1 of the original time series (x), the second

embedding dimension [x(t + τ)] would start at data point 11, and the

third embedding dimension [x(t + 2τ)] would start at data point 21. A

two-dimensional phase space could be constructed the same way that

one plots a two-dimensional scatterplot to evaluate the relationship

between two variables in correlation or regression, the two variables in

this case being x(t) and x(t + τ). One could add a third (or higher)

dimension to the phase space in the same fashion (see Figure 4.4 in

Shockley, Chapter 4). Previous studies in which COP data were

sampled at 100 Hz used time delays ranging between 0.04 s and 0.09 s.

Considering the sampling rate of the force plate used in the present

study (50 Hz), delays of 2 to 5 data points corresponded to time delays
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of 0.04 to 0.10 seconds. We decided to examine RQA output for delays

ranging between 2 and 10 data points.

Radius. The radius parameter defines the Euclidean distance

within which points are considered neighbors in the reconstructed

phase space. Said differently, the radius sets the threshold for

recurrence. The larger the radius, the more points will be considered

recurrent. As a general guideline, a radius should be selected such that

%REC remains low (see Webber & Zbilut, Chapter 2). We wanted a

radius that was small enough to yield relatively low %REC (no larger

than 5%), but not so small as to produce a floor effect with values of

%REC near or at 0.0%. Other investigators have used a radius of 10 or

11 in the analysis of COP data (Balasubramaniam et al., 2000; Riley et

al., 1999; Riley & Clark, 2003). We decided to examine RQA output for

radius settings ranging between 10 and 26.

Norm. The norm parameter determines the method used for

computing distances between vectors in the reconstructed phase

space. We selected Euclidean normalization, which is consistent with

previous studies using RQA to examine COP data (see Riley et al., 1999;

Riley & Clark, 2003).

Rescale. The rescale parameter determines the method used to

rescale the distance matrix. Although rescaling to maximum distance is

a typical choice, we decided to rescale relative to mean distance. Mean

distance rescaling minimizes the influence of an outlier, which can be a

problem when rescaling to maximum distance. An assumption of

rescaling to mean distance, however, is that the distribution of the

distances is Gaussian.
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Range. The range of data points included in the recurrence

analysis is specified by setting the first point, Pstart (the data point in the

time series at which the analysis will start), and the last point, Pend (the

data point in the time series at which the analysis will end). We wanted

to include as many of the data points in the time series as possible in

our recurrence analysis. For that reason, we input the first point as 1

and the last point as 1410, thereby selecting the largest range possible,

given constraints due to the number of data points in the time series (N

= 1500), maximum embedding dimension (M = 10), and maximum

delay (τ = 10). The last data point was determined by Pend = N – (M – 1)

× τ. This guarantees the use of the maximal number of data points and

the same number of data points within each surrogate dimension in the

phase space. When using RQS.EXE, however, we did not actually have

to compute Pend, because when the program prompts the user to input

LAST (Pend), it specifies the last possible point in the time series that

could be used. We simply input that last possible point, 1410, as our

value for Pend.

Line Length. Line length specifies the number of consecutive

recurrent points required to define a line segment. Often, line length is

set at two points. Specifying a line length of more than two points yields

increasingly conservative estimates of the deterministic structure in the

system. In the present study, line length was set to three points.

Having determined a range of parameter settings for embedding

dimension, delay, and radius, and selected settings for norming

method, rescaling method, range, and line length, our next step was to

choose (at random) a few trials from each experimental condition and

use program RQS.EXE to compute recurrence measures for the selected
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parameter ranges. To reiterate, our purpose in running RQS.EXE on a

sample of experimental trials was to generate recurrence measures for

a range of embedding dimensions, delays, and radius values. As noted

above, we set minimum embedding dimension at 7, maximum

embedding dimension at 10; minimum delay at 2 samples, maximum

delay at 10 samples; and minimum radius at 10, maximum radius at 26.

Table 3.1 lists each of the parameter settings we selected in running

RQS.EXE.

We inspected the recurrence measures that were generated by

RQS.EXE for our sample of trials to decide on specific settings for

embedding dimension, delay, and radius to be used in carrying out the

RQA of all experimental trials. To recap, we were looking for small

changes in parameter settings yielding smooth changes in output

measures, %REC values ranging between 1% and 5%, and absence of

ceiling or floor effects on %DET. We created in Matlab (Mathworks,

Inc., Natick, MA) a series of surface plots to visualize changes in %REC

as a function of embedding dimension, delay, and radius. A separate

plot was created for each of the four embedding dimensions under

examination (see Figure 3.11), with radius on the x-axis, delay on the yaxis, and the dependent variable %REC on the z-axis. The surface plots

in Figure 3.11 illustrate well that in spite of the fact that increasing

values of radius yielded higher %REC, each of the plots looks

qualitatively similar. That is, there are no qualitative differences in the

patterns of %REC (i.e., the shape of each surface) for this range of

parameter settings. The fact that incremental changes in parameter

values yield smooth (not abrupt) changes in %REC (e.g., steady

increases in %REC with increases in radius or steady decreases in
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Table 3.1. Parameter settings selected when prompted by program RQS.EXE. MIN =

minimum; MAX = maximum; RANDSEQ = randomize data sequence. See README.TXT

file accompanying software for further explanation of parameters listed above (Webber.

2004). NORM value of 3 corresponds to Euclidean normalization. Selection of RANDSEQ

n is a “no” response to the option of randomly sequencing points in the data set, thereby

retaining the original order of points in the time series. Rescale value of 2 instructs the

program to rescale the matrix to mean distance.

Parameter Setting

DELAY MIN 2

DELAY MAX 10

EMBED MIN 7

EMBED MAX 10

NORM 3

FIRST 1

LAST 1410

RANDSEQ n

RESCALE 2

RADIUS MIN 10

RADIUS MAX 26

RADIUS STEP 1

LINE 3

%REC with increases of embedding dimension) suggests that using a

set of parameters within the selected range will not yield notable

changes in %REC that are artifacts of parameter selection. For

additional information about surface plots, see Shockley’s (Chapter 4)
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Figure 3.11. Surface plots for embedding dimensions 7-10, showing steady increase in

percentage of recurrent points (%REC) with increasing values of radius, but no apparent

difference in the pattern of %REC across the four plots.

application of cross-recurrence analysis. Based on our inspection of

the surface plots and the numerical recurrence output generated by

program RQS.EXE, we selected the following parameter settings:

Embedding dimension of 7, time delay of 3 samples (corresponding to

a 0.06 s lag), and radius of 16. Our decision to set the radius parameter

at 16 means that points falling within 16% of the mean Euclidean

distance of each other would be considered recurrent. As can be seen

in the surface plots, this radius ensures that our %REC values will be in
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our target range of 1-5%. It is important to note that had we selected

slightly different parameters, we would still have seen the same basic

pattern in the results, although the particular magnitudes of recurrence

measures would have scaled up or down.

Our next step was to run the RQA with the selected parameter

settings on the entire set of experimental trials. Program RQE.EXE was

used to compute the five recurrence variables of interest, %REC,

%DET, MAXL, ENT, and TND. As a practical note, recurrence analysis

can take a long time (hours) to run, depending on file size, number of

trials, and processor speed. An advantage of using RQE.EXE as

opposed to RQD.EXE, for example, is that the former allows multiple

analyses to be executed in batch mode, rather than waiting for each file

to be analyzed and typing the next command for the next file to be

analyzed. Computations were performed using the following parameter

settings: Delay = 3, embedding dimension = 7, range = 1–1482, norm =

Euclidean, rescaling = mean distance, radius = 16, and line length = 3.

The program RQEP.EXE was used to generate a parameter file, to be

called by the batch file commands, containing those parameter

selections. An ASCII (text), tab-delimited batch file (filename.bat) was

set up such that each row corresponded to the MSDOS command for

analyzing one file using RQE.EXE. The number of rows corresponded to

the number of files to be analyzed (see README.TXT file for complete

instructions). Program run time for the present data was approximately

four hours. Mean values for the recurrence measures were calculated

for the two trials in each condition. Separate ANOVAs were conducted

on each recurrence measure for AP COP and ML COP time series.
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After the RQA was complete for all of the experimental trials, we

reran the RQA for six randomly chosen trials using the same parameter

settings, but selecting the option to randomize the order of the data

points. Comparing the RQA findings of the randomly shuffled data and

the normally sequenced data provides the means to confirm our choice

of parameter settings as appropriate for revealing the deterministic

structure present in the original time series (see Webber & Zbilut,

Chapter 2).

Figure 3.12 depicts the recurrence plots generated with random

shuffling of data from the time series in Figure 3.4. Although the values

in the time series of Figure 3.12 (just below the recurrence plot) are

exactly the same as those for the time series in Figure 3.5, because they

are randomly shuffled, nearness in time no longer necessarily means

nearness in value. For example, in a typical time series, the value for

the 10th data point will be reasonably close to the value for the 11th data

point, simply because a person cannot instantaneous move the body

across large distances. However, when the values are randomly

shuffled, the 100th data point from the original time series could end up

next to the 10th data point of the original time series. When the data

points from the new, randomly shuffled time series are connected by a

line for plotting, the time series now looks extremely densely packed

as compared to the original, in spite of the fact that none of the values

have changed. This, however, is simply an artifact of “connecting the

dots,” as it were.

What is more important than comparing the time series of

Figures 3.5 and 3.12 is comparing the recurrence plots. Recall that only

recurring points are plotted in a recurrence plot. Visual comparison of
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Figure 3.12. Recurrence plots for the AP COP (top) and ML COP (bottom) data sets shown in

Figure 3.4, but randomly shuffled. The data series (following random shuffling) are shown at the

bottom of the figure; recurrence parameters and recurrence output are listed to the left of the

plot. Note that the same recurrence parameters were used to generate recurrence plots for the

original times series (see Figure 3.5) and the randomly shuffled data sets.
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the recurrence plots in Figures 3.5 and 3.12 shows that fewer points are

recurrent for the randomly shuffled data and that almost none of those

recurrent points form diagonal lines. This qualitative change is

reflected quantitatively by the fact that, for the randomly shuffled data,

%REC < 0.01% and %DET < 0.001%. Randomizing the data reduced

the number of recurrent points, but, perhaps more importantly, it

eliminated the deterministic structure of the original time series. The

interested reader can reproduce the plots shown in Figure 3.12 by

using the data that accompany this chapter and running program

RQD.EXE with the parameter settings listed previously (and depicted at

the left side of the plots in Figure 3.12) and selecting ‘y’ for the

randomize data sequences option.

Results of RQA

For AP COP, ANOVA revealed a main effect of cognitive task

condition on %DET, F(3, 57) = 3.52, p < .05, which is shown in Figure

3.13a. %DET was greater for counting back by 3s (M = 88.70) than quiet

standing (M = 85.65) and digit reversal (M = 83.61). This finding

suggests that the temporal structure of AP postural fluctuations became

more regular as the attentional demands of the cognitive task

increased. Recalling the results of the traditional analysis of AP COP

data (Figures 3.3b and 3.3d) in view of the observed changes in %DET

for the AP COP time series, we see that although the amplitude and

variability of postural sway increased with greater attentional demands

of the concurrent cognitive task, the postural fluctuations became more

deterministic (regular). ANOVAs on %REC, MAXL, ENT, and TND did

not reveal any other effects of cognitive task condition for the AP COP

time series.
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82

84

86

88

90

Quiet

Standing

Reversal Classification Back by 3s

Cognitive Task

%DET AP COP

a

2.8

3

3.2

3.4

3.6

Quiet

Standing

Reversal Classification Back by 3s

Cognitive Task

%REC ML COP

b

1100

1150

1200

1250

1300

Quiet

Standing

Reversal Classification Back by 3s

Cognitive Task

MAXL ML COP

c

4.55

4.6

4.65

4.7

4.75

4.8

Quiet

Standing

Reversal Classification Back by 3s

Cognitive Task

ENT ML COP

d

Figure 3.13. Results of RQA for all experimental trials. (a) Percent determinism (%DET) for AP

COP as a function of cognitive task condition. For ML COP, (b) Percentage of recurrent points

(%REC), (c) maxline (MAXL), and (d) entropy (ENT) as a function of cognitive task condition.

For ML COP, ANOVA on %REC showed that the effect of

cognitive task condition approached significance, F(3, 57) = 2.64, p <

.06 (see Figure 3.13b). Pair-wise comparisons revealed a lower

percentage of recurrent points (p < .05) for the counting back by 3s

cognitive task (M = 2.89) than for digit classification (M = 3.25) and

quiet standing (M = 3.47). Generally, ML COP fluctuations were less

recurrent when performing concurrent cognitive and postural tasks

than when simply standing.

ANOVA on MAXL of the ML COP time series revealed a main

effect of cognitive task condition, F(3, 57) = 2.83, p < .05. MAXL was

shorter for counting back by 3s (M = 1142.9) than for digit classification

(M = 1286.8) and digit reversal (M = 1278.3). This finding, depicted in
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Figure 3.13c, suggests that the temporal structure of the ML COP was

less mathematically stable when the cognitive task required higher

attentional demands.

Figure 3.13d shows the significant main effect of cognitive task

condition on ENT of the ML COP data, F(3, 57) = 4.25, p < .01. ENT was

lower for counting back by 3s (M = 4.60) than for digit classification (M

= 4.73) and quiet standing (M = 4.78). This finding suggests the

deterministic structure of the ML COP was less complex for the

cognitive task of highest attentional demand.

ANOVAs for the ML COP data did not indicate an effect of

cognitive task condition on the recurrence measures of %DET and TND.

GENERAL DISCUSSION

The present research highlights the utility of RQA for the study of

postural fluctuations. Using a traditional approach to the analysis of

COP data, we found that total COP excursion, as well as the range and

variability of AP and ML COP motion, were impacted by attentional

demands in a similar manner. Results of the traditional analyses showed

that performing a concurrent cognitive task increased the magnitude

and variability of postural sway. These findings could lead one to

conclude that carrying out an unrelated but concurrent cognitive task

compromises postural stability.

The RQA results suggest an alternative interpretation. Attentional

demands impacted postural sway, but not necessarily in the form of a

decline in the effectiveness of the postural control system. Examination

of the temporal structure of postural fluctuations revealed that

attentional demands influenced AP COP and ML COP in different ways.
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Whereas with higher attentional demands fluctuations in ML COP

became less recurrent, less stable, and less complex, AP COP

fluctuations became more deterministic. What might explain these

observed differences in recurrence patterns in the two COP component

directions? One possibility is that the RQA findings reflect a strategy

being used by the central nervous system (CNS) to optimize postural

control. Recall the stance position maintained by study participants,

which is shown in Figure 3.2. In keeping with instructions to stand with

feet together, participants adopted a narrow base of support during

testing. Thus, it is likely that the perceived limit of stability (PLOS—the

distance an individual can sway without losing balance or taking a

protective step) was smaller for ML motion than for AP motion.

Increases in ML COP motion may have presented a greater threat to

postural stability than AP COP motion, since increased ML motion

would bring the ML COP closer to the PLOS. The changes in recurrence

patterns of ML COP data series may follow from the increased COP

motion in that direction. Of note, spontaneous ML postural sway, rather

than AP sway, has been shown to be predictive of fall risk in older

adults (Lord, Rogers, Howland, & Fitzpatrick, 1999; Maki, Holliday, &

Topper, 1994).

Why wouldn’t the CNS simply reduce sway range and variability

as a way of promoting postural stability under conditions of greater

attentional demand? The observed increase in the deterministic

structure of AP COP may be a more efficient and more effective means

of optimizing postural control. Sway range and variability are important

aspects of exploratory postural behavior—sway is “exploratory”

because it generates stimulation regarding the current state of postural
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stability. The regularization of AP motion could be a strategy that

simplifies the problem of postural control without sacrificing the pickup of perceptual information made available through spontaneous

postural sway. In short, rendering AP COP motion more deterministic

may be one approach by which the postural system adjusts to the

attentional demands of a concurrent cognitive task.

To summarize, it is difficult to interpret the results of the

traditional analysis of COP data in terms other than a classical dual-task

effect, in which concurrent performance of a cognitive task brings

about a decrement in the ability of the CNS to control posture. Results

of RQA, however, suggest adaptation of the postural system (perhaps

proactively as well as reactively) to changing task demands. Although a

full understanding of the findings reported here must await further

research, a few points are clear. The results of the RQA brought to light

dynamical processes inherent in postural control that are not evident in

summary measures of COP path magnitude and variability. In addition,

our findings offer further evidence that AP and ML COP motion can be

affected differentially in response to varying task requirements

(Balasubramaniam et al., 2000). Most importantly, the present study

supports the notion that the response of the postural control system to

dual-task requirements is one of adaptation not deterioration.

CONCLUSION

A conventional approach to the analysis of COP data revealed

increases in measures of COP path magnitude and variability during

performance of a concurrent cognitive task. Those findings are

consistent with the notion that dual-tasking compromises postural
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136

control. Also of note, AP COP and ML COP summary measures were

impacted in a similar manner. In a second stage of data analysis, RQA

revealed changes in the dynamical properties of postural sway brought

about by concurrent performance of cognitive and postural tasks.

Among the observed changes were differential effects on the AP and

ML components of postural fluctuations. The results of RQA suggest that

the postural control system adapts, rather than deteriorates, in

response to changing attentional requirements. The analytic tools

available through RQA promise insight into the mechanisms and

processes underlying postural control not accessible with a

conventional approach to the study of postural sway.
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AUTHOR NOTE

This work was supported in part by a Patrick and Catherine Weldon

Donaghue Medical Research Foundation grant awarded to G. L.

Pellecchia. The authors thank Michael Riley for many helpful

discussions about RQA parameter settings for COP data.

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CHAPTER 4

Cross Recurrence Quantification of

Interpersonal Postural Activity

Kevin Shockley

Department of Psychology

University of Cincinnati

Cincinnati, OH 45221-0376

U. S. A.

E-mail: kevin.shockley@uc.edu
Interpersonal Postural Activity

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Measuring interpersonal coordination in the context of

conversation has been a challenging problem in psychology for at least

three decades. This is an interesting problem because observations of

interpersonal coordination in the context of cooperative conversation

appear to index the coordination that is required to complete the goals

of a particular interaction (Clark, 1996). This type of coordination has

been indexed in a number of ways, including the convergence of

speaking rate (Street, 1984), vocal intensity (Natale, 1975), pausing

frequency (Cappella & Planalp, 1981), and even in the convergence of

conversational partners’ dialects (Giles, Coupland, & Coupland, 1991).

Conversational partners have also been observed to mirror or mimic

each others’ postures (LaFrance, 1982). Until very recently, however,

such indices of coordination have been based on fairly subjective

observation procedures. For example, Condon and Ogston (1971)

assessed interpersonal coordination by hand scoring video-taped

interactions to evaluate the timing of listeners’ movements with

reference to the rhythmic properties of a speaker.

More recent techniques of quantifying interpersonal

coordination have made the visual scoring approach somewhat more

systematic by sketching joint angles from video tapes and quantifying

the number of joint angle changes (Newtson, 1994; Newtson, Enquist, &

Bois, 1977; Newtson, Hairfield, Bloomingdale, & Cutino, 1987). Spectral

profiles of periodicity of joint angle changes were then compared

across conversers. This strategy has revealed apparent coupling of the

behavioral waves of conversers. While the Newtson et al. approach is

certainly an improvement over previous methods, problems with

distortions from scoring 2-dimensional video tapes of 3-dimensional
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movements cannot be avoided. The magnitude of angle changes of

joints based on visual estimates of a 2-D video is dependent upon the

angle of the video-taped person relative to the 2-D viewing plane of the

video screen. Thus, visual estimates of angle changes from video will

be distorted unless the movements are always aligned with the viewing

plane. An additional drawback is that the degree of joint angle change

is not measured—all that is measured is that joint angles changed.

Shockley, Santana, and Fowler (2003) introduced a strategy for

evaluating the degree of interpersonal coordination that involves

submitting measurements of postural sway to cross recurrence

quantification (CRQ) analysis. The advantage of CRQ over

conventional linear methods is that it requires no assumptions about the

nature of the data in question, and it offers an objective method for

studying interpersonal coordination. The purpose of this chapter is to

provide a tutorial for how to apply this recently developed analysis to

postural data using the method and data of Shockley et al. (2003).

Before discussing the CRQ technique and its theoretical foundations,

however, some understanding of postural sway is necessary to make it

clear why such an approach is warranted.

POSTURAL SWAY

Although standing upright seems straightforward and effortless

to most of us, it is actually quite challenging to explain how we are able

to accomplish this task. Consideration of the underlying anatomical

constraints and surrounding physiological processes reveals that the

apparently simple act of standing upright is quite complex and yields a

correspondingly complex behavior that is not straightforward to
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quantify. The fact that postural activity is complex motivates the need

for measures that are not restricted by the assumptions of linear

methods.

Operationally, upright stance is best understood as the

maintenance of the horizontal position of the center of mass of the body

within the boundaries of the base of support of the body. Imagine a line

drawn straight down to the ground along the direction of gravity from

the center of mass of the body (generally in the region of the

abdomen). In order to avoid falling down, the projected line from the

center of mass must remain within the base of support of the body

(usually the anterior-posterior and lateral extents of the feet).

The simplest image of this control requirement is to keep an

inverted pendulum upright. To build your intuitions, try to balance an

inverted broomstick on one finger. Note that in order to keep the

inverted broomstick upright, quick adjustments to the position of the

supporting finger are required as the broomstick begins to fall in one

direction or another. This is a useful image, but the inverted pendulum

image is certainly not an adequate model of postural control, however,

given that most animals are of the multi-segmental variety. The

consequence of this fact is that the center of mass must effectively be

balanced over the base of support across many joints.

One may imagine that simply freezing all joints in a particular

position would achieve the desired goal of standing upright. The

physiological activity surrounding all of our actions, however, must also

be considered. For example, instability in the position of the center of

mass is introduced by physiological processes such as the inherent

tremor of muscular tensile states, heart compressions, and the
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expansion and compression of the chest cavity involved in breathing.

These inherent perturbations result in instability of the position of the

center of mass over time; this is generally referred to as postural sway

(see Figure 4.1).

Postural sway occurs even during so-called quiet stance

(standing without engaging in other activities; Collins & De Luca, 1994;

Newell, Slobounov, Slbounova, & Molenaar, 1997), but occurs

especially when supra-postural tasks (e.g., reading, talking, pointing,

reaching) are added to the demands of maintaining upright stance

(Balasubramaniam, Riley, & Turvey, 1997; Belen’kii, Gurfinkel’, &

Pal’tsev, 1967; Fel’dman, 1966; Riccio & Stoffregen, 1988; Riley, Mitra,

Stoffregen, & Turvey, 1997; Stoffregen, Pagulayan, Bardy, & Hettinger,

2000). For example, the seemingly benign act of raising one’s arm

compromises postural stability and requires concurrent (and often

prior) compensation of the muscles of the thighs, hips, and trunk to

keep the center of mass within the base of support (Belen’kii,

Gurfinkel’, & Pal’tsev, 1967; Pal’tsev & Elner, 1967). Speaking and

gesturing are ubiquitous in conversation and, accordingly, add to the

instability of the location of the body’s center of mass.

CAPTURING THE DYNAMICS OF UNKNOWN NONLINEAR SYSTEMS

The inherent instability of postural activity described above has

made the quantification of postural sway quite challenging. The most

direct methods for measuring postural sway involve the measurement

of the center of pressure of the body using a force platform or the

measurement of displacement of the center of mass using a motion
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Figure 4.1. Sample postural sway time series during quiet stance. The abscissa corresponds to

time (sec) while the ordinate corresponds to anterior-posterior displacement (cm).

tracking system.1 These measures yield time series that are typically

irregular, non-stationary (i.e., there is drift in the mean and/or standard

deviation of the time series over time), and non-periodic (Carroll &

Freedman, 1993; Collins & De Luca, 1993). Thus, conventional (linear)

analyses that assume normal distributions and stationarity, such as

correlational or spectral methods, are not appropriate for postural sway

 1 The center of pressure corresponds to the point of application of the sum of forces acting

between the feet and the surface of support (see Pellecchia & Shockley, Chapter 3).
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data (see Riley, Balasubramaniam, & Turvey, 1999; see also Pellecchia

& Shockley, Chapter 2, for a comparison of linear and nonlinear

methods of analyzing postural sway).

Linearity vs. Nonlinearity

By definition, linear time series analysis methods assume

independence and additivity of the multiple degrees of freedom that

contribute to a given observable. Nonlinear systems have degrees of

freedom that interact multiplicatively. Relatively recent investigations

into nonlinear systems have inspired the development of analysis

methods that capitalize on the interactive nature of nonlinear systems

(see Abarbanel, 1996). In some cases, the variables that contribute to

the dynamics of some nonlinear systems are known and can be

indexed. Most often, however, the variables that contribute to the

dynamics of a system under investigation are not known. The

advantage of a nonlinear system, however, is that the dynamical

variables interact. Thus, the influence of unknown (or perhaps

unmeasurable) variables can be indexed by variables that are readily

measurable. To illustrate this peculiar feature of nonlinear systems, I

will show how the measurement of a single variable from a system with

known dimensions can be used to capture the dynamics of the whole

system.

The Lorenz Attractor: A Nonlinear System With Known Dynamics

The Lorenz (1963) system is a model of convection (i.e., heat

transfer) in the atmosphere. The dynamics (i.e., changes in states) of

the Lorenz model can be characterized with three first order differential

equations,
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X = a(Y − X ) & [4.1]

Y& = X (b − Z) −Y [4.2]

Z& = XY = cZ [4.3]

where X, Y, and Z correspond to the three dynamical variables

(corresponding to two temperature measures and a velocity measure),

the over-dot corresponds to the rate of change (i.e., derivative) of the

variable in question, and a, b, and c are constant parameters. What

makes the Lorenz system a complex, nonlinear system is the interaction

of the three dynamical variables. As can be seen in Equations 4.1-4.3,

changes in X are dependent not only upon the value of X, but also upon

the values of Y and Z. Therefore, the influences of the variables X, Y,

and Z on the current state of the system are not independent and

additive, but are instead mutually dependent and multiplicative. The

interactive nature of dimensions along which a system may change

embodies the complexity of nonlinear systems and is also the key to

quantifying systems with unknown or unmeasured dynamical variables.

Phase Space Reconstruction and the Embedding Theorem

Often, the interplay of dynamical variables is best characterized

using a phase space (see Figure 4.2). A phase space is essentially a

multidimensional scatterplot of each dynamical variable with respect to

the other dynamical variables. As is often the case, however, the

particular variables that comprise a given nonlinear system may not be

known and perhaps may not even be knowable, a priori. Fortunately,

lack of identification of the dynamical variables does not preclude one
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from gaining access to the underlying dynamics of the system in

question.

Figure 4.2. A three-dimensional plot (phase space) of the time evolution of the three dynamical

variables (X, Y, & Z) of the Lorenz system (Equations 4.1-4.3).

Takens (1981) introduced the embedding theorem, which

revealed that the preferred relations of the dynamical variables in a

nonlinear system (its attractors) may be discovered by reconstructing a

phase space for the system in question using time-delayed copies of a

single, observable dynamical variable of the system. That is, the time

series of a single dynamical variable can be used to reveal the

underlying dynamics of the entire system by using time-delayed copies
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of the measured time series. This is possible because (as described

previously) in nonlinear systems the multiple dynamical variables

interact with one another. The interactive nature of the dynamical

variables comprising the nonlinear system generally yields quite

messy and unpredictable time series of the variables in question.

However, the interactive nature of the degrees of freedom of a

nonlinear system dictate that the activity of a single variable will be

influenced by the activity of all other dynamical variables. Access to

one of the variables of a dynamical system can, accordingly, allow the

dynamics of the entire system to be evaluated by unfolding the time

series into the appropriate number of dimensions to reveal the

underlying dynamics. In the next section I will illustrate how the

dynamics of the Lorenz system may be unfolded using time-delayed

copies of a single variable.

Distortions Due to Projection

Consider the time series of X in Figure 4.3. The evolution of this

variable does not resemble the clearly defined attractor in Figure 4.2.

This is because the time series of the observed variable is represented

by only one dimension (X), whereas the Lorenz system requires three

dimensions to reveal the influence of the three dynamical variables.

While the time series depicted in Figure 4.3 does embody the dynamics

of the entire system, one cannot see the dynamics unfold properly

because the dynamics of the 3-dimensional system are ‘projected’ onto

a single dimension. Such a distortion can be illustrated using a less

abstract example.

Imagine positioning your two hands such that both hands are

between a light source and a wall. Both hands are some distance apart
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Figure 4.3. The time series X(t) generated from Equations 4.1-4.3. Given that the data are

simulated, time (t) corresponds to data points with no particular time units.

along the dimension separating the light source and the wall, but the

two hands occupy the same position along the horizontal and vertical

dimensions of the wall. In this configuration, the projected shadows of

the two hands on the wall will appear to occupy the same space. This

shadow example illustrates a distorted, two-dimensional projection of

three-dimensional space.

Unfolding the Time Series

The lesson to be learned from Takens’ theorem is that the

dynamics of a nonlinear system of multiple degrees of freedom (e.g.,
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Figure 4.4. A reconstructed phase space of the Lorenz attractor using X(t) to create timedelayed copies to serve as surrogate dimensions. The delay (τ) used was 55 data points.

the Lorenz system) may be seen properly by unfolding the system into

the appropriate number of dimensions using time-delayed copies of

one measured dimension as surrogate dimensions in reconstructed

phase space. This can be accomplished by using the original time

series, X(t), as the first dimension, X(t + τ) as the second dimension, and

X(t + 2 τ) as the third dimension, and so on for four and higher

dimensions. In this example, I used a delay of 55 data points. So, the

first dimension, X(t), begins at data point 1 of the original time series.

The second dimension, X(t + 55), begins at data point 56 of the original

time series. The third dimension begins at data point 111 of the original
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time series. What Takens demonstrated is that the reconstructed phase

space is isomorphic to the true phase space of the system and,

accordingly, allows the system in question to be evaluated in the

appropriate number of dimensions (compare Figures 4.2 and 4.4). The

purpose of the Lorenz example is to show that a phase space of a

system can be reconstructed even with access to only one of the many

possible dimensions of change.

QUANTIFICATION OF POSTURAL SWAY

How can one take advantage of the technique of phase space

reconstruction to quantify what appears to be terribly complex postural

activity? Recent efforts have made headway in this regard and have

demonstrated that nonlinear methods of quantification are useful in

differentiating among postural sway time series that correspond to

subtly different activities of the person standing (see Riley et al., 1999;

Riley & Clark, 2003). For example, Riley et al. (1999) detected

deterministic structure in postural sway time series that cannot reliably

be differentiated from stochastic noise using conventional methods.

The method they used was recurrence quantification analysis (RQA;

Webber & Zbilut, Chapter 2), which capitalizes on Takens’ embedding

theorem.

Recurrence Analysis

Webber and Zbilut (1994) introduced the basic RQA strategy,

which involves reconstructing a phase space for a given time series, in

the manner described above for the Lorenz system, and then tallying

the number of instances that an unfolded time series visits each location

in reconstructed phase space (i.e., how often a value recurs). The
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degree of deterministic structure in the system may be assessed by

quantifying how many sequences of recurrent points are repeated (i.e.,

repeating patterns of recurrent points). The variety of differing lengths

of these sequences of recurrent points (number of data points forming a

recurring sequence) may be used to determine the complexity of the

signal by computing the Shannon entropy (see below) of the

distribution of the lengths. The stability of the system may be

measured by the longest of these sequences of recurrent points

(maxline) (Eckmann, Kamphorst, and Ruelle, 1987). Finally, the degree

of stationarity of the system may be assessed by determining the slope

of the density of recurrent points as the points become more separated

in time (trend). This procedure is known as auto-recurrence since a

time series is compared to itself. See chapters in this volume by

Webber and Zbilut (Chapter 2) and Pellecchia and Shockley (Chapter

3) for more extensive discussion and illustration of RQA measures.

CROSS RECURRENCE Q UANTIFICATION AS A M EASURE OF

COUPLING

Cross Recurrence Quantification (CRQ) was introduced by

Zbilut, Giuliani, and Webber (1998) as an extension to RQA (see also

Webber & Zbilut, Chapter 2). This extension involves effectively

embedding two synchronous time series in a reconstructed phase

space. Rather than tallying the recurring locations of a single

embedded time series (auto-recurrence), the number of instances for

which locations are shared by the two time series is tallied in CRQ (see

Figure 4.5). Measures comparable to those of RQA are available with

CRQ. Percent recurrence (%REC) in CRQ corresponds to the ratio of
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Figure 4.5. Illustration of time series collection, phase space reconstruction, and CRQ

measures %REC and MAXL. Blue corresponds to data from one member of the participant pair

and red corresponds to data from the other member of the participant pair. %REC quantifies

shared locations in reconstructed phase space of two points from the two time series. MAXL

quantifies the longest of parallel trajectory of the two time series in reconstructed phase space.

the number of shared location relative to the number of possible shared

locations (see Figure 4.5). Percent determinism (%DET) is the ratio of

the number of shared locations that are part of a sequence of shared

locations relative to the total number of shared locations. Maxline

(MAXL) is the longest shared trajectory and is a measure of the stability

of the shared activity (see Figure 4.5). Entropy (ENT) is the Shannon

entropy of the distribution of lengths of sequences of shared locations.

It was recently demonstrated that CRQ is a useful measure of the

coupling of two signals by evaluating physically coupled oscillators

(Shockley, Butwill, Zbilut, & Webber, 2002). An apparatus was
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constructed that immersed a rotor with a paddle into a container filled

with viscous fluid. The container (i.e., the driver) was then oscillated in

a translational fashion at a fixed frequency while the immersed rotor

was allowed to spin in the fluid at its natural frequency. The idea was to

vary the strength of the coupling of the two oscillators by changing the

viscosity of the fluid. In this coupled-oscillator system, the coupling

strength of the signals was directly manipulated so that the efficacy of

CRQ for detecting shared activity could be evaluated. Very subtle

couplings between two signals were detected by CRQ measures that

remained undetected by conventional, linear measures of coupling,

such as cross-spectral analysis. For example, in medium and low

coupling conditions, CRQ was able to detect the influence of the driver

tray on the rotor oscillation to which linear spectral analysis was blind.

Having CRQ as a tool for quantifying the shared activity between two

signals allowed Marie-Vee Santana, Carol A. Fowler, and I to turn our

attention to the problem of quantifying interpersonal coordination

(Shockley et al., 2003). We endeavored to evaluate the utility of CRQ in

detecting subtle postural coupling that may exist between two people

engaged in conversation.

V ERBAL C OORDINATION OF INTERPERSONAL POSTURAL

ACTIVITY—THE EXPERIMENT

The Task

The strategy of Shockley et al. (2003) was to track the postural

activity of two people engaged in a conversation. In order to generate

conversation, a puzzle task was used that required participants to

determine the differences between two similar cartoon pictures.
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Similar types of puzzles can often be found in puzzle books or

newspapers. Typically, however, the task involves a single person

comparing two pictures. The innovation we introduced was to give

each member of a participant pair one of the two similar cartoon

pictures. We did not allow participants to visually inspect each other’s

picture during the course of a trial. This constraint left only verbal

interaction to facilitate finding differences between the two pictures.

This method was quite effective in generating normal conversation.

Furthermore, the participants expressed genuine interest in the task

both during and after data collection, as indicated by their reluctance

to stop at the end of a trial and their unsolicited positive comments

following the data collection session.

In recognition that visual interaction may influence postural sway

in addition to the verbal interaction required by the task, two

independent variables were factorially combined, Task Partner and

Body Orientation. There were two levels of each variable (see Figure

4.6). A participant’s Task Partner could be either the other member of

the participant pair (Participant) or one of the experimenters

(Confederate), who was seated out of view. The Body Orientation of the

participant pairs involved either facing each other (Facing) or facing

away from each other (Away). In all conditions, participants were

instructed to discuss their pictures with their task partner and

determine as many differences as possible between the two pictures

over the course of two minutes. In all trials, the dependent measures

were the postural sway of the two members of a given participant-pair,

irrespective of the Task Partner on a given trial. Thus, the shared

postural activity between two persons engaged in conversation with
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Figure 4.6. Method for evaluating interpersonal postural constraints involved in cooperative

conversation. Participants faced toward or away from one another and conversed with each

other or a confederate (C). Figure adapted from Shockley, Santana, and Fowler (2003).

each other could be compared to the shared postural activity of the

same two persons engaged in conversation with others. Greater

shared postural activity (i.e., greater %REC) was expected when

participant pairs were conversing with each other than when

participant pairs were conversing with confederates.

Data Collection and Reduction

A Polhemus FasTrak magnetic motion capture system (Polhemus,

Inc., Colchester, VT) was used with 6-D Research System software (Skill

Technologies, Phoenix, AZ) to track the participants’ movements in the

anterior-posterior direction. Sensors were placed using Velcro straps

at the waist and the forehead. For the purposes of this chapter I am

only reporting data measured at the waist. Participants stood on

opposite sides of the magnetic field emitter and were each
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approximately 18 inches away from the emitter. Data columns of

displacement in the anterior-posterior direction were extracted from

the data file recorded by the motion capture software.

Unless otherwise indicated, for the subsequent analyses

Recurrence Quantification Analysis software was used (the package is

available free of charge from http://homepages.luc.edu/~cwebber/).

Detailed instructions for how to use the recurrence analysis software

can be found in the README.TXT file, which is provided with the

software. It is assumed that the user has a working knowledge of an

MS-DOS environment. The program ZSCORE.EXE was used to convert

our displacement data into z-scores to ensure a common scale for all

participants.

CRQ Parameters

Prior to calculating recurrence quantities, one must select the

settings for seven parameters (embedding dimension, delay, range,

norm, rescaling, radius, and line length). Embedding dimension,

delay, and radius are among the most challenging parameters to

determine. When the system in question is of unknown dimensionality

and periodicity (e.g., postural sway data), one method for estimating

these parameters is to evaluate the number of recurrent points for a

range of these parameter settings (Zbilut & Webber, 1992; Riley et al.,

1999; see Pellecchia & Shockley, Chapter 3). For a range of parameter

settings it is important that there are smooth changes in the number of

recurrent points in response to small changes in parameter values (see

Trulla, Giuliani, Zbilut, & Webber, 1996). A large, discontinuous

change in the number of recurrent points may correspond to a change

in the scale of activity in the system to which the recurrence measures
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are sensitive. Selecting parameters that are near such a threshold of

sensitivity could, therefore, yield changes in recurrence values due to

crossing the threshold, rather than changes due to experimental

manipulations. Thus, it is safer to use parameter values within a range

that exhibits smooth changes in %REC.

The program KRQS.EXE may be used to calculate %REC for a

range of the three parameters in question. For each execution of the

program, one inputs the file names corresponding to the two time

series to be compared as well as an output file name to which the

recurrence measures are saved. In our case, the input files

corresponded to the z-score time series for Person A and Person B,

respectively, of a given participant pair for a given 2-minute trial. After

entering the command to execute KRQS.EXE, one is prompted for the

minimum and maximum values for delay, embedding dimension, and

radius.

Embedding Dimension. Embedding dimension specifies how

many dimensions will be used in reconstructing the phase space. As

discussed previously, the goal is not to determine exactly how many

dimensions the system has. The goal is to be confident there are

sufficient embedding dimensions to allow the dynamics of the system to

be revealed without distortions. Webber (2004) suggested, for

physiological data, starting with embedding dimensions between 10

and 20 and working downward. Based on Webber’s suggestion and

previous investigations of postural activity using recurrence analysis

(e.g., Balasubramaniam, Riley, & Turvey, 2000; Riley et al., 1999), I

elected a range of 8 through 14 embedding dimensions for the present

discussion.
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Delay. The delay parameter specifies the time lag to use for the

time-delayed copies of the original time series (i.e., the surrogate

dimensions in the reconstructed phase space). For illustrative purposes

of this chapter and data processing economy, I selected delays ranging

between 15 and 25 data points. A larger range, however, would

certainly be appropriate. The sampling rate of the motion capture

system used in the present study was 60 Hz. This means that delays of

15 to 25 data points would correspond to time delays of 0.25 to 0.42 s.

Radius. The radius parameter specifies the Euclidean distance

within which points from the two time series are considered neighbors

(i.e, recurrent) in the reconstructed phase space. The selected radius

should yield a sparse recurrence matrix. That is, %REC should remain

low (no larger than 5%), but not so small as to produce a floor effect

(%REC at or near 0%). I selected radii ranging between 20 and 40, with

a step size of 2. Given that mean distance rescaling will be selected

(see below), the radius corresponds to the percentage of the mean

distance separating points in reconstructed phase space.

Norm. The norm parameter specifies how distances are

normalized in reconstructed phase space. I selected Euclidean

normalization, which is consistent with previous studies using

recurrence analysis to evaluate postural activity (see Riley, et al., 1999;

Riley & Clark, 2003).

Rescale. The rescale parameter determines the method used to

rescale the distance matrix (i.e., matrix of all distances among postural

data points of person A and person B). Given that each participant pair

may have had different distance magnitudes, Shockley et al. (2003)

elected to rescale the distances among the points to the mean distance.
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This served to intrinsically define the inclusion radius (see above) to a

percentage of the average distance separating the postural trajectories

of a given participant pair, rather than using an arbitrarily defined

threshold (e.g., an absolute Euclidean distance). A distance

corresponding to the mean distance separating postural locations in

reconstructed phase space would, therefore, have a value of 100. Mean

distance rescaling also serves to minimize the influence of an outlier

value (e.g., large postural excursion due to laughing or sneezing)—as

compared to rescaling to the maximum distance, for example.

Range. Each time series consisted of 7200 data points (120

seconds of data at 60 Hz). In the present study, there is no reason not to

include as many data points as possible. Therefore, the first data point

should be 1 and the last point should be 6875 (given a maximum

embedding dimension of 14 and a maximum delay of 25). These values

guarantee the use of the maximal number of data points and the same

number of data points within each surrogate dimension in the phase

space (see Pellecchia & Shockley, Chapter 3, for a detailed explanation

of how to determine the last data point). The value for the last data point

does not need to be determined, however. The software will provide a

range of values, and the user can simply enter the maximum in the

range presented by the software.

Line Length. Line length specifies the number of consecutive

recurrent points required to define a line segment. This parameter

should be set to two points (the minimum option) unless one has reason

to believe that a more conservative estimate of a line is warranted.

The next step in determining parameters for CRQ is to randomly

select a few trials from each experimental condition and execute the
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program KRQS.EXE to compute recurrence measures for the selected

parameter ranges for the selected trials. For illustrative purposes, I

selected four sets of data to submit to KRQS.EXE.

Surface Plots for Parameter Selection in CRQ

One method of evaluating the %REC values for the selected

range of parameter values is to use surface plots, which can be

generated in the MatLab software environment (Mathworks, Inc.,

Natick, MA). For each embedding dimension, one can plot the %REC

values returned from KRQS.EXE for a given data set as a surface (see

Figure 4.7). A surface corresponds to a numeric matrix (stored in a

MatLab variable that I arbitrarily named “rec”). The number of rows of

the matrix corresponds to the number of values of radius used by

KRQS.EXE and the columns of the matrix correspond to the number of

values of delay used by KRQS.EXE. The values in the matrix correspond

to the %REC values for one set of data (generated by KRQS.EXE) for

each radius (rows) and delay (columns) for a given embedding

dimension. To review, the extents of the two sides of each surface

correspond to the range of radius and delay, respectively.

To create a surface plot in MatLab, one needs two vectors—one

of extents corresponding to the numbers of columns and rows of the

%REC matrix, respectively, and the matrix that I called “rec” in MatLab.

First, create two MatLab vectors—“radius” [20 22 24 26 28 30 32 34,36

38 40] and “delay” [15 16 17 18 19 20 21 22 23 24 25]—to be used for all

surfaces plotted. As discussed above, the matrix should be created

from the %REC values returned from KRQS.EXE. The values of %REC

will, therefore, be different for each trial analyzed. The matrix will,

accordingly, need to be re-created for each surface to be plotted. Each
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Figure 4.7. Shared postural locations (%REC) in reconstructed phase space for a randomly

selected subset of data from Shockley, Santana, & Fowler (2003) for a range of values of the

following parameters: Embedding dimension, delay, and radius. Each surface in a given plot

corresponds to a different data set and each plot represents the same data sets for the

parameter ranges.

surface plot is created by the following command: surf(delay, radius,

rec). To plot multiple surface plots on one graph (as seen in Figure

4.7), type ‘hold’ after the first surf command. I have found that it is most

useful to create a different plot for each embedding dimension. To

generate new plots, open a new figure window and repeat the

procedure. For illustrative purposes I have plotted four of the eight

embedded dimensions (see Figure 4.7).
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Inspection of Surface Plots

Embedding Dimension. Notice in Figure 4.7 that the recurrence

values for the sampled data files tend to bottom out (%REC equals 0)

for higher values of the delay and low values of the radius for

embedding dimensions 12 and 14. Thus, an embedding dimension of

10 allows sufficient unfolding of the time series and still yield a sparse

recurrence matrix (i.e., recurrence values just above 0%).

Delay. The particular delay selected is often arbitrary for

postural data. What is most important is that the patterning of

recurrence measures is consistent across a range of delay values. If so,

one can be confident that any observed differences across

experimental conditions are not artifacts of the delay. Shockley et al.

(2003) selected a delay of 25 data points, which fits these criteria.

Radius. As discussed above, the radius is the maximum

Euclidean distance by which points can be separated in reconstructed

phase space and still be considered recurrent points. One should

choose a value of radius that ensures that the number of recurrent

points is reasonably low (so as to avoid global recurrence, i.e., all

points recurrent). Webber (personal communication, June 2000)

suggested that recurrence should be approximately 1%. Given that

CRQ must be performed on multiple trials of multiple participants,

however, Webber’s prescribed recurrence range tends to be a bit too

low. This is so because data from some trials may yield %REC = 0 for

the same parameter settings that yield 1% recurrence for other

participants. For biological movement data, I have found that

recurrence of around 3%-5% for a randomly selected subset of data
Interpersonal Postural Activity

167

tends to yield non-zero %REC values for all subjects, but still offers

sufficiently low recurrence.

In summary, Shockley, et al. (2003) selected a delay of 25 data

points, an embedding dimension of 10, and a radius of 30% of the mean

distance separating points in reconstructed phase space. Modest

changes of each of the values of those parameters does not change the

patterning of results found by Shockley et al. (2003).

Full Analysis

After the parameters to be used for the analysis have been

selected, the next step is to run the CRQ with the selected parameter

settings on the entire set of experimental data. Shockley et al. (2003)

used KRQE.EXE to compute the recurrence variables—%REC, %DET,

MAXL, and ENT—for each participant pair. The reason for using

KRQE.EXE, rather than KRQD.EXE, for example, is that the former allows

multiple analyses to be executed in batch mode, rather than waiting for

each file to be analyzed and then typing the next command for the next

file to be analyzed.

To review, the following parameter settings were used with

KRQE.EXE: delay = 25, embedding dimension = 10, range = 1-6875,

norm = Euclidean, rescaling = mean distance, radius = 30, and line

length = 2. The program KRQEP.EXE was used to generate a parameter

file to be accessed by the commands in the batch file. This obviates the

need to type in the parameters for each file to be analyzed. An ASCII

(text), tab-delimited batch file (filename.bat) was set up such that each

row corresponded to the MSDOS command for analyzing one file using

KRQE.EXE. The number of rows corresponded to the number of files to

be analyzed (see README.TXT file for complete instructions). Program
Shockley

168

run time for the present data was several hours. Mean values for the

recurrence measures were calculated for the four trials of each

condition. Separate ANOVAs were conducted on each recurrence

measure for postural time series pairs.

Results of Shockley, Santana, & Fowler (2003)

Among the measures derived from the tallied cross recurrence

values, %REC (the ratio of the number of recurrent points to the

number of possible recurrent points) and MAXL (the length of the

longest trajectory of consecutive parallel recurrent points; a measure of

the stability of the shared activity) were found to be significantly

influenced by the experimental manipulations. Representative cross

recurrence plots are provided in Figure 4.8A and 4.8B. The plots are

organized such that the postural time series of one member of a

participant pair (A) is indexed along the abscissa while the postural

time series of the other member of a participant pair (B) is indexed

along the ordinate. Points are plotted in the cross recurrence plot when

the trajectories of the unfolded time series, A and B, occupy the same

area of the reconstructed phase space within some radius of inclusion.

That is, if a given position (i) in time series A occupies the same position

in reconstructed phase space as the position (j) of time series B (i.e.,

A[i] = B[j]), then the point is considered recurrent (see Figure 4.5).

Shockley et al. (2003) found greater shared postural activity among

pairs of participants engaged in conversation with each other as

compared to the activity shared among the same participants engaged

in conversation with confederates, regardless of visual interaction.

They also found that trajectories of participant pairs stayed

parallel longer when the participant pairs were engaged in
Interpersonal Postural Activity

169

Figure 4.8. (A) Sample cross recurrence plot of the postural activity of two persons engaged in

conversation with confederates. (B) Sample cross recurrence plot of the postural activity of two

persons engaged in conversation with each other. Both are examples of the participant-pair

facing away from each other. Indices along the abscissa (i) and the ordinate (j) corresponds to

data points of person A and B, respectively of a participant-pair. Illuminated pixels correspond

to shared postural locations in reconstructed phase space.
Shockley

170

conversation with each other than when participant pairs were

engaged in conversation with others. As far as the authors could

determine, their observations mark the first objective measures of

interpersonal postural coordination. Representative examples of a

linear method of shared signal activity—coherence analysis—are

provided in Figure 4.9. In Figure 4.9A, the participants were speaking

to each other. In Figure 4.9B, the participants were each speaking to a

confederate. Coherence analysis is effectively a correlation of Fourier

spectrum power estimates compared across two signals for a range of

frequencies. Note that there are not distinct frequencies at which

power is correlated in either plot. Furthermore, on average the

coherence for the two plots is the same (~0.085). This is consistent with

the Shockley et al. (2002) finding that CRQ is a more sensitive measure

of coupling than linear measures. Perhaps of greater importance,

however, is the fact that CRQ is a more appropriate method for

quantifying shared postural activity than linear methods, given that

CRQ does not require the assumptions of normal distributions and

stationarity of the data required by linear methods. That is, any

significant effects that may have been discovered with linear methods

would be suspect due to violations of the assumptions of those methods.

It is encouraging that recurrence strategies continue to prove

their usefulness for the study of postural control. It remains to be seen

how sensitive of a measure CRQ may prove to be. For example, is the

degree of postural coordination scaled by the degree of cooperation of

a given verbal interaction? Giles (1973) recognized changes in

coordination among conversers with his observation that conversers in

a cooperative and friendly setting show convergence of dialects, while
Interpersonal Postural Activity

171

Figure 4.9. (A) Representative example of a coherence plot of the postural activity of two

persons engaged in conversation with confederates. (B) Representative example of a

coherence plot of the postural activity of two persons engaged in conversation with each other.

Both are examples of the participant pair facing away from each other. Frequency (Hz) is

plotted along the abscissa and the ordinate corresponds to the coherence between the power

of the Fourier spectra of the individual time series of the participant pair.
Shockley

172

hostile conversations show divergence of dialects. Will %REC

decrease in a less cooperative situation as compared to a more

cooperative situation? Furthermore, it remains to be seen what is the

mechanism of coupling that was observed by Shockley et al. (2003).

Investigations are currently under way to determine what aspects of the

interaction (e.g., speaking rhythms, conversational turn-taking, word

similarity) facilitate the type of shared postural activity that was

observed. I leave these questions to future investigations.
Interpersonal Postural Activity

173

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199-203.
CHAPTER 5

Introduction to Fractals

Larry S. Liebovitch

Center for Complex Systems and Brain Sciences, Center for Molecular

Biology and Biotechnology, & Departments of Psychology and

Biomedical Sciences

Florida Atlantic University

777 Glades Road

Boca Raton, FL 33431

U. S. A.

E-mail: liebovitch@clifford.ccs.fau.edu

http://walt.ccs.fau.edu/~liebovitch/larry.html

Lina A. Shehadeh

Center for Complex Systems and Brain Sciences

Florida Atlantic University

777 Glades Road

Boca Raton, FL 33431

U. S. A.
Liebovitch & Shehadeh

179

INTRODUCTION

This chapter is a Word version of the PowerPoint presentation

given by Dr. Larry S. Liebovitch at the NSF Nonlinear Methods in

Psychology Workshop, October 24-25, 2003 at George Mason

University, Fairfax, VA. The PowerPoint presentation itself is also

available as a part of this web book. Here the notes which can be seen

on the PowerPoint presentation by using “Normal View” are presented

as text around their respective PowerPoint slides. The concept here is

to try to reproduce the look and feel of the presentation at the

workshop. Therefore, this is not, and is not meant to be, your usual

“print” article. The form of the language here is more typical of

spoken, rather than written, English. The form of the graphics is

sparser, larger pictures captioned with larger fonts, that is more typical

of PowerPoint presentations than printed illustrations. We hope that

this experimental format may provide a simpler introduction to fractals

than that of a more formal presentation. We also hope that the

availability of the PowerPoint file will be of use in teaching these

materials and may also serve as a starting point for others to customize

these slides for their own applications.

This chapter is about “fractals”. Objects in space can have fractal

properties. Time series of values can have fractal properties. Sets of

numbers can have fractal properties. Much of the statistics that you are

familiar with deals with the “linear” properties of data. Fractals can

help us describe some “non-linear” properties of data.
Introduction to Fractals

180

Most data are characterized by the mean and standard deviation, like

45.3 ± 0.3. You’ll learn here that if the data are fractal, those means and

standard deviations are meaningless! A pretty basic change in the

simplest way we handle data.

Fractals are important because

they CHANGE the most basic

ways we analyze and understand

experimental data.
Liebovitch & Shehadeh

181

We’ll start with objects. Let’s first see the difference between the nonfractal and fractal objects.

Properties of Objects

in Space

Non-Fractal and Fractal Objects are different.
Introduction to Fractals

182

As we enlarge a non-fractal object, no new details appear.

Non-Fractal

Properties of Objects in Space
Liebovitch & Shehadeh

183

But, as we enlarge a fractal object we keep seeing ever smaller pieces.

For example, this series of pictures could show first the inside of the

intestine, then the crypts between the cells, then the microvilli on each

cell. The smaller pieces are copies of the larger pieces. They are not

exact smaller copies, but they are smaller replicas that are kind of like

the larger pieces.

Fractal

Properties of Objects in Space
Introduction to Fractals

184

A non-fractal object has most pieces that are about the same size.

Non - Fractal

Size of Features

1 cm

1 characteristic

scale

Properties of Objects in Space
Liebovitch & Shehadeh

185

A fractal object has pieces of all different sizes. The variation in the size

of the pieces of fractal objects is much larger than the variation in the

size of the pieces of non-fractal objects. Typically, there are a few big

pieces, some medium-sized pieces, and very many tiny pieces.

Fractal

Size of Features

2 cm

1 cm

1/2 cm

1/4 cm

many different scales

Properties of Objects in Space
Introduction to Fractals

186

Fractal objects have interesting properties. Here we describe those

properties very briefly. Then later, we will describe them in more

detail.

Properties of Fractal Objects

Self-Similarity.

The little pieces are smaller copies of the larger pieces.

Scaling.

The values measured depend on the resolution used to

make the measurement.

Statistics.

The “average” size depends on the resolution used to

make the measurement.
Liebovitch & Shehadeh

187

A tree is fractal. It has a few large branches, some medium-sized

branches, and very many small branches. A tree is self-similar: The

little branches are smaller copies of the larger branches. There is a

scaling: The length and thickness of each branch depends on which

branch we measure. There is no average size of a branch: The greater

the number of smaller branches we include, the smaller is the

“average” length and thickness.

This tree is from http://www.feebleminds-gifs.com/trees23.jpg.

Example of a Fractal

A tree is fractal

from http://www.feebleminds-gifs.com/trees23.jpg
Introduction to Fractals

188

The pattern of lightning in the sky is fractal. It has a few large

branches, some medium-sized branches, and very many small

branches. The lightning pattern is self-similar: The little branches are

smaller copies of the larger branches. There is a scaling: The length of

each branch depends on which branch we measure. There is no

average size of a branch: The greater the number of smaller branches

we include, the smaller is the “average” length and thickness.

from http://bobqat.com/Mazama/Sky/013.html

Example of a Fractal

Lightning is fractal
Liebovitch & Shehadeh

189

The pattern of clouds in the sky is fractal. They are made up of a few

big clouds, some medium-sized clouds, and very many small clouds.

The cloud pattern is self-similar: The little clouds are smaller copies of

the larger clouds. There is a scaling: The size of each cloud depends

on which cloud we measure. There is no average size of a cloud: The

greater the number of smaller clouds we include, the smaller is the

“average” size of a cloud.

Example of a Fractal

Clouds are fractal

From http://www.feebleminds-gifs.com/cloud-13.jpg
Introduction to Fractals

190

The pattern of paint colors in a Jackson Pollack painting is fractal. The

pattern is made up of a few big swirls, some medium-sized swirls, and

very many small swirls. The pattern is self-similar: The little swirls are

smaller copies of the larger swirls. There is a scaling: The size of each

swirl depends on which swirl we measure. There is no average size of

a swirl: The greater the number of smaller swirls we include, the

smaller is the “average” size of a swirl.

Example of a Fractal

A Pollock Painting is Fractal

From R. P. Taylor. 2002. Order in Pollock’s Chaos, Sci. Amer. Dec. 2002
Liebovitch & Shehadeh

191

Fractals

Self-Similarity

Self-similarity: Objects or processes whose small pieces resemble the

whole.
Introduction to Fractals

192

The coastline, the fractal border between the land and the sea, has

many bays and peninsulas. As you magnify the coastline you see ever

smaller bays and peninsulas. The structure at a large scale is similar to

the structure at a small scale. It is similar to itself at different scales.

This is called self-similarity.

Water

Land

Water

Land

Water

Land

Self-Similarity

Pieces resemble the

whole.
Liebovitch & Shehadeh

193

This is the Sierpinski Triangle. In this mathematical object each little

piece is an exact smaller copy of the whole object.

Sierpinski Triangle
Introduction to Fractals

194

The blood vessels in the retina are self-similar. The branching of the

larger vessels is like the branching of the smaller vessels. The airways

in the lung are self-similar. The branching of the larger airways is like

the branching of the smaller airways. In real biological objects like

these, each little piece is not an exact copy of the whole object. It is

kind of like the whole object which is known as statistical self-similarity.

Branching Patterns

blood vessels

Family, Masters, and Platt 1989

Physica D38:98-103

Mainster 1990 Eye 4:235-241

in the retina

air ways

in the lungs

West and Goldberger 1987

Am. Sci. 75:354-365
Liebovitch & Shehadeh

195

Let’s try to understand statistical self-similarity. Here is an

unrealistically simplified picture of the blood vessels in the retina. If

we ask how many vessels are there of each different size we see that

there is one that is 40mm long, two that are 20mm long, four that are

10mm long, and eight that are 5 mm long.

Blood Vessels in the Retina
Introduction to Fractals

196

We can plot how many vessels there are of each size. This is called the

Probability Density Function (PDF). A power law distribution is

evidenced in a straight line on a plot of log (number) vs. log (size).

PDF - Probability Density Function

HOW OFTEN there is THIS SIZE

Straight line on log-log plot

= Power Law
Liebovitch & Shehadeh

197

The PDF of the large vessels is a straight line on a plot of log (number)

vs. log(size). There are a few big-big vessels, many medium-big

vessels, and a huge number of small-big vessels.

The PDF of the small vessels is also a straight line on a plot of

log(Number) vs. Log(size). There are a few big-small vessels, many

medium-small vessels, and a huge number of small-small vessels.

The PDF of the big vessels has the same shape (i.e., is similar to) the

PDF of the small vessels. The PDF is a measure of the statistics of the

vessels. So, the PDF (the statistics) of the large vessels is similar to the

PDF (the statistics) of the small vessels. This is statistical self-similarity.

The small pieces are not exact copies of the large pieces, but the

statistics of the small pieces are similar to the statistics of the large

pieces.

Statistical Self-Similarity

The statistics of the big pieces is the same

as the statistics of the small pieces.

Number

1 10 100 1000

1

10

100

1000

1 10 100 1000

1

10

100

1000

size in μm size in mm

SMALL Number

blood

vessels

BIG

blood

vessels
Introduction to Fractals

198

Fractals are not only objects in space, but can also be processes in

time. There are proteins, called “ion channels,” in the fatty membranes

of living cells that let ions, like sodium and potassium, enter or exit the

cell.

Fractal Properties in Time: Currents

Through Ion Channels
Liebovitch & Shehadeh

199

A small pipette can suck up a small piece of cell membrane with only

one ion channel in it, and even it tear it off and away from the cell. The

movement of sodium or potassium through the ion channel produces an

electrical current that can be measured. It’s a pretty small current, a

picoAmp, which is about one billionth (1/1,000,000,000) of the current

from a “D” battery. This is called the “Patch Clamp.” What’s really

interesting is that these ion channel proteins act like little electrical

switches. Either they are either fully open or fully closed to the

movement of sodium or potassium. They switch, all the time, between

these fully open and fully closed states. It’s impressive to watch this

technology measure the changes in a single molecule at a time.

Fractal Properties in Time: Currents

Through Ion Channels
Introduction to Fractals

200

These open and closed times are fractal! If you record them and play

them back slowly you see a sequences of open and closed times. But if

you take one segment of time, and play it back at higher resolution, you

see that it actually consists of many briefer open and closed times. It is

self-similar in time.

Currents Through Ion Channels

ATP sensitive potassium channel in

cell from the pancreas

Gilles, Falke, and Misler (Liebovitch 1990 Ann. N.Y. Acad. Sci. 591:375-391)

5 sec

5 msec

5 pA

FC = 10 Hz

FC = 1k Hz
Liebovitch & Shehadeh

201

Here is a histogram of the times (in ms) that one channel was closed.

The recording was made at the fastest time resolution, allowing the

briefest closed times to be recorded. The PDF is mostly a straight line

on this log (number) versus time (t) plot, but with an occasional longer

closed time. Data with fractal properties often have unusual events that

occur more often than expected from the usual “Bell Curve.” Those

occasional longer closed times are a hint that these data might be

fractal.

Closed Time Histograms

potassium channel in the

corneal endothelium

Number

of

closed

Times

per

Time

Bin in

the

Record

Liebovitch et al. 1987 Math. Biosci. 84:37-68

Closed Time in ms
Introduction to Fractals

202

Here is another histogram of the closed times (in ms) of that same ion

channel. This recording was made at a little slower time resolution and

so longer closed times were recorded. The PDF is mostly a straight line

on this log (number) versus time (t) plot, but with an occasional longer

closed time.

Closed Time Histograms

potassium channel in the

corneal endothelium

Number

of

closed

Times

per

Time

Bin in

the

Record

Liebovitch et al. 1987 Math. Biosci. 84:37-68

Closed Time in ms
Liebovitch & Shehadeh

203

Here is another histogram of the closed times (in ms) of that same ion

channel. This recording was made at an even slower time resolution

and so even longer closed times were recorded. The PDF is mostly a

straight line on this log (number) versus time (t) plot, but with an

occasional longer closed time.

Closed Time in ms

Number

of

closed

Times

per

Time

Bin in

the

Record

Closed Time Histograms

potassium channel in the

corneal endothelium

Liebovitch et al. 1987 Math. Biosci. 84:37-68
Introduction to Fractals

204

Here is another histogram of the closed times (in ms) of that same ion

channel. This recording was made at a much lower time resolution and

so only the longest closed times were recorded. The PDF is mostly a

straight line on this log (number) versus time (t) plot, but with an

occasional longer closed time. The PDF looks similar at different time

resolutions. The PDF is a measure of the statistics. So, the statistics is

similar to itself at different time resolutions. This is statistical selfsimilarity in time.

Closed Time Histograms

potassium channel in the

corneal endothelium

Number

of

closed

Times

per

Time

Bin in

the

Record

Liebovitch et al. 1987 Math. Biosci. 84:37-68

Closed Time in ms
Liebovitch & Shehadeh

205

Each of those histograms of the closed times is measured at its own time

resolution, the time width of each bin. Wouldn’t it be nice to see all

those different time scales at once? We can’t do that with a histogram,

but we can covert each histogram into its PDF and then combine those

PDFs. Here is the PDF of all those histograms combined. Now we can

see that there is a simple relationship (red line) between all the

different closed times. Thus, there is a relationship between the closed

times as short as a millisecond and those as long as a second. This

relationship is called a scaling relationship.

Closed Time PDF

potassium channel in the

corneal endothelium

Liebovitch et al. 1987 Math. Biosci. 84:37-68
Introduction to Fractals

206

Fractals

Scaling

Scaling: The value measured depends upon the resolution used to

make the measurement.
Liebovitch & Shehadeh

207

If we measure the length of the west coast of Britain with a large ruler,

we get a certain value for the length of the coastline. If we now

measure it again with a smaller ruler, we catch more of the smaller bays

and peninsulas that we missed before, and so the coastline

measurement is longer. The value we measure for the coastline

depends on the size of the ruler that we use to measure it.

Scaling

 The value measured depends

on the resolution used to do

the measurement.
Introduction to Fractals

208

Here is a plot of how the length of the west coast of Britain depends

upon the resolution that we use to measure it. There is no one value

that best describes the length of the west coast of Britain. It depends

upon the scale (resolution) at which we measure it. As we measure it at

a finer scale, we include the segments of the smaller bays and

peninsulas, and the coastline is longer. This is one of the surprising

way in which fractals change the most basic way that we analyze and

understand our data. There is no one number that best describes the

length of the west coast of Britain. Instead, what is important is how the

length depends upon the resolution that we use to measure it. The

more smaller bays and peninsulas, the more the length of the coast

increases when it is measured at a finer resolution, and the steeper the

slope on this plot. This plot therefore shows that the coast of Britain is

rougher than that of Australia, which is rougher than that of South

Africa, which is rougher than that of a plain circle.

How Long is the Coastline of Britain?

Richardson 1961 The problem of contiguity: An Appendix to Statistics of

Deadly Quarrels General Systems Yearbook 6:139-187

Log10 (Total Length in Km)

AUSTRIALIAN COAST

CIRCLE

SOUTH AFRICAN COAST

GERMAN LAND-FRONTIER, 1900

WEST COAST OF BRITIAN

LAND-FRONTIER OF PORTUGAL

4.0

3.5

3.0

1.0 1.5 2.0 2.5 3.0 3.5

LOG10 (Length of Line Segments in Km)
Liebovitch & Shehadeh

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Iannaccone and his colleagues study how organisms develop in order

to understand and cure cancer in kids. They mix cells from another

animal into an embryo so that the fate of these marker cells can be

traced out as the animal develops. The cells that are added have a

different enzyme, which attaches to a radioactive marker that blackens

a photographic film to make a picture. On the following page are some

of those pictures of the liver. Look, the added cells are not in one

clump. They are in islands of all different sizes.

There is no one area that best describes the size of these islands. The

area measured depends on the resolution used. This scaling

relationship is a straight line on a plot of log (area) versus log

(resolution).

There is no one perimeter that best describes the size of these islands.

The perimeter measured depends on the resolution used. This scaling

relationship is also a straight line on a plot of log (perimeter) versus log

(resolution).

This is one of the surprising way in which fractals change the most basic

way that we analyze and understand our data. There is no one number

that best describes the area or perimeter of these islands. Instead,

what is important is how the area or perimeter depends upon the

resolution that we use to measure it.
Introduction to Fractals

210

Genetic Mosaics in the Liver

P. M. Iannaccone. 1990. FASEB J. 4:1508-1512.

Y.-K. Ng and P. M. Iannaccone. 1992. Devel. Biol. 151:419-430.
Liebovitch & Shehadeh

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So far, we’ve seen fractal scaling in space. There are also fractal

scaling in time. The usual way to measure the switching of an ion

channel is the “kinetic rate constant.” That tells us the probability that

the ion channel switches between open and closed states. But the ion

channel must be closed (or open) long enough for us to see it as closed

(or open). A more appropriate measure is the probability that the ion

channel switches between open and closed states, given that it has

already remained in a state for a certain amount of time. That certain

amount of time defines the time resolution at which we measure the

switching probability. We called that probability the “effective kinetic

rate constant” (keff),

keff = Pr (T=t, t+Δt | T > teff) / Δt, [5.1]

which is the probability (Pr) for the ion channel to open (or close)

during the time interval T = (t, t+Δt), given that it has already remained

closed (or open) for a time T ≥ teff. In the branch of statistics called

renewal theory, keff is called the “age specific failure rate,” for

example, the probability that a light bulb fails in the next second given

it has already burned for teff hours. In the branch of statistics used in

epidemiology and insurance, keff is called the “survival rate,” for

example, the probability that a patient dies of cancer this year, if they

have already had cancer for teff years.
Introduction to Fractals

212

Kinetic Rate Constant:

k = Prob. to change states in the next dt.

Effective Kinetic Rate Constant:

keff = Prob. to change states in the next dt,

given that we have already remained

in the state for a time keff.

k = Pr ( T=t, t+dt | T > t ) / dt eff eff

age-specific failure rate

= – d

dt

ln P(t)

P(t) = cumulative dwell time distribution

Fractal Kinetics

Liebovitch et al. 1987 Math. Biosci. 84:37-68.
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We measured the open and closed times for an ion channel in the cells

in the cornea, the clear part in the front of the eye that you look through

to see these words. The effective kinetic rate constant is a straight line

on a plot of log (effective kinetic rate constant) versus log (effective

time used to measure it). This is a fractal scaling relationship in time.

The faster we could look, the briefer open and closed times we would

see.

70 pS K+ Channel

Corneal Endothelium

Liebovitch et al. 1987 Math. Biosci. 84:37-68.

keff in Hz

effective time scale

teff in msec

effective

kinetic

rate

constant

100

1000

10

1

1 10 100 1000

keff = A teff

1-D
Introduction to Fractals

214

Fractals have given us a new way to analyze data from the patch clamp

measurements of the open and closed times of ion channels. Instead of

measuring a property (the kinetic rate constant) at one time scale, we

measure how a property (the effective kinetic rate constant) changes

when we measure it at different time scales. We have been using the

information in this fractal scaling relationship to give us clues about the

structure and motions in ion channel protein. Specifically, we have

been using the scaling relationship to calculate the energy difference

between the open and closed states of the ion channel protein and how

that energy difference varies in time. The picture of ion channels

before fractals analysis was that they are firm, sharp, uptight things that

go click, click, click, between a few, very different static states. The

picture of ion channels after fractal analysis is that they are complex

dynamic things, with many pieces of different size that move over

different time scales, whose new shapes and movements determine

what it’s going to do next.

Fractal Approach

New viewpoint:

Analyze how a property, the effective kinetic

rate constant, keff, depends on the effective

time scale, teff, at which it is measured.

This Scaling Relationship:

We are using this to learn about the structure

and motions in the ion channel protein.

Liebovitch 1989 Math. Biosci. 93:97-115.

Liebovitch and Tóth 1991 Bull. Math. Biol. 53:443-455.

Liebovitch et al. 2001 Methods 24:359-375.
Liebovitch & Shehadeh

215

We have seen examples of scaling relationships for measurements in

space and time. There can also be scaling relationships for the

correlations between measurements. Like the scaling relationships for

measurements, the scaling relationship for the correlations between the

measurements is often a power law, that is, a straight line on a

logarithmic-logarithmic plot. For example, at the left in the figure on

the following page is a measurement in time. It is self-similar—there

are ever larger fluctuations over ever longer times. We can measure

the dispersion, the variation in the value, over different windows of

time. The dispersion is ever larger over ever longer time windows.

The slope of this scaling relationship on a plot of log (dispersion) versus

log(window size) is called the Hurst Exponent, H. When H = 0.5, the

measurements are not correlated. When H > 0.5, the measurements

are positively correlated. This is called persistence. An increase now

is more likely followed by an increase at all time scales later. When H

< 0.5, the measurements are negatively correlated. This is called antipersistence. An increase now is more likely followed by a decrease at

all time scales later. There are many different ways to find the

correlational scaling relationship. One method is the Hurst Rescaled

Range Analysis. Another method is Detrended Fluctuation Analysis.
Introduction to Fractals

216

Correlations

window 1

window 2

window 3

dispersion 1

dispersion 2

dispersion 3

Measures of disperson:

Hurst Rescaled Range Analysis: R/S

Detrended Fluctuation Analysis: DFA

log (window size)

H = 1/2

no correlation

anti-persistent

H < 1/2

persistent

H > 1/2

log (dispersion)
Liebovitch & Shehadeh

217

On the left, the Hurst rescaled range analysis was used to measure the

correlations in the open and closed times of an ion channel protein

(open circles). At short times, H = 0.6, and at along times H = 0.9.

These are very persistent correlations. The correlations disappear

(black circles) when the order of the open and closed times was

randomly shuffled. This means that there is a long term “memory,”

which gets stronger with time, in how the shape of the ion channel

protein changes in time. Previous models of ion channels, as shown on

the right, assumed that the channel switched between a few, discrete

shapes, without any memory. This fractal analysis tells us that ion

channels do not behave that way. Instead, the fractal analysis has

enabled us to see that there are important, continuous dynamical

processes, with memory, going on inside the ion channel protein.

C8 C7 C6 C5 C4

Ca++ Ca++ Ca++ Ca++

C3 C2 C1

Ca++ Ca++

8-state Markovian Model

“memoryless”

H = 0.5

Data

“a process with memory”

H = 0.6

H = 0.9

Kochetkov, et al. 1999. J. Biol. Phys. 25:211-222.

Fractal Kinetics
Introduction to Fractals

218

Here, the detrended fluctuation analysis was used to measure the

correlations in the time between footsteps. This scaling relationship is

also a power law, a straight line on a logarithmic-logarithmic plot. The

scaling exponent of that power law is different for the young and the

elderly person. These studies have given us insight into how the brain

controls coordination and walking, and how that control depends on

age and is changed by disease.

Fractal Walking

Hausdorff et al. 1997. J. Appl. Physiol. 82:262-269.
Liebovitch & Shehadeh

219

This is the take-home lesson: We are used to thinking that there is one

measurement that best describes a property of an object. For a fractal

object that extends over many scales, in space or time, a property

depends on the scale at which it is measured. There is no one

measurement that best describes the object. The object is best

described by how the property measured depends upon the resolution

at which it is measured. This relationship is characterized by a

parameter called the fractal dimension. The fractal dimension can be

calculated from the slope of this logarithmic-logarithmic graph.

one measurement:

not so interesting

slope

ScalingLogarithm of the measuremnt

Logarithm of the measuremnt

one

value

Logarithm of the

resolution used to make

the measurement

Logarithm of the

resolution used to make

the measurement

scaling relationship: much more

interesting
Introduction to Fractals

220

Fractals

Statistics

Fractals have some unique statistical properties. The “average” size

depends on the resolution used to make the measurement. What is

important is not the average, but how the average depends on the

resolution used to make the measurement.
Liebovitch & Shehadeh

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Here is a set of numbers; maybe they are the values measured from an

experiment. I have drawn a circle to represent each number. The

diameter of the circle is proportional to the magnitude of the number.

Here is a non-fractal set of numbers. Most of them are about the size of

an average number. A few are a bit smaller than the average. A few

are bit larger than the average.

Not Fractal
Introduction to Fractals

222

Here is the PDF of theoe non-fractal numbers. The PDF is how many

numbers there are of each size. The PDF here is called a “Bell Curve,”

a “Gaussian Distribution,” or a “Normal Distribution.” It’s strange that

someone chose to call this a “normal” distribution. We are about to see

that much of the world is definitely not like this kind of “normal.”

Not Fractal
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223

Here is a picture of Gauss on the old 10 Deustche Mark German bill. He

has now been replaced by the 5 Euro. You can see his curve and even

the equation for it on this bill! There are no equations on American

money. (There is a scientist on American money. Do you know who it

is?)

Gaussian

Bell Curve

“Normal Distribution”
Introduction to Fractals

224

Here is a set of numbers from a fractal distribution. The diameter of

each circle is proportional to the size of the number. These numbers

could be from the room around you. Look around your room. There

are few big things (people and chairs), many medium-sized things

(pens and coins), and a huge number of tiny things (dust and bacteria).

It is not at all like that “Normal” distribution. Sets of data from many

things in the real world are just like this. We call this a fractal

distribution of numbers because it has the same statistical properties as

the sizes of the pieces in fractal objects.

Fractal
Liebovitch & Shehadeh

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Here is the PDF of these fractal numbers. The PDF is how many

numbers there are of each size. There are a few big numbers, many

medium sized numbers, and a huge amount of tiny numbers. The PDF

is a straight line on a plot of log(How Many Numbers; the PDF) versus

log(value of the numbers).

Fractal
Introduction to Fractals

226

The statistics of a fractal set of numbers is very different from the

statistics of “normal” numbers that they taught you about in Statistics

101. The statistics you learned in Statistics 101 is only about non-fractal

numbers. Take the average of a sample of non-fractal numbers. This is

called the Sample Mean. As you include ever more data, the sample

means, shown here as μ, get ever closer to one value. We call that

value the Population Mean, shown here as μpop. We think that the

population mean is the “real” value of the mean.

Mean

Non - Fractal

More Data

pop
Liebovitch & Shehadeh

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The statistics of fractal numbers is very different. Take the average of a

sample of fractal numbers. This is called the Sample Mean. As you

include ever more data, the sample means do NOT get ever closer to

one value. Either the sample means keep increasing OR the sample

means keep decreasing as you include more data. THERE IS NO

Population Mean. There is NO one value that best describes the data.

The data extend over a range of many different values.

The Average Depends on the

Amount of Data Analyzed
Introduction to Fractals

228

Here is why that happens. Again, here is a set of fractal numbers. The

diameter of the circles are proportional to the size of the numbers. As

you include ever more numbers one of two things will happen:

1. If there is an excess of many small values, the sample means 

get smaller and smaller.

2. If there is an excess of a few big values, the sample means get

larger and larger

Whether 1 or 2 happens depends on the ratio of the amount of small

numbers to the amount of big numbers. That ratio is characterized by a

parameter called the Fractal Dimension.

The Average Depends on the

Amount of Data Analyzed

each piece
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229

Let’s play a non-fractal game of chance. Toss a coin, if it comes up tails

we win nothing, if it comes up heads we win $1. The average winnings

are the probability of each outcome times how much we win on that

outcome. The average winnings are (1/2) x ($0) + (1/2) x ($1) = 50¢.

Let’s go to a fair casino to play this game. Fair casinos exist only in

math textbooks; “fair” means the bank is willing only to break even and

not make a profit. We and the casino think it’s fair for us to be charged

50¢ to play one game. That seems reasonable; half the time we win

nothing, half the time we win $1, so if it costs 50¢ to play each time, on

average, we and the casino will break even.

Ordinary Coin Toss

Toss a coin. If it is tails win

$0, If it is heads win $1.

The average winnings are:

2-1.1 = 0.5

1/2

Non-Fractal
Introduction to Fractals

230

Here is the PDF of that non-fractal game of chance. It shows how often

(the PDF on the vertical axis) you will win how much money (the x value

on the horizontal axis) if you play 100 times. It’s a Bell Curve—a

Gaussian, Normal distribution—just the kind of distribution they taught

you about in Statistics 101.

Ordinary Coin Toss
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231

Here’s what happens when I played that non-fractal game, over and

over again. A computer (actually a Macintosh Plus running Microsoft

BASIC!) picked a random number to simulate flipping the coin. Here,

the average winnings per game is shown after n games. For a while I

(the Mac) was lucky. I was winning more than an average 50¢ in each

game. But, as you might suspect (this is called the Law of Large

Numbers), after a while my luck ran out. In the long run, I was winning

exactly an average of 50¢ in each game.

Ordinary Coin Toss
Introduction to Fractals

232

Now, let’s play a fractal game of chance. This game was invented by

Niklaus Bernoulli who lived in St. Petersburg, Russia, and was

published by his uncle Daniel Bernoulli who lived in Germany, about

350 years ago. Here, we toss a coin UNTIL it comes up heads. If it

comes up heads on the first toss, we win $2. If it comes up tails first,

and then heads on the second toss, we win $4. If it comes up tails twice,

and then heads on the third toss, we win $8. And so on.

The average winnings are the probability of each outcome times how

much we win on that outcome. The average winnings are (1/2) × ($2) +

(1/4) × ($4) + (1/8) × ($8) + (1/16) × ($16) + .... = 1 + 1 + 1 + 1 ... = ∞.

We say to the casino, “Half the time we’ll win $2; the median winnings

of this game is $2” because half the time the coin comes up heads on

the first toss and we win $2. “So, we think it is very fair to put up twice

the median winnings, $4, to play each game”. To our surprise, the now

angry casino owner says, “No!” He adds, “The average winnings of this

game are infinite, you must put up more than all the money in the

universe to play this game, even once!”

This game became known as the St. Petersburg Paradox, because we

and the casino cannot agree on the fee to play this game. It was called

a “paradox” because it was so surprising, and difficult for many people

to believe, that the player and the casino owner could disagree on what

is fair for such a simple game. This game is well known amongst

mathematicians, which means it’s well known amongst the people who

know it well. But, because of its unusually mathematical character, it

was not one of the threads of probability theory that was woven into the
Liebovitch & Shehadeh

233

fabric of statistics that became popular in the natural sciences and that

is taught in Statistics 101. Now, with the popularity of fractals, it is being

rediscovered and its importance in analyzing and understanding real

data increasingly appreciated.

St. Petersburg Game (Niklaus Bernoulli)

Toss a coin. If it is heads win $2,

if not, keep tossing it until it

falls heads.

If this occurs on the N-th toss

we win $2N.

With probability 2-N we win $2N.

H $2

TH $4

TTH $8

TTTH $16

The average winnings are:

2-121 + 2-222 + 2-323 + . . . =

1 + 1 + 1 + . . . = Fractal
Introduction to Fractals

234

Here is the PDF of that fractal game of chance. It shows how often (the

PDF on the vertical axis) you will win how much money (the x value on

the horizontal axis). It’s NOT a Bell Curve, Gaussian, or Normal

distribution. Most often you win only a small amount, more often you

win a bigger amount, very rarely you win a huge amount. It is just like

the fractal PDF of the blood vessels in the retina, or any fractal object! It

is a straight line on a plot of log (How Often) versus log(How Much).

St. Petersburg Game (Niklaus Bernoulli)
Liebovitch & Shehadeh

235

Here’s what happens when I played that fractal game over and over

again. Here, the average winnings per game is shown after n games.

The more I played, the more often there was sometimes a lot of tails

before that first head. When there are a lot of those tails, I won a huge

jackpot. As more and more of those jackpots happened, the average

winnings per game kept increasing. There is no average (population

mean) for this game. The more I played, the more the average kept

changing. They told you in Statistics 101 that the more data you have,

the closer the sample means are to the population mean. Not here!

There is no population mean. The more data we have (the more games

I played) the more the sample means keep changing. The few

exponentially large wins keep pushing the sample mean up, which is

very different than what you learned in Statistics 101. Welcome to

fractals.

St. Petersburg Game (Niklaus Bernoulli)
Introduction to Fractals

236

Here is a non-fractal object. It is a checkerboard. Actually, I’m only

showing you a piece of it; it should really extend forever in each

direction. Place a circle on it. Count all the black pixels in that circle,

and divide by the total number of pixels. That is the average density

within that circle. The graph shows how that density changes as the

circles get bigger and bigger. The average density fluctuates a bit;

after all, we are putting a round circle over a square grid. But, as the

circles get bigger and bigger, the average density gets closer and

closer to 1/2. This seems reasonable because the checkerboard is 1/2

black and 1/2 white.

Non-Fractal

Log avg

density within

radius r

Log radius r
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237

The figure on the next page is a fractal object. It is called a Diffusion

Limited Aggregation (DLA). It is statistically self-similar. It has little

spaces between its little arms, medium spaces between its mediumsized arms, and large spaces between its large arms. We’re only

showing you a piece of it, but it should also really extend forever in

each direction. Place a circle on it. Count all the black pixels in that

circle, and divide by the total number of pixels. That gives the average

density within that circle. The graph shows how that density changes as

the circles get bigger and bigger. As the circles get bigger we catch

more of the ever larger spaces between the arms, and so the density

gets smaller. As the circles get ever bigger, the density gets ever

smaller. There is no one density that describes this object. What’s

more, the local density on a big arm is very high. The local density

between big arms is very low. Yet, the same mechanism makes the

arms and the spaces between them. Based upon our Statistics 101

training, we are used to thinking that when the local average changes,

when there is a difference in the mean value between an experiment

and a control, or between now and then, that the system must have

changed. Here, fractals, with infinite variance, have moments, such as

the mean, that can be very different in space and time or between

experiments and controls, even though the basic process has not

changed at all!
Introduction to Fractals

238

Fractal

Log avg

density within radius r

Log radius r

.5

-1.0

-2.0

-1.5

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

-2.5

0

Meakin 1986 In On Growthand Form: Fractal and Non-Fractal Patterns in

Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp. 111-135
Liebovitch & Shehadeh

239

Here is yet another example of fractal data. Data from many

experiments have fractal properties. Here are the action potentials, the

little electrical sparks, that encode information sent down the nerves in

your body. Teich et al. measured them in the auditory nerve, which

brings information about sounds from your ear to your brain.

Electrical Activity of Auditory Nerve

Cells

Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52

voltage

time

action potentials
Introduction to Fractals

240

They divided the time record into windows and counted the number of

action potentials in each window. Here, two in the first window, six in

the second, and so on. The “firing rate” is the number of action

potentials in each window, divided by the time duration of that window.

Electrical Activity of Auditory Nerve

Cells

Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52

2

Count the number of action potentials in

each window:

6 3 1 5 1

Firing Rate = 2, 6, 3, 1, 5,1

Divide the record into time windows:
Liebovitch & Shehadeh

241

Here they made the windows twice as long in time, and counted the

number of action potentials in each window. Again, the “firing rate” is

the number of action potentials in each window, divided by the time

duration of that window.

Electrical Activity of Auditory Nerve

Cells

Teich, Johnson, Kumar, and Turcott 1990 Hearing Res. 46:41-52

Repeat for different lengths of

time windows:

8 4 6

Firing Rate = 8, 4, 6
Introduction to Fractals

242

In Statistics 101 they taught you that as you collect more data, the

fluctuations average out. You were taught to expect that the fluctuations

in the firing rate should be less as the time windows get longer. But

look here—the variations don’t change much as the time windows go

from 0.5 s to 5.0 s to 50.0 s! [Actually, the real deal here is that the

variance of the fluctuations falls much slower than 1/sqrt(n)]. You

include more data, but you don’t get any closer to the real firing rate.

There is no one single value, like a population mean, that best

describes the firing rate. The increase in variation at longer time

windows is real. It represents correlations in the action potentials

which may tell how information is encoded in the timing of the action

potentials.

Electrical Activity of Auditory Nerve Cells

Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52

0

The

variation

in the

firing rate

does not

decrease

at longer time

windows.

4 8 12 16 20 24 28

70

60

80

90

100

120

130

140

110

150

T = 50.0 sec T = 5.0 sec

T = 0.5 sec

FIRING RATE

SAMPLE NUMBER (each of duration T sec)
Liebovitch & Shehadeh

243

Fractals

Power Law PDFs

PDFs: Fractal data have a characteristic PDF form called a Power Law.
Introduction to Fractals

244

These are electrocardiograms (EKGs) that record the voltage from the

heart. The first chart strip shows a normal heart. The second chart strip

shows a heart that is beating dangerously fast (ventricular tachycardia).

This is dangerous because it can lead to ventricular fibrillation, where

the heart no longer contracts in a regular way and can result in death in

3 minutes.

Heart Rhythms
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245

To prevent ventricular tachycardia from leading to fibrillation and

death, a small device called a “cardioverter defibrillator” can be

placed just under the skin in the chest. Small wires are snaked around

through the blood vessels into the heart. It listens, electronically,

carefully to the heartbeat. When the heart beats 3 times its normal rate

for 5 beats, it sends an electrical shock strong enough to kick the heart

back into a normal rhythm. But, it’s also a computer, with a small

memory. It can remember when it is triggered. Back in the hospital, a

small coil is placed on the patient’s chest. Over this radio link, the

cardioverter defibrillator can play back when it was triggered. We

have been analyzing the times between when the device was

triggered. Just to be clear, we are not analyzing the time between

heartbeats. We are analyzing the durations between consecutive

triggerings of the cardioverter defibrillator.

Inter-event Times

Episodes of Ventricular

Tachycardia (v-tach)

t1 t2 t3 t4 t5

time ->

Cardioverter Defibrillator

Liebovitch et al. 1999 Phys. Rev. E59:3312-3319.
Introduction to Fractals

246

Here, from one patient, is the relative frequency, the number of

occurrences of a given duration between the defibrillator events. The

PDF is proportional to the relative frequency. The plot is a straight line

on a log (how often) versus log (time between events) scale. This is

fractal scaling. The events of rapid heart rate happen with a fractal

timing.

Interval (in days)

Relative

Frequency

10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 10 -4 10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

Relative Frequency =

(9.8581) Interval-1.0988

Patient #33
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247

Here again, from another patient, is the relative frequency. The PDF is

proportional to the relative frequency. The plot is also a straight line on

a log (how often) versus log(time between events) scale. This is also

fractal scaling. The events of rapid heart rate for this patient also

happen with a fractal timing.

Interval (in days)

Relative

Frequency

Relative Frequency =

(3.2545) Interval-1.3664

10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 10 -4 10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

Patient #53
Introduction to Fractals

248

In fact, for many patients events of ventricular tachycardia happen with

a fractal pattern. Most of the times between events are short,

sometimes they are longer, and very rarely they are very long, which is

typical of fractals. There is no average time between events. If you

measured the “average rate” of events you would get a different

answer if you measured them over a day or a week or a year. There is

no one number that best describes the time between these events. The

time between events happens over many different time scales. We are

working on other ways, fractal ways, of characterizing these times to

assess the status of patients and the effectiveness of medical therapies.

For example, we are seeing whether the slope or intercept of these

PDFs is a good indicator of diagnosis or treatment outcome.

6 Patients

Liebovitch et al. 1999 Phys. Rev. E59:3312-3319.
Liebovitch & Shehadeh

249

We are also analyzing the times at which different e-mail viruses arrive

at the gateway into an internet service provider. On the picture on the

following page are the events—the arrival times of e-mail viruses. We

are looking at the duration of times between the arrival of each virus.

We have studied 4 viruses:

1. AnnaKournikova doesn’t have a picture of her, it’s a file that 

you wouldn’t want to open.

2. Magistr can erase sectors on your hard disk or your 

cmos/bios. If you don’t know what the cmos/bios is, you don’t 

want us to tell you what happens if it gets erased.

3. Klez puts together messages by joining fragments of phrases 

that it contains.

4. Sircam tempts you to open and execute its attached file.

Much is known about the structure of the Internet. Less is known about

the dynamics of the Internet. The arrival times of these viruses depend

on both the structure and dynamics of the Internet. We are hoping that

our study of these arrival times will tell us how the structure interacts

with the dynamics if the Internet.
Introduction to Fractals

250

Inter-arrival Times of E-mail Viruses

t1 t2 t3 t4 t5

time ->

Liebovitch and Schwartz 2003 Phys. Rev. E68:017101.

AnnaKournikova

"Hi: Check This!” AnnaKournikova.jpg vbs.

Magistr

Subject, body, attachment from other files: erase disk, cmos/bios.

Klez

E-mail from its own phrases: infect by just viewing in Outlook Express.

Sircam

“I send you this file in order to have your advice.”
Liebovitch & Shehadeh

251

We have data, unfortunately, from many, many virus events. Here are

the PDFs from AnnaKournikova and Magistr, how often the different

times between the arrivals occurred. These plots are straight lines on a

log (how often) versus log(time between arrivals) scale. These are

fractal scalings. The arrival of these viruses happens with a fractal

timing.

E-mail Viruses

10 1 10 0 10 -1 10 -2 10 -3 10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

Interval

PDF AnnaKournikova

10 1 10 0 10 -1 10 -2 10 -3 10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

Interval

PDF

Magistr.b

d=1.51 d=3.19

20,884 viruses 153,519 viruses

Liebovitch and Schwartz 2003 Phys. Rev. E68:017101.
Introduction to Fractals

252

Here are the PDFs from Klez and Sircam, how often the different times

between the arrivals occurred. These plots are also straight lines on a

log (How Often) versus log(time between arrivals) scale. These are

also fractal scalings. The arrival of these viruses also happens with a

fractal timing.

E-mail Viruses

413,183 viruses 781,626 viruses

10 1 10 0 10 -1 10 -2 10 -3 10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

Interval

PDF Klez.e

10 1 10 0 10 -1 10 -2 10 -3 10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

Interval

PDF

Sircam.a

d=2.40 d=2.96

Liebovitch and Schwartz 2003 Phys. Rev. E68:017101.
Liebovitch & Shehadeh

253

We made a simple model to study how the structure and the dynamics

of the Internet are interrelated.

Structure: There are networks of computers of different sizes.

Dynamics: When each network sends out viruses, the number 

and time between them depend on the size of the network.

Model: Structure + Dynamics

Structure

Dynamics

K n(k) = k-a units of size k

e(k) = kb viruses

t(k) = k-c apart

In one burst:

time

Liebovitch and Schwartz 2003 Phys. Rev. E68:017101.
Introduction to Fractals

254

There were some interesting results from these models. First, the

model predicts a power law distribution in the PDF of the times

between the arrival of the viruses. Second, the exponent of that power

law tells us whether relatively more viruses are sent from the small

number of larger networks or the larger number of small networks.

Model: Results

The relative number of viruses from all units of size k ~ kb-a

(b-a) = c (d-1)

When d > 1: (b-a) > 0,

relatively more viruses come from the larger units,

as seen in the data.

When d < 1: (b-a) < 0,

relatively more viruses come from the smaller units.

d = 1 - a/c + b/c

PDF(t) = t

-d

where d = 1 - a/c + b/c
Liebovitch & Shehadeh

255

Fractals

Methods for Determining

the PDFs

The PDF is an important tool in determining if experimental data have

fractal properties. A power law PDF is characteristic of fractal

behavior. The standard method for evaluating the PDF is to make a

histogram of the data. That method is very good at determining the

PDF when the data are not fractal. It is less good at determining the

PDF when the data are fractal. Next, we’ll see other ways of

determining the PDF, and how they compare to the histogram method.
Introduction to Fractals

256

It’s not so easy to make a good PDF from the histogram of How Many

events there are in each time bin. If we choose the bins narrow, we get

good resolution for the small events. But, at long times, because the

bins are so narrow, we may see only 1 or 2 events, or even 0 events, in

some of those bins. If we try to fix this by making the bins wider, we do

get more events in the bins at longer times. But, at short times, we now

have poor resolution.

Determining the PDF

from a Histogram

Bins dt Small

Good at small t.

BAD at large t.

t

PDF

Bins dt Large

BAD at small t.

Good at large t.

t

PDF
Liebovitch & Shehadeh

257

We figured out a nice algorithm to get a better PDF. Narrow bins are

good at short times. Wide bins are good at long times. So, we make

histograms of different bin sizes. But, we cannot combine histograms of

different bin sizes. However, we can compute the PDF from each

histogram and then combine the PDFs. For each histogram, the PDF(t)

is N(t), the number of values in the bin that covers (t, t+dt), divided by

Nt, the total number of values in that histogram, divided by dt, the width

of the bins in that histogram. The histograms with narrow bins give us

good resolution in the PDF at short times. The histograms with wide

bins give us good values in the PDF at long times. We’ve found that this

method yields accurate and reliable PDFs for tails of many different

kinds of distributions. See Liebovitch et al. 1999 for details.

Determining the PDF

Liebovitch et al. 1999 Phys. Rev. E59:3312-3319.

Solution:

Make ONE PDF

From SEVERAL Histograms of DIFFERENT Bin Size

Choose dt = 1, 2, 4, 8, 16 ... seconds

PDF = N(t)

Ntotdt

N(t) = number in [t, t+dt]

Ntot = total number in each histogram

dt = bin size
Introduction to Fractals

258

Here, PDFs were measured from a set of fractal data. The red boxes

indicate the PDF made in the usual way from one histogram. You can

see where there are only 1 or 2 events in the largest bins. The black

boxes indicate the PDF generated from the same data using the new

multi-histogram method to make the PDF. Pretty impressive difference.

New multi-histogram

Standard fixed dt

10 4 10 3 102 10 1 10 0 10 -1 10 -2 10 -6

10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

Values

PDF

Determining the PDF
Liebovitch & Shehadeh

259

Fractals

Summary
Introduction to Fractals

260

SELF-SIMILARITY

Definition: Pieces of an object in space, or parts of a process in time, are

smaller versions of the whole object or process.

Examples: The Sierpinski Triangle in space and the times between the

arrival of e-mail viruses.

Methods: A power law distribution of the PDF of the pieces of an object

in space or the parts of a process in time is indicative of fractal

behavior.

Importance for data analysis: There is no single scale, in space or time,

that characterizes such data that extends over many scales.

Summary of Fractal Properties Summary of Fractal Properties

Self-Similarity

Pieces resemble the

whole.
Liebovitch & Shehadeh

261

SCALING

Definition: The value measured for a property depends on the scale, in

space or time, over which it is measured.

Examples: The length of the west coast of Britain and the closed times of

ion channel proteins in the cell membrane.

Methods: A power law scaling of the measured values or the correlation

between the measured values is indicative of fractal scaling behavior.

Importance for data analysis: Since no one value properly characterizes

the data, what is important is how the value measured depends on the

resolution used to make the measurement.

Summary of Fractal Properties Summary of Fractal Properties

Scaling

 The value measured

depends on the

resolution.
Introduction to Fractals

262

STATISTICS

Definition: The PDF is a power law. The population mean and

population standard deviation don’t exist.

Examples: The winnings in the St. Petersburg game and the variation in

the times between action potentials recorded from auditory nerve cells

in the ear.

Methods: A power law distribution of the PDF or a power law scaling

relationship for the moments is indicative of fractal behavior.

Importance for data analysis: When the mean depends on the spatial

scale, the temporal scale, or how much data we analyze, then the mean

is meaningless. What is meaningful is how the sample means, or

another scaling property, depend on the spatial scale, the temporal

scale, or how much data we analyze, which is described by the fractal

dimension.

Summary of Fractal Properties Summary of Fractal Properties

Statistical Properties

 Moments may be zero

or infinite.
Liebovitch & Shehadeh

263

Probability theory started from solving gambling problems about 400

years ago. About 200 years ago, those results were used to develop

basic statistics. Most of the statistical tests we use were developed less

than 100 years ago. We show you this to emphasize that statistics is

NOT a dead science, although it’s often presented like that in Statistics

101. It has changed a lot. It is still changing. It will change even more

in the future. The statistical properties of fractals are examples of new

ideas that are now being incorporated into and are changing statistics.

400 years ago:

Gambling Problems Probability Theory

200 years ago:

Statistics How we do experiments.

100 years ago:

Student’s t-test, F-test, ANOVA

Now:

Still changing

Statistics is NOT a dead science.
Introduction to Fractals

264

The take-home lesson here is not that fractals some arcane supersophisticated mathematical tool that only needs to be used in some

strange circumstance. Fractals change the most basic way we look at

experimental data. They allow us to analyze and make sense out of the

huge amount of real data that “just ain’t a bell curve.” The most

common use of mathematics and statistics in all science is means ±

s.e.m. Fractals tell us that if the data are fractal, those means are

meaningless! That’s a pretty basic change in the simplest way we

handle data. That’s what revolutions in science are about—not about

changing the complex stuff, but about changing the simplest stuff. The

stuff that we were taught so firmly that we never thought it would

change.

Fractals CHANGE the most basic ways we

analyze and understand experimental data.

Fractals

Measurements over many scales.

What is real is not one number, but how the measured

values change with the scale at which they are measured

(fractal dimension).

No Bell Curves

No Moments

No mean ± s.e.m.
Liebovitch & Shehadeh

265

TO LEARN MORE ABOUT FRACTALS

1. A book called Fractals and Chaos Simplified for the Life Sciences

(Liebovitch, 1998). This book consists of facing pages, where the left

page is text and the right page is a picture. It leads you, one concept at

a time, through the material.

2. A CD-ROM of curricula materials for a mathematics course for

college students who never liked and never did well in math (funded,

in part, by the National Science Foundation, Division of Undergraduate

Education). The materials emphasize what mathematics is, how

mathematicians do mathematics, and how mathematics is used in

science. We’re almost finished with it and would be happy to send you

a free demo (contact information is on the first page of this chapter). References: References:

Fractals and Chaos and

Simplified for the Life

Sciences

Larry S. Liebovitch

Oxford Univ. Press, 1998

The Mathematics and

Science of Fractals

Larry S. Liebovitch and

Lina Shehadeh

www.ccs.fau.edu/~liebovitch/

larry.html

CD ROM

NSF

DUE-9752226

DUE-9980715
Introduction to Fractals

266

We have concentrated here (and in the references noted on the

previous page) on providing an introduction to fractal concepts, their

importance, and what can be learned from them. Here are some books

that describe the mathematical details of these methods and give

examples of how scientists have used them.

Technical Details

J. Feder. 1988. Fractals. Plenum Press.t.

J. B. Bassingthwaighte, L. S. Liebovitch and B. J. West.

1994. Fractal Physiology. Oxford University Press.

P. M. Iannaccone and M. Khokha. 1996. Fractal

Geometry in Biological Systems. CRC Press.

A. Bunde and S. Havlin, eds. 1994. Fractals in

Science. Springer-Verlag.

ACKNOWLEDGEMENTS

The development of some of these materials was supported, in part, by

NSF grants DUE-9752226 and DUE-9980715.
CHAPTER 6

Gauging the Fractal Dimension of

Response Times from Cognitive

Tasks

John G. Holden

Department of Psychology

California State University, Northridge

Northridge, CA 91330-8255

U. S. A.

E-mail: jay.holden@csun.edu
Holden

268

An unexpected and exotic brand of variability resides in the

trial-by-trial fluctuations of human judgments of passing time. The

pattern, called 1/ƒ or pink noise, is a construct from fractal geometry.

Pink noise is associated with complex systems whose components

interact on multiple time scales to self-organize their behavior

(Bassingthwaighte, Liebovitch, & West, 1994; Jensen, 1998; Van Orden,

Holden, & Turvey, 2003; see also Aks, Chapter 7). This chapter

describes the phenomenon of pink noise, and explains how to conduct

statistical analyses that identify it in response time data from

elementary cognitive tasks.

It was Gilden, Thorton, and Mallon (1995) who first reported pink

noise in response time variability during the fundamentally cognitive

task of estimating fixed intervals of time. Gilden et al.’s temporal

estimation task required participants to repeatedly estimate fixed

intervals of time—in essence to “become a clock”—by pressing a

button at each instant they believed a specific time interval had

elapsed. Separate laboratory sessions were administered for each of

several fixed target time-interval conditions. In each session 1000 timeinterval judgments were collected in succession. The time-interval

conditions ranged from 1/3 s up to 10 s. Of course, no participant’s

succession of time-interval judgments was exactly the same on every

trial. Instead they varied from trial to trial. Lining up the series of

successive time estimates in the strict order in which they were

collected (i.e., trial 1, 2, ... 1000) yielded a trial series of response

times, which was treated very much like a standard time series in

Gilden et al.’s statistical analyses. Pink noise was revealed in the

intrinsic residual variability that remained after the average time
Fractal Variability in Cognitive Performance

269

interval that each participant produced, for each target interval, was

removed from each trial series. Thus, pink noise emerged in the

structure of the “background noise” of cognitive performance—the

intrinsic variability of a person’s judgments of passing time.

To begin to understand the phenomenon of pink noise it is

perhaps easiest to simply examine it visually. Figure 6.1A displays an

example of an individual participant’s trial series from a temporal

estimation task that used a method similar to that of Gilden et al. (1995).

The x-axis depicts the successive trials in the experiment and the y-axis

records the participant’s time estimate on each trial, in terms of

standard deviations from their average time interval, taken across all

trials (i.e., the time estimates are normalized and graphed as z-scores).

The overall pattern of trial-to-trial fluctuation is consistent with

pink noise. Notice the undulating “waves” of relatively longer and then

shorter time estimates that travel across the series. One shorthand way

to describe the overall rising and falling trends is to say that the first

500 or so trials follow a giant inverted-U shape or an arc. It is fair to say

that a similar large arc begins around the 500th trial and continues to

the end of the series of observations. Now, look within each large arc,

and similarly shaped arcs which run across fewer trials, perhaps only

50 to 100 trials at a time, can be discerned. Inside the smaller arcs are

even smaller ones, and so on. Loosely speaking, the trial series is

comprised of a progression of nested, similarly shaped arcs or patterns

of fluctuation. Of course, there is nothing special about the arcing

inverted-U shape; you could imagine M or W shapes, or even right-side

up Us, for example. What is important in this example is the concept of

a shape or pattern that is comprised of smaller copies of essentially the
Holden

270

Figure 6.1. (A) A trimmed, detrended and normalized trial-series of 690 ms time interval

estimates for a single participant. The x-axis indexes the successive trials in the experiment,

the y-axis indexes the time interval judgments, relative to the overall mean and standard

deviation of the trial series. (B) Results of a power spectral analysis, in linear units. Frequency

is plotted on the x-axis, from lowest (near the origin) to highest. The y-axis indexes the power

or relative energy of each frequency. Larger values indicate more power. (C) Results of the

same spectral analysis plotted in (B), on double-logarithmic axes. The negatively accelerated

linear relation, with a slope less than 0 but greater than –1, is consistent with pink noise. (D)-(F)

The same succession of plots as (A) through (C) for a randomly reshuffled “surrogate” version

of the data depicted in (A). Random reshuffling destroys the natural trial ordering and yields

white noise. (E) depicts the spectral analysis of the white noise on a linear scale. There is no

systematic variation in power as a function of frequency. This power spectrum is characteristic

of white noise; all frequencies have roughly equal power. (F) depicts the same power spectrum

on double-logarithmic coordinates. The slope of the regression line is approximately zero,

which is consistent with white noise.

0 500 1000

−4

−2

0

2

4

Trial Number

Normalized Time Estimates

A

0 0.25 0.5

0

2

4

6

8

Frequency

Power

B

−3 −2 −1 0

−1

0

1

2

Frequency in Log 10 Units

Power in Log 10 Units

SLOPE = −.59

C

0 500 1000

−4

−2

0

2

4

Trial Number

Normalized Time Estimates

D

0 0.25 0.5

0

2

4

6

8

Frequency

Power

E

−3 −2 −1 0

−1

0

1

2

Frequency in Log 10 Units

Power in Log 10 Units

SLOPE ≈ 0

F
Fractal Variability in Cognitive Performance

271

same shape—the notion of a nested structure of similar-shaped

fluctuations.

If you are not sure you see this structure, simply compare Figure

6.1A with Figure 6.1D, which represents exactly the same data set

depicted in Figure 6.1A, but where the order of the successive data

points was randomly shuffled. The shuffling procedure destroyed the

nested, statistically self-similar pattern of trial-to-trial fluctuations

characteristic of pink noise. The random rearrangement of the series

yields a pattern called white noise. Notice that just about any portion of

the shuffled data series depicted in Figure 6.1D could be used as

relatively good “stand-in” for any other portion of the series. This is not

true for the trial-ordered data plotted in Figure 6.1A, for which most of

the observations between trials 500-600 fall below the overall mean,

while the majority of the observations between trials 800-900 fall above

the overall mean.

The two different arrangements of the same data set are quite

distinct, illustrating the difference between pink noise and white noise.

For pink noise, the local means and standard deviations depend on

where in the series the sample was taken. White noise indicates

statistical independence from observation to observation, and local

sample means and standard deviations do a good job of describing

other local samples, and an overall population mean. This fact about

white noise forms the cornerstone of inferential statistics, such as ttests, analysis of variance (ANOVA), and regression.

In the context of response time research pink noise refers to a

statistically self-similar (see Liebovitch, Chapter 5) pattern of trial-to-
Holden

272

trial variability. The pattern is structured such that persistent, longterm fluctuations across several hundred trials nest within themselves

progressively smaller, proportionately scaled fluctuations across a few

decades of trials. Nested within those fluctuations, one finds even

smaller patterns of fluctuation, and so on. Pink noise is unexpected

from the perspective of conventional statistical intuitions inherited from

the standard linear statistical tools of behavioral science research.

Those intuitions lead to an expectation that successive individual time

judgments, decisions, or other elementary cognitive performances

should vary unsystematically or randomly from trial to trial. That is,

individual observations are assumed to be statistically independent.

After all, it is properties of the presented stimulus that are

conventionally assumed to be driving a participant’s response, not

aspects of the previous response.

Since Gilden et al.’s (1995) report, pink noise was uncovered in

the trial-by-trial variability, or trial-series, of a wide range of standard

cognitive psychology laboratory tasks that collect response times.

Examples include simple reaction time, speeded word naming, choice

reaction time, lexical decision, and mental rotation, among many others

(e.g., Gilden, 1997; Kelly, Heathcote, Heath, & Longstaff, 2001; Van

Orden et al., 2003). The pattern is not limited to cognitive activities;

pink noise appears in measurements of human performance that use

dependent measures other than response time, such as patterns of eye

movements (Aks, Zelinsky, & Sprott, 2002), postural sway (Riley, Wong,

Mitra, & Turvey, 1997), and self-reports of changes in mood over time

(Delignières, Fortes, & Ninot, in press; see Gilden, 2001, and Van

Orden et al., 2003, for reviews).
Fractal Variability in Cognitive Performance

273

The goal of this chapter is to provide a primer to the geometric

concepts and statistical techniques that are necessary to characterize

pink noise in trial series of response time measurements derived from

cognitive performances. The first step is to introduce three interrelated

concepts from fractal geometry: Self-similarity, scaling, and fractal

dimension. Those ideas motivate statistical analyses that are aimed at

the identification of fractal patterns in data from empirical phenomena.

A description of a simple temporal estimation task, modeled in large

part after the method used by Gilden et al. (1995), follows the

introductory sections. A trial-series resulting from the illustrative

temporal estimation task is used to provide a practical context for a

tutorial presentation of the statistical procedures involved in a fractal

analysis of response time data, including spectral density estimation

and fractal dimension estimation. Potential theoretical implications of

pink noise are briefly discussed in the final section of the chapter.

FRACTAL PATTERNS

Self-Similarity and Scaling

Two key constructs in fractal geometry are pattern and selfsimilarity of pattern. The parts of fractal objects are composed, in some

way, of copies of the whole object (Feder, 1988; Mandelbrot, 1982; see

Liebovitch, Chapter 5). Ideal geometric fractals may be composed of

exact replicas of the whole object—they are strictly self-similar. By

contrast, statistical fractals are self-affine, or statistically self-similar; they

are composed of statistically equivalent replicas of the whole object.

Naturally occurring fractals usually exhibit statistical self-similarity.
Holden

274

A contrast between the left and right sides of Figure 6.2

illustrates this distinction. The left side of the figure depicts a

Sieripinski Gasket, a classic mathematical fractal. It was generated by

removing a white, smaller triangle, with vertices that fall at the

midpoints of the sides of the largest, gray outer “initiator” triangle.

Next, smaller similar white triangles were removed from the three new

triangles that were formed, and their centers were, in turn, removed.

The construction process can be continued indefinitely (see Peitgen,

Jürgens, & Saupe, 1992, for details). The Sieripinski Gasket is

comprised of smaller, exact copies of itself—it is strictly self-similar.

The top panel on the right side of Figure 6.2 depicts 8192 observations

of idealized pink noise, a statistical fractal. The middle panel “zooms

in” on the center 4096 observations of the same series depicted in the

top panel, and the bottom panel depicts the center 2048 observations

that appeared in both the middle and top panels. When the x- and yaxes are appropriately scaled, the pieces of the series are visually and

statistically indistinguishable from the overall series—pink noise is

statistically self-similar, or self-affine.

Fractal patterns in nature are composed of nested forms that

cannot be measured on a single scale of measurement. The result of a

measurement depends on the scale, or size of the increment used to

take the measurement (Bassingthwaighte et al., 1994; Mandelbrot,

1982; Schroeder, 1991; see also Liebovitch, Chapter 5). For example,

the measured length of the British coastline increases proportionally as

the scale of the "yardstick" used for measurement is shortened from

kilometers to meters. An even shorter, centimeter scale of

measurement would result in a further proportional increase in the
Fractal Variability in Cognitive Performance

275

Figure 6.2. The object depicted on the left side of the figure is a classic self-similar

mathematical fractal called a Sieripinski Gasket. It is generated by removing successive

generations of triangles (white) from the centers of the gray triangles. The three panels on the

right side of the figure illustrate how idealized pink noise is statistically self-similar. The middle

panel zooms in on a piece of the series depicted in the top panel, and the lower panel zooms in

on a piece of the series depicted in the middle panel. Each magnification of the pieces of the

series results in a new series that looks essentially the same as the original series.

measured length of the coastline. The changing measurements arise as

a consequence of using regular line segments, the yardstick, to

approximate the irregular, nested, self-similar structure of coastal bays

0 4096 8192

−4

−2

0

2

4

Observation Index

Observation Value

0 2048 4096

−2

−1

0

1

2

Observation Index

Observation Value

0 1024 2048

−1

−0.5

0

0.5

1

Observation Index

Observation Value
Holden

276

and peninsulas. The length measurement increases when a bay or

peninsula that was not captured at a lower resolution adds length at a

higher resolution. Thus, as smaller and smaller sub-bays and subpeninsulas are resolved, they add to the length of the coastline.

When measurements change as a function of measurement scale,

there is no "true" or characteristic value for the measurement. The

length of the British coastline grows in proportion to the precision of the

yardstick used to measure length. This proportional power-law scaling

relation between the size of the yardstick and the costal length implies

that results of a measurement procedure depend on the measurement

scale or sampling unit used to take the measurement (over a finite

range of scales). A power-law scaling relation, a linear relation

between the logarithm of the scale and the logarithm of the

measurement result, is commonly observed in natural fractal

phenomena, and is symptomatic of self-similar patterns

(Bassingthwaighte et al., 1994, Peitgen, Jürgens, & Saupe, 1992). It is

the functional form of the scaling relation that, in turn, is used to

describe and even model subtly different aspects of coastlines (e.g.,

Mandelbrot, 1982).

Scaling in Statistics

What is measured with a statistical sample depends on what

statistic is computed on the sample of observations. A sample mean is

a measure of location—the center or balance point of a distribution of

observations. If the variability in the sample is unsystematically but

symmetrically dispersed about the mean, and the observations are

statistically independent of one another, then a sample mean identifies
Fractal Variability in Cognitive Performance

277

a location on the number line of the dependent measure that best

characterizes the level or amount of the measurement in the context of

sampling error. For instance, mean response time is used to estimate

the duration of time that passes between the presentation of a stimulus

and the collection of a response in a response time task. However,

response time distributions are typically positively skewed, and the

pattern of pink noise implies that successive observations are not

statistically independent.

Thus, patterns of variability constrain the utility of a sample mean

as a point estimate of a location or amount. Dispersion measurements,

like the standard deviation, are intimately linked to location

measurements. They impart information about disagreement among

the individual measurements, and indicate how much a measure of

location, such as the mean, can be trusted.

Scale magnification makes intuitive sense for an object such as

the coastline of Great Britain. The statistical counterpart to a

measurement scale is sample size. Obtaining fewer statistical samples

corresponds to using measurements at a lower resolution; larger or

more numerous statistical samples correspond to measurements at a

higher resolution.

At first, the relation between sample size and scale may not be

obvious, but remember the example of Great Britain’s coastline. The

example helps to develop an intuitive analogy about what it means that

no characteristic amount of variability may exist in response time data.

The coastline scaling relation indicates that smaller spatial features are

nested within larger spatial features. The key point to keep in mind
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about the coastline is the measurement result (the length) depends on

the scale used to take the measurement.

Statistically self-similar patterns of fluctuation affect statistical

estimates of location and dispersion, like the mean and standard

deviation or variance (see also Bassingthwaighte et al., 1994, pp. 33-

41). The answer to the question “how long” for a coastline is supplied

in terms of a length measurement. The answer to the same question for

a response time task is supplied by a parameter estimate, such as mean

response time. In contrast to length scales, where the smallest units

yield the most accurate measurements, larger sample-sizes—a larger

N—corresponds to a more precise measurement scale in statistics. This

may seem counterintuitive, but it is conventional to assume that

certainty in estimates of a population parameter—parameter

resolution—increases as sample sizes increase (i.e., the central limit

theorem).

As mentioned, the ability of a statistic such as the mean to act as a

gauge of location depends crucially on the inherent patterns of

variability in the data. If the measurements emerge from a process with

outputs that conform to standard statistical assumptions, those of the

central limit theorem, for example, a sample mean can be trusted to

reliably penetrate the variability and reveal increasingly reliable

estimates of characteristic values—population parameters—as sample

size is increased.

Nevertheless, another possibility exists. Suppose trial-by-trial

response time measurements are comprised of a statistically-self

similar pattern of positive correlation, where local patterns of
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correlation, across just a few successive trials, are nested within

increasingly more global and proportionally scaled (enlarged) patterns

of positively correlated fluctuations, as broader and broader runs of

consecutive trials are spanned. This is essentially a description of pink

noise, and just as for the coastline, a change to a more detailed or

inclusive scale yields essentially the same pattern that was observed at

the less inclusive, lower-resolution scale.

Keeping in mind the analogy to the coastline, it is easier to begin

to understand how nested patterns of fluctuation lead to counterintuitive

statistical properties. The value of any given lower-resolution locally

computed mean or standard deviation, that includes just a few adjacent

samples, depends crucially on where in the trial-by-trial series it was

taken—did it come from a waning “bay” or a waxing “peninsula,” for

instance. Increasing the resolution of the sample by including more

and more adjacent observations results in the inclusion of more and

larger bays and peninsulas. The implication of each increase in sample

resolution is the existence and inclusion of even larger scale

fluctuations that reach well beyond the scope of each new, larger scale

of resolution. Larger samples simply admit a broader range of

variability. The resulting increase in variability created by widening

the window of observation may outpace a sample statistic’s ability to

stabilize about a particular characteristic value, in the normal way,

through the process of aggregating larger and larger samples.

As such, the utility of the mean and standard deviation as simple

measures of location and dispersion may be foiled because of the

proportional, nested patterns of fluctuation. Different sample means

and standard deviations, taken at different locations or times, would
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tend to disagree in their location and dispersion estimates (up to the

limits of the system or sample), resulting in a persistent heterogeneity,

or disagreement among the sample statistics themselves. In this way,

patterns of variability that are comprised of nested, interdependent,

statistically self-similar fluctuations hamper the ability of these statistics

to provide a uniform summary or gauge of certain kinds of data sets.

Simply put, the concept of a summary statistic is not neatly applicable to

a heterogeneous process.

All this is not to say that descriptive statistics such as the mean

and standard deviation are not useful or applicable to fractal or

nonlinear science—in many cases they are essential analytical tools. It

is really only the semantics of those descriptive measures that is altered

in the analysis of a fractal process. Additional and crucial statistical

information resides at a higher level, in the manner in which the values

of descriptive statistics change as sample size (i.e., measurement

resolution) is systematically changed, rather than at the level of any

particular summary value that uses a particular sample size. That is, the

manner of disagreement across different sample sizes becomes a

primary statistical gauge for a fractal analysis.

In the context of response time research, a fractal dimension

analysis describes how response-time variability scales with sample

size. Essentially, it is a statistical analysis that is analogous to taking

measurements of a coastline’s length using different ruler sizes and

reporting how the length changes as the ruler size changes—the fractal

dimension analysis determines a scaling relation between sample size

and sample variability. The goal of the fractal dimension analysis is to

describe the changes in the variability of a measurement across a
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range of sample sizes (measurement resolutions) in terms of a powerlaw scaling relation.

Fractal Dimension

Self-similarity and self-affinity across multiple scales of

resolution often gives rise to objects or patterns that occupy noninteger

or fractal dimensions. Essentially, a fractal dimension refers to the

spatial dimension of an object whose dimension falls between the

standard Euclidean integer dimensions of one, two or three (see

Bassingthwaighte et al., 1994). Mandelbrot (1982) explained how the

dimension of an object is partly determined by the perspective of the

observer (the entry level of the analysis). A tautly stretched piece of

thread closely resembles a line, an ideal one-dimensional Euclidean

object. Tightly weaving the thread, back and forth, results in a piece of

fabric, an ideal two-dimensional object. Thus, a line can be rearranged

so that it occupies area. Rolling the thread onto a spool yields an object

that occupies volume in 3-D space. Of course, if the spool of thread is

viewed from a great distance, its dimension appears to collapse to zero,

a point.

One way to understand the link between Euclidean geometry

and fractal geometry conceptually is to think of fractal geometry as a

generalization or elaboration of the standard Euclidean geometry of

lines, squares, cubes, and so on. Euclidean objects only occupy

integer dimensions, 1 for a line, 2 for a square, and so on. Fractals,

however, may occupy noninteger dimensions, dimensions that fall in

between 1 and 2 or between 2 and 3. How can this be? Refer again to

Figure 6.1A, the normalized series of temporal estimates graphed in the
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order in which they were collected. They are points connected by a

line. Clearly, if every successive time estimate was identical,

connecting the points would form a line, and the series would have a

Euclidean dimension of one. But any departure from the ideal form of a

line begins to occupy or “leak into” the next higher (second, in this

case) Euclidean dimension. It is in this spirit that fluctuations in trial-bytrial response times may be said to partly occupy or leak into the next

higher Euclidean dimension. In a sense, the variability of time

estimates results in the series occupying area, and it will have a

dimension between an ideal one-dimensional line and an ideal twodimensional area. The more jagged and irregular the graph of

response times, the more area it occupies. It turns out that this intuitive

continuum of relative jaggedness can be characterized formally with

the help of a statistical procedure called dispersion analysis.

Dispersion analysis results in an estimate of the fractal dimension of the

trial series. The fractal dimension characterizes the structure of the

intrinsic variability in the trial series.

Conventional statistical analyses presuppose that intrinsic

variability is white noise. White noise yields a jagged and irregular line

with a fractal dimension of 1.5, because successive observations are

statistically independent of each other. Its fractal dimension indicates

the extent to which white noise occupies 2-D space. White noise is

uncorrelated noise. By contrast, the successive observations of pink

noise tend to be positively correlated. This results in a less jagged trial

series, and lower fractal dimensions that fall in the interval between 1

and 1.5.
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A SIMPLE TIMING EXPERIMENT

Data from a timing experiment is now used to demonstrate how

to conduct statistical analyses that may uncover fractal patterns in trial

series of response times. Except for the setting of trimming criteria, the

required statistical techniques are the same for a host of standard

cognitive tasks that record response time as a dependent measure.

The laboratory protocol was modeled after Gilden et al.’s (1995)

temporal judgment task. At the beginning of the experimental session

each participant was presented with about one minute’s worth of

examples of a particular time interval. The time interval was illustrated

by repeatedly flashing a simple visual stimulus on a standard PC

monitor for the specified period of time. Each participant was then

asked to attempt to replicate the example time interval as best as they

could, 1100 times, in succession. Participants never received feedback

about how accurate they were in their time estimates.

The method section that follows describes a study that replicated

Gilden et al.’s (1995) essential finding of pink noise in trial series of

temporal estimates. Since the purpose of this chapter is not to

disseminate empirical results, but to supply a detailed “how to” tutorial

on methods of fractal analysis, just a single participant’s trial series is

described in the results section (the series that appears in Figure 6.1A

was used). The particular series was explicitly selected because it is a

very clean and clear example of empirical pink noise; it lacks artifacts

that sometimes appear in real-world data and that threaten to further

complicate this introductory discussion.
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Method

Participants. One undergraduate psychology student

participated in exchange for course credit.

Procedure. The participant was given a one-minute sample of a

target temporal interval. The sample intervals were constructed by

presenting a visual stimulus (#######) at the center of a standard CRT

monitor controlled by a PC running DMASTR software (Forster &

Forster, 1996). The target interval duration was 690 ms or 50 monitor

raster refresh cycles. The samples of the 690 ms time interval were

generated by displaying the visual stimulus on the monitor for exactly

690 ms, then the monitor went blank for 690 ms, at which point the

visual signal again appeared for 690 ms, and so on. The visual stimulus

flashed on and off for about one minute. (There is nothing special about

the 690 ms duration; the monitor’s vertical raster-refresh rate was 72

Hz, or once every 13.8 ms, and 13.8 times 50 equals 690).

On each temporal estimation trial, the visual stimulus (#######)

was displayed until the participant responded by saying “/ta/” into a

microphone, or for a maximum of 10 s. The experimental task was

paced by a computer. Each response was followed by a 690 ms intertrial interval in which the computer monitor was blank. The participant

was told to pace her responses so that the visual signal was displayed

for the same time interval she saw during the one minute sample time

interval session. The participant completed 25 practice trials

immediately prior to completing 1100 experimental temporal

estimation trials. The entire task took about 30 minutes to complete.
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Results

Standard statistical analyses such as regression and ANOVA are

typically used in a manner that either ignores the temporal order of the

trials in an experiment, or that treats order as a nuisance factor. By

contrast, the patterns of fluctuation that unfold across the successive

experimental trials are the main focus of the fractal techniques

introduced here. Thus, the analysis begins with the participant’s trial

series of temporal interval estimates arranged in the order in which they

were collected (Trial 1, Trial 2 ...Trial 1099, Trial 1100).

As a practical matter, it is best to present enough trials in an

experiment to be left with at least 1024 observations after any timed-out

trials, extreme times, and outliers are removed. Presenting 1100 timeestimation trials left a healthy 76 trial “buffer.” While it is possible, and

potentially informative, to apply fractal techniques to data sets shorter

than 1024 observations, the results of the analysis become less reliable

as fewer and fewer data points are used. For example, the spectral

slopes and the fractal dimension estimates, explained shortly, tend to

become more variable as progressively shorter data sets are used

(Cannon et al., 1997; Eke, Hermán, Kocsis, & Kozak, 2002).

Additionally, the measurements should be collected as regularly in

time as possible. A “lined up” series of measurements that were

actually collected across different experimental sessions distorts the

time scale, and the fractal analysis may not accurately characterize the

temporal structure of the series.

Trimming and Detrending. Data trimming procedures are often

required to bring the series of time estimates (a finite, irregular natural
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object) more in line with the assumptions of spectral and dispersion

analyses. The mathematics of spectral analysis assume an ideal,

stationary, strictly periodic process of infinite duration. Dispersion

techniques are less assumptive and more robust than spectral

techniques but nevertheless ultimately assume that the measured

process is at least weakly stationary—that its mean and standard

deviation remain essentially the same over time (Caccia et al., 1997;

Chatfield, 1996). If the trial series in hand happens to be a good

example of pink noise, then its mean and standard deviation probably

do fluctuate as a function of time, or trial.

In general, response time distributions are notorious for the fact

that they often contain extreme observations, and the trial series of

temporal estimates are essentially response time trial series. No matter

their origin, a few extreme measurements or simple long-term trends

will likely distort the outcome of a fractal analysis. It is important to

note that the main issue surrounding the decision to remove extreme

valued data points is not so much whether or not they represent

legitimate measurements, as they certainly may. The issue is whether

their inclusion will dominate and thus distort the outcome of the

analysis.

Response time trial series typically require two censorship

passes. The first pass eliminates times that exceed fixed extreme

truncation values; different cognitive tasks require the use of different

fixed truncation values. Reasonable truncation points can be identified

by consulting the relevant literature for typical censorship values.

Adopt conservative truncation values from that range as a starting point

(err on the side of including more data). When truncating an
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observation, just delete it and “close up” the series so that the deleted

observation’s two immediate neighbors themselves become

neighbors. While this procedure slightly disrupts the time-ordering of

the series, its overall impact on the analysis is usually minimal. To best

preserve the trial order, response times on trials that produced an

error in, for example, a choice response time task, should be included

in the trial series (see Gilden, 1997).

The main purpose of fixed-value censorship is to facilitate a

second pass through the data, which uses the series mean and standard

deviation as a censorship origin. For temporal interval estimates, nine

observations were less than 100 ms or greater than 3500 ms and were

removed. Then the series mean and standard deviation were

computed. On the second censorship pass, 13 observations that fell

beyond ±3 standard deviations from the series’ mean were eliminated.

The dispersion analysis, as described below, requires the

number of observations to be an integer power of 2 (e.g., 210 = 1024).

While spectral techniques do not always strictly require a series to be

an integer power of 2 in length, the algorithms work faster when the

data series is an integer power of 2, and some computer

implementations of spectral routines do require the series to be an

integer power of 2 in length. The two censorship passes eliminated 22

observations, for a total of 1078 remaining observations. The first 54

observations were then eliminated to yield a series that was 1024

observations in length.

Trial series that display self-similar patterns of fluctuation are

expected to display nonstationary drift (i.e., trends) at all scales. It can,
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however, be difficult to distinguish simple long term trends, or a verylow frequency periodic oscillation from a nested, fractal pattern of long

range fluctuations in empirical data sets (Hausdorff et al., 1996). This

difficulty again arises from the fact that real data sets have a finite

length. A linear trend at the scale of the whole data set could be either

a simple linear trend, or a small piece of a fractal pattern of fluctuation

that expresses itself across scales that run far beyond the duration of

the particular sample of the process at hand. What appears as a linear

trend across 1024 observations could be just that, or it could be part of

a proportionately scaled fractal fluctuation that runs across 2000-3000

observations. Without the extra data, it is impossible to tell which

option is a better description of reality. Only fluctuations that live on

scales somewhat smaller than the full length of the series can be

resolved clearly enough (i.e. statistically) to determine whether they

are consistent with a nested fractal pattern of fluctuation.

Most importantly, simple long-term trends not only have the

potential to bias estimates of spectral slopes and fractal dimension,

they may also overwhelm the analyses, and yield spurious spectral

slopes and fractal dimension statistics (Caccia et al., 1997; Hausdorff et

al., 1996). As such, it is prudent to remove at least linear and quadratic

trends before conducting the analysis. As a general rule, if the trial

series has fractal structure, progressively more liberal detrending

procedures will not result in dramatic changes in the overall fractal

dimension estimates (Hausdorff et al., 1996). Nevertheless, detrending

does eliminate variability at the larger scales, in the neighborhood of

the size of the entire series, and any fractal dimension estimation

procedure, like those presented later, must be tuned to accommodate
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this fact. Otherwise, the detrending introduces its own bias. For the

timing trial series trends up to a quadratic were removed. The basic

detrending procedure involves generating a least squares linear and

quadratic fit to the series, using the index of observation as the x

variable and the temporal estimate as the y variable. One way to do

this is to use the method of Powered Vectors, in conjunction with

Hierarchical Regression (Keppel & Zedeck, 1989), which can be coded

on a spreadsheet or accomplished with standard statistical software.

The final preparatory step is to normalize the series to have a

mean of zero and a variance of one. That is, transform the data points

into z-scores by subtracting the series mean from each observation,

and dividing each observation by the series standard deviation (SD).

Use the population formula for the SD and divide by N, the number of

data points in the series, rather than the usual bias-corrected N – 1. The

descriptions of the spectral analysis and the dispersion analysis that

follow assume the data sets are already in this format.

Spectral Analysis. Spectral analysis techniques provide a general

way of characterizing the correlational structure of fluctuations in a

series of successive response time measurements (Gilden et al., 1995;

Gilden, 1997). There are several kinds of spectral techniques; what is

referred to here as a spectral analysis is a particular method called

power spectral density estimation, which yields a power spectrum of a

trial series.

Successful applications of spectral methods require a certain

amount of care, sophistication, and background knowledge.

Accessible introductions to spectral techniques are provided by
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Gottman (1981) and Chattfield (1996). Press, Teukolsky, Vetterling,

and Flannery (1992) describe "how to" information as well as provide

source computer code for the analyses. The information presented in

Press et al. is rather technical but nevertheless very helpful. In fact,

much of the information presented below is adapted from more formal

treatments of the same topics in Press et al.

With the exception of the censored observations, each

participant’s trial series of time estimates is now ordered according to

the trial on which the observation was collected—the order of the

successive trials in the experiment. Connecting the points that

represent the successive time estimates forms a complex waveform, as

in Figure 6.1A. In a sense, a power spectral density analysis

decomposes a trial series much as a prism breaks white light into its

basic wavelengths, or colors. Spectral analyses decompose a trial

series into a set of regular oscillations, component waves with

particular frequencies and amplitudes. Taken together, the component

waves mimic the overall pattern of oscillation in the observed trial

series. An intuitive grasp of spectral analysis may be gleaned by

thinking of it as a multiple regression analysis that fits a large set of

simple sine (and/or cosine) waves to the complex response time

waveform. The period of oscillation (the inverse of frequency) and the

amplitude (relative height) of each component wave can vary.

Oscillations corresponding to quickly changing trial-to-trial "jitter" map

to a high frequency wave. Persistent excursions in one direction or the

other from the mean of the trial series, over the course of, say, tens to

hundreds of trials, map to lower and lower frequency oscillations. The

output of a spectral analysis is a set of coefficients that characterize the
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relative amplitudes of all the wave forms, ordered from lowest to

highest frequency. This output is called the power spectrum of the

signal. Loosely speaking, the spectral density coefficients correspond

to a relative sum of squares for each frequency of sine wave that is used

to fit the trial series. This is a bit like an r

2 for each sine wave that was

passed through, or fit to, the series. Frequencies with larger amplitude

coefficients imply more of the total variability was attributed to that

particular frequency of oscillation.

As described, this basic recipe for a spectral decomposition

procedure yields estimates of the amplitude (relative strength or

energy) of many sinusoidal frequencies, but there is little statistical

certainty in the magnitudes of any one of them. Each amplitude

estimate is derived from just a single pass or "fit" of an individual ideal

sine wave. Like any other statistical sample, the coefficient resulting

from that fit can be unduly influenced by idiosyncratic properties of the

data set in hand. Put differently, the standard deviation of each

amplitude estimate for each particular frequency is huge—100% of its

value (Press et al., 1992). Increasing the number of data points by

using progressively longer data series only allows more and more

frequencies to be estimated. A straight spectral density analysis

always yields about half as many frequencies as there are data points

(the highest resolvable frequency oscillates back and forth on every

other data point). Thus, analyzing the entire data sequence at once

yields maximum frequency resolution (many different sized sine waves

are approximated), but does not lower the variability in the estimate of

the amplitude for any particular frequency (see Press et al.). In terms of

statistical certainty, the output is about as trustworthy as a factorial
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ANOVA that has many, many experimental cells, but each cell mean is

based on just one data point.

For data sets as variable as response time trial series, the

procedures for computing a power spectrum must be adapted to

balance the need to identify fluctuations over a suitably wide range of

frequencies while simultaneously minimizing error variance in the

estimation of the magnitude of any particular frequency. This is

accomplished by breaking a single long 1024-trial series into several

shorter, overlapping series of response times. This procedure is called

data segmenting or data blocking. The individual power spectra

derived from each short sub-series are then averaged. The averaging

reduces the variability in the power spectral density estimate at each

frequency; the cost is a reduction in the maximum number of

frequencies that may be estimated.

Finally, the mathematics that govern the translation from the time

domain (the trial-by-trial representation of the data) into the frequency

domain (the frequency-by-frequency representation of the data)

require the use of a procedure called data windowing. Essentially, a

difficulty emerges from the fact that real data sets have distinct

beginnings and ends, but the mathematics of spectral analysis assumes

a data set of infinite length that is, more or less, strictly periodic. The

consequence of this mismatch is a tendency for the power or energy

associated with any particular frequency to "reverberate" or be blurred

into the amplitude estimates of nearby frequencies. A typical data

window applies a weighting function to the segment of the trial series

that is undergoing spectral analysis. Data windows are designed to

smooth the transition into and out of the data, and work something like
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slowly turning up “volume” of the data starting at the beginning of the

series, until it is at a maximum at the center of the series, and then

slowly turning it down to zero again by the end of the sample (see Press

et al., 1992 for details). The data blocking, spectral density averaging,

and data windowing procedures, or similar statistical fixes that address

the same issues, are often available as standard options in many

spectral analysis computer routines.

If you are using a spectral density routine that returns the

frequencies and power (the square of the absolute value of each

amplitude) in linear units, the first step is to delete the highest (Nyquist)

and lowest frequency (DC) coefficients, and then to transform the

remaining coefficients by taking the log, base 10, of each frequency

and its corresponding power estimate. Plot log-frequency against logamplitude in a scatter plot. Reasonable evidence for inverse power-law

scaling in the form of pink noise appears as a negatively sloped linear

relation between the two variables on the log-log scatterplot. The linear

relation must span a range of at least 2 decades of frequency (i.e., 2 log

units, or 100 frequencies; Eke et al., 2000, 2002). Since natural fractals

exhibit self-affinity across only a finite range of scales, the inverse

power-law scaling relation may break down at either the highest or

lowest frequencies, or both. It is also notable that the strength of the

linear relation trades off with the number of frequencies that are used in

the analysis. For a given series length, estimating the amplitudes of

more frequencies typically yields a wider scatter of points in the loglog scatter plot. This is a consequence of the issues relating to the

statistical certainty in the spectral coefficients, as was discussed

previously.
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The next step is to determine the value of the scaling exponent,

the α in the scaling relation 1/ƒα (where ƒ denotes frequency). The

scaling exponent describes how the amplitude of the fluctuations

change or scales as a function of their frequency. The easiest way to

estimate α is to determine the slope of a least-squares regression of

power as a function of frequency, using the logarithmically transformed

values. (Technically, it is more correct to fit a least-squares power law

in the linear domain but few researchers feel this extra step is critical).

The slope of the regression line is the scaling exponent α. Response

time series usually yield negatively accelerated slopes (recall that 1/ƒα

=ƒ-α) that, within certain boundary conditions, discussed later, indicate

pink noise, or slopes that are statistically equivalent to zero, which

suggests white noise.

One additional difficulty with the spectral method is that the high

frequency portion of the power spectrum can sometimes be

“whitened,” which appears either as flattening to zero slope at the

highest frequencies or as combination of linear and U-shaped quadratic

trends in the log-log regression that looks a bit like a tilted and mirrorreversed J. Figure 6.1C and Gilden et al.’s (1995) plots for the shorter

time estimates display evidence of this pattern. The flattening at high

frequencies may simply indicate a breakdown in the scaling relation at

the highest frequencies, but in response time trial series it was linked

to issues related to experimental design and measurement procedures

that add sources of white noise to the signal (Gilden et al., 1995; Gilden,

1997; Gilden, 2001; Van Orden et al., 2003). For instance, very high

frequency oscillations that unfold on a pace faster than the trial-by-trial

pace of measurement may be “aliased” into the power of measured
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frequencies. That tends to whiten the high-frequency end of a 1/ƒ

spectrum. Aliased frequencies refer to oscillations that live outside of

the measured frequency range, and that are misinterpreted by the

spectral analysis as different frequencies that fall within the measured

range of frequencies (Press et al., 1992). It is similar to the way a

person dancing in relative dark under a regularly flashing strobe light

can be perceived as not moving, or to the apparent, but false,

appearance of backward rotation of spoked wheels that one sometimes

notices in old films. Excluding the highest frequencies in the log-log

regression is often recommended to avoid the whitening of the highfrequency end (Eke et al., 2000, 2002).

The details of how the spectral analysis on the temporal

estimates were conducted on the trial series appearing in Figure 6.1A

can be summarized as follows. The spectral analyses resulted from

averages of seven successive power spectra computations taken across

successive sub-blocks of 256 trials. Each sub-block was multiplied by a

triangular (Bartlett) window and the power spectrum was computed.

The trial series was then shifted by 128 trials (1/2 the sub-block length)

and a new power spectrum was computed. This process was repeated

until the end of the series was reached. Thus, the power spectrum

resulting from each participant's trial series was based on an average of

seven (semi-independent) samples of the data set. This process

yielded estimates of 129 (n/2 + 1) frequencies, but the highest and the

lowest frequency were dropped, resulting in a total of 127 frequencies.

Figure 6.1B displays the results of the spectral analysis on linear

scales. The x-axis depicts frequency, ranging from low to high. The yaxis depicts power, the square of the absolute value of each amplitude.
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Figure 6.1C displays the results of same spectral analysis, now

depicted on log-log scales. The approximately linear relation between

the two variables in the log-log domain implies an inverse power-law

scaling relation, consistent with pink noise. The slope of the regression

line is –0.59, which corresponds to a 1/ƒ0.59 scaling relation. Note,

however, that the highest frequencies in Figure 6.1C seem to be

slightly “whitened,” which introduces a slight quadratic trend to the

power spectrum and a slight bias in the slope of the regression line

toward a shallower value. This pattern could justify eliminating the

highest frequencies from the regression line by fitting only the lowest

25% of the frequencies, for instance. As expected, excluding the

whitened higher frequencies yields a steeper spectral slope of –0.86.

As an additional check, it is important to recompute the spectral

analysis using both more and fewer frequencies, and see comparable

results. Analyses that use fewer frequencies should better resolve the

linear nature of the scaling relation, and analyses using more

frequencies should suggest that the scaling relation reaches into the

lower frequencies, but it is critical to eliminate the detrending steps

when examining the coefficients for lowest frequencies. The presented

analysis, using 127 frequencies, reflects a compromise between the

need for a satisfactory level of statistical certainty in the spectral

coefficients (by examining frequencies that correspond to scales no

larger than 1/4 the length of the series) and the need to establish the

scaling range across at least 2 decades of frequencies.

Assuming a lack of evidence for white noise, the main reason for

conducting a spectral analysis is to determine whether the value of the

slope of the log-log regression line lies very near or less than –1, which
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marks the boundary between a stationary fractional Gaussian noise and

a nonstationary fractional Brownian motion. Dispersion analysis more

accurately characterizes the fractal structure of a trial series than

spectral analysis, but it can only be used on approximately or weakly

stationary data sets. In this regard, idealized pink noise, with a spectral

slope of –1, marks the boundary between mathematically stationary

and nonstationary trial series. Fractional Brownian motions are

nonstationary, and require other techniques, such as detrended

fluctuation analysis (Peng, Havlin, Stanley, & Goldberger, 1995), which

can be used on fractional Gaussian noise as well, or rescaled range

analysis (Cannon et al., 1997). Those techniques are closely related to

dispersion methods, but they will not be covered in this chapter.

Response times often yield scaling exponents suggesting pink noise,

and indicate that dispersion analysis is appropriate to determine the

fractal dimension of the series. Next, the spectral slopes that define

fractional Gaussian noise and fractional Brownian motions are

described in detail.

Variability Categories. The magnitude of the power-law scaling

exponent circumscribes at least two general classes of temporal

variability that are called fractional Gaussian noise (fGn) and fractional

Brownian motion (fBm) (see Mandelbrot & Wallis, 1969a/2000; Eke et

al., 2000, 2002). Collectively, they are often referred to as 1/ƒα noise.

The original theoretical development of these ideas used Gaussianshaped probability density functions, but the classifications generalize

to empirical data that have non-Gaussian density functions (Mandelbrot

& Wallis, 1969c/2000).
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Figure 6.3 illustrates how fGn and fBm can be described using

the spectral slope of the power-law scaling relation. fGn exhibits loglog spectral slopes that range between 1 and –1. A slope of 0 indicates

no historical dependence—independent sources of random variation,

or white noise. Slopes reliably greater than 0 and less than 1 indicate a

tendency for positive data values to be followed by negative values,

which is termed anti-persistence; this is sometimes called blue noise.

Slopes less than 0 and greater than –1 indicate persistence—positive

data values tend to be followed by positive data values. This is the

domain of pink noise, the main topic of this chapter. Assuming the loglog regression is linear across 2 or more decades (log units) of

frequency, pink noise is simply a statistically reliable departure from

white noise in the direction of persistence, evaluated using a

combination of spectral and fractal analyses.

Spectral slopes less than –1 and greater than –3 describe a

related but fundamentally different kind of variability called fractional

Brownian motion. This is the domain of random walks. A spectral slope

of –2 indicates idealized Brownian motion. Slopes less than –1 but

greater than –2 indicate anti-persistent fractional Brownian motion, in

which successive increments tend to have opposite signs. Slopes

between –2 and –3 indicate persistent fractional Brownian motion, in

which successive increments tend to have the same sign (this is

sometimes called black noise). Notably, a single parameter

characterizes this entire family of noises.

Idealized pink noise, or 1/ƒ1 noise, is special mainly because

when the x- and y-axes of a 1/ƒ1 noise are enlarged in like proportions;

the enlarged portion of the series is statistically indistinguishable from
Fractal Variability in Cognitive Performance

299

Figure 6.3. The figure adapted from Eke et al. (2000) portrays an idealization of spectral slopes

that distinguish fractional Gaussian noises (fGn) and fractional Brownian motions (fBm) (see also

Cannon et al., 1997). Idealized pink noise, or 1/ƒ1 noise, is special as a mathematical way-point;

it marks the boundary between stationary (fGn) and nonstationary (fBm) data series, two

categorically distinct kinds of variability (Eke et al., 2000, 2002).

Log 10[Relative Frequency]

Log 10[Relative Power]

← White Noise

Persistent fGn

"Pink Noise"

Antipersistent fGn

"Blue Noise"

← Brownian Motion

Antipersistent fBm

Persistent fBm

 f −2

 f −1

 f 0

 f 1

 f −3

Fractional Gaussian Noise (fGn) Fractional Brownian Motion (fBm)
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300

the original series. Pink noise that has a scaling exponent that falls

between 0 and 1 requires the x- and y-axes to be enlarged in different

proportions to yield the same effect (Eke et al., 2000, 2002). Idealized

pink noise is also important as a mathematical way-point, as it marks

the analytic boundary between stationary and nonstationary data

series. Idealized pink noise, and data series with scaling spectral

slopes less than –1, have in theory, and in practice, infinite variance.

Pink noise with nontrivial spectral slopes less than –1 has infinite

variance in practice, in the sense that the interdependence of finite

statistical samples yields unreliable population parameters

(Bassingthwaighte et al., 1994; Mandelbrot & Wallis 1969b/2000). Thus,

a spectral slope of –1 marks an important boundary between two

categorically distinct kinds of variability, fGn and fBm (Eke et al., 2000,

2002).

Many natural systems emit pink noise, but spectral slopes of

exactly –1 are not usually observed. Heart rate variability can exhibit a

spectral slope very near –1 (e.g., Eke et. al., 2002), but many

established examples display slopes between 0 and –1. A large

sample of yearly tree ring indices has average spectral slopes of –0.43.

Annual precipitation statistics have average slopes of –0.48. The classic

Nile River yearly minimum series yields a spectral slope of –0.82 while

measurements of the Nile’s yearly maximum levels display a slope of

about –0.68 (Mandelbrot & Wallis, 1969b/2002). Natural phenomena

entail sources of unsystematic external variability, in addition to

sources of intrinsic 1/ƒ scaling, which results naturally in pink noise

with spectral slopes greater than –1.
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301

Typically, response times from elementary cognitive tasks, such

as temporal estimation, yield scaling exponents that fall within the

range of 0-1, and thus contain a nested, statistically self-similar pattern

of positive correlation across successive observations. Spectral slopes

that lie near –1 suggest that the nested structure of positive correlation

dominates the series. Spectral slopes that lie closer to zero indicate a

less prominent, “whitened” structure of nested positive correlation

across the series.

Next, a statistical technique called dispersion analysis is

introduced. It more accurately characterizes the pattern of statistical

self-similarity than spectral analysis. It yields a fractal dimension

statistic (FD), which is closely related to the slope of the power

spectrum. In fact, FD = 1 + (S + 1)/2, where S is the spectral slope, of

the log-log regression line. This formula assumes the spectral slope of

the series falls between –1 and 0. Note, however, that since the two

analyses “break up” the trial series in mathematically different ways

(e.g., Fourier analysis versus means and standard deviations) they will

not typically output exactly the same fractal dimension for the exact

same signal, although they should yield reasonably similar outcomes.

If there is a strong disagreement between the two methods, examine

the signal, and the steps in the analysis, carefully for potential artifacts.

Dispersion Analysis. Dispersion analysis yields the fractal

dimension of a trial series and gauges the change in variability due to

changing sample sizes. Dispersion analysis determines whether the

trial series variability converges fast enough, as sample size increases,

to yield stable statistical estimates of population parameters. If not,

then the process that produced the variability is, in practical terms,
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302

scale free in the sense that it has no characteristic “quantity” or scale of

variability.

There are several ways to compute the fractal dimension, and

dispersion techniques are among the most accurate (Bassingthwaighte

et al., 1994; Caccia et al., 1997; Eke et al., 2000, 2002). Spectral

analyses yield less reliable fractal dimension estimates than dispersion

methods. A practical advantage of dispersion analysis is that familiar

statistical constructs, means and standard deviations, are used for the

analysis. A version of the standard technique of relative dispersion

analysis is presented here. It allows for the use of normalized data

instead of raw data—call this standardized dispersion analysis to avoid

confusion with other methods. Note that the standardized dispersion

analysis yields dispersion measurements that are in units of the

standard error of the mean; the standard deviation of a sampling

distribution of means, comprised of means of (adjacent) samples of

specified sizes. By contrast, the original method described by

Bassingthwaighte et al. is based on the relative dispersion statistic that

is comprised of a ratio of the standard deviation and the mean (i.e. RD =

SD/M). The outcomes of the two techniques are identical. However,

the spectral technique presented earlier assumes a normalized trialseries and standardized dispersion analysis allows the same detrended

and normalized data set to be submitted to both analyses.

When computing the dispersion statistics in the subsequent

steps, compute the standard deviation using the population formula

(i.e. use N, the number of data points, in the calculation, rather than the

usual bias corrected N – 1).
Fractal Variability in Cognitive Performance

303

A dispersion analysis repeatedly resamples the trial series using

sampling units of different sizes to estimate the fractal dimension of a

trial-series. In the steps that follow, variability is gauged using the

standard deviation of means of progressively larger adjacent samples.

That is, the analysis tracks how variability in sample means decrease as

progressively larger samples of adjacent data points are aggregated

together in a sample mean. If the samples are statistically independent,

then it should not matter that adjacent samples are being grouped and

regrouped to form samples of different sizes.

To perform the analysis first construct a table, like Table 6.1.

Begin the table by recording a 1 in the points-per-bin column and

another 1 in the dispersion column. The overall standard deviation (SD

= 1) of the normalized series represents the overall dispersion of the

series, given that the data points are treated individually. Essentially,

the overall trial series standard deviation is treated as a population

parameter, and for this initial step, N is also 1. The overall standard

deviation is identical to the variability of a sampling distribution of 1024

“means,” computed across single, individual observations, which is just

the raw score standard deviation.

The next step involves grouping the data points into adjacent

pairs, which makes it more obvious that variability in the means of

sample bins is being tracked. Compute the mean for each successive

pair of points—each bin. This yields the new set of data points; a

sampling distribution of means that contains the 512 values of each 2-

point mean. Compute the standard deviation of this new distribution.

Enter a 2 in the points-per-bin column of the table, because 2 point

sample means were used. Then enter the standard deviation of the
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sample means in the dispersion column. If the trial series were

composed of statistically independent observations, then the expected

standard deviation of the sampling distribution of the 512 two-point

means would be 1/√2 or about 0.71. Interested readers may consult

Van Orden et al. (2003) for an explanation of the relation between the

standardized dispersion statistic and the equation for the standard error

of the mean.

Table 6.1: Standardized dispersion as a fucntion of sample bin size in linear and

logrithmic units.

Bin Size Standardized

Dispersion

Log10

(Bin Size)

Log10 (Standardized

Dispersion)

1 1 0 0

2 0.88 0.30 –0.06

4 0.79 0.60 –0.10

8 0.73 0.90 –0.14

16 0.68 1.20 –0.17

32 0.60 1.51 –0.22

64 0.54 1.81 –0.27

128 0.47 2.11 –0.32

256 0.15 2.41 –0.82

512 0.02 2.71 -1.68

Repeat the previous step until only two data points are left (i.e.,

the third iteration will use 256 bins of every four successive data points,

the fourth iteration uses 128 bins of size eight, and so on, until there are

two bins of size 512). At the culmination of each step, enter the number
Fractal Variability in Cognitive Performance

305

of points that comprises each bin, and the standard deviation of the

distribution of the sample means into the table. In the final repetition,

the final two data points come from a bin containing the first half of the

original trial series and a bin containing the last half. To summarize:

Each step in constructing the table generates an N that is equal to the

bin size and a standard deviation that estimates dispersion at that bin

size. If the trial series was a series of statistically independent data

points, then the standard deviation should diminish very nearly as a

function of 1/√N, as the size of the sample bin sizes are progressively

increased.

Finally, plot the logarithm of the numbers in the points-per-bin

column against the logarithm of the numbers in the standard deviation

column as in Figure 6.2. Base-10 logarithms were used here, but other

bases also work. For instance, using base 2 represents the number of

samples in the bins as integer powers of 2—just be sure to use the same

base for taking the log of the dispersion values as well. The relation

between the two variables should be linear on double-log scales,

except perhaps for the three or four points that correspond to the

largest bins. Typically, the last few relative dispersion measurements

that correspond to the very largest bin sizes are excluded at this point

(Cannon et al., 1997); this is a critical adjustment when detrending is

used. Here, the three largest bins were excluded because the

detrending procedures removed the variability at these scales. Natural

fractals exhibit scaling relations across a finite range of scales, so the

linear relation is expected to break down at some point, for either (or

both) the smallest or largest bin sizes. (Points excluded in the log-log

regression in Figure 6.2 appear as open circles.) If using a
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standardized series the dispersion values at the largest bin size

approach zero (and negative infinity when the log transformation is

performed). As such, they bias the slope of the regression line (see

Caccia et al., 1997, for additional refinements of this technique,

especially for shorter data sets).

A linear relation with a negative slope in log-log coordinates

establishes an inverse power-law scaling relation and indicates that the

trial series is a simple fractal (Bassingwaighte et al., 1994). The fractal

dimension of the series is given by subtracting the slope of the leastsquares regression line from one, the Euclidean dimension of the

series. The relation illustrated in Figure 6.4 is an inverse power-law

scaling relation. The slope of the log-log regression line is –0.14, and

the fractal dimension of this trial series is therefore 1.14. Transforming

the spectral slope of –0.59 into a fractal dimension yields 1.21; although

those values are not identical, they are in the same neighborhood. Also

recall that the spectral plot revealed evidence that the high-frequencies

were “whitened,” which tends to bias the spectral slope towards

shallower slopes, and thus, larger fractal dimensions. For instance,

returning to the spectral coefficients and fitting only the lowest 25% of

the power spectrum coefficients yields a spectral slope of –0.86, which

translates to a FD of 1.07.

At this point, a reader might ask him or herself, which

characterization is the correct one? It is important to be mindful of the

fact that each method of analysis has strengths and susceptibilities. For

example, the spectral methods are sensitive to a host of artifacts that

affect the high-end of the frequency range, but dispersion analysis is

not as susceptible to these influences since it is not based on a Fourier
Fractal Variability in Cognitive Performance

307

Figure 6.4. Standardized dispersion is depicted as a function of sample-bin size, on double

logarithmic scales. The x-axis indexes the base-10 logarithm of the number of adjacent data

points in each adjacent sample bin. The y-axis indicates the base-10 logarithm of the standard

deviation of the mean standardized dispersion measures, across all the sample bins. The solid

line is a least-squares regression line for the first 7 data points, represented by the solid points.

The three points depicted as open circles correspond to the three largest bins and were not

included in the regression analysis because the detrending procedures tends to eliminate

variability at those larger scales. The open circle corresponding to the 2 largest 512 point bins

is not shown because it fell below the limit of the y-axis.

transform. Moreover, the earlier discussion of spectral analysis

illustrates that the raw data is subjected to a series of transformations to

yield a spectral slope. Dispersion analysis, on the other hand, can be

unduly influenced by simple linear trends that span the full length of

the series. Dispersion analysis has been shown to lose some of its

accuracy in characterizing time series that display very strong

0 1 2 3

−1

−0.5

0

Log10 [Bin Size]

Log10 [Standardized Dispersion]

Trial Series Slope = −.14

White Noise Slope = −.5

Fractal Dimension = 1 − Slope
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interdependence, such as pink noise with a spectral slope near –1, (Eke

et al., 2000, 2002), which are essentially nonstationary signals in the

case of limited sample sizes. Dispersion analysis should not be used on

fBm series, which are truly nonstationary signals.

Most importantly, both methods should yield solid, converging

evidence of the presence of a power-law scaling relation. All other

things being equal, the fractal dimension based on the dispersion

method is more accurate, but it would not be unreasonable to report an

average across these two, or perhaps additional methods in research

reports. Notice the complementary nature of the two analyses; spectral

methods are reliable at the intermediate and larger scales (the

intermediate and lower frequencies), while dispersion methods are

reliable at the intermediate and smaller scales (medium and higher

frequencies).

The take-home point is that both spectral and dispersion

methods decompose the raw data in different ways, and thus interact

with the idiosyncrasies of a given empirical signal in slightly different

ways. In addition, both analyses require the researcher to make a

number of choices about a range of parameters, such as the manner of

detrending, the number of spectral coefficients or bin sizes to fit with a

regression line, and so on. Each choice will impact the outcome of the

analysis in some way, at least compared to other choices that could

have been made. As such, both methods should agree qualitatively,

and yield similar, but not necessarily identical, fractal dimension

estimates. Redoing the analysis using other parameter choices may

change the value of the fractal dimension statistics somewhat, but it will

typically do so in a systematic manner (e.g., fitting the whole spectrum
Fractal Variability in Cognitive Performance

309

often yields shallower spectral slopes). As long as the fractal

description survives trivial changes in the parameters that are used to

govern the analysis (e.g., dropping the 3 or 4 largest bins in the

dispersion analysis or fitting the entire spectrum versus the lowest

25%-30% of the spectrum), it is likely accurate. It is then up to the

researcher to present a succinct, conservative analyses that, in his or

her best judgment, accurately portrays the data at hand.

Significance Testing. There are two general kinds of statistical

tests that one may want to conduct on the fractal dimension or spectral

slopes resulting from the analyses discussed earlier. The first situation

involves experimental research designs that entail contrasts between

two or more conditions, and standard linear statistical methods such as

a t-test or ANOVA (or nonparametric equivalents) can be used to

establish differences in fractal dimension across groups. However,

such comparisons must be carefully considered. Simple issues of

measurement affect the fractal dimension and spectral slope of trial

series arising from different tasks (e.g., Van Orden et al., 2003), so the

fact of a statistically significant difference across tasks may not point to

a theoretically interesting difference.

The second, and perhaps more novel, approach is to determine

whether white noise or pink noise better describes an observed trial

series. There are a number of ways that one or several trial series can

be statistically distinguished from white noise. If a reasonably sized

distribution of slope or fractal dimension estimates is available from

several subjects, a single sample t-test could be used to determine if

the sample mean of the spectral slopes is significantly different from 0,

the expected slope of white noise (or if the fractal dimension differs
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from 1.5). If only a single trial series is available, then one can easily

test for a significant departure from a pattern of white noise by

contrasting the observed data with surrogate data sets (e.g., the

reshuffled data in Figure 6.1D.). Simply randomly reshuffle the order of

the original trial-series, 10 or 20 times, each time computing the fractal

dimension and/or spectral slope of each reshuffled version of the trialseries. Then calculate the fractal dimension mean and SD of the

surrogate data sets and compare the mean to the observed values of

the original series. If the fractal dimension and spectral slope of

original series is in the pink noise range, and more than 3 SD away from

the mean of the surrogates, then a null hypothesis of white noise can be

rejected (Hausdorff, 1996; see Theiler et al., 1992, and Efron &

Tibshirani, 1993, for more details and options).

DISCUSSION

At first glance, Gilden et al.’s (1995) original temporal estimation

task appears to be directed at understanding a person’s ability to

replicate a range of fixed time intervals. Thus, one might expect it to

include an extensive discussion of how well or poorly people

performed, but accuracy was barely mentioned in their report. The

focus was instead on the structure of intrinsic patterns of trial-by-trial

variability.

Conventional research methods in psychology emphasize a

“cause-effect” metaphor as the only sensible route to scientific

understanding. Experimental manipulations are introduced as a way of

brining empirical phenomena under experimental control. Thus,
Fractal Variability in Cognitive Performance

311

factorial manipulations are designed to control empirical “effects.”

Perhaps one experiment is designed to amplify an experimental effect,

and another design may diminish the same effect. All hope of a

plausible scientific explanation hinges on reducing experimental

manipulations to causal factors inside the mind or brain (Van Orden,

Holden, Podgornik, & Aitchison, 1999). As such, one impulse that

cognitive psychologists sometimes have, especially those with solid

training in conventional statistics, is to think of pink noise in terms of a

statistical nuisance. If pink noise causes problems for inferential

statistics (and, therefore, for the aforementioned causal logic), then

perhaps pink noise can be removed, modeled, or otherwise controlled.

This is a natural move for one taking a purely statistical perspective on

data. However, finding pink noise can lead to fundamentally different

scientific perspectives on the data. For example, one may wonder how

the human mind and body might be organized to give rise to statistical

interdependence, and to interact across different time scales. From a

scientific perspective it seems to make sense to ask what the empirical

fact of pink noise in cognitive and other human performance implies

about the fundamental coupling of bodily processes.

For cognitive science, Gilden’s approach was revolutionary

because it supplied the seed of an alternative format for scientific

explanations of cognitive performance, and for psychology in general.

That alternative began with attempts to establish the presence of pink

noise despite a range experimental manipulations (e.g., Gilden et al.,

1995), and it was subsequently expanded to include experiments in

which pink noise survived changes in cognitive tasks (e.g., Cayton &

Frey, 1997; Gilden, 1997, 2001; Van Orden & Holden, 2002; Van Orden
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312

et al., 2003). Thus, the alternative approach was directed at the

identification of ubiquitous performance phenomena, in the hope that

those phenomena may illustrate something very general about human

performance. Gilden was perhaps one of the first to suggest, on the

basis of an empirical finding, that pink noise implied that complex

systems theory may be relevant to cognitive psychology (Gilden et al.,

1995). The link is implied, in part, because pink noise is an important

(but not sufficient) footprint of self-organization and self-organizing

systems in nature (see also Aks, Chapter 7, and Van Orden et al., 2003).

More generally, the widespread finding of pink noise in trial series

from exhaustively studied, standard cognitive tasks underscores the

potential for analyses of patterns of variability in cognitive performance

to be informative.
Fractal Variability in Cognitive Performance

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AUTHOR NOTE

I gratefully acknowledge Vahe Amirian for his help in collecting the

data for the timing experiment, and the helpful comments and feedback

I received from three reviewers while preparing this chapter.

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CHAPTER 7

1/f Dynamic in Complex Visual

Search: Evidence for Self-Organized

Criticality in Human Perception

Deborah J Aks

Department of Psychology

University of Wisconsin-Whitewater

Whitewater, WI 53190

U. S. A.

E-mail: aksd@uww.edu
Visual Search and 1/f Dynamics

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Consider the “search and select” problem we face each day.

We constantly set goals for ourselves that require us to scan our

environment until we find a target. Whether we are looking for a face in

a crowd, our keys in a cluttered environment, or a tumor in an x-ray,

we rely on this ability to search the environment. What type of

process is involved? Most researchers argue that some form of

memory serves as a guide to search. Attention also plays an important

role (e.g., Kowler, Anderson, Dosher, & Blaser, 1995; McPeek,

Maljkovic, & Nakayama, 1999), and is arguably what drives saccadic

eye movements to be effective. For now, I will focus on the role of

memory in guiding search. A popular view is that a record accumulates

(e.g., Townsend, 1974) and persists across fixations to guide search

(e.g., Irwin, 1992). Once items are visited they are tagged so as to

inhibit unnecessary subsequent visits—a phenomenon known as

inhibition-of-return (e.g., Posner & Cohen, 1984; Klein, 1988). Surely,

search would be more efficient if we only needed to check each item

once until the target was found. While this theory has empirical

support (see Shore & Klein, 2000, for a review), views of what

characterize this memory are widely varied, as are the experiments

that attempt to test them (Irwin, 1992; Posner & Cohen, 1984; Klein,

1988; Treisman & Gelade, 1980; Wolfe, 1994). Moreover, and

perhaps surprisingly, studies have shown that under some natural

search conditions memory does not seem to play much of a role in

guiding search (e.g., Ballard, Hayhoe, & Petz, 1995; Horowitz & Wolfe,

1998). Eye movements are often sloppy when scanning a natural

environment with our eyes returning repeatedly to objects and

locations that have already been visited (Ballard et al., 1995). Instead
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of systematically retrieving information from memory, search may rely

on the external world to serve as its guide (O’Regan, 1992).

RECONSIDERING HOW WE STUDY VISUAL SEARCH

To help reconcile why such widely varying search behavior has

been seen in those different experiments, let us consider the different

methodological approaches that have been taken. Typically, the

analytical focus is on comparing performance speed, accuracy, or

amount of information recalled across conditions (e.g., experimental

and control). In studies of inhibition-of-return, for example,

researchers are interested in the effect of prior exposure on

subsequent search performance. Once the eyes fixate and “take

note” of a particular item, there should be no need to revisit this item.

Similarly, in priming experiments, the critical comparison is between

“new” vs. “repeated” information. Positive priming experiments show

benefits accrued from repeated exposure to target and distractor

information, and negative priming produces interfering effects on

search performance (e.g., Maljkovic & Nakayama, 1994).

In these traditional experiments, RTs and error rates are

combined across trials of the same conditions. The variability of the

behavior, which inevitably emerges in all conditions, is attributed to

extraneous noise. That variability is isolated and removed from the

purported impact of the independent variable. By contrast, similar to

Gilden (1996) and Gilden, Thornton, and Mallon (1995), we focus on

the variability over time. As is commonly done in the dynamical

systems approach in psychology, we start by asking how a behavior

changes over time (within a single condition). We no longer assume
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erratic fluctuations are noise in the system (Gilden, 1996). Rather,

these fluctuations may contain important information about how search

history may have an important influence on future behavior. In this

chapter I assess whether a dynamic is driving visual search by

analyzing the temporal properties of eye movements. This approach

may help us understand whether a simple iterative process is behind

visual search.

PATTERNS OF SEARCH

Studying the pattern of search behavior is key to understanding

what drives search, and it will help us understand essential

characteristics of any mediating memory mechanism. An absence of

memory will be signaled by random search. The opposing extreme of

random search would be a highly systematic pattern of search, for

example, looking sequentially from left to right and top to bottom.

Such a systematic search guarantees coverage of the visual field until

the target is located. Although this increases the likelihood of

detecting a (fixed) target, there are costs involved. Perhaps most

important is the substantial cost in time. A speedy search is imperative

under many real world conditions, such as when the target is not fixed,

its identity or context changes, when the perceiver is constrained by

competing goals, or when an aggressive predator is pursuing the

perceiver. Such time constraints need to be incorporated into any

realistic model of human visual search.

Could a rapidly implemented, unsystematic, even haphazard

search be better than a systematic one? Literature that documents

examples of erratic yet effective search adds plausibility to this idea
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(e.g., Ellis & Stark, 1986; Engle, 1977; Inditsky & Bodmann, 1980;

Krendel & Wodinsky, 1960). Similarly, visual search has been found to

be “nearly random” (Scinto, Pillalamarri, & Karsh1986), outright

“random” (Groner & Groner, 1982; Horowitz & Wolfe, 1998), and a

“random walk” (i.e., brown noise; Scinto et al., 1986). Likewise,

studies focusing on the type of memory guiding search find little

relation to the recall of objects (e.g., Melcher & Kowler, 2001; Ballard

et al., 1995) or their locations in the scene (Zelinsky & Loschky, 1998).

Contrary to what we might expect, and to what much conventional

theorizing holds, visual search often is unsystematic and not

necessarily related to explicit memory for what and where our eyes

have just visited.

Computational theory as well as empirical findings (e.g., Megaw

& Richardson, 1979; Locher & Nodine, 1974) have begun to clarify

these notions and illustrate how a pseudo-random search can afford

better coverage and more efficient search than many systematic ones.

As we will see, a long-term memory across fixations—one that does not

necessarily contain explicit information about the identity or locations

of objects—may be instrumental in driving behavior. Although the

complicated search behavior appears inefficient, the efficiencies come

in the form of significant cognitive savings. For instance, cognitive load

may be reduced since complex long-memory search behavior may

require minimal resources for coding, retrieval, and recall. As

described later in discussion of the SOC model, the only cognitive load

required involves iteration of a very simple set of rules.
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PATTERNS OF SCALING , COLORED N OISE, & SELFORGANIZATION

When we look at the statistical properties of many systems’

behavior we often find dynamics with well-defined scaling properties

(e.g., Bak, Tang, & Wiesnfeld, 1988; Jensen, 1998). Scaling behavior

(see Liebovitch & Shehadeh, Chapter 5, and Holden, Chapter 6) is a

sign of long-term influences on system behavior and may be the

product of a simple yet flexible process. The scaling behavior itself

implies that no single characteristic scale is best suited to describe the

behavior of the process. There is not just one time scale that controls

the evolution of these systems; the means and variances change

depending on the size of the sampling resolution. If given unlimited

time, these “scale-free” distributions can stretch on indefinitely

without encountering a cut-off. This stretching property is key to

scaling and can be quantified by power laws. We can succinctly

express some quantity N as some power (an exponent, e) of another

quantity, s: N(s)= s

–e. Therefore, by examining the exponent of the

power law we know how the distribution changes as a function of some

underlying variable, which in this case is time.

Importantly, systems characterized by power laws often

produce complex behavior that appears random. In a particular form of

power scaling, one that emerges frequently in complicated behavior,

slower (i.e., low-frequency) behavior dominates. The temporal

phenomenon scales as the inverse of the frequency (f), or as 1/f noise.

Bak et al. (1987) suggested that these systems, with a power spectral

exponent α = –1.0 (i.e., f

α

), consist of many interacting components,

are ubiquitous in nature (see Bak et al., 1987 for examples), and,
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under many conditions, are dynamical systems which organize

themselves into a state with a complex structure. Self-organization

(see, e.g., Kelso, 1995) implies that patterns develop without a need

for a controlling agent. The patterns emerge from a decentralized set

of interactions that are intrinsic to the system.

Recognizing that difficult visual search tasks are often

unsystematic, yet effective, has led to my belief that the oculomotor

system uses subtle, self-organizing properties that produce erratic

fluctuations in search behavior. Because the process involves very

simple, iterated rules, only a minimal cognitive load is needed to carry

out the complicated search behavior. The amount of information

needed to be stored is reduced to a simple iterative function. This

highly compact code may suffice to guide search. Evidence for such a

self-organizing, complex system would reflect determinism inherent to

the system and support the notion that a simple memory persists

across fixations. It is a memory quite different from that of conventional

thinking—the use of “memory” here does not imply “memory” in the

everyday sense—and one that shares known properties of

neurophysiology (e.g., spreading activation and inhibition).

SELF-ORGANIZED CRITICALITY AS A MODEL OF VISUAL

SEARCH

One candidate model of visual search is Self-Organized

Criticality (SOC1; Bak et al., 1987). In the SOC model, dramatic change,

 

1 Despite the controversy regarding Bak’s original conception of SOC as a reliable model of

1/f dynamics, slight variants have proven to be reliable (Jensen, 1998). Alternative models

maintain many similar properties including simple rules producing complex behaviors and

self-organization. Thus, SOC or similar alternatives could account for described data trends.
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or criticality, arises from the local interactions of the system’s

component parts. Such changes give rise to both the complex

behavior as well as the self-organization within the system. The

simplicity of the local rules according to which neighbors interact

implies the cognitive load for driving eye-movements is minimal. In

previous work, we have illustrated how SOC can easily be generalized

to a neural network capable of evoking perceptual changes (Aks &

Sprott, 2003). Here I describe how local interactions can occur

through lateral inhibition and excitation across neurons. Together with

simple threshold rules (as are typical in SOC models), these

interactions can produce perceptual changes (i.e., Stassinopoulos &

Bak, 1995). Similarly (as illustrated later in Figure 7.12 and the

Discussion), we can conceive of eye movements being driven by the

interaction of neurons across an underlying network of neurons.

Applications of dynamical approaches to other cognitive and

perceptual phenomena (Gilden et al., 1995; Kelso, 1995; Port & Van

Gelder, 1995; Pressing, 1999; Ward, 2002) show great promise for

extending that approach to the visual search system. My proposal that

a simple deterministic process may drive the human visual system has

been tested in a challenging visual search task (Aks, Zelinsky, &

Sprott, 2002). As described in the Method section in this chapter, our

analysis focused on the impact of time on the resulting probability

distributions and power spectra. We looked for scale-invariance in

eye movements by evaluating whether the means and variances of

those data distributions changed over time, and whether power laws

emerged in the power spectra. Finding a scale-invariant perceptual

system, characterized by a power law, would suggest that there is
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determinism and compact coding of information in visual search (c.f.

Voss, 1992). Furthermore, evidence of SOC or a similar such iterative

mechanism in the perceptual system (as indicated by 1/f power laws)

would present another illustration of a complex system with a simple

underlying dynamic—one that can potentially account for the flexibility

of our visual system in adapting to novel environments.

METHOD

The visual search task, illustrated in Figure 7.1, consisted of

eighty-one 0.43° T shapes. Targets and distractors differed in

orientation by 90°. Items were presented in a pseudo-random

arrangement so that all locations had an equal probability of being

searched. The participant’s task was to search the array and press a

hand-held button when the target was located.

A B

Figure 7.1. (A) Sample display from the search task. The subject searched for an upright ‘T.’

(B) The correct target is circled in red

The experiment consisted of 400 target-present trials and lasted

approximately 2.5 hours. Every effort was made to emulate a
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continuous search but factors such as fatigue and actual target

detection forced brief discontinuities in the data series. The two types

of discontinuities that occurred in this study were due to (1) inter-trial

intervals between when the participant found the target and a new trial

began, and (2) subdividing the trials into eight sessions separated by

five-minute rest periods. Eye movements were sampled using a

Generation V dual purkinje-image (DPI) tracker that was controlled by

a computer.

The duration and x and y positions of the eyes were recorded at

each fixation. Each measure was treated as a set of data points whose

spatial and temporal properties were analyzed over the course of

search. Additional parameters of the eye movements were used to

map the trajectory of the eyes as they moved from fixation to fixation.

These included differentiation of consecutive eye positions (e.g., xn –

xn+1), eye movement distance (x2 + y2)1/2, and eye movement direction

[arctan (y/x)].

ANALYSIS STRATEGY

Many of the tools of complexity theory involve formalizing (with

mathematical and visual representations) the interactions that occur

within a network. A typical aim is to look for a statistical pattern that

might emerge in data that have been collected over time. In our

analysis of the eye fixations we looked for scaling and other patterns

across data points in the series. Here I will focus on three sets of

analyses—spectral analysis (FFT), power laws, and the Iterated

Functions System (IFS) clumpiness test.
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Spectral and Fourier analyses2 are well-established methods to

test for correlations within a time series. Jagged data series (appearing

superficially as a random series) are often produced by natural

complex systems. The data series can be described as a complex

waveform best estimated by the composite of simple regular (sine)

waves that span a range of frequencies. A Fourier transform involves

first decomposing the observed series into simple waves and then

plotting the power against frequency to describe what combination of

waves best describes the observed waveform. The analysis in our

study used a Fast Fourier transform (FFT; Press, Flannery, Teukolsky,

& Vetterling, 1986) and the resulting plot of the power (mean square

amplitude) against frequency. For an introduction to Fourier analyses

see one of many tutorials, such as Peak and Frame (1994) and Sprott

(2003).

Power Laws in Spectral Analyses

Can the eye movement data be described by a power law, and

do the data possess scale invariance associated with a self-organizing

complex system? A linear function on a double-log plot produced by

FFT indicates the presence of a power law. The regression slope of

this function, denoted by α, is the power exponent. When the

exponent of the power spectrum is α = –1.0 (as shown in the middle

 

2 Spectral analysis is a linear method that can be effectively used to characterize a nonlinear

(fractal scaling) process. A linear method can be very useful in detecting correlated structure

in noisy data regardless of the process that produced it. For example, the eye movement data

that we have analyzed can be the output of either a linear or nonlinear process. I argue that the

1/f structure—estimated by FFT spectral analysis—may suggest that a SOC-type process

drives eye-movements. SOC is a simple nonlinear process (iterated many times over) which

can operate in a neural network—perhaps to produce complicated eye movements. The linear

Fourier analysis is an effective tool to uncover (and estimate) correlated structure regardless

of whether it has emerged from a linear or nonlinear process.
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Figure 7.2. Spectral analysis of three types of colored noise. White (1/f0) noise with a flat

spectrum indicates no correlation across data points; Brown (1/f2) noise has a steep slope

indicating short-term correlation; and Pink (1/f) noise has a shallow slope indicating

extremely long time correlation. The white noise is a Gaussian distribution of scores with all

frequencies equally represented. Sprott and Rowland (1995) included this sample data set in

their Chaos Data Analyzer software. Brown noise is a simple integration of the white noise. To

generate the pink noise I took the Fourier transform of the white noise data to isolate the

frequency domain, multiplied the amplitude of the frequency components by 1/f, and

performed an inverse Fourier transform. See also Equations 9.35 and 9.36 in Sprott (2003).
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panel of Figure 7.2), the given temporal phenomenon scales as the

inverse of the frequency (f), or as 1/f noise. In this particular form of

scaling, fluctuations occur in the same proportion at all scales (i.e., they

are self-similar and scale invariant), and there exists a great deal of fine

structure in the data.

An important aspect of spectral analysis is that it serves as a

useful measure of the strength of memory across the system.

Revealing temporal correlation is important not only in assessing

whether memory exists across eye-movements, but the magnitude of

the exponent also quantifies memory strength. The steepness of the

slope (on a log-log scale) reflects the duration of memory (i.e.,

correlation across points). As shown in Figure 7.2, Brown (1/f2) noise

has a steep slope, indicating short-term correlation. Pink (1/f1) noise

has a shallow slope, indicating extremely long time correlation, and

white (1/f0) noise, with its flat spectrum, indicates no correlation across

data points. Implications for these trends are described in the

Discussion section, including how pink noise hints at a process that has

important self-organizing properties.

Iterated Function Systems (IFS) provides an interesting

alternative to determine whether temporal correlations exist across

fixations (Peak & Frame, 1994; Jeffrey, 1992; Sprott & Rowlands, 1995;

Mata-Toledo & Willis, 1997). This technique is used to create a pattern

that helps to visually characterize the color of noise. It does so by

producing clumped patterns for colored noise while producing

homogeneously filled spaces when the data are uncorrelated. At a

minimum we learn whether our data deviates from a random pattern,

but there is also the potential for learning about the degree of
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correlation present in noisy data. The different degrees of temporal

correlation correspond to the different forms of coloring to the noise.

As can be seen in Figure 7.3, white (1/f0), pink (1/f1, or 1/f), and

brown (1/f2) noise are easily distinguishable. White noise produces a

pattern that uniformly fills its representation space. At the other

extreme, brown (1/f2) noise produces a pattern in which the dots

accumulate over the diagonals and some of the sides of the square,

leaving most of the representation space empty. Pink (1/f) noise

produces self-similar repeating triangular structures of different sizes,

and accumulates, albeit in a dispersed way, near the diagonals. These

examples illustrate how to visualize the fine structure of time series

using the IFS test to help distinguish the color of noise in a system.

The IFS Procedure

First, take either the x or y fixation series (or some derivative as

shown in Figure 7.4), and sort the data from the minimum to the

maximum value. Then, subdivide the series into four quartiles, in such

a way that each group contains the same number of points. The

original unsorted data set is then normalized and grouped into one of

four values, 1 to 4, representing the quartile to which the data belong.

As shown in Figure 7.5, the representation space is a square that

provides a 2-D picture of the correlation structure present in the

trajectory of eye movements. The four corners are labeled 1 to 4 in a

clockwise direction (starting in the lower left corner) to represent the

quartile of that fixation. The first fifteen fixations from Figure 7.4 are

used here to demonstrate the IFS procedure.
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Figure 7.3. Output of the IFS test performed on white, pink, and brown noise (the same

simulated data used in the analysis of Figure 7.2). Each case is easily distinguishable. White

noise uniformly fills the representation space. Brown noise produces a pattern in which the

dots accumulate over the diagonals and some of the sides of the square, leaving most of the

representation space empty. 1/f (pink) noise produces self-similar, repeating, triangular

structures of different sizes, and accumulates, albeit in a dispersed way, near the diagonals.

Because previous points determine the position of each

subsequent point, the plot represents a trajectory of the eye

movements. Each point gives a short-term history of eye movements,

since the influence of previous fixations diminishes over time. The
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Figure 7.4. The first 15 data points for the difference across the y coordinate of each fixation.

In the right column is the quartile to which each data point belongs.

result is a scattering of points in the plane as shown in Figure 7.3. Any

departure from a uniform distribution of points is evidence for

correlated structure and possibly a deterministic mechanism driving

the behavior. Clustering along the diagonals in the figure reveals the

short-term, highly correlated pattern associated with brown noise. The

additional fractal microstructure reflects longer-term, but weaker,

correlations often associated with pink noise.

RESULTS

Visual search produced, on average, 24 fixations (SD = 15) per

trial, with each trial lasting 7.6 s (SD = 6.9 s). Mean fixation duration

was 212 ms (SD = 89 ms) with 10,215 fixations across the complete

search experiment. The number of fixations decreased from 1888 to

657 across eight sessions, with the average duration increasing from
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Figure 7.5. (A; top-left) A point is plotted halfway between the center of the IFS square and the

quartile of the first point of the series. The first point falls into the 2nd quartile of the full data

series. The quartile of the data point is circled in red. (B; top-middle) A second point (falling

into the 1st quartile of the full data series) is plotted halfway between the first plotted point and

the second point in the fixation series. (C; top-right) A third point (falling into the 2nd quartile) is

plotted halfway between the second plotted point and the third point in the series, and so forth.

(D)-(O) (smaller panels, from top-left to bottom-right) show the remaining evolution of the IFS

map when applied to the first fifteen trials.
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206 to 217 ms. Mean deviation from last trial fixation to new target

location was 0.4°. indicating a high degree of accuracy in actual target

detection.

Figure 7.6 shows a representative sample of the first

differences across eye position (yn-1 – yn). These erratic trends were

similar for x- and y-coordinate positions, except for the overall

direction in which the eye position changed over time. While

differences across y positions gradually increased over time,

differences across x positions tended to decrease over time. The

same trends occurred with relative dispersions (SD/M), a measure

which reflects system contingencies as function of sampling resolution

(Liebovitch, 1998). These changes in mean and variance with fixation

duration are characteristic of fractal structures and scale-invariant

systems.

Figure 7.6. Representative fixation series for the first differences of eye position (yn+1 – yn).

Only fixations along the vertical coordinate are shown. The erratic pattern in the fixation

series is similar for horizontal eye positions. The spikes are typical of 1/f behavior.
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The first method to assess temporal correlations (i.e., memory)

in search involved FFT and power spectral analysis. The mean

regression slope of the power spectra for the x and y eye-position

coordinates was α = –1.7 (i.e., brown noise). Differentiated (x and y)

data showed reduced regression slopes, α = –0.23. The spurious lowfrequency regions of the spectra flattened the slope. In the highfrequency region of the curve α = –0.7 (for y, α = –0.7; for x, α =

–0.6). An example of the 1/f trends is illustrated in Figure 7.7.

Figure 7.7. Power spectra of first differences of x and y fixation series produced pink (1/f)

noise. Spectra of differences across y fixations are shown here. Mean regression slope of the ydifference power spectrum is α = –0.7 in the high frequency region. Also shown is a line

depicting an exact 1/f power spectrum.
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The second method, the IFS clumpiness test (Jeffrey, 1992),

evaluates memory in complex search (i.e., temporal correlations and

deviations from randomness). Clustering along the diagonals reveals

short-term, highly correlated consecutive data points typically found in

brown noise. Such a pattern was observed in the analysis of raw eye

fixations for the x and y coordinates. Additional fractal microstructure

appeared in the IFS test when fixation differences were analyzed. This

trend, shown in Figure 7.8, reflects long-term, but weaker, correlations

often associated with pink noise.

Figure 7.8. Results of the IFS clumpiness test of differentiated vertical (y) fixations. A similar

pattern emerged for horizontal (x) fixations. The diffuse fractal microstructure reflects longerterm correlations appearing weaker than those in the raw data. Both cases resemble patterns

associated with pink noise.
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A combined measure of distance across eye fixations (Δx2 +

Δy2)

1/2 produced power spectra with 1/f trends dominating the lower

frequency range and 1/f2 trends dominating the high frequency range

(Mean α = –0.47; see Figure 7.9). The corresponding IFS test, shown

in Figure 7.10, produced a clear but distinct colored noise pattern with

more diffuse clustering of data points than those found in the raw and

differentiated data sets. Random shuffling of x, y, and distance data

sets produced white noise. Thus, differences in these fixations

possessed a potentially important, long-term dynamic characterized

by ~1/f pink noise.

Figure 7.9. Power spectra of distance across eye fixations (Δx2 + Δy2)

1/2. Pink (1/f) trends

were dominant in the lower frequency range, and 1/f2 trends were dominant in the high

frequency range (mean α = –0.47). Also shown is a line depicting an exact 1/f power spectrum.
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Figure 7.10. Results of the IFS clumpiness test on the time series of distances between fixations

(Δx2 + Δy2)

1/2. A unique colored noise pattern emerged with more diffuse clustering of data

points than those found in the raw and differentiated data sets.

DISCUSSION

I examined complicated eye movements to gain insight into the

underlying mechanism guiding visual search. Using a challenging

conjunction search task, we generated and then analyzed the resulting

eye movements (Aks et al., 2002). My focus on the dynamic of the

fixation series offers a unique perspective to the broader study of eye

movement behavior. It contributes new insights to the debate in the

visual-cognitive literature on whether memory plays a role in guiding

visual search (e.g., Horowitz & Wolfe, 1998; Kristjansson, 2000;

Melcher & Kowler, 2001; Ballard et al., 1995; Shore & Klein, 2003).
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According to conventional theory and common sense, visual

search utilizes information from previous fixations to guide subsequent

search. Memory models of visual search incorporate mechanisms such

as inhibitory tagging (Klein, 1988) or identification of previously

searched items (e.g., Irwin, 1992; Jonides, Irwin, & Yantis, 1981).

Horowitz and Wolfe (1998) is a recent example of research that

challenges the assumption that search is guided by memory from

previous fixations. Doubts about a memory-based guidance emerge

from their findings of RTs being unaffected by randomly repositioned

items, together with recent research showing that visual memory is

often surprisingly poor (Melcher & Kowler, 2001; Rensink, O’Regan, &

Clark, 1997; Simons & Levins, 1997) and that the visual system retains

little information about the locations (or identity) of objects over time.

Instead, the visual system seems to act on fleeting neural

representations that are overwritten by a change in the visual scene.

Horowitz and Wolfe’s (1998) claim that visual search does not keep

track of previously searched locations comes from their examination of

overall RTs to complete visual search. However, relying on such a

coarse measure of behavior means that subtle contingencies in

scanning behavior can be overlooked. Direct analyses of the eye

movements revealed a less obvious form of memory.

The key finding that a sequence of fixations can be represented

by a power law function confirmed our prediction that search might be

guided by a memory of previous fixations. Contrary to Horowitz and

Wolfe (1998), we found that search behavior was not random and that

contingencies did in fact exist across fixations. While much cognitive

theory implicates search mechanisms such as tagging and inhibition of
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return (to previously visited items), the form of memory we have

found involves general contingencies across fixations. The 1/f power

law is a signature of these contingencies. The power law further

indicates that the system has scale-invariant properties typically

associated with a system optimized to adapt to a changing

environment. Since systems characterized by power functions are

known to be flexible, this suggests that the contingencies guiding

search may play an important role in selection of appropriate

information in an array of constantly changing environmental

information. The IFS clumpiness tests confirmed the results of the

power spectral analyses, showing that differences across fixations

revealed pink (1/f) noise. These results suggest a long-term memory

is maintained across complicated search in a manner that may involve

the use of a simple set of rules with self-organizing properties (i.e.,

variants of Bak et al., 1988; Jensen, 1998).

IMPLICATIONS OF 1/f NOISE

Simple neuronal interactions can produce complex, selforganizing behavior. To understand how simple rules can produce

complex eye movements let us consider Bak’s SOC model in the

context of a neural network. Figure 7.11 depicts such a network that

could be used by the human visual system to represent a scene.

Activation of different neural sites on the network can serve both as a

means to represent the scene and a means to guide eye movements.

Figure 7.12 shows how the network is represented as a twodimensional grid of interacting cells. Each cell possesses a certain

degree of activation represented by a numerical value, Z(x,y). Activity
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Figure 7.11. Representation of a visual search scene mapped onto a neural network. The

scene is depicted in the top-right portion of the figure and shows one search display used in

the experiment. Shifting attention to a particular stimulus produces an increase in the

activation at the corresponding site on the neural net. Each node’s numbers and colors

represent the level of neural activation at that site. Nodes with the highest activation pull the

eyes to that location. The resulting path of the eye movements is shown as a shifting black line

superimposed on the network.

can be induced by any of a number of factors such as a salient visual

feature, shifting attention to a feature of interest, or random activity

that is produced even at rest. Neural activity can also be triggered by

movement of the eyes to different locations. As individual neurons are

activated beyond a threshold of, say, 4 arbitrary units, the activity in

the original site is dispersed to surrounding cells, incrementing the

activity in these regions by 1, {Z(x,y) ➞ Z(x,y) + 1}, thus depleting

the activity in the original site to zero, Z(x,y) ➞ Z(x,y) – 4. In the
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A B

C D

E F

Figure 7.12. (A) A neural network. Numbers and colors indicate the relative activation of

neurons. High numbers and orange colors indicate a high level of activity. Low numbers and

blue colors indicate low levels of activity. (B) A one-unit increment in neural activity at the

central site. This increase in neural activity can be due to any one of a number of factors,

including the appearance of a salient feature, shifting attention to a feature of interest, or

random activity that is produced at rest. (C) Neural activity in the original central site is

depleted to zero after the threshold of 4 activation units is reached. (D) Neural activity in the

immediately surrounding sites is increased by one unit. (E)-(F) Neural activity in subsequent

surrounding sites is increased by one unit.
Aks

345

absence of useful environmental information during visual search, the

eyes may be guided to sites that contain the highest level of activity

among immediately surrounding cells, and evade local sites depleted

of neuronal activity. The global result can be a complicated search

pattern that could easily be mistaken for a random search (see Figure

7.13).

A B C

D E F

Figure 7.13. Eye-movements being pulled to sites of greatest neural activity, following the

same progression of neural activity in Figure 7.12. When two sites have an equal level of

activity (a tie), as in (A), the eyes traverse an intermediate path. (B) and (F) show another

example of tied activity, but here the tie is on opposing sites. Rather than remaining fixed, the

eyes are pulled to the site where the prevailing activity surrounding the recipient site is

greater. Aside from these cases of equal activity, the general rule is that eyes are pulled to a

single adjacent site of greatest activity.
Visual Search and 1/f Dynamics

346

The key finding of 1/f noise in eye movements has some

important implications. First, we know that search is not random.

Instead the eyes are guided by their history. A simple form of

temporal memory exists across the sequence of eye movements. It is

possible that the 1/f search pattern is produced from guidance of eye

movements by changes in the intensity of neuronal activity across a

network of neurons. These changes may be best described as the

output of iterating a simple set of threshold-based rules such as those

associated with SOC models.

The 1/f eye movements may also involve a cognitive mechanism

such as attention-based sampling and selection of useful information

from a complicated environment. It is an open question whether

neuronal interactions and their spread of activation drives this

selection process. Could this process produce the rapid and effective

search known to occur in humans? The answer may relate to the

general finding that 1/f systems offer an optimal compromise between

efficient recovery of information and the tendency to err (Voss, 1992).

The significance of these complex yet adaptive behaviors remains

open to future scientific inquiry.
Aks

347

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CHAPTER 8

Categorization and Learning in

Speech Perception as Dynamical

Processes

Betty Tuller

Center for Complex Systems and Brain Sciences

Florida Atlantic University

Boca Raton, FL 33431

U. S. A.

E-mail: tuller@ccs.fau.edu
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354

The perception and production of speech unfolds in time.

Although this seems like an obvious and perhaps trivial statement, this

defining quality is not well captured by linguistic descriptions of

speech, most of which are fundamentally static. Over the past few

decades, both the theoretical and mathematical foundations for

understanding organized behavior that emerges in time have been

more fully developed and have infiltrated the study of many different

human behaviors. In general, nonlinearity is a hallmark characteristic

of these behaviors. That is, small changes in context or constituents can

produce large behavioral effects and large changes in context or

constituents might, in other conditions, produce little or no behavioral

effect. The conditions that reveal the nonlinearities are often exactly

those conditions that are excluded from experiments since they are

more difficult to analyze and understand.

One nonlinear aspect of speech perception that has been the

subject of a large number of studies is the phenomenon known as

categorical perception. Within certain ranges of an acoustic parameter

it is extremely difficult to discriminate between different stimuli that are

labeled as the same speech segment. At the same time, stimuli with the

same-size acoustic difference but in a different part of the parameter

range are easily discriminated (e.g. Liberman, Cooper, Shankweiler, &

Studdert-Kennedy, 1967; Liberman, Harris, Hoffman, & Griffith, 1957).

As an example, consider the words “say” and ”stay”. When a short

silent gap is introduced between the “s” noise and the vowel in “say”

(e.g., from 0-20 ms), listeners continue to perceive the word “say.”

Similarly, listeners perceive the word “stay” when the silent gap after

the “s” in the original stimulus is as long as 60 ms to 80 ms. But when
Speech Perception Dynamics

355

the gap ranges from 30-50 ms, the same absolute difference in gap

duration as in the previous two examples (i.e., 20 ms), listeners

perceive an abrupt shift in the stimulus from “say” to “stay” (Best,

Morongiello, & Robson, 1981). Importantly, when listeners identify two

stimuli as belonging to the same phonetic category, they often have

great difficulty discriminating between them. As the “category

boundary” nears, stimuli are more easily discriminated from each

other. Obviously, stimuli that are identified as different words or

syllables are also easily discriminated. This means that there is acoustic

variability but phonetic/perceptual stability in some ranges of the

acoustic parameter (here gap duration) but perceptual change

accompanies the same degree of acoustic change for other values of

the acoustic parameter. In other words, the relationship between

acoustics and phonetic perception does not change in a linear fashion.

The perceptual boundaries between categories, or “critical

points,” are not hard-wired by neurophysiology, or set indelibly by

one’s native language, but adjust flexibly with factors such as phonetic

context, the acoustic information available, speaking rate, speaker, and

linguistic experience (see Repp & Liberman, 1987, for review). This is

not simply a laboratory demonstration: Listeners recognize the same

word produced by different speakers (males, females, speakers of

different ages) and by the same speaker in markedly different

linguistic and intentional contexts, even when the listener has had no

prior experience with the other individual’s speech patterns. Thus,

perceptual stability coexists with perceptual flexibility.

About a decade ago, Pam Case, Mingzhou Ding, Scott Kelso and I

considered seriously the ideas that speech categorization is inherently
Tuller

356

nonlinear, and that the nonlinearity can serve as a window into the

dynamics of speech perception (i.e., the equations of motion that

characterize the intrinsic organization of speech perception). My

charge by the organizers of the NSF workshop “Nonlinear Methods in

Psychology” was to re-cap that work as an example of how these ideas

can help guide empirical studies to allow a deeper understanding of

the phenomenon under study. In exploring the process of speech

perception in a non-traditional way, two things are extremely

important. First, it is important not to ignore the decades of previous

research on how people perceive speech. Any alternative theoretical

views should be compatible with that body of work. Second, theory and

experiment are related in a mutually informative fashion. The

investigator’s theoretical viewpoint guides not only what she or he

chooses to examine, but also how it is examined. In turn, experimental

results guide theory development, which then suggests the next

empirical step. In the present case, since the focus is on evaluating

whether the identification of speech sounds is itself characterized by

what has been termed perceptual dynamics (characterized by

multistability, loss of stability, flexibility, etc.; cf. Kelso, 1994a, 1995),

the methodology chosen for the experiment must be one that allows for

the possibility to see those signature characteristics. In what follows, I

give a brief description of the main features of nonlinear dynamical

systems relevant to this enterprise and describe how these features

offered strategic guidance for the specific experiments and the

theoretical model developed.

The first step was to conceive of perceptual space as a dynamical

system with context, experience, and learning (among other things) as
Speech Perception Dynamics

357

processes that can modify this dynamical system. Briefly, a dynamical

system is one that evolves over time such that its present state always

depends in some rule-governed way on previous states. Differential

equations or maps (equations that dictate how a system evolves in

discrete time steps) of relevant variables offer a mathematical

description of the system's behavior as time passes and parameters

change. Typically, one observes the stable behaviors of a system,

referred to as its attractors. The attractor layout, or set of possible

behaviors of a system, may change over time in such a way that

observed behaviors change gradually or abruptly. Abrupt, or

qualitative, changes (called phase transitions or bifurcations) may be

thought of as the spontaneous emergence of new forms of organization

(a self-organized pattern formation process) under specific boundary

constraints (e.g., Haken, 1977; Nicolis & Prigogine, 1977). In a speech

perception experiment, qualitative change in categorization of the

stimuli allows a clear differentiation between patterns; there is no

ambiguity as to what are the stable patterns for a given listener. Note

that the qualitative change (here the shift in categorization) is

informationally meaningful (Kelso, 1994). Although in any experimental

situation there are many variables likely to be changing, the key is to

discover the ones that bring about this qualitative categorical change.

As Kelso (1995) has pointed out, situations where qualitative change

occurs are also regions of dynamic instability and dynamic instability is

the generic mechanism underlying self-organized pattern formation

(Haken, 1977; Nicolis & Prigogine, 1977). Without the dynamic

instability, no change in pattern would occur. In turn, if one can see

evidence of growing dynamic instability, then one can study the
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358

emergence of the new pattern. We will return to the idea of dynamic

instability when we describe the experiments evaluating speech

categorization as a dynamical phenomenon.

Although this description of qualitative pattern change as some

parameter varies bears a strong similarity to the results of speech

categorization tasks, the similarity may be only superficial. Empirical

work on speech categorization, in order to maintain the independence

of treatment levels required by most parametric statistical techniques,

typically presents the stimuli to listeners in random order. Such

experiments thus describe the location of a statistically defined

phoneme boundary (most often, the point corresponding to the 50%

crossover of the response function for a two-category set; see Ganong

& Zatorre, 1980, for a comparison of different methods for defining

boundary location). Unfortunately, this traditional methodology is far

from optimal for revealing the dynamical characteristics being

evaluated, because the randomization of stimuli destroys the footprints

of any underlying dynamical process that may govern the transition

between speech sounds. So one’s theoretical viewpoint must influence

experimentation from the initial design stage.

The strategy in our experiments was to use a stimulus continuum

for which categorical perception has often been demonstrated but to

vary the acoustic parameter sequentially, i.e., as a control parameter. A

control parameter is one that, when the appropriate range of values is

used, takes the subject from one perceived categorization to another.

For some behaviors, finding control parameters is non-trivial. However,

the literature on categorical perception gives us many plausible control

parameters for different speech categorizations. In what follows, I will
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359

review the observed dynamical effects and delineate some of the

factors responsible. A model of the results was proposed and is

discussed, and unique predictions of the model tested. Lastly, I will

describe how viewing the speech perception process as a nonlinear

dynamical system forces, as a natural extension, a re-examination of the

process that occurs when learning to hear non-native phonemic

distinctions. Our experiments demonstrate the fruitfulness of the

approach and reveal that speech perception and perceptual learning in

speech are characterized by rich underlying dynamics.

In 1994, Tuller, Case, Ding, and Kelso examined speech

categorization when an acoustic parameter—the length of the silent gap

between a natural “s” and a synthetic “ay”—was varied in a stepwise

fashion. We used this particular stimulus continuum because it had

already been shown that listeners perceive “say” at short silent gaps

but they perceive “stay” at long silent gaps (e.g., Best et al., 1981).

Thus, the gap duration after the “s” was a possible control parameter

by which we could explore the mechanism of switching between

categorizations. However, a major difference between our experiment

and those of others was that we presented the stimuli in order. That is,

gap duration either increased systematically from 0-76 ms, then back to

0 ms, in 4-ms steps, or decreased from 76 ms to no gap, then back to 76

ms in 4 ms steps. There were 5 trials of each of these two sequences.

We also randomized the stimuli and presented 10 randomizations to the

listeners. The subject’s task was to indicate whether they perceived the

word “say” or the word “stay” by pressing appropriately labeled keys

on a computer keyboard. First, we determined that the randomized

stimuli resulted in the same perceptual identification function as
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360

reported previously in the literature. This ensured that our stimuli (and

listeners) were equivalent to those used by others. Because the point at

which categorization shifts as a function of the direction of changes in

gap duration is considered a theoretically important juncture, the next

analysis focused on that point.

Logically, there are only three possible patterns of switching: (1)

A subject will switch between “say” and “stay” at the same gap

duration regardless of direction of gap change (a critical boundary); (2)

A subject’s percept will change at a larger gap duration as gap

increases than when gap decreases (an effect know as hysteresis or

assimilation); or (3) A subject’s percept will change at a larger gap

duration when gap decreases than when gap increases (a contrastive

effect). All three patterns were observed, with critical boundary being

much less frequent than hysteresis or contrast, which occurred equally

often. Thus, the perceptual changes in this speech identification task

show quite complicated dynamics when a relevant acoustic parameter

is sequentially varied. Closer analysis revealed that the incidence of

hysteresis and contrast was not simply random fluctuation around a

critical boundary, because their relative frequency changed in

predicted ways over the course of the experiment. These patterns of

change reveal that dynamic instability is playing a role in perceptual

switching, thereby linking phonemic categorization to self-organized

pattern formation.

How do you begin to connect experimental data to a generic

dynamical model? Quite simplistically, since we have two reproducibly

observed states—here the two categorizations “say” and “stay”—we

identify the categorizations with attractors, or stable states in
Speech Perception Dynamics

361

perception. We use differential equations to define systems with

attractor properties that fit the observed experimental data.

Differential equations allow us to model quantities that change

continuously in time. We can find stable solutions of the differential

equations by finding equilibrium points, values of x for which the

derivative dx/dt=0 (see Equation 8.1; by definition, if the derivative of

some variable is zero, that means the variable is unchanging, which is

what it means for that value to correspond to a stable state).

Trajectories (solutions to the differential equation) may be "attracted" to

an equilibrium point or "repelled." We call the first case a stable

attractor (also called a sink) and we call the second case an unstable

attractor (also called a source or repeller).

If listeners perceived only a single perceptual category, a

theoretical model of a single attractor, a fixed-point, would be

adequate. A situation in which two states, or categories, occur requires

that the model contain at least two stable attractors that change with the

control parameter. In our case, the model must be able to account for

the fact that at some gap durations a listener perceives only “say” and

for other gap durations the listener perceives only “stay.” The

presence of hysteresis and contrast is also informative, indicating that

more than one stable percept can coexist for a given acoustic

stimulus—either “say” or “stay” might be perceived. In this case the

stimulus is bistable—the two attractors must coexist for some range of

the control parameter.

These results were modeled concisely by the following

dynamical system (Tuller et al., 1994), written as a differential equation:
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362

dx/dt = –dV(x)/dx = –k + x – x3 [8.1]

Differential equations may be rewritten in the form of a potential

function (Equation 8.2), in which the attractors are geometrically

obvious when the potential is plotted. Here x is a variable

characterizing the perceptual form and k is a parameter specifying the

direction and degree of tilt for the potential. This allows visualization of

the behavior of the system as the parameter k is manipulated.

V(x) = kx – x2/2 + x4/4 [8.2]

Think of Equation 8.2 as describing the motion of a viscous point

mass (a “sticky” ball) moving in the potential landscape V(x) (such as

one of those shown in Figure 8.1). The minima of the potential, the

valleys in the landscape, are the attractors corresponding to the two

perceptual categories.

Figure 8.1. Potential landscape defined by Equation 8.2 for five values of k (adapted from Case

et al., 1995).

Figure 8.1 shows how the landscape changes for several values

of k. With k = minimum only one stable point exists corresponding to a
Speech Perception Dynamics

363

single category (e.g., “say”). As k increases, the potential landscape

tilts but otherwise remains unchanged. However, when k reaches a

critical point k = –kc, a qualitative change in the attractor layout takes

place. In other words, a bifurcation occurs. The particular change at k =

–kc is a saddle-node bifurcation in which a “saddle” (the point repeller,

or maximum, at x = 0) and a “node” (the point attractor at x < 0) are

simultaneously created. Thus, where there was once only a single

perceptual category there are now two possible categories. This

bistability, the co-existence of both categories, continues until k = kc

where the attractor corresponding to “say” ceases to exist via a reverse

saddle-node bifurcation (where the qualitative change is from two

available categories to one), leaving only the stable fixed point

corresponding to “stay.” Further increases in k only serve to deepen

the potential minimum corresponding to “stay.” Thus, the model

captures the three observed states of the system: At the smallest values

of the acoustic parameter only “say” is reported, for an intermediate

range of parameter values either “say” or “stay” are reported, and for

the largest values of gap duration only “stay” is reported.

An accurate portrait of any real-world problem must take into

account the influence of random disturbances. In the present work, we

considered factors such as fatigue, attention, and boredom to

correspond to random disturbances because we could not measure the

changes in those factors over time. Further experimental work may

elaborate whether these factors are indeed random or predictable

modifiers of perceptual space. Mathematically, spontaneous switches

among attractive states occur as a result of these fluctuations, modeled

as random noise. For a given point attractor, the degree of resistance to
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364

the influence of random noise is related to its stability, which, in

general, depends on the depth and width of the attractor (i.e., its basin

of attraction). As k is increased successively in Figure 8.1, the stability

of the attractor corresponding to the initial percept decreases (the

minimum becomes shallower and flatter), leading to an increase in the

likelihood of switching to the alternative percept. This implies that

perceptual switching is more likely with repeated presentations of a

stimulus near the transition point than with repetition of a stimulus far

away from the transition point, a prediction confirmed in Tuller et al.

(1994).

In order to account for the three response patterns observed

(critical boundary, hysteresis, and contrast), the behavior of k must

have multiple determinants. One influential factor suggested by earlier

research is the number of repetitions perceived from each category.

Repetitive presentation of a speech stimulus has long been known to

shift the location of adjacent phoneme boundaries in a predictable

direction (see Darwin, 1976, and Eimas & Miller, 1978, for early

reviews). Taking this factor explicitly into account we proposed the

following equation describing the behavior of k as a function of the gap

duration:

k(λ)= k0 + λ + ε /2 + εθ (n–nc) (λ–λf), [8.3]

where the value of k0 specifies the percept at the beginning of a run, λ

is linearly proportional to the gap duration, λf denotes the final value of

λ (i.e., at the other extreme from its initial value), and n is the number of

perceived stimulus repetitions in a run. The influence of the last term
Speech Perception Dynamics

365

depends on a step function, θ(n–nc). Before a critical number of

accumulated repetitions nc is reached, θ(n–nc) = 0. That is, in the first

half of each run, the tilt of the potential is only dependent on gap

duration and the initial configuration. When n ≥ nc (during the second

half of each run) θ(n–nc) = 1. This means that each step change in gap

duration λ will produce a larger change in tilt k than it did in the first

half of the run. An additional parameter, ε, represents cognitive factors

such as learning, linguistic experience, and attention. Note that the

importance of cognitive processes is well-established, for example,

attention and previous experience play a large role in synergetic

modeling of perception of ambiguous visual figures (Haken, 1990;

Ditzinger & Haken, 1989, 1990) and contribute to factors that determine

adaptation level in Helson's work (Helson, 1964).

Although the additional term was needed to incorporate contrast

effects into the same model that described hysteresis and a critical

boundary, it gave rise to unexpected predictions. For example, if the

subject is presented with a run with gap duration first systematically

increasing (from 0-76 ms) then systematically decreasing (from 76 ms

back to 0 ms), the percept is predicted to be more stable—the potential

would have a locally steeper slope—when the same stimulus appeared

as the last item in the run than as the first item in the run. This is

because the rate of change of tilt of the potential is faster in the second

half of the run for the same amount of acoustic change. This prediction

is unexpected given the literature on selective adaptation effects in

speech. In selective adaptation, a standard identification task is first

used to locate the “category boundary,” or point of subjective equality,

for the test continuum. Next, the subjects listen to the stimulus from one
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366

end of the continuum presented many times over. After a second

identification test with the original stimulus continuum, the position of

the perceived category boundary moves towards the repeated

stimulus. For example, in a [ba]-[pa] continuum varying in the lag of

voicing onset after the initial consonant release burst, if the stimulus

with the longest voicing lag is repeatedly presented after the first

identification test, listeners then require a longer voicing lag for a

stimulus to be perceived as a [pa] (Eimas & Corbit, 1973)—in our

terms, perception of [pa] has destabilized. Somewhat

counterintuitively, our model predicts that when a word is perceived

many times over, its stability will increase.

This prediction was confirmed by experiment (Case, Tuller,

Ding, & Kelso, 1995). In that work, we used the same “say”-“stay”

stimulus continuum but asked listeners not only to categorize the

stimulus as either “say” or “stay” but also to rate how good an

exemplar of the category the stimulus was. The goodness rating was

used as an index of the stability of the percept (the local steepness of

the potential function). As predicted, regardless of whether the stimuli

were presented with gap duration between the “s” and the “ay” first

increasing from 0-76 ms and then decreasing back to 0 ms, or in the

opposite direction, the same physical stimulus presented at the end of a

sequence was judged a better exemplar of the category than was the

identical stimulus presented at the beginning of the sequence (Figure

8.2). One crucial difference between the work of Case et al. (1995) and

the earlier work on selective adaptation concerns the repeated

stimulus. In the former, the stimuli were changing systematically, albeit

at a subcategory level; in the latter, the identical stimulus (typically an
Speech Perception Dynamics

367

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mean Difference in Judged Goodness

random 76 ms

random 0 ms

sequential 0 ms

sequential 76 ms

First-Last Two Middle Stimuli

Figure 8.2. Mean differences in judged goodness versus position in sequence as a function of

sequential vs. random stimulus order. When stimuli are presented sequentially (solid symbols),

the last stimulus presented is judged as a better exemplar than the same stimulus when

presented first in the sequence. This occurs for both 0 ms (square) and 76 ms (circle) gap

stimuli and does not occur with random stimulus orders intervening (open symbols) or when

the same stimuli are the “turnaround” stimuli in the middle of the trial (adapted from Case et

al., 1995).

end-point stimulus) was repeated. In fact, when Case and colleagues

presented stimuli with an intervening set of stimuli with randomly

changing gap durations, no differences in judged goodness were

observed. This result confirmed one prediction of speech

categorization as a context-sensitive, pattern-forming system.

Another difference between this empirical confirmation of the

model’s predictions and the literature on selective adaptation

motivated additional research. The model implies that the temporal

evolution of the alternative forms, and hence switching between them,
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368

depends on how the stimuli move through perceptual space. This was

supported by Case et al. (1995), described above, at least for the

judged goodness of the stimuli as members of the identified category.

Thus, systematic change in an acoustic control parameter, and not

solely the number of stimulus repetitions, is crucial. This was tested

directly by presenting subjects with a single “say”-“stay” trial with gap

duration either increasing or decreasing (again, in 4-ms steps between

0-76 ms silent gap). The second trial was adjusted for individual subject

responses to the first trial. If, for example, a subject heard a switch from

"say" to "stay" on the 6th stimulus in the first trial, then in the second

trial stimulus #1 was presented 5 times, then stimulus #6 was

presented, then the trial continued to the end, with each successive

stimulus presented once. Selective adaptation leads one to expect that

repeating the initial stimulus in trial 2 should cause listeners to switch

earlier, or at the same stimulus, as in trial 1 (contrast or critical boundary

should increase in observed frequency). Similarly, if the

preponderance of hysteresis observed previously reflects only a

response perseveration, then the incidence of critical boundary should

increase markedly because both trials present the same number of

instances of the initial category. Identical predictions are made by

Helson’s (1964) Adaptation Level Theory, which holds that all stimulus

inputs in a given domain are pooled and their running average

determines the level of stimulation to which the person is adapted.

Alternatively, if the underlying nonlinear dynamic model has validity,

then subcategorical sequential acoustic change, not simply perceived

repetition, enhances hysteresis. Results confirmed overwhelmingly that

only sequential acoustic change increases the frequency of hysteresis
Speech Perception Dynamics

369

(Figure 8.3), a result that was later shown to generalize to the

perception of directional pitch (Giangrande, Tuller, & Kelso, 2003).

0

20

40

60

80

100

critical

boundary

% Sequences Across Subjects

contrast hysteresis

Systematic vs. Random Order

Figure 8.3. Comparison of switching behavior in sequences that contain systematic acoustic

change with matched sequences that instead repeat the end-point category (see text). Percent

of sequence pairs perceived as switching at the same stimulus (critical boundary; white bar),

systematic stimulus change switching earlier than random change (contrast; gray bar), and

systematic stimulus change switching later than random change (hysteresis; black bar).

Yet another test of the model’s predictions began to address the

role of learning and experience. Recall that enhanced experience (of

which stimulus repetition is one example) causes the potential to

change more rapidly. Minimizing learning and experience should lead

to a majority of hysteresis response patterns; contrast should occur

much less often. To evaluate this prediction, we presented subjects with
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370

a single run of the “say”-“stay” continuum with gap duration first

increasing from 0-76 ms then decreasing back to 0 ms. Another group

of subjects was presented with a single run of stimuli that began at 76

ms gap duration, decreased in 4-ms steps to no gap, then increased

back to 76 ms gap duration. The task was to identify each stimulus as

"say" or "stay." A subject's pattern of responding (hysteresis, critical

boundary, or contrast) was determined by comparing the gap duration

at which the perceptual switch occurred in the increasing vs.

decreasing portion of a run. Results confirm that when experience with

the stimuli is minimized, the proportion of hysteretic responses is far

greater than either contrast or critical boundary. In fact, hysteresis is

over 3 times more prevalent than any other response pattern and is

independent of the direction of change in gap duration. When the first

trial for each subject from Tuller et al. (1994) and Case et al. (1995) is

examined, results are statistically identical to those obtained when

subjects were presented with only a single trial (Figure 8.4).

Obviously, these experiments consider only a very restricted

definition of “phonological learning” in adults. Typically, when adults

attempt to learn new speech sounds, they do so in the context of the

phonology of their native language. From the perspective we have

been taking, it makes sense to think of perceptual space as a dynamical

system that is modified by learning. In other words, learning a new

phonological category (when a range of acoustic objects acquires a

common meaning) is viewed as the creation of an attractor that

modifies the existing dynamics. This allows us to predict how learning

will proceed, depending on how the stimuli are initially perceived by

the individual. In non-speech perceptuomotor tasks, evidence that
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371

Initial Gap: 0 ms 76 ms

Averaged

across initial

gap

TCDK

 CTDK

 1st trial only

1-trial experiment

0

20

40

60

80

100

% sequences across Ss

100

 80

 60

 40

 20

 0

Figure 8.4. Percent of sequences perceived as having a critical boundary (white bar), contrast

(gray bar), and hysteresis (black bar) when for only one trial per subject, or the first trial per

subject. TCDK: Tuller et al. (1994). CTDK: Case et al. (1995).

learning consists of the interaction between pre-existing constraints

that the subject brings into the learning situation and the behavior to be

learned has been provided by Schöner and Kelso (1988; see also

Schöner, Zanone, & Kelso, 1992). In their model, behavioral information

(such as the task to be learned) acts as a parameter of the attractor

dynamics, attracting behavior toward the required behavior. When the

former does not correspond to a stable attractor of the existing,

intrinsic dynamics, learning is predicted to take the form of a phase

transition: A new behavioral attractor is found that alters the entire

dynamics. When the required task is close to, or coincides with, an

existing stable pattern, cooperative mechanisms ensure that learning
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372

will proceed rapidly and smoothly (Zanone & Kelso, 1992; 1994; 1997;

see also Kelso, 1990).

How might these ideas impact upon the acquisition of new

phonological categories that a person has never used? If a listener

initially can perceive a non-native sound as "different" from a native

one, although perhaps still acceptable as an exemplar of the native

category, the existing perceptual landscape cooperates with the sound

to be learned. Operationally, the rate of change of the landscape to

include the sound to be learned, the progressive stabilization of the

new sound, should be relatively smooth and fast. In contrast, if a

listener initially perceives the non-native sound as indistinguishable

from a native one, then learning to recognize the non-native sound

competes with the existing perceptual organization. In this case, the

strength of the attraction of the to-be-learned sound increases until a

qualitative change (a bifurcation, or phase transition) reflects the

emergence of a new attractor. The rate of change of the perceptual

space to the new sound should be slower than when the initial

perceptual landscape cooperates with the new sound. In addition,

because this competition entails destabilization of the existing attractor,

the bifurcation should be marked by high variability.

In order to test these ideas, it is necessary to modify the standard

experimental techniques used in phonological learning tasks in two

ways. First, it is not sufficiently informative simply to note whether

learning occurs with a particular stimulus set and training régime.

Observations of the changes in each listener’s behavior as learning

proceeds must supplement measures of whether the trained distinction

was finally learned to some criterion. Second, the focus of analysis must
Speech Perception Dynamics

373

be the individual, not the language. As an example, consider Iverson

and Kuhl's (1996) investigation of native English speakers' perception of

English /r/ and /l/ in which multidimensional scaling analyses of

individual listener's similarity ratings of stimulus pairs revealed that the

warping of perceptual space corresponded best to the listener's own

identification patterns. Similarly, Aaltonen, Eerola, Hellström,

Uusipaikka, and Lang (1997) showed individual differences in mismatch

negativity EEG patterns depending on how the subject categorized the

stimulus sequence. In other words, perceptual learning as a result of

language training must be assessed relative to the individual's

perceptual space as it exists before training begins. To do this,

appropriate probes, or maps, of the latter should be conducted prior

to, and during, the learning process.

In a doctoral thesis that embodied these attributes, Case (1996)

used the voiced Hindi dental stop consonant /d/, which is acoustically

similar to the American English alveolar stop consonant /d/, as the

category to be learned. The major articulatory distinction between

these two sounds is in place of articulation—in /d/ the tongue tip is

placed against the upper front teeth, and in /d/, the tongue tip is

against the alveolar ridge. There is no phonemic contrast between the

dental and alveolar place of articulation in either Hindi or American

English, although it is contrastive in at least a half dozen languages

(including Malayalam and several Australian and African languages;

Jongman, Blumstein, & Lahiri, 1985).

Here I will concentrate on the following questions: What are the

dynamics of the learning process itself? Does the form that learning

takes depend on the relationship between the sounds to be learned and
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374

how the individual initially perceives them? What are the effects of

learning a new speech sound on an acoustically/articulatorily close

native speech sound? That is, does an individual's phonetic system

reorganize during learning by modifying native categories (e.g., Flege,

1995)?

To answer these questions, we used a “perceptual mapping”

procedure that included three different tasks (identification, judged

goodness, and difference ratings). These tasks together allow a more

complete assessment of each listener's perceptual space than use of

any of the tasks alone. Each of the tasks taps somewhat different aspects

of speech perception. Identification tasks encourage phonetic coding,

and a variable stimulus context that includes different speakers,

utterances, and phonetic contexts facilitates robust category formation

with training (Lively, Logan, & Pisoni, 1993; Pisoni & Lively, 1995). The

judged goodness task examines the internal structure of a category in a

way that an identification task obscures, allowing the listener to

determine how good an exemplar of a category a given stimulus is and

focusing attention on differences among stimuli. Data from the

difference-rating task allow one to investigate the internal structure of

one or more categories simultaneously. Incorporating the results of all

three tasks gives a fuller picture of how a given listener perceives the

stimuli.

A group of monolingual American English listeners first

completed the three-task perceptual mapping procedure and then

participated in a 15-session training program distributed over a threeweek period. Their progress was monitored throughout training.

Following training, the perceptual mapping procedure was repeated.
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Pre-training/post-training comparisons as well as daily assessments

during the training process were performed to assess whether learning

occurred and, if so, to reveal its dynamics. Persistence of learning was

evaluated by follow-up testing administered a few weeks after the

training was completed. This methodology stems from the scanning

probes of the dynamics employed during the learning process by

Zanone and Kelso (1992, 1997) in order to understand how, in their

case, pre-existing coordination tendencies were modified by

practicing a new skill.

The training stimuli, a list of /CV/ syllables and /αCV/

disyllables, were produced by four native speakers of Hindi (H) and

two native speakers of American English (AE). The consonant was

either /d/ or /d/ and the vowels were those in "hot," "heat," "hoot," and

"hut." Hindi speakers were instructed in the production of the alveolar

stop, and AE speakers were instructed in the production of the dental

stop. Three native speakers of AE rated all intended alveolar

productions and three native speakers of H rated all intended dental

productions. Only productions judged to be acceptable by all native

listeners were used in training. The final training set was acoustically

diverse in that it included 3 tokens each of the 16 different syllables (8

dental, 8 alveolar) from four H speakers and two AE speakers.

The test stimuli were a synthetic continuum of eleven syllables

with an initial stop consonant followed by the vowel /α/. The consonant

spanned a range from the Hindi dental /d/ to the American English

alveolar /d/ by manipulating the second (F2) and third (F3) formant

onset frequencies. Hindi listeners judged stimuli from the dental end of
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376

the continuum to be better exemplars of their native category than

stimuli from the alveolar end of the continuum.

Monolingual speakers of American English (AE) participated in

two pre-training sessions of about one hour each. In the first session,

they performed the judged goodness and identification tasks. In the

second session, they performed the difference-rating task. For the

judged goodness procedure, subjects were presented with a

randomized set of ten tokens each of the eleven unique synthetic

stimuli. The task was to rate from 1 to 7 (poorest to best) how good an

exemplar of /d/ the stimulus was.

For the identification task, subjects were presented with a

differently randomized set of ten tokens each of the eleven stimuli.

Subjects were told that stimuli would be either a synthesized version of

an American English alveolar /d/ or a Hindi dental /d/. Differences in

how the two sounds are produced were described and examples of the

endpoint stimuli from the continuum representing the two sounds were

presented. The two-alternative forced-choice task was to identify the

stimulus as either alveolar or dental.

In the difference rating task subjects heard all possible pairs of

stimuli from a 6-stimulus subset of the continuum (stimuli 1, 3, 5, 7, 9,

and 11). Pairs were rated on a scale from 1 to 7, with 1 being “exactly

the same” and 7 being “most different.”

After the initial perceptual mapping subjects participated in 15

training sessions within a 3-week period, a second perceptual mapping

just after training, and another mapping at least two weeks later. Each

daily training session consisted of (in order) an initial free exploration

period, a two-alternative forced-choice identification task (with
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377

feedback) for a training set of 48 natural speech stimuli randomly

chosen from the full set of 288 natural speech stimuli, the difference

rating test, an identification task with feedback for a different 48-item

subset of the natural speech stimuli, and a second difference rating test

with a new randomization of stimulus pairs. If subjects had not been

paid for participating I doubt anyone would have completed the

experiment!

Although every subject showed some improvement in

differentiating dental from alveolar stop consonants in natural speech,

in what follows, I will discuss two subjects’ learning patterns in order to

address the questions posed above.

In the pre-training identification task with voiced stimuli, our first

learner showed some ability to identify the four extreme dental-end

stimuli as dental (Figure 8.5). Nevertheless, he still rated all stimuli as

relatively good members of the alveolar category (Figure 8.6). These

results are intriguing in that stimuli consistently identified as dental

were still judged as relatively good alveolars. This underscores not

only the poverty of using only a single measure of an individual's

phonetic perception but also the flexibility of perception.

In both the post-training and follow-up identification tasks, the

identification functions partition the stimuli into two clear categories

with more stimuli now being identified as dental (Figure 8.5). In

contrast to the pre-training mapping, however, stimuli on the dental

end of the continuum are now judged to be poor exemplars of the

alveolar category and the stimulus judged as the “best” alveolar moves

toward the alveolar end of the continuum (Figure 8.6).
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378

Learner #1: Identification

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11

Stimulus

Percent Alveolar Identification

Pre-training Post-training 3 weeks after training

Figure 8.5. Identification functions pre-training (solid line), post-training (dotted line), and

three weeks after training (dashed line) for learner #1.

Multidimensional scaling (MDS) analyses based on the

difference ratings were also calculated. MDS is a technique used to

uncover and visualize proximities in a low dimensional space and is

strongly related to methods such as principal component analysis and

cluster analysis. Although in many perceptual studies order of

presentation of stimuli in a pair is presumed to have no effect
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379

Stimulus

Mean Judged

Goodness as Alveolar

Figure 8.6. Mean judged goodness (error bars indicate one standard deviation) as an

exemplar of the alveolar /d/. Pre-training (solid line), post-training (dotted line), and three

weeks after training (dashed line) for Learner #1.

(Schiffman, Reynolds, & Young, 1981), our earlier data suggested that

order of pair elements might indeed influence difference ratings (in

other words, the initial condition, or initial categorization, matters). In

the pre-training data, when the first stimulus in a pair is identified as the

subject's native category, stimuli that are acoustically closest to the best

exemplar are attracted or pulled in; dental-end stimuli cluster

separately from the alveolar-end stimuli. When the acoustically more

dental stimulus is presented first, there is little if any evidence of

stimulus grouping before training. In the post-training and follow-up

testing, the dental-first pairs also show an attractive effect, although the

effect is still weaker than that observed for the pairs in which the native

sound, the alveolar, is presented first. When the day-to-day variability
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380

of the MDS solutions is calculated, total variability is relatively low from

the beginning of training and quickly decreases over the first six days,

remaining low thereafter. The initially higher variability in the total is

exclusively due to the degree of clustering across the alveolar-first

pairs (Figure 8.7).

Learner #1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Day

sd (alveolar 1st)

Figure 8.7. Total variability in the MDS analysis, as a function of day of training.

Our second learner showed a very different initial perceptual

mapping from learner #1, and a markedly different pattern of learning

over time. Pre-training, only stimuli 7 and 8 are identified at levels

different from chance (both as alveolar; Figure 8.8) and stimulus 8 is

judged as the "best" alveolar (although all stimuli were judged as

acceptable members of the alveolar category; Figure 8.9). After

training and in follow-up testing, this subject’s identification functions

showed clear categorization of the stimuli into alveolar and dental, with
Speech Perception Dynamics

381

stimuli on the alveolar end of the continuum now judged to be better

exemplars of the alveolar category than stimuli from the dental end.

Stimulus 11, judged the best alveolar after training, was also judged a

better alveolar than before training (Figures 8.8 and 8.9).

Learner #2: Identification

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11

Stimulus

Percent Alveolar Identification

Pre-training Post-training 3 weeks after training

Figure 8.8. Identification functions pre-training (solid line), post-training (dotted line), and

three weeks after training (dashed line) for Learner #2.
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382

Stimulus

Mean Judged

Goodness as Alveolar

Figure 8.9. Mean judged goodness (error bars indicate one standard deviation)as an exemplar

of the alveolar /d/. Pre-training (solid line), post-training (dotted line), and three weeks after

training (dashed line) for Learner #2.

The MDS analyses based on difference ratings (taking order into

account) revealed that the pre-training solution does not respect

acoustic ordering, consistent with the initial identification results. By the

time of the post-training evaluation, difference ratings of the alveolarfirst pairs showed a tight clustering of stimuli into two groups

corresponding to alveolars and dentals; dental-first pairs also grouped,

although somewhat more weakly. Grouping of stimuli was tighter in the

follow-up as well, with less of an order effect. The total variability in the

MDS solutions is shown in Figure 8.10. Total variability was initially

much higher than for learner #1 and showed a steady decline until,
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383

around Day 5, an increase in variability occurred through Day 9. This

increase preceded a sharp drop in total variability at Day 10 to levels

equivalent to those observed for learner #1. Note that the peak in

variability in judging the alveolar-first pairs may be interpreted as a

destabilization of the attractor corresponding to the alveolar category.

Learner #2 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Day

sd (alveolar 1st)

Figure 8.10. Total variability in the MDS analysis, as a function of day of training.

To summarize, learner #1 showed an initial ability to distinguish

some of the dental-end stimuli from the alveolar, even though they

were still acceptable as alveolars. The pre-, post-, and follow-up test

results all indicate a smooth and rapid learning process occurring over

the first six days of training and the decrease in variability in his MDS

profile was smooth and fast. This is congruent with the initial prediction:

If a listener initially can perceive a non-native sound as "different" from
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384

a native one, the existing perceptual landscape cooperates with the

sound-to-be-learned and learning should be relatively smooth and fast.

This pattern is consistent with the idea of progressively stabilizing an

already existing stable pattern.

Learner #2 showed little evidence for an initial ability to hear

dental-end stimuli as different from alveolar-end stimuli. Variability of

the MDS solutions also began at a level nearly three times greater than

initial variability for learner #1, and the rate of contraction of the stimuli

into groups was slower than for learner #1. After the variability began

to decrease, it reversed direction and peaked again just prior to

reliable clustering of the MDS solutions. This local increase in

variability occurred almost exclusively in alveolar-first pairs and can

be considered analogous to critical fluctuations that often precede

bifurcations (Schöner, Haken, & Kelso, 1986). Again, these results were

congruent with predictions: If a listener initially perceives the nonnative sound as indistinguishable from a native one, then learning to

recognize the non-native sound competes with the existing perceptual

organization. This process is slower than when the initial perceptual

landscape cooperates with the new sound, and because this

competition entails destabilization of the existing attractor, the

bifurcation is marked by high variability.

One aspect of the data that has not yet been highlighted is that

learning the non-native category modified perception of the native one

(cf. Flege, 1992, 1995), especially for listeners who did not initially

parse the stimulus continuum. After learning, not only did the stimulus

judged as the best alveolar exemplar shift away from the dental group,

but the best exemplar was also a better exemplar post-training than
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385

pre-training. Thus the pre-existing phonological organization is

malleable. Learning does not entail simply an addition of a new

category but in fact changes the existing attractor layout (see also

Sancier & Fowler, 1997).

In the cognitive, behavioral, and brain sciences, large strides

have been made in understanding pattern formation using the concepts

of self-organization and the mathematical tools of nonlinear dynamical

systems (e.g., see Haken & Stadler, 1990, for a variety of different

contributions in this context; Kelso, 1995). Explicitly dynamical

investigations of speech include attempts to identify phonological units

with dynamically specified gestures (Browman & Goldstein, 1986, 1989,

1992; Kelso, Saltzman, & Tuller, 1986; Kelso, Tuller, & Harris, 1983), to

construct a topology of vowels (Wildgen, 1990) and consonants (PetitotCocorda, 1985) in terms of a landscape of attractors and repellers

within an articulatory or acoustic space, and to model the phonological

system of artificial languages as a self-organized solution of talkerbased and listener-based constraints (Lindblom, MacNeilage, &

Studdert-Kennedy, 1983). In our own work (Tuller et al., 1994; Case et

al., 1995; Tuller, 2003), we demonstrated that changes in perception

that occur as the acoustic signal is altered are indicative of a patternformation process in perception. A model of the results was proposed

and unique predictions of the model were tested and confirmed.

The approach also provides a theoretically motivated way to

understand the process of learning to perceive non-native speech

sounds (and perhaps the emergence of categories in development).

Fundamental to this approach is a methodological stance: Instead of

studying features of objectively existing prototypes (either as abstract
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linguistic entities or as stored multiple exemplars) in a group of

listeners, focus on the interaction of an individual perceiver with

speech stimuli in context. In this way, we have observed changing

patterns of categorization that parallel those observed in perceptuomotor learning (Kelso, 1990; Kelso & Zanone, in press; Schöner,

Zanone, & Kelso, 1992; Zanone & Kelso, 1992, 1994, 1997) and are

consistent with the notion that reliably categorizing a new speech

sound depends on whether the new category cooperates or competes

with an individual's initial perceptual capabilities and that learning

serves to reorganize the perceptual space.

In summary, I have described a program of research in which

the tenets of dynamical systems and empirical research on speech are

mutually informative and directive. In this, I have followed the basic

strategy identified by Kelso (in press), but applied to the study of

speech perception. This strategy entails (1) Choosing a level of analysis

and description that captures the behavior you are studying. (So if I’m

interested in how people learn to change their perceptual

categorization of speech, it would not be fruitful to choose to describe

the behavior in terms of the phasing of harmonics in the signal.); (2)

Prune away complications so that the essence of your question remains

foremost in the experimental design; (3) Focus on finding the

conditions that yield qualitative changes in behavior. Qualitative

change allows one to define the perceptual categories clearly as well

as to exploit the patterns of change as a key to the mechanisms

underlying pattern formation (e.g., dynamic instability); and (4)

Explore both the coordinative and the component levels as well as the

relation between them. How one defines the coordinative level and
Speech Perception Dynamics

387

“one level down” depends on the experimenter’s insights into step

(1)—choosing the level of description. This last step, deriving the

coordinative level dynamics from the usually nonlinear coupling

among individual components, is as yet the weakest link in

understanding the self-organizing nature of speech dynamics.

Finally, the empirical and modeling strategy described here is

both speech-specific and generalizable. The approach has also been

fruitfully applied to the verbal transformation effect (Ditzinger, Tuller,

Haken, & Kelso, 1997; Ditzinger, Tuller, & Kelso, 1997) and more

recently, auditory streaming (Almonte, Jirsa, Large, & Tuller,

submitted). It also shares much with studies of the effects of attention on

behavioral patterns (e.g., Temprado, Zanone, Monno, & Laurent, 1999),

and with studies of learning from behavioral, theoretical, and

neurophysiological perspectives (Jantzen, Fuchs, Mayville, & Kelso,

2001; Kelso & Zanone, in press; Kelso, 1995; Schöner, Zanone, & Kelso,

1992; Sporns & Edelman, 1993; Zanone & Kelso, 1992, 1994, 1997). More

recently, neural correlates of the stability and change of behavioral

coordination have been uncovered using several methods that reveal

brain function, such as high density SQuID, multichannel EEG, and

functional MRI and PET (Daffertshofer, Peper, & Beek, 2000; Frank,

Daffertshofer, Peper, Beek, & Haken, 2000; Fuchs, Jirsa, & Kelso, 2000;

Fuchs, Kelso, & Haken, 1992; Fuchs, Mayville, Cheyne, Weinberg,

Deecke, & Kelso, 2000; Kelso, Bressler, Buchanan, DeGuzman, Ding,

Fuchs, & Holroyd, 1992; Kelso, Fuchs, Holroyd, Lancaster, Cheyne, &

Weinberg, 1998; Mayville, Bressler, Fuchs, & Kelso, 1999; Mayville,

Fuchs, Ding, Cheyne, Deecke, & Kelso, 2001; Meyer-Lindenberg,

Ziemann, Hajak, Cohen, & Berman, 2002; Ullen, Ehrsson, & Forssberg,
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388

2000; Wallenstein, Kelso, & Bressler, 1995). Behavioral investigations

have been spurred by, and have spawned, theoretical work at the

neural level (Fuchs & Jirsa, 2000; Haken, Kelso, & Bunz, 1985; Jirsa, Fink,

Foo, & Kelso, 2000; Jirsa, Friedrich, Haken, & Kelso, 1994; Jirsa & Haken,

1996, 1997; Schöner, Haken, & Kelso, 1986; Schöner, Jiang, & Kelso,

1990; Treffner & Turvey, 1996) that is rapidly becoming more

neurobiologically grounded (Frank et al., 2000; Fuchs et al., 2000; Jirsa,

Fuchs, & Kelso, 1998; Jirsa & Haken, 1997).

Despite this wealth of information concerning the dynamics of

behavior, the specific boundary conditions and control parameters that

establish the context for speech phenomena, the coordinative and

component levels that makes sense in speech, are specific to speech

and must be identified within the speech context. “Dynamics” in and of

itself will not give us the answers—it must be fleshed out for each

system under study with conceptual content and implementation via

experiment, simulation, modeling, and theory development.
Speech Perception Dynamics

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