Evaluating Indices That Measure 

Departure of Current Landscape 

Composition from Historical Conditions

Robert E. Keane, Lisa Holsinger, and Russell A. Parsons

United States Department of Agriculture / Forest Service

Rocky Mountain Research Station

Research Paper RMRS-RP-83

February 2011
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Acknowledgments

This project was supported, in part, by funding from USDA Forest Service Fire and 

Aviation Management Fire Modeling Institute and the LANDFIRE prototype project. We thank 

Issac Grenfell for his assistance in programming key statistical algorithms and two anonymous 

reviewers for their comments.

Keane, Robert E; Holsinger, Lisa; Parsons, Russell A. 2011. Evaluating indices that measure 

departure of current landscape composition from historical conditions. Res. Pap. RMRSRP-83. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, Rocky Mountain 

Research Station. 19 p.

Abstract

A measure of the degree of departure of a landscape from its range of historical conditions can 

provide a means for prioritizing and planning areas for restoration treatments. There are few statistics 

or indices that provide a quantitative context for measuring departure across landscapes. This study 

evaluated a set of five similarity indices commonly used in vegetation community ecology (Sorenson’s 

Index, Chord Distance, Morisita’s Index, Euclidean Distance, and Similarity Ratio) for application in 

estimating landscape departure (where departure = 1 – similarity). This involved comparing composition (vegetation type by area) of a set of reference landscapes to the compositions of 1,000 simulated 

historical landscapes. Stochastic simulation modeling was used to create a diverse set of synthetic 

reference and historical landscapes for departure index evaluation. Five reference landscapes were 

created to represent various degrees of expected departure from historical conditions. Both reference 

and historical landscapes were created to contain four important factors that could potentially influence departure calculation: (1) number of classes defining landscape composition, (2) dominance of 

the classes, (3) variability of area with the classes, and (4) temporal autocorrelation. We found that 

most evaluated indices are useful but not optimal for calculating departure. The Sorenson’s Index 

appeared to perform the best with consistent and precise behavior across the ranges of the four 

treatments. The number of classes used to describe vegetation had the strongest influence on index 

performance; landscape composition defined by few classes had the least accurate, most imprecise, 

and most highly variable departure estimates. While results from this study show the utility of similarity 

indices in evaluating departure, it is also evident that a new set of statistics are needed to provide a 

more comprehensive analysis of departure for future applications.

Keywords: ecosystem management, similarity indices, landscape ecology, historical ecology, 

historical range and variability, HRV, neutral landscapes
Research Summary

The relatively new concept of Historical Range and 

Variability (HRV) provides a spatial and temporal foundation for developing benchmark conditions for evaluating 

difference in today’s landscape, and HRV also brings an 

understanding of past spatial and temporal ecological 

variability into ecosystem management. However, an 

important limitation in the application of HRV is the lack 

of comprehensive analysis techniques to compare the 

multi-observational historical time series landscape data 

with the one observation of contemporary conditions. A 

comprehensive measure is needed for computing departure of current landscape conditions from the historical 

conditions. 

The objective of this study was to evaluate a set of five 

similarity indices commonly used in vegetation community 

ecology (Sorenson’s Index, Chord Distance, Morisita’s 

Index, Euclidean Distance, and Similarity Ratio) for application in estimating landscape departure. This involved 

comparing landscape composition (vegetation type by 

area) of reference landscapes that represents current 

conditions to the compositions of 1,000 simulated historical neutral landscapes. Stochastic simulation modeling 

was used to create a diverse set of synthetic reference 

and historical landscapes for departure index evaluation. 

Five reference landscapes were created to represent 

various degrees of expected departure from historical 

conditions. Both reference and historical landscapes 

were created to reflect four important factors that may 

potentially influence departure calculation: (1) number of 

classes defining landscape composition, (2) dominance 

of the classes, (3) variability of area with the classes, and 

(4) temporal autocorrelation. 

We found that:

•	 Most similarity indices are useful but not optimal 

for calculating departure. 

•	 The Sorenson’s Index appeared to perform the 

best for HRV comparisons with consistent and 

precise behavior across the ranges of the four 

treatments. 

•	 The number of classes had the strongest influence 

on performance of most indices with landscapes 

that were described by few vegetation classes 

had the least accurate, most imprecise, and most 

highly variable departure estimates. 

•	 Temporal autocorrelation had the least effect on 

the behavior of the departure index. 

Results from this study show that similarity indices, 

especially Sorensen’s Index, can be used to evaluate 

departure, but it is also evident that a new set of statistics 

are needed to provide a more comprehensive analysis 

of departure for future applications.

i

Contents

Research Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Generation of autocorrelated series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Series modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

The Simulation Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Historical time series factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Reference landscapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

Time Series Synthesis Method YTY Base Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Other Time Series Synthesis Methods (AAY, MMY, QTY) . . . . . . . . . . . . . . . . . . . . 10

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

iii

USDA Forest Service Res. Pap. RMRS-RP-83. 2011 1

Introduction

	 The era of ecosystem management ushered in the

concept that land should be managed as a whole by

considering all organisms, and their pattern, abundance,

and connectivity of their habitats, and the ecological

processes that influence these organisms on the landscape (Bourgeron and Jensen 1994;Crowand Gustafson

1997; Kaufmann and others 1994). But, to effectively

plan, design, and implement ecosystem management

treatments,managersrequired benchmarks orreference

conditions to fully represent functional ecosystems or

landscapes(Cissel and others 1994; Laughlin and others

2004;Swanson and others 1994).Today’slandscape conditions can be evaluated against benchmark conditions

to determine status, detect change, and design effective

treatmentstoprovidesocietywithsustainableandvaluable

resources while also returning declining ecosystems to a

morehealthyandresilient condition(EganandHowell2001;

Hessburg and others 1999; Swetnam and others 1999). It is

also critical that these benchmark conditions represent

the dynamic character of ecosystems and landscape as

they vary over time and space (Morgan and others 1994;

Swanson and others 1994).

	 The relatively new concept of Historical Range and

Variability (HRV) was introduced to provide a spatial

and temporal foundation for developing benchmark

conditions and to bring understanding of past spatial

and temporal ecological variability into ecosystem

management (Landres and others 1999; Swetnam and

others 1999). HRV can be used to plan and implement

possible treatments to improve ecosystem health and

integrity by assuming thatrecent historical variation representsthe broad envelope of conditionsthatsupportsresilient

and self-organizing landscapes (Harrod and others 1999;

Hessburgandothers1999). One major use of HRV in land

management involves comparing a time series of historical landscape conditions(benchmark) to the current

landscape condition to detect change and trend (Keane

and others 2009). While easily understood, the concept

of HRV can be difficult to implement due to scale, data,

and analysis limitations (Keane and others 2009; Wong

and Iverson 2004). Another important limitation in the

application ofHRVisthe lack of comprehensive analysis

techniquesto compare themulti-observational historical

time series landscape data with the one observation of

contemporary landscape conditions (Keane and others

2009; Steele and others 2006).

	 There are few statistical or ecological analysis techniquesthat perform well acrossthe temporal and spatial

scales ofmostHRVlandscape analyses(Landres andothers 1999).ManyHRVstudies compare landscape composition (vegetation type by area) using indices of departure

that are comparable to the similarity indices commonly

employed in community ecology to qualitatively evaluate

differences in species composition across sample plots

(Huhta 1979; Mueller-Dombois and Ellenberg 1974;

Wolda 1981). Similarity is often computed as an index

or number between zero (dissimilar) and one (identical),

and in this paper, we define the converse ofsimilarity as

departure (in other words, departure = 1 – similarity).

Sorenson’sindex,for example, was used by Hann (2004)

as a measure to compute landscape departure and fire

regime condition class and by Keane and others (2008)

to determine changes in landscape composition HRV

caused by climate change. Holsinger and others (2006)

used a variation of the Sorenson’s index to compute

departure for the LANDFIRE project. Alternatively,

Steele and others (2006) used a statistical regression

technique to compute a landscape departure measure

with a test of significance, but this measure did not

perform well across large regions because the range of

departure statistics was quite small making it difficult

to identify subtle differences.

	 Community ecology-based indices of departure have

been used for HRV analyses, but several problems have

Evaluating Indices That Measure 

Departure of Current Landscape 

Composition from Historical Conditions

Robert E. Keane, Lisa Holsinger, and Russell A. Parsons
2 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

been encountered. Keane and others (2008) found that

departure measures based on similarity indices are

highly influenced by the number of classes used to describe landscape composition, and they are insensitive

to subtle, yet important, changes in landscape composition. Major landscape changes can occur, but important

disturbance effects were offset by successional changes

over space, thereby resulting in minor changes in the

departure statistic. Moreover, there are ranges of the

departure index (close to zero and one) that are rarely

encountered. The inherent variability of classes in the

time series, along with their temporal autocorrelation

(degree to which previous landscape conditions influence future landscape conditions), can also influence

the comparison of departure measures.

	 The challenge, then, is to identify a comprehensive

measure for computing departure of current landscape

conditions from the historical time series of past conditions. This measure should perform well regardless of

how many mapping units (landscape classes) are used,

how the area is distributed acrossthese classes, and how

correlated these classes are across time and space. This

is complicated by the fact that land managers desire

simple measures that are easy to calculate, understand,

and interpret, yet are still scientifically credible for

describing departure from historical conditions (Hann

2004). The objectives of this study were to evaluate

the performance of the community ecology similarity

indices for computing departure from historical conditions and to examine how these metrics are influenced

by various landscape characteristics. Ultimately, the

goal of this study was to determine an optimum measure of departure that can be used in HRV applications

for mapping departure across large regions (Steele and

others 2006). This study did not attempt to develop new

departure indices;instead,it evaluatedexisting similarity

measures developed in the field of community ecology

(Mueller-Dombois and Ellenberg 1974; Wolda 1981).

Because of this, the findings from this study could also

be applied to vegetation classification and community

similarity analysis.

Background

	 In this study, the landscape is an area of any size or

resolution that is composed of unique areas of homogeneous ecological conditions. Each unique area is assigned a class or category that describes a compositional

characteristic that is usually specific to the objective of

the landscapemapping effort.The class often represents

a species dominance classification category, such as a

covertype.Landscape compositionisthe areaorpercent

area covered by each class across the landscape. HRV

analysis involves the comparison of one observation of

the landscape composition (area by class) for the current

landscape with the historical landscape time series 

represented by numerous sequential observations of

historical landscape compositions (fig. 1). In this study,

we refer to the reference landscape as a landscape

that represents current conditions, but we have altered

these current conditions to represent various degrees of

departure. Departure is calculated by comparing the

areas occupied by the landscape composition classes

common to both the reference landscape averaged across

all the instances in the historical landscape time series.

The term departure index refers to the metric used to

calculate departure from historical and it is typically

the opposite of similarity.

	 Manytypesofsimilarityindices areusedincommunity

ecological analyses with mixed success (Bloom 1981;

Faithandothers1987;Kobayashi1987;Mueller-Dombois

Figure 1—An illustration of the HRV for landscape composition for a 50,000 ha landscape in southwestern Montana 

(Keane and others 2008) showing an example of the historical 

range of landscape composition on a Montana USA landscape 

using five cover types, the average proportion area occupied 

by each of the five types over the simulation run, and the 

composition of the current landscape, which contains only 

open tall Douglas-fir. 
USDA Forest Service Res. Pap. RMRS-RP-83. 2011 3

andEllenberg 1974;Orloci 1967).We assessed a number

of common similarity indices (see “Methods” section)

and selected five commonly used ecological indices to

evaluate in this study. These indices have been used in

many vegetation classification and plant diversity studies

(Gauch 1982; Ghent 1963; Wolda 1981):

1. Sorenson’s Index (SI) is calculated using the following formula:

i i

i i

h c

Min h c

SI

2 ( , ) (1)

where hi refers to the area (m2

) of the historical

reference class within the landscape, ci is the area

(m2) of the same class on the current landscape, i

is class index number, and Min is a function that

picks the minimum of two values. Since SI is an

index of similarity, it was inversed as SI = 1 – SI

to obtain departure.

2. Morisita’s Index (MI) is modified from Horn

(1966) and is calculatedwith the following formula:

2 2

2 ( )( )

i i

i i

h c

h c

MI (2)

where hi refers to the area (m2

) of the historical

reference class i within the landscape, ci is the area

(m2) ofthe same class on the current landscape, and

i is class index number. Again, MI is actually an

index ofsimilarity so it wasinversed as MI = 1 – MI

to obtain the MI departure index.

3. Similarity Ratio (SR) emphasizes the differences

using the square of the products in the following

formula:

i i i i

i i

h c h c

h c

SR 2 2 (3)

where hici is the product of the areas of the historical and current landscape for class i. Again,

we subtracted the SR from 1.0 to get a departure.

4. Chord Distance (CD) is calculated with the following formula from Faith and others (1987):

s

i s

k

ik

i

s

j

ij

i

c

c

h

h CD

1

2

1

2

1

2

(4)

where i, j, k are the classindex numbers.We rescaled

the CD range from zero to √2 to zero to 1.0, and

then subtracted from 1.0 to get departure.

5. Euclidean Distance (ED)ismuch like the SRusing

the squares of the historical and current landscape

areas but the index emphasizes the differences

between the two areas as taken from Ludwig and

Reynolds (1988):

2 2

2

( )

i i

i i

h c

h c

ED (5)

	 This paper deals only with departure in landscape

composition (area by landscape class) at the landscape

level, but it is important to mention that departure can

be calculated for many attributes across many scales.

Landscape structural attributes,such as contagion, patch

density, and largest patch index, can also be used as the

response variablesto determine departure.It is also possible to determine departure for stand attributes such as

basal area, tree density, and timber volume. However,

this paper deals only with landscape composition because it is used most by managers (Hann 2004; Keane

and others 2006) and it is a complex landscape attribute

composed of many descriptor variables (area by class)

whose values all sum to the area of the landscape.

Methods

	 We explored the suitability of the above measures for

evaluating departure using simulation modeling where

synthetic historical time series and synthetic reference

conditions were created for a partially nested factorial

statisticaldesignusingamodelwebuilt calledTSG(Time

Series Generator) (fig. 2). In summary, the TSG model

created the historicaltime series of 1,000 landscapes and

it also created the set of five reference conditions each

with four unique characteristics that were considered

factorsin oursimulation experimental design:(1) number

classes, (2) dominance distribution, (3) class variability,

and (4) autocorrelation. The five synthetic reference

landscapes were designed to represent increasing degrees of departure (expressed as a delta statistic) from

the historical time series. We stochastically created an

historical landscape time series of 100 landscapes and

then replicated this process ten times to obtain 1,000

observations (historical landscapes). We also reduced

the historical time series into a simplified set of maps

using three synthesis techniques. Since the area of the

landscape remains constant overtime, we relativized all

class areas to sum to 1.0. It is important to note that the
4 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

Figure 2—Tasks involved in this simulation experiment where synthetic historical time series are created using five factors using the TSG model and compared against a set of four reference conditions that represent 

degrees of departure from historical conditions. Five departure indices are used to perform the comparison. 

The behavior of these departure indices are evaluated across all factors and reference conditions. See table 

1 for symbol definitions and simulation details.

historical time series really doesn’t represent historical

conditions; we used stochastic simulation models to

create neutral landscapes with the properties that we

wish to evaluate (factors in our experimental design).

We named them “historical time series” to be consistent

with terminology in HRV analysis, but these are not

historical landscapes.

Model Description

	 TheTSGmodelwasdesignedtostochasticallygenerate

landscape composition time serieswith known statistical

properties to facilitate objective comparison and evaluation of different measures of departure. Our modeling

approach reflects a set of design criteria emphasizing

simplicity, flexibility, and computational feasibility.

Landscape compositional time series may extend for

centuries or millennia, and may exhibit temporal autocorrelation over various time scales. Depending on the

detailwithwhichthe landscape composition is described,

the number of vegetation classes over time may range

from few to many. In real ecological systems, classes

of vegetation may interact over time and space with a

diversityof biophysicalfactors(e.g., climate, disturbance,

soils) to create complex landscapes. To explicitly model

these interactions mathematically would have required

an extensive mechanistic process modeling, requiring

a large number of inputs and computational resources.

As our purpose was to compare the various measures

of departure across a wide spectrum of straightforward

comparison cases, we decided to simulate the landscape

composition time series as a set of independent series

and then assemble them into synthetic landscape time

series.

	 Each simulated landscape composition time series is

described by (1) a series length (total number of years

for which time series are to be generated), (2) lag length

(maximumlength in time overwhich lagged correlations

will be considered), and (3) number of classesthatrepresent vegetation states. These three factors were selected

based on the results of a previous study in landscape

departure (Keane and others 2008). For each class, we

specified three parameters, which, when used together,

provided a great deal of flexibility and allowed for the

generation of a wide range of landscape composition

time series across classes. The first parameter, TCORR

or temporal correlation, controls the temporal autocorrelation within the series for a class and is described by

an ACF (AutoCorrelation Factor) value thatrangesfrom

zero (no autocorrelation) to one (highly correlated in

time). The second parameter, PDOM or percent dominant, sets the target dominance, or the degree to which

a landscape is dominated by a few classes. PDOM is

defined as the expected mean proportion of total landscape area occupied by a given class over the length of
USDA Forest Service Res. Pap. RMRS-RP-83. 2011 5

the time series. Since these are proportions, the sum

of the PDOM parameter values across all classes for a

given simulated landscape composition time seriesmust

be equal to 1.0. The third parameter, CVAR or class

variation controls the amount of variability in the time

seriesfor that class and it isscaled from 0.3 to 1.2 in this

study based on analyses of other mapped landscapes.

	 The simulation of a landscape composition time series

consists of two steps: (1) the generation of autocorrelated series creates independent autocorrelated time

series for each class and (2) the series modification 

that rescales those independent classes to conform to

the PDOM and CVAR parameters, and relativizes all

classes such that they represent proportions of total

landscape area by class for each point in time.

Generation of autocorrelated series—For each

class, an independent, stationary, autocorrelated time

series is generated using the ACF parameter. The ACF

parameter sets the slope of a simple exponential decay

function of the form:

r(t) = e(–λt) (6)

where r(t) is the correlation coefficient between the

value at a given point and the value at lag time t, and λ 

is the ACF parameter. This exponential decay function

representsthe coefficientsof a theoretical autocorrelation

function (ACF). Higher values of ACF result in a lower

correlation,while lower valuesmaintain correlation over

a longer period of time. Thistheoretical autocorrelation

function isthen used to generate an autoregressive series

of the specified length following a processthat has been

used extensively since the late 1960s (Box and Jenkins

1990). Given an autocorrelation function ρ(k) = ρk for

k = 1, ... , m, an autoregression function can be obtained

in the following way:

	 Let

.. .. .. 1

.. .. .. 1 ..

.. .. 1 ..

1 ..

1 .. ..

1

1

1 1 2

1 1

m

m

m

Pm (7)

	 And let

m

m ..

1

	 And

m

m ..

1

Then the equation

φ = Pm

–1ρ m (8)

will satisfy the autoregressive process:

2

0 1 1 2 2 .. p p  a              (9)

Thus t t t m t m t z z z .. z a 1 1 2 2 where at is a

random shock with variance 2  a , in which the variance

of the output will reflect the character and structure of

the autocorrelation function used as input as follows:

σ σ a z ρ φ ρ φ ρ φm m 2 2

1 1 2 2 = −1 − −..− (10)

The autocorrelated series for a given vegetation class

(Equation 8) is then arranged as a set of linear equations that are solved through a process of Lower-Upper

decomposition followed by forward and then backward

substitution using standard numerical routines (Press

and others 2002).

Series modification—Once all of the independent

series of each class have been generated according to

the process described above, the series are modified

so they can represent landscape composition across all

vegetation classes.The valuesfor each series are rescaled

to the range [L, H], with the following relation:

Xmod = (X − m)(H − L)

(M − m) + L (11)

where X is the unmodified series values, Xmod is the

modified series, m isthe minimum value ofthe unmodified series and M is the maximum of the unmodified

series, and

L = D – DV (12)

H = D + DV (13)

where D is the PDOM input parameter and V is the

CVAR parameter. Finally, each class value for each

year is divided by the sum over all classes for that year

to ensure that they sum to 100 percent of the total area.

Simulation Experiment

	 The goal ofthissimulation experimentwasto compare

the behavior of departure indices across a wide range of

conditions of historical time series and reference landscapes with known statistical properties to determine

if any index is superior and if any of the indices have
6 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

undesirable qualities that may influence the interpretation of HRV departure results (Keane and others 2002,

2008).Initially, we wanted to take an ANOVA (Analysis

of Variance) approach and conduct a fully nested factorial experiment using extensive simulations from the

TSG model, but preliminary resultsrevealed that, while

the departure indices differed across all treatments, the

small number of treatments within a factor (less than 5

treatments) did not provide sufficient detailto effectively

evaluate the behavior of each index acrossthe full range

of possible historical and reference conditions.Therefore,

we re-designed our experiment so that a wide range of

values (treatments) for each factor were simulated with

TSG to determine critical thresholdsin departure index

behaviors. Because there were so many treatments that

spanned the full range of behaviors, it was difficult to

simulate allfactor-treatment combinationswith theTSG

model. Therefore, we simulated the range of values for

one factor while holding all other factors to a base case

(noted in underlined bold in table 1) and then used oneway ANOVA to evaluate differences among treatments

by factor.

	 We alsowanted to include a number of othersimilarity

indices in the evaluation, but results from an extensive

preliminary analysis revealed that some similarity indices had virtually identical results, while others had

obvious undesirable behaviors across a minor subset

of the important factors mentioned above. The Jaccard

Table 1—Factors evaluated for testing the response of departure indices to several historic time series and reference conditions. 

Values in italics and underlined are those held constant while varying a different factor (base condition).

Factor

Levels

1 2 3 4 5 6 7 8 9 10 11 12 13 14

HRV landscape composition factors

NCLASSnumber of 

classes

10 20 30 40 50 60 70 80 90 100 200 300 400 500

PDOM -

Percent 

dominant

10 20 30 40 50 60 70 80 90 — — — — —

TCORRACF slope 0.00005 0.0005 0.001 0.002 0.004 0.01 0.016 0.032 0.064 0.1 — — — —

CVARTime series 

variability

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 — — — —

Time series synthesis methods

TSSMTime series 

synthesis 

methoda

YTY AAY MMY QTY — — — — — — — — — —

Reference conditions

DELTAdegree of 

departure 

from 

historical

0.2 0.4 0.6 0.8 1.0 — — — — — — — — —

Departure Indices

Departure 

indicesb SI MI SR CD ED — — — — — — — — —

a

Time series synthesis methods are defined as the comparison of the reference condition to (1) all instances in the historical time series and an 

average departure is computed (YTY-year to year); (2) one synthetic historical landscape that is the average of all classes across all landscapes in 

the historical time series (AAY); (3) two historical landscapes that are the minimum and the maximum, respectively, of each class in the historical 

series (MMY); and (4) one synthetic historical landscape created from the 90% quantile of class area (QTY).

b

Departure indices are computed from the following similarity indexes: SI-Sorenson’s Index, MI-Morista’s Index, SR-Similarity Ratio, CD-Chord 

Distance, and ED-Euclidean Distance.
USDA Forest Service Res. Pap. RMRS-RP-83. 2011 7

(Mueller-Dombois and Ellenberg 1974) and Gleason

(Huhta 1979) indices, for example, performed poorly in

nearly all cases, while the FRCC(Hann 2004)index was

acceptable, but was highly correlated to the Sorenson’s

index. Therefore, we reduced the number of indices to

the set of five discussed above (table 1).

Historical time series factors—We had previously

identified four factors that commonly influence the

calculation of current landscape composition departure from historical conditions (table 1) (Keane and

others 2008). The number of classes (NCLASS) used

to describe the landscape composition (cover types,

for example) is influential in departure calculations

because major changes in departure are difficult to

detect when a large number of classes are used (Keane

and others 2008). To test this, we varied the number

of classes from 10 to 500 to encompass the full range

of possible mapping units that are used to describe

landscapes. Percent dominance (PDOM) describes

the percent of the number of classes that are dominant

in a time series. We define dominant as comprising at

least 80 percent of the landscape area. For example, 10

percent dominance for 10 classes would mean that one

class comprises 80 percent of the landscape while the

remaining 20 percent of the area isspread over the other

nine classes.

Class variability (CVAR) was used to explore the

effect of the variation within classes on departure calculations. In TSG, the variance depends on the percent

dominance (PDOM) value where the PDOM value is

multiplied by theCVARvalue (in thisstudy 0.2 <CVAR

< 1.2; table 1) to compute a range around the historical

mean (area in a class+PDOM*CVAR*area).LowCVAR

values(<0.5)indicate lowvariability in classfluctuations

across sequential landscapes, while high CVAR values

(>0.8) represent a significant increase in the noise in the

HRV time series. Temporal autocorrelation (TCORR

as represented by ACF parameter) is also an important

factor because departure may be difficult to detect in

landscape time series that are strongly autocorrelated

in time. The TSG program uses the ACF (see “Model

Description” section) to create time series with low to

high degrees of autocorrelation (0.0 to 0.3,respectively).

We simulated several landscapes with the LANDSUM

landscape simulation model (Keane and others 2006)

to identify reasonable ranges of ACF values and found

that the simulated time series ACF value ranges from

0.00002 (uncorrelated) to 0.1 (somewhat correlated)

(table 1).

	 We also included another analysisto evaluate the time 

series synthesis method (TSSM).Historical time series

of landscape compositions can be summarized in a

number of waysto simplify departure calculations. The

most obvious and commonmethod involves no synthesis

at allwhere each landscape in the time seriesis compared

to the reference, and in our simulation experiment, this

Year-to-Year (YTY) calculation computes departure

by comparing each historical map in the series to the

reference map, and then averages all departures across

every comparison in the historical time series (1,000

comparisons in this study). While the YTY approach is

the most intuitive and provides the greatest detail in the

HRV analysis, it is sometimes impractical because, in

an operational setting, it requires that digital maps for

each time step in the historical series be stored for each

landscape, requiring extensive disk space. To minimize

computer storage, simplify computational demands for

large regions, and provide a simple and easy historical

reference for managers (Hann 2004), we created three

simplified historical time series that may have application in landscape management. An average historical

year (AAY) was created by averaging the area for each

class across all instances in the historical time series to

create one average historical condition to compare with

the reference. We also reduced the time series to two

instances by taking the minimum area for each class

across all years in the series and the maximum area

for each class in the series to create two observations

in the “time series.” We then averaged departure across

both the minimum and maximum years (MMY). Last,

we took the 90 percentile of class area over all years

(QTY) to create one historical year to compare with

the reference. This study concentrated on the base case

of YTY (average across all historical combinations) for

the evaluation analysis, but we also report results on the

effectiveness of reducing historical time series.

Reference landscapes—The reference condition

in this study represents the condition of the current landscapes in the calculation of departure from

HRV. Reference conditions that are highly disparate

from the historical time series test the magnitude of

departure that the index can capture, while similar

reference and historical conditions test the sensitivity

of the index. In this study, we created a series of five

different reference conditions that spans a range of

departure from the set of historical time series conditions (fig. 2). The degree of departure is represented

by a parameter called DELTA (a number ranging from
8 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

0 to 1) that describes the proportion of area in a reference landscape class that will be shifted away from

the average historical value for that class. A value of

1.0 would mean maximum departure or 100 percent of

the area for each reference class is in a different class

than the historical time series average. Five reference

conditions were created using DELTA values from

0.2 (minimum expected departure) to 1.0 (maximum

expected departure) in 0.2 increments (Table 1).

Analysis—The response variables in this analysis

are values of the five departure indices (SI, MI, CI,

SR, EI) for each of the 10 replicates for the four factors

(NCLASS, CVAR, PDOM, TCORR) across all treatment levels per factor, compared to all five reference

(DELTA) conditions.We stratified this analysisinto two

phases. The first phase involved using the TSG program

to simulate 1,000 historicallandscapesto create the set of

ten historical time series. We ran TSG for all treatments

within a factor while holding all other factors at the base

case (table 1). We then performed ANOVA techniques

to detect differences between treatments (NCLASS,

PDOM, CVAR, TCORR) but not differences across

factors (values of treatments) for only the YTY time

series synthesis method (YTY or year to year involves

averaging departures computed by comparing a reference condition to each landscape in the ten historical

time series).In the second phase, we conducted the same

simulations across the remaining time series analysis

methods (AAY, MMY, QTY) (fig. 2). Again, ANOVA

was used to test differences across treatments.

	 As mentioned, we used ANOVA techniques to determine significance across treatments within the factors,

but index performance was mostly evaluated based on

visual inspection of the box and whisker plots of the

simulation results using the following criteria:

	 1.	Low and consistent levels of variability across

replicates (boxes small and same size).

	 2.	Consistentbehavior acrossthe range offactorvalues;

no decreasing or increasing trends.

	 3.	Distance from expected departure. We expected

a certain departure value based on the specified

DELTA value input for each reference condition,

and included this value as a line on all box plot

graphs.

4.	 Similar behavior across all values of DELTA. The

properties of the variance across replicates as described in 1 and 2 above should be the same across

all departure reference sets.

An idealindexwould have lowreplicate variability (small

boxes and whiskers) and the replicate mean would be

identical to the expected departure.

Results

	 This study illustrated that there is a wide variety of

behaviors for all departure indices across all types of

historical time series and reference conditions for the

YTY time series synthesis technique (other synthesis

methods will be discussed in the last part of this section). We first discuss the behavior of departure indices

using the averaging synthesis method (YTY) from a

qualitative perspective and in the context of a statistical

evaluation using ANOVA.

Time Series Synthesis Method 

YTY Base Case

	 The Sorenson’s Index (SI) performed well across

a wide range of classes (NCLASS) regardless of the

degree of departure in the five reference sets (fig. 3a).

There was little variability in this index across the full

range of number of classes (10-500 classes). However,

at low class numbers (10-30), the variability seemed to

increase and the ability of the index to detect departure

decreased (greater distance fromthe expected departure

line).SI also performedwell acrossmostlevels of percent

dominance (PDOM), except when reference and historical conditions were highly departed (high DELTA) even

though there were significant differences across most

levels of expected departure (see asterisks in fig. 3b).

When DELTA was high (high expected departure),

the SI performed well, even when the landscape was

dominated by a few classes (10-30% dominance), but

performance decreased as landscape area was evenly

distributed across all classes because both variability

across replicates (bigger boxes and whiskers) and distance fromexpected departure increased.SI also seemed

unaffected by the inherent variability (CVAR) in the

historical time series (fig. 3c). In general, when there

was higher departure (DELTA near 1.0), SI was more

variable, less precise, and less accurate (distance from

expected departure increases); however, when expected

departure waslow (DELTA <0.5), SI had low variability

and consistent behavior across all levels of variability.

Temporal autocorrelation (TCORR) appeared to have

little effect on SI performance, except maybe when the

similarity of reference and historical landscapes was

low (DELTA >0.6; fig. 3d).
USDA Forest Service Res. Pap. RMRS-RP-83. 2011 9

Figure 3—Sorenson’s Index (SI) performance as a departure measure. Box and whisker plots of 10 replicates of 100 historical 

landscapes were compared to five reference landscapes that represent similar (DELTA = 0.2) to highly departed (DELTA = 1.0) reference conditions. We evaluated four factors: (a) number of classes, (b) percent dominance, (c) variability of classes, and (d) temporal 

autocorrelation. In the boxplots, the lower boundary of each box is the first quartile (25th percentile), the upper boundary is the 

third quartile (75th percentile), and the line within the box represents the median of the distribution. The upper and lower whiskers 

represent the 10th and 90th percentile and the dots below and above the whiskers represent outlying values. Asterisks within each 

plot indicate significant differences in the Departure Index across all treatments at the 0.05 (*), 0.01 (**), and 0.001 (***) levels.
10 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

	 The Morisita Index (MI) was quite sensitive to

NCLASS with values farther from expected departure

when there were fewer classes (less than 30) at high expected departures (DELTA >0.8), or when there were a

high number of classes(greater than 50) at low expected

departures (DELTA <0.6) (fig. 4a). As with SI, the differencesin MI across all classes were highly significant

(three asterisks in fig. 4a). MI seemed completely ineffectual at describing departure as NCLASS increased

for reference conditions that were similar to historical

conditions.However,MI performedwellwhen expected

reference departure was high (DELTA >0.8) with low

replicate variability (except for lowest NCLASS) and

consistent behavior across all classes. The level of class

dominance (PDOM) in the HRV time series had little

effect on MI behavior except at high expected departure reference conditions with high variability across

replicates. There was reduced replicate variability and

consistent MI behavior across all levels of dominance

when expected reference departure reference conditions

were low (DELTA <0.2) (fig. 4b). The same findings

were true across the levels of class variability (CVAR)

with MI values lower than expected and more variable

acrosslevels of PDOM andCVAR (fig. 4c). MI behavior

seemed consistent across all levels of class variability,

but the replicate variability (box and whisker ranges)

increased when references were dissimilar to historical

conditions(DELTA>0.8).Aswith SI, temporal autocorrelation did not seem to affect MI results (fig. 4d).

	 The Similarity Ratio (SR) had a mixed performance

across most factors (fig. 5). SR performance was especially poor when the number of classes and the degree

of similarity between reference and historical were

high (DELTA <0.5). Replicate variability was high and

somewhat variable when NCLASS was low (less than

60) and reference-historical landscapes were similar

(DELTA <0.4) (fig. 5a). SR appeared to perform well

across alllevelsofPDOMwhenreference-historicalsimilarity was high, but when expected departure was high

(DELTA >0.5), replicate variability was high, distance

fromexpected departure increased, and overall behavior

was inconsistent (fig. 5b). Class variability (CVAR) and

temporal autocorrelation (TCORR) had the same effect

on SR with dissimilar reference-historical comparisons

(DELTA >0.5) having greater replicate variability, less

accuracy (fartherfromexpected departure line), and less

precision (inconsistent behavior) (figs. 5c,d).

	 ChordDistance (CD) had the samemixed performance

as SR for nearly every factor (fig. 6). CD performance

was poor when NCLASS was high (>50) and the degree

ofsimilarity between reference and historical was great

(DELTA <0.5) because the replicate variability was

high and inconsistent, especially when NCLASS was

low (<100) (fig. 6a). CD performance was mixed across

most levels of PDOM with high replicate variability and

imprecise behavior, except when reference-historical

similarly was high (DELTA <0.5), when the CD behavior stabilized and became highly consistent and

somewhat accurate (fig. 6b). Class variability (CVAR)

effects on CD were strongest when DELTA was high

(when reference-to-historical landscapes are dissimilar)

with CD becoming more variable and inconsistent (fig.

6c). Temporal autocorrelation had the least effect on CD

with only the least departed landscapes fostering the

best CD behaviors (fig. 6d).

	 The last index, Euclidean Distance (ED), had only

marginally better performance than CD or SR (fig. 7).

The effect ofthe number of classes onED was especially

pronounced when DELTA was low (minimal expected

departure from historical conditions) as the distance

fromexpecteddeparture linedramaticallyincreasedwith

increasing number of classes(fig. 7a), but when DELTA

was high, ED performed well except when there were

a low number of classes (<10). As with the last three

indices, the ED behavior is highly inconsistent across

the gradient of dominance (PDOM) and class variability (CVAR) with distance from the expected departure

line and replicate variance quite variable (figs. 7b, c).

Again, temporal autocorrelation didn’t seem to affect

ED behavior as much as the similarity of the referencehistorical conditions (fig. 7d) with the most consistent

ED behavior occurring when expected departures were

low.

Other Time Series Synthesis 

Methods (AAY, MMY, QTY)

	 Simplifyingthehistoricaltime seriesbycomparingone

or two historical conditions to one reference condition

instead of calculating a departure for each landscape in

the historical time series is probably more desirable for

land managers, but this may create greater uncertainty

and more inaccurate HRV evaluations. In this section,

we presentresults ofthe remaining time seriessynthesis

methods, but only for the Sorenson’s Index (SI) since it

performed the best, and only for two reference conditions (highly similar where DELTA = 0.2 and highly

departed where DELTA = 1.0).
USDA Forest Service Res. Pap. RMRS-RP-83. 2011 11

Figure 4—Morista’s Index (MI) performance as a departure measure. Box and whisker plots of 10 replicates of 100 historical landscapes were compared to five reference landscapes that represent similar (DELTA = 0.2) to highly departed (DELTA = 1.0) reference 

conditions. We evaluated four factors: (a) number of classes, (b) percent dominance, (c) variability of classes, and (d) temporal 

autocorrelation. In the boxplots, the lower boundary of each box is the first quartile (25th percentile), the upper boundary is the 

third quartile (75th percentile), and the line within the box represents the median of the distribution. The upper and lower whiskers 

represent the 10th and 90th percentile and the dots below and above the whiskers represent outlying values. Asterisks within each 

plot indicate significant differences in the Departure Index across all treatments at the 0.05 (*), 0.01 (**), and 0.001 (***) levels.
12 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

Figure 5—Similarity Ratio (SR) performance as a departure measure. Box and whisker plots of 10 replicates of 100 historical landscapes were compared to five reference landscapes that represent similar (DELTA = 0.2) to highly departed (DELTA = 1.0) reference 

conditions. We evaluated four factors: (a) number of classes, (b) percent dominance, (c) variability of classes, and (d) temporal 

autocorrelation. In the boxplots, the lower boundary of each box is the first quartile (25th percentile), the upper boundary is the 

third quartile (75th percentile), and the line within the box represents the median of the distribution. The upper and lower whiskers 

represent the 10th and 90th percentile and the dots below and above the whiskers represent outlying values. Asterisks within each 

plot indicate significant differences in the Departure Index across all treatments at the 0.05 (*), 0.01 (**), and 0.001 (***) levels.
USDA Forest Service Res. Pap. RMRS-RP-83. 2011 13

Figure 6—Chord Distance (SR) performance as a departure measure. Box and whisker plots of 10 replicates of 100 historical landscapes were compared to five reference landscapes that represent similar (DELTA = 0.2) to highly departed (DELTA = 1.0) reference 

conditions. We evaluated four factors: (a) number of classes, (b) percent dominance, (c) variability of classes, and (d) temporal 

autocorrelation. In the boxplots, the lower boundary of each box is the first quartile (25th percentile), the upper boundary is the 

third quartile (75th percentile), and the line within the box represents the median of the distribution. The upper and lower whiskers 

represent the 10th and 90th percentile and the dots below and above the whiskers represent outlying values. Asterisks within each 

plot indicate significant differences in the Departure Index across all treatments at the 0.05 (*), 0.01 (**), and 0.001 (***) levels.
14 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

Figure 7—Euclidean Distance (ED) performance as a departure measure. Box and whisker plots of 10 replicates of 100 historical landscapes were compared to five reference landscapes that represent similar (DELTA = 0.2) to highly departed (DELTA = 

1.0) reference conditions. We evaluated four factors: (a) number of classes, (b) percent dominance, (c) variability of classes, 

and (d) temporal autocorrelation. In the boxplots, the lower boundary of each box is the first quartile (25th percentile), the upper 

boundary is the third quartile (75th percentile), and the line within the box represents the median of the distribution. The upper 

and lower whiskers represent the 10th and 90th percentile and the dots below and above the whiskers represent outlying values. 

Asterisks within each plot indicate significant differences in the Departure Index across all treatments at the 0.05 (*), 0.01 (**), 

and 0.001 (***) levels.
USDA Forest Service Res. Pap. RMRS-RP-83. 2011 15

Figure 8—Performance of Sorenson’s Index for the four time series synthesis methods when reference landscapes are similar to 

historical conditions (expected departure low; DELTA = 0.2). Results for the average departure calculation across all years (YTY) 

is shown in (a) while the time series synthesis methods reduced the 1,000 historical landscapes into three simplified series: (b) 

average across all years (AAY), (c) a minimum and maximum values across all year, and (d) the 90th percentile value across all 

years. Box and whisker plots of 10 replicates and the lower boundary of each box is the first quartile (25th percentile), the upper 

boundary is the third quartile (75th percentile), and the line within the box represents the median of the distribution. The upper 

and lower whiskers represent the 10th and 90th percentile and the dots below and above the whiskers represent outlying values. 

Asterisks within each plot indicate significant differences in SI across all treatments at the 0.05 (*), 0.01 (**), and 0.001 (***) levels.
16 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

	 In general, the synthesis methods did not affect the

behavior of SI across the four factors (fig. 8) for the

similar reference-historical conditions (DELTA = 0.2),

but the variability across replicates (box size) did decrease. The AAY method (average historical landscape)

seemed to have nearly the same result as our detailed

base case YTY method. The MMY method was the

least desirable with some major shifts in behavior for

NCLASS and PDOM. The QTY method seems to yield

acceptable results but this method should only be used

when degrees of departure must be integrated into the

HRV analysis in the context of the stated management

objective.

	 None of the synthesis methods produced acceptable

resultswhen the reference landscapewas highly departed

from the historical representation (DELTA=1.0) (fig. 9).

Acceptable behavior occurred fortheAAYmethodwhen

class numbers were high (greater than 50 classes) and

percent dominates were low (less than 10 percent; 10

percentlandscape classes have 80 percent ofthe area).All

synthesis methods performed poorly across all levels of

class variability and temporal autocorrelation with high

variability, high expected departure, and inconsistent

behaviors (fig. 9).

Discussion

	 Whilemostofthevegetationcommunityecologybased

departure indices evaluated in this study performed

acceptably for determining departure from HRV, none

of them seemed to contain the optimum qualities for

comprehensive assessments of departure across all possible landscape conditions.Those indicesthat performed

well for one factor (number of classes, for example),

performed poorly for other factors(percent dominance,

for example). The Sorenson’s Index (SI) appeared to

perform the best in this study when it was qualitatively

compared with the other indices using a qualitative

scale of 1 to 5; the SI score is more than 10 above all

other indices (table 2). SI was least affected by the four

simulated factors but it still had unacceptable behavior

when the number of classes was low and the expected

departures were high. While many studies have used

the Sorenson’s Index or some variant with acceptable

results (Holsinger and others 2006; Keane and others

2008; Wimberly and others 2000), it appears that the

simplest landscapes(low number of classes, few classes

dominate, low variability) yield themostsuspectresults,

especially when reference conditions are similar to the

historical time series. This is also true for all the other

evaluated indices.

	 The synthesis of an extensive historical time series

of landscape maps into simplified historical references

(average, minimum-maximum, and 90th percentile)

did not create the high degree of uncertainty that we

expected (greater replicate variability). Instead, it appearsthatreducing historicaltime series could be a viable

option for land managers who cannot store or manage

large historical data sets. Thisis especially important as

more land managers obtain their HRV time series from

simulation modeling (Keane and others 2002; Nonaka

and Spies 2005; Wimberly and others 2000) because

models can produce extensive outputs of simulated

historical conditions. However, these synthesis options

could result in less variability in the historical record

that may make it difficult to detectsubtle changesin the

reference landscape. While we advocate that the entire

historical time series be used to determine departure,

we recognize that the average historical landscape may

also be a useful alternative for computing departure.

	 One situation that was not explored in this study is

when classesthat did not exist in the historical reference

conditions were present in the current conditions or vice

versa. Today’s landscapes are highly departed from

historical conditions partially because of increases in

exotic plant communities, which were absent on historical landscapes.Thisimportant phenomenon (addition of

new classes) could also affect the behavior of similarity

indices. A related unevaluated situation isthe extinction

of an historical class or the absence of a class in the

reference landscape (missing classes). The main effect

of these additional factors on departure measures may

be to increase variability across replicates.

	 The departure indexes used in this study have many

limitations that influence their use and interpretation

in HRV landscape analyses. First, these indices only

describe the magnitude and trend of change at the landscape level and do not incorporate geographical change

in their derivation. Decreases in class area in one portion of the landscape may be offset by increases in that

class in another portion resulting in no net change and

a stable departure index, yet there may be a great deal

of change at finer scales. Second, inherent landscape

properties, such as topography, soils, and climate, may

restrict the magnitude of change making it difficult to

evaluate the range of change for unique landscapes across

large geographical regions (Hessburg and others 1999).

For example, natural fuel barriers, such as scree fields

and rocklands, might restrict fire spread in some landscapes,resulting in relatively small changesin landscape

composition. The large number of historical observations (1,000 in this study) needed to compute a stable
USDA Forest Service Res. Pap. RMRS-RP-83. 2011 17

Figure 9—Performance of Sorenson’s Index for the four time series synthesis methods when reference landscapes are highly 

departed from historical conditions (expected departure high; DELTA = 1.0). Results for the average departure calculation 

across all years (YTY) is shown in (a) while the time series synthesis methods reduced the 1,000 historical landscapes into 

three simplified series: (b) average across all years (AAY), c) a minimum and maximum values across all year, and (c) the 

90th percentile value across all years. Box and whisker plots of 10 replicates and the lower boundary of each box is the first 

quartile (25th percentile), the upper boundary is the third quartile (75th percentile), and the line within the box represents 

the median of the distribution. The upper and lower whiskers represent the 10th and 90th percentile and the dots below 

and above the whiskers represent outlying values. Asterisks within each plot indicate significant differences for SI across all 

treatments at the 0.05 (*), 0.01 (**), and 0.001 (***) levels.
18 USDA Forest Service Res. Pap. RMRS-RP-83. 2011

departure index may restrict the creation of the HRV

time series to only using simulation modeling, rather

than using historical maps and data. The resolution and

extent of the analysis landscape can also influence the

range and variability of compositional classes thereby

limiting departure estimation (Keane and Karau 2006).

It is often difficult to detect departure when landscapes

are large and their resolution is fine.

	 Results of this study illustrate the great need for the

development of new, more comprehensive metrics for

examining landscape departure (Barrett and others

2006; Steele and others 2006). These new metrics

should performwell acrossthe factors used in thisstudy,

but they should also contain the following attributes:

(1) easily understood and interpreted by land management, (2) statistical tests for significance, (3) utility for

other purposes, (4)sensitive to subtle differences across

landscapes, (5) contain the ability to incorporate geographic change in departure estimation, and (6) easily

computed by hand or with computer. In the meantime, it

appears that the community ecology similarity indices,

especially SI, can be useful for estimating landscape

departure from historical conditions.

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Departure

Index

Factors

NCLASS PDOM CVAR TCORR Totals

∆<0.5 ∆>0.5 ∆<0.5 ∆>0.5 ∆<0.5 ∆>0.5 ∆<0.5 ∆>0.5

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Euclidean 

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Rocky

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