Mathematical Modeling and Data Analysis in Psychology: Principal Results, Obtained in 200105

1
Mathematical Modeling and Data Analysis in
Psychology Principal Results, Obtained in
2001-05
MOSCOW STATE UNIVERSITY OF PSYCHOLOGY AND
EDUCATION (MGPPU)
Computer Science Faculty Department of Applied
Informatics
L.S. Kuravsky
? MGPPU, 2006
2
Backgrounds
3
Psychology as an exact science

• Tendency for creating new exact (mathematical)
  branches in classical descriptive sciences
• For example besides the classical descriptive
  biology its new exact branches were created
  (molecular biology, bioinformatics, etc.)
• The same process in psychology is under way
  mathematical methods are sometimes the main part
  of research
• This seminar is a good proof of such point of
  view

4
Scientific work forms of organization

• Scientific seminar to ensure coordination of
  scientific efforts
• Joint projects with scientific centers
  responsible for solution of applied problems of
  interest
• Students qualification works

5
Scientific seminar

• Considers technologies of computer and
  mathematical modeling and methods of data
  analysis that are applied to solve different
  problems in psychology, biology, sociology and
  adjacent disciplines

6
Scientific research results
7
Scientific research results
8
Success of our students at scientific events
9
Contacts with classics (1998-2005)
Prof Galushkin (Russia) Prof Malykh (Russia) Prof
Kasatkin (Russia) Prof Stroganova (Russia) Prof
Hammond (UK) Prof Gelman (UK) Prof White (UK) Dr
Neale (USA) Dr Maes (USA) Dr Eaves (USA) Dr
Martin (Australia) Dr Bulanova (Russia) Prof
Boomsma (Holland) Prof Sidahmed (France)
Last 8 years we had contacts with Foreign and
Russian specialists getting a possibility to
discuss different components of our techniques
and to learn their opinions
10
Principal results (2001-2005)
11
List of results plan of the lecture

• Analysis and forecasting of development of
  psychological characteristics with the aid of
  Markov networks
• Analysis of interaction and evolution of
  psychological characteristics using multifactor
  Markov networks
• The study of information dependences between
  psychological characteristics of twins and their
  evolution in the course of ontogenesis
• Confirmatory factor analysis of psychological
  twin data
• Trained structures to solve the problems of
  psychological diagnostics
• Forecasting of inclination to psychoneurology
  pathologies using the parameters of early age
  development
• Neuronet technology of recognition of diseased
  conditions on the basis of anomalies of
  Electroencephalograms
• Identification of quantitative trait loci using
  mutual information contained in physiological and
  psychological twin data

12
Markov Networks Concept and Applications
13
Analysis and forecasting of development of
psychological characteristics

• Carried out on the base of accumulated
  observations
• The given system conditions are considered as
  separate discrete states in which the analyzed
  system has some probability to find itself. In
  due course transitions between the states are the
  case
• Probability dynamics is described by continuous
  time, discrete state Markov random processes
  (parametric models)

14
The model to describe transition dynamics

• represented by a graph
• nodes (see rectangles) correspond to the states
• branches (see arrows) correspond to transitions
• the process of evolution may be imagined as a
  random walk along the graph from one state to
  another following the arrows
• state-to-state transitions are instantaneous and
  take place at random time points

15
State-to-state transitions

• State-to-state transitions meet the following two
  properties of Poissons flows of events
• ordinariness
• independence of increments.
• Only stationary flows (a(t,?)??, ?const) will
  be taken up here. Parameter ? is the rate of a
  stationary flow (mean number of events per unit
  time interval).

16
Time-domain dynamics of state probabilities is
described by the Kolmogorov set of ordinary
differential equations
17
Goodness-of-fit measure
Pearson statistic is distributed asymptotically
according to a chi-square distribution.

• Its large values correspond to bad fit and small
  values correspond to good fit
• The given technique is called the method of
  chi-square minimum (for the problems under
  consideration, it usually yields estimations,
  which are close to ones of the maximum likelihood
  method)
• This statistic is minimized at the specified time
  points in which observed data are available.

18
The procedure of parameter estimation

• Some numerical integration scheme for the
  Kolmogorov differential equations is coded to
  calculate all probability functions using a
  spreadsheet
• A numerical procedure for multidimensional
  non-linear optimization is run as a macros to get
  required values of free parameters
• The inverse problem is solved here, viz. we
  determine coefficients of differential equations
  using the given solution characteristics
• Obtained values of free parameters are considered
  as system characteristics that have become
  apparent during observations

19
Example Analysis and forecasting of IQ
development
20
Multifactor Markov networks
21
What is the result?

• Developed are the trained multifactor Markov
  networks and their identification technology,
  which make it possible to reveal the evolution
  features for joint distributions of psychological
  characteristics under the influence of systematic
  factors.

22
Common remarks

• To take into account more details of an observed
  phenomenon, it is frequently expedient to apply
  forecasting multifactor Markov networks
• Some factors of different nature, which are
  essential for the phenomenon under study and must
  be revealed by a researcher, are represented by
  the model
• Each of them corresponds to one of the general
  orthogonal directions horizontal, vertical, etc
• Every row (column, etc.) of this network
  corresponds to some factor level.

23
General form of a forecasting 2-factor Markov
network

• Flow rates may be considered as functions of
  position at the given global diagram
• ?ij?(i,j), ?ij?(i,j),
• ?ij?(i,j), ?ij?(i,j)
• Such functions make it possible to represent
  subtle features of the system development.

24
Applications

• Capabilities of the approach under study have
  been demonstrated using the example of
  longitudinal analysis of verbal and nonverbal IQ.

25
Applications representation of system behavior

• by dynamics of probability column diagrams at
  different time points
• by identified transition flow rates

26
The 3-factor network

• Flow rate ? describes transitions in the 3rd
  direction
• To represent evolution acceleration, additional
  parameter ? is inserted
• ?i,k1(1?)?i,k ?ij,k1(1?)?ij,k, k1,2,
• where 3rd index corresponds to the number of 3rd
  factor levels.

27
Information Connection Analysis
28
The information approach for studying balance and
evolution of genetic and environment influences
on psychological characteristics

• New and effective way of estimating the balance
  of genetic and environment influences on
  psychological characteristics
• Based on methods and terms of the information
  theory (in particular, on the concept of entropy)
• Features and advantages were shown using the
  example of longitudinal twin psychometric
  intelligence study
• Obtained data were compared with analogous
  results derived from traditional confirmatory
  factor analysis

29
Scheme of the method
30
Main results and conclusions

• The simple dependencies, which allow estimating
  the influences of genetic and environment factors
  on psychological characteristics using standard
  descriptive statistics resulted from twin and
  longitudinal studies, have been derived.
• It has been investigated how the ratio of genetic
  components of the complete mutual information
  contained in the psychological characteristics of
  mono- and dizygotic twins depends on the alleles
  frequency distribution in a population.
• The estimations that have been carried out show
  comparability of results obtained with the aid of
  the analysis of information connections and the
  traditional confirmatory factor analysis, with
  agreement of the corresponding derived
  qualitative conclusions being the case.

31
Further development
32
Advanced genetic information techniques
!

• New effective method of chromosome quantitative
  trait loci identification, which is based on
  methods and terms of the information theory (in
  particular, on the concept of entropy) and uses
  the mutual information contained in twin
  psychological characteristics, was developed
• Under study are the loci associated with
  different psychological and physiological
  characteristics
• The technique was used to reveal gender
  influences on amplitudes of EEG alpha- and
  delta-rhythm amplitudes for baby twins

33
TraditionalConfirmatory Factor Analysis
34
Notes

• Traditional way to solve a wide range of
  problems, especially in behavior genetics
• Not an object or research this technique is
  widely used by our students in their work
• Ways of the most useful application twin
  research and studying evolution of psychological
  characteristics as well as revealing the most
  significant factors acting on observed
  characteristics

35
Notation in path diagrams
Observed variable
Latent variable
Causal dependence
Covariance connection
36
Confirmatory factor model
37
Basic genetic model DZ twin pairs
38
Genetic simplex-model

• Measurement model yiAiEi?i
• Structural equations Ai?aiAi-1?ai,
  Ei?eiEi-1?ei
• A genetic, E environment components.

39
Psychological Diagnostics Application of Trained
Structures
40
Trained structures to solve the problems of
psychological diagnostics

• The concept of discriminant networks with
  supervised learning, which were elaborated on the
  base of Bayesian variant of discriminant
  analysis, was developed, implemented and applied
  to solve some problems of psychological
  diagnostics (viz., to reveal symptoms of
  childrens deviant behavior)

41
Discriminant analysis discriminant networks

• The purpose of discriminant analysis to find
  rules for classifying multivariate observed cases
  to the beforehand specified classes
• Discriminant analysis sample estimates of mean
  values and elements of covariance matrices are
  determined with significant errors, especially
  for small samples
• Discriminant network recognition quality may be
  improved if one consider these quantities as free
  parameters and obtain their values as a result of
  optimization process for a training sample, which
  yields the minimum of classification errors.

42
Discriminant network architecture

• Discriminant functions are calculated in the
  corresponding hexagonal elements
• Elements S1, S2,,SN (rhombs) make a decision
  about belonging to the given classes
• Triangles correspond to input variables

43
Neural networks forecasting capabilities
44
Problem formulation

• Purpose
• Forecasting of inclination to psychoneurology
  pathologies using the parameters of early age
  development
• Studying of forecasting sensitivity to different
  prognostic factors
• Motivation revealing hard-hitting
  characteristics on the basis of easily
  accessible ones to ensure preliminary prophylaxis
  and correction
• Toolkit neural networks

45
Comparison of different forecasting techniques

• Multilayer Perceptrons
• Radial Basis Function networks
• Kohonen networks
• Linear networks
• Discriminant analysis

46
Neuronet Recognition Technology diagnostics of
pathologies (epilepsy) using EEG anomalies
47
Recognition technology

• Converting a current lead signal realization of
  some prescribed standard length into the relevant
  bipolar/binary representation
• Representation entering either the Hamming
  network or the Hopfield network with discrete
  states and time
• Network convergence to the so-called attractor,
  which is some reconstructed pattern or its
  number, after cyclic computations
• Identification of input signal status

48
Signal bipolar / binary conversion

• Wavelet spectrum is obtained from a current lead
  signal realization (discrete wavelet transform is
  used)
• Selecting simplified representation containing
  sufficient quantity of information and being
  sufficient for subsequent analysis. Discrete
  approximation of so-called skeletons (wavelet
  spectrum local extrema points) is acceptable.
• Recoding to bipolar representation local extrema
  points are coded as 1 and other points as -1
  (or 0).

Bipolar / binary representation
49
Software implementation
50
Wavelet Analysis reference
Applications to analysis of psychophysiologic data
51
Wavelet transforms

• Signal representation is created with the aid of
  wavelet transforms
• These transforms make it possible to reveal
  differences in process characteristics for
  diverse scales
• Wavelet-spectra are calculated by using wavelets,
  which are able to shifts along this axis and as
  well as to scaling (stretching/contraction).

52
Wavelet transforms

• If signal is a usual one-variable function,
  resulting wavelet-spectra is the function of 2
  arguments scale parameter characterizes
  oscillation time cycles whereas shift parameter
  time displacements.

• Wavelet-analysis has superiority over
  traditional spectral analysis
• it yields correct representation in case of
  non-stationary processes and
• keeps more useful information.

53
Hopfield networksreference
Applications to analysis of psychophysiologic data
54
Relaxation networks features

• Relaxation neural networks are characterized by
  direct and inverse information distribution
• Data circulation takes place until balance state.
• Synapse weights are computed only once, before
  the network operation
• The matrix of synapse weights is calculated with
  the aid of available patterns corresponding to
  different damage types
• This process may be considered as training that
  is carried out using information about
  recognition patterns.

55
Hopfield network structure

• Hopfield networks have a single neuron layer,
  where each neuron has synaptic connections with
  the other ones as well as one input synapse
• Number of outputs equals to the number of inputs
• In classical case, bipolar signal representation
  is used each input equals -1 or 1 .

The network finds relevant output damage pattern
having arbitrary imperfect signal at the entry or
draws a conclusion about the absence of
corresponding patterns.
56
References list 1

• Kuravsky L. S., Malykh S. B. On the application
  of queuing theory to analysis of twin data.
  Twin Research, Vol. 3, Issue 2, June 2000, Pp.
  92-98.
• Kuravsky L.S., Baranov S.N. and Kravchuk T.E.
  Structure condition diagnostics based on the
  wavelet transform and relaxation networks. In
  Proc. Condition Monitoring 2005, Cambridge,
  United Kingdom, July 2005, pp. 119-126.
• Kuravsky L.S. and Baranov S.N. The concept of
  multifactor Markov networks and its application
  to forecasting and diagnostics of technical
  systems In Proc. Condition Monitoring 2005,
  Cambridge, United Kingdom, July 2005, pp.
  111-117.
• Kuravsky L. S., Malykh S. B. Application of
  Markov models for analysis of development of
  psychological characteristics. - Australian
  Journal of Educational Developmental
  Psychology, 2004, Vol 2, pp 29-40.
• Malykh S.B., Zyrianova N.M., Kuravsky L.S.
  Longitudinal Genetic Analysis of Childhood IQ in
  6- and 7-year-old Russian Twins. Twin Research,
  2003, Vol. 6, No 4, pp. 285-291.

57
References list 2
58
References list 3
59
Thank you for attention!