Informative Psychometric Filters

322 Pages·2006·4.455 MB·English

by  Gregson, Robert A M

InformatIve
PsychometrIc fIlters
RobeRt A. M. GReGson
Previous Books by the Author
Psychometrics of Similarity
Time Series in Psychology
Nonlinear Psychophysical Dynamics
n-Dimensional Nonlinear Psychophysics
Cascades and Fields in Perceptual Psychophysics
InformatIve
PsychometrIc fIlters
RobeRt A. M. GReGson
Published by ANU E Press
The Australian National University
Canberra ACT 0200, Australia
Email: [email protected]
Web: http://epress.anu.edu.au
National Library of Australia
Cataloguing-in-Publication entry

Gregson, R. A. M. (Robert Anthony Mills), 1928- .
Informative psychometric filters.
Bibliography.
Includes index.
ISBN 1 920942 65 3 (pbk).
ISBN 1 920942 66 1 (online).
1. Psychometrics - Case studies. 2. Human beings -
Classification. 3. Psychological tests. 4. Reliability.
I. Title.
155.28

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photocopying or otherwise, without the prior permission of the publisher.
Cover design by Teresa Prowse.
The image on the front cover, that resembles a scorpion, is the Julia set in the
complex plane of the Complex Cubic Polynomial used as a core equation in Nonlinear
Psychophysics; it is employed in its equation form in some chapters in this book,
and is discussed in more detail in the author’s three previous books on nonlinear
psychophysics and in some articles in the journal Nonlinear Dynamics, Psychology
and Life Sciences
Printed by University Printing Services, ANU
This edition © 2006 ANU E Press
Preface
There has been a time gap in what I have published in monograph form,
duringwhichsomeveryimportantandprofoundshiftshavedevelopedin
thewaywecanrepresentandanalysethesortsofdatathathumanorgan-
ismsgenerateastheyprogressthroughtheirlives.InthatgapIhavebeen
fortunateenoughtomakecontactwithandexchangeideaswithsomepeo-
plewhohavechosentoexplorenonlineardynamics,andtoseeiftheideas
that are either pure mathematics or applications in various disciplines
other than psychology can be usefully explored as filters of behavioural
timeseries.
Three strands in my previous published work have come together
here;theyaretimeseries,psychophysics,andnonlinearnearlychaoticdy-
namics.Also,thewidthinscaleofthedataexamplesexaminedhasbroad-
enedtoincludesomepsychophysiologicalandsocialprocesses.Therehas
evolvedaproliferationofindices,hopefullytoidentifywhatishappening
in behavioural data, that can be regarded as partial filters of information
withveryvaryingefficiency.
Thetitleofthiseffortwasselectedtoemphasisethatwhatismeasured
isofteninformationorentropy,andthatthefocusisonproblemsanddata
thatareill-behavedascomparedwithwhatmightbefoundin,say,physics
or engineering or neurophysiology. The object is to model in a selective
way,tobringoutsomefeaturesofanunderlyingprocessthatmakesome
sense,andtoavoidmisidentifyingsignalasnoiseornoiseassignal.
Recent developments in psychophysiology (Friston, 2005) have em-
ployed networks of mixed forward, reverse and lateral processes, some
ii INFORMATIVEPSYCHOMETRICFILTERS
of which are linear and some nonlinear. That form of construction brings
us closer to neurophysiological cortical structures, and takes theory fur-
ther than is pursued here, though there is in both approaches an explicit
assumption of nonlinear mappings playing a central role in what is now
calledtheinverseproblem;thatistosay,workingbackfrominput-output
datatotheidentificationofamost-probablegeneratingprocess.
There is no general and necessary relationship between identifiabilty,
predictability and controllability in processes that we seek to understand
astheyevolvethroughtime.Inthephysicalsciencessometimesthethree
are sufficiently linked that we can model, and from a good fitting model
we can predict and control. But in many areas we may be able to control
without more than very local prediction, or predict without controlling,
because the process under study is simple and linear, and autonomous
fromenvironmentalperturbations.Thatdoesnotholdinthelifesciences,
particularlyinpsychologyoutsidethepsychophysicallaboratory.
The unidentifiabilty or undecidabilty in identification, prediction and
controlcanbeexpressedininformationmeasures,andinturn,usingsym-
bolic dynamics, can be expressed theoretically in terms of trajectories of
attractorsonmanifolds.Itistheextensionofourideasfromlinearautocor-
relation and regression models to nonlinear dynamics that has belatedly
impactedonsomeareasofpsychologyexploredhere.
Arelatedproblemthatisunresolvedinthecurrentliterature,forexam-
ple in various insighful studies in the journal Neural Computation, is that
of defining complexity. The precise and identifiable differences between
complexity and randomness have been a stumbling block for those who
wanttoadvanceverygeneralmetricsfordifferentiatingtheentropyprop-
erties of some real data in time-series form. I have not added to that dis-
putehere,but soughtsimplytoillustrate whatsortsofcomplicated, non-
stationaryandlocallyunpredictablebehaviourareubiquitousinsomear-
easofpsychology.Theapproachismoreakintoexploratorydataanalysis
thantoanalgebraicformalism,withoutwishingtodisparageeither.
Theproblemsofdistinguishingbetweenthetrajectoriesofdeterminis-
tic processes and the sequential outputs of stochastic processes, and con-
sequently the related problem of identifying the component dynamics of
iii
mixturesofthetwotypesofevolution,hasproducedaveryextensivelit-
eratureoftheoryandmethods.Onemethodthatfrequentlyfeaturesisso-
called box-counting or cell-mapping, where a closed trajectory is trapped
in a series of small contiguous regions as a precursor to computing mea-
sures of the dynamics, particularly the fractal dimensionality (for an ex-
ample,whichhasparallelsintheanalysisofcubicmapsinnonlinearpsy-
chophysics,seeUdwadiaandGuttalu,1989).
Serious difficulties are met in identifying underlying dynamical pro-
cesses when real data series are relatively short and the stochastic part is
treated as noise (Aguirre & Billings, 1995), it is not necessarily the case
that treating noise as additive and linearly superimposed is generically
valid(Bethet,Petrossian,Residori,Roman&Fauve,2003).Thoughdiverse
methodsaresuccessfullyinuseinanalysingthetypicaldataofsomedis-
cipines,asinengineering,therearestillapparentlyirresolvableintractabil-
ities in exploring the biological sciences (particularly including psychol-
ogy),andaproliferationoftentativemodificationsandcomputationalde-
viceshavethusbeenproposedinthecurrentliterature.
Thetheoreticalliteratureisdominatedbyexamplesfromphysics,such
asconsiderationsofquantumchaos,whicharenotdemonstrablyrelevant
for our purposes here. Special models are also created in economics, but
macroeconomics is theoretically far removed from most viable models in
psychophysiology. Models of individual choice, and the microeconmics
of investor decisions, may have some interest for cognitive science, but
the latter appears to be more fashionably grounded, at present, in neural
networks, though again the problem of simultaneous small sample sizes,
nonlinearity, non-stationarity, and high noise have been recognised and
addressed(Lawrence,Tsoi&Giles,1996).
One other important social change in the way sciences exchange in-
formation has in the last decade almost overtaken the printed word. For
any one reference that can be cited in hard copy, a score or more are im-
mediatelyidentifiableininternetsourcessuchasGoogle,andthechanges
and extensions of ideas, and perhaps also their refutation, happens at a
ratethatbypassestheprintedtextevenunderrevisionsandneweditions.
Forthisreason,therearesomeimportanttopicsthatarenotcoveredhere,
iv INFORMATIVEPSYCHOMETRICFILTERS
tools such as Jensen-Shannon divergence are related to entropy and to
metric information and could well be used to augment the treatment of
nonlinearandnon-stationarypsychologicaldatabutsofarhavenotbeen
considered. We urge the reader to augment and criticise the present text
bycheckingdevelopmentsintheelectronicsources,particularyfocussing
onworksuchasthatbyFugledeandTopsøe(www.math.ku.dk/topsoe/)
on Jensen-Shannon divergence, or Nicolis and coworkers (2005) on dy-
namicalaspectsofinteractionnetworks,thathaverelevanceandpromise.
Jumps between modes of dynamical evolution even within one time se-
ries essentially characterise psychological processes, and transient states
suchaschimera(AbramsandStrogatz,2006)mayyetbeidentifiedinpsy-
chophysiology.
Iwanttothankvariouspeoplewhohaveencouragedorprovokedme
to try this filtering approach, and to bring together my more recent work
that is scattered over published and unpublished papers, conference pre-
sentations,invitedbookchapters,andeveninbookreviews.Oneverycon-
genial aspect of the modern developments in applied nonlinear dynam-
ics is the conspicuously international character of the activity. Professors
Stephen Guastello and Fred Abraham in the USA, Hannes Eisler in Swe-
den, John Geake in England, Ana Garriga-Trillo in Spain, and Don Byrne
and Rachel Heath in Australia, have all offered me constructive help or
encouragementoverthelastdecade.
SchoolofPsychology
TheAustralianNationalUniversity
Contents
1 Introduction 1
Filters and Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Event Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Markov and Hidden Markov models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Stimulus-response sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Limits on Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Some peculiarities of psychophysical time series . . . . . . . . . . . . . . . . . . . . . . 24
Measure Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Other Special Cases of Transition Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Information, Entropy and Transmission 33
Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Fitting a Model plus Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Preliminary Data Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Relaxation of Metric Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Fast/Slow Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Filtering Sequential Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Transients onto Attractors 53
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Identification of local manifold regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Treating as Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Tremor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Higher-order Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
vi InformatIve PsychometrIc fIlters
4 Inter- and Intra-level Dynamics of Models 71
Intermittencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Synchrony and Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Bounded Cascades in 6-d partitioned NPD . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Comparison Control Conditions for the Dynamics . . . . . . . . . . . . . . . . . . . . 92
Serial Hypercycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
The extension to 2D and 3D Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 A Bivariate Entropic Analogue of the Schwarzian Derivative 105
The Schwarzian derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Coarse Entropy Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A Schwarzian derivative analogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
Extending to the bivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114
The manifold of the lagged BESf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Appendix: Bernstein economic data and Physionet data . . . . . . . . . . . . . . 125
6 Tribonacci and Long Memory 129
The Tribonacci Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Higher-Order Derived Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Classical ARMA analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
ApEn modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
ESf Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Classical ARMA analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Human Predictive Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Postscript on the Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173