From: platt@watson.ibm.com (Daniel E. Platt)
Subject: Re: What is meant by the lack of a Characteristic scale?
Date: Thu, 17 Dec 1992 15:35:07 GMT
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM T.J. Watson Research Center

In article <42665@sdcc12.ucsd.edu>, hkhatuya@sdcc3.ucsd.edu (Haripada Khatuya ) writes:
|> My question concerns the application of Multi-resolution stochastic
|> processes in the area of signal processing.  
|> Most of the papers say this kind of analysis is
|> useful for signals where there is no one single characteristic
|> scale.
|> 
|> o What exactly IS meant by a time series of signals having no
|> characteristic scale?  

A time series with some feature that shows up over some time T,
which is not evident over some time T/10^4, or also does not
show up in equipment with resolution T*10^4, then you have 
identified some time T over which a signal is dominant.

A really simple example not in the time domain would be the problem
of detecting elephants.  Microscopes are too fine grained to pick
out the large-scale morphology of elephants (you won't see tusks
and trunks, and other features, such as the shape of the legs
and the relationship between leg shape and mechanics of size), though
you could pick out details such as hair.  The elephants body is on a
different length scale from its hair which is on a different
length scale from its blood vessels.  On the other hand, you would
not detect elephants if you had imaging equipment designed to detect
continents.

Something without a length scale shows features over a wide
range of length scales (many 'decades' or factors of 10), and
these features look similar on all those length scales.  A
famous example is a coastline.  As you draw maps of continental
shorelines and then go down to smaller scales to see coastlines
of countries, then to states, then to towns, then to shoreline
parks in towns, you'll see wiggles in the coastline at all of those
length scales.  

|> o Does the lack of a unique characteristic scale imply the
|> non-existence of statistical moments?

Usually, the moments are computed at some characteristic resolution
length.  If you could resolve the signal to an infinite precision,
the moments would tend to diverge or go to zero.  The actual moments
you measure will depend on the resolution length scale to some power.

	M_n = <p^n> = a(n)*L^nD(n)

for some exponent D and resolution n.  So, on a log-log plot of M_n
as a function of resultion L, you'll see a straight line.  This kind
of 'scaling' relationship is dominated by a scaling exponent D(n),
often called a 'Dimension.'  (Holder exponent, Hausdorf dimension, etc
depending on circumstances).  If the D(n) are all the same, it is a
simple 'fractal.'  If they are different, the signal is 'multifractal.'
(This is a sort of oversimplification of details, but communcates
some of the general feel).

|> o Is there any body of statistical theory existent to detect 
|> the presence or absence of mutliple scales?

See the above.

|> o Is it true that multiplicity of scales imply signals 
|> with power-law (as opposed to exponential) correlation decay?

Not necessarily.  You can have a multiplicity of scales that
does not scale according to the resolution length L.

|>  
|> I know that the issues of long-range range dependence (slow decay of
|> correlations), 1/f noise, self-similarity, fractal supports, non-analytic
|> distributions, power laws, wavelets, multi-resolution stochastic
|> processes etc., are interlinked, but I haven't been able to find any
|> rigorous (especially in terms of statistical theoretic models)
|> links.  For eg., Multi-fractal measures are now widely used in
|> Physics, but I have yet to see them in any journal of statistics.
|> There have been many journal articles on the application of Chaos
|> theory to statistical analysis, but all of these concentrate on
|> invariants like dimension and Lyapunov exponent estimates.  I have
|> seen NOT ONE article on the consequences on standard statistical tests
|> and estimation procedures of substituting ,say, fractal spaces for
|> the standard sample spaces of classical probability analyses. 

I saw something from the University of Waterloo's statistics department
around 1987 on that topic, but didn't really follow it up.  You might
try calling them to find out who is interested in it there.  Perhaps they
can give you a start on a literature search.


|> 
|> Also, can anybody expound on the link of the new theory of Self
|> Organised Criticality to the presence of multiple scales,if any at
|> all exist?  Does the SOC model imply that the lack of a
|> characteristic scale causes a breakdown of the traditional concept
|> of "Causality", ie., a given input causes a definite response in
|> output?  This is beginning to sound wrong, but I'd still love to
|> hear more on this. 

If you have a sand pile and you're dropping sand at one spot, that
spot will tend to get higher and higher until you get an avalanch.
The avalanche occurs because a critical condition dependent on the
local mechanical characteristics of the sand translate into an inability
for the whole pile to support the height with such a small base.  It
will collapse and spread out (avalanch).  Critical conditions (phase
transitions, etc) tend to reflect scale-invariant (no particular
length scale) behavior (you see little avalanches and bigger avalanches,
and the moments will show power law resultion dependence).  Any time
an avalanche pushes the sand pile away from criticality, the mechanism
of the sand-dropping tends to push it back to criticality.

One point of correction: there's a difference between causality and
determinism.  Causality basically means that an effect cannot occur
before a cause.  More particularly, energy can only be transmitted
in a way that all reference frames see the energy move from a cause
to an effect.  Determinism means that you can figure out what all
the effects will be to some cause.

If you have a good model for avalanches (modeling sand-grain cohesiveness,
stress/strain behavior of the pile, etc, based on Newton's laws), then
ideally, you should be able to calculate exactly when each avalanche
will occur given precisely how each grain was dropped on the pile.
This turns out to be very sensative to initial conditions.  That means,
if you take two systems, and change something about just one of the 
sand grains just a little bit, the avalanche sequence will diverge
between the two sand piles.  That divergence is measured by the
Lyapunov exponent.

|> 
|> Any leads, clues, explanations, pointers, papers, references, etc.
|> highly appreciated.  Sorry about the long rather rambling post.
|>

This is a wide area.  You can look at Field Theory and Renormalization
Group work, as well as critical phenomena.  Daniel Amit has a good book
on the topic.  You might also look at the fractal literature, which
ranges from rather technical to very popular.  Then there's the literature
on chaos.  That too covers a broad range.  There's a book by DuVaney (sp?)
on that topic.  The most recent Scientific American talks a bit about
SOC (I didn't look at it too closely).  There are obvious points of
contact between all of these areas...

Hope this is a help.

|> Hari.

Dan

-- 
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Daniel E. Platt                                      platt@watson.ibm.com
The views expressed here do not necessarily reflect those of my employer.
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