Dynamic Systems methods

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Dynamic Systems methods

in the study of development


A practice
-
oriented approach






An introduction to the ISED Workshop on

Dynamic systems methods in development

Groningen, May 14
-
16 2007

Dynamic systems of Development

2

Dynamics of Development: knowledge map

Development and
dynamic systems

Dynamic systems


What it is and what it is
not not

Development


Basic components


Basic mechanism(s)

Application of dynamic
systems to development

Theory
formation

Model
building

Empirical
design

Statistical
methods

Dynamic systems …

1

Dynamic systems of Development

4

Dynamics: basic definition and properties


an approach to the description and
explanation of
change


what it is not: misunderstandings in
social science


it is not a model with time as predictor (as in
multilevel growth model, for instance)


social science and psychology has focused on
static ergodic models


whereas it should have been focusing on
dynamic non
-
ergodic models if it wants to really
understand change


Dynamic systems of Development

5

Two equations: the static and dynamic models


Dynamic system: x
t+1

= f ( x
t
)


The value of a variable x is a function of its
preceding state


Static system: x
i

= f ( y
i
)


The value of a variable x


Is a function of the variable y


Or any set of such variables, y
a
, y
b
, y
c
, …

x
t+1

= f ( x
t

)


time

Variable x

Give me a value of x and I will tell
you what the
next

value of x will
be

The model generates a
time
series

Dynamic systems of Development

6

x
i

= f ( y
i

)


Variable y

Variable x

Give me a value of y and I will tell
you what the corresponding
value of x will be

The model generates a
population sample

wobbles, humps and sudden jumps
-

theoretica lreflection

7

Dynamic system and geometric space


Mutual preference

similarity

start

End
attractor

Example:
emergence of
friendship in
function of
mutual
preference and
similarity;
applies to
dyad

Dynamic systems of Development

8

Application of Dynamics to Social Sciences


dynamic

model


of
complex

systems


that are
non
-
ergodic


Explain
complex system

Explain
ergodic

Dynamic systems of Development

9

ergodicity


An informal definition


Imagine a statistical analysis over the entire ensemble
of people at a certain moment in time


and a statistical analysis for one person over a certain
period of time


an ensemble is ergodic if the two types of statistics give
the same result, and non
-
ergodic if this is not so


ergodicity hardly ever applies to behavioral
data!


Molenaar


considerable consequences for research methodology


Dynamic systems of Development

10


A complex system is any system featuring
a large number of interacting components
(agents, processes, etc.)


whose aggregate activity is nonlinear (not
derivable from the summations of the
activity of individual components)


and typically exhibits … self
-
organization



Rocha, 1999

What is a complex system?

Development …

2

Dynamic systems of Development

12

What does
development

mean?
(1)


meaning unwrapping



like the unwrapping of a bookroll,


notion of an inner logic in the sequence


notion of finality


life span more than just development

Dynamic systems of Development

13

What does
development

mean?
(2)


there's also education,
learning and
teaching


learning: having experiences that make
you change


teaching: giving someone experiences that
make him change in a particular direction


there is maturation and aging


biologically governed processes of change,
including the aspects of rising and falling,
deterioration

Dynamic systems of Development

14

What does
development

mean?
(3)


developmental viewpoint: development is
the overarching term


Encompassing learning, teaching, niche
-
seeking,
maturation, aging, ….


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15

The classical developmentalists’ view


Piaget, Vygotsky, Wallon, Werner. …


all changes of the system occur through
information that is "moderated" through the
system


“moderated” means that


the system encodes the information and adapts only in
function of this encoding (as in Piaget's assimilation and
accommodation)


the system selects its own niche, i.e. preferred and
adapted environment (also biologically and genetically
-
Plomin)


the caring environment (educators, parents) adapt the
environment to the system's level and possibilities (as in
Vygotsky's ZPD)


Developmental mechanisms
of change …

3

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17

The basic mechanism
(1)


all changes of the system occur through
information that is "moderated" through
the system


it is a fundamentally recursive notion


next step is a function of the preceding step


and thus a direct expression of a dynamics in
the fundamental sense (see basic definition)


Dynamic systems of Development

18

The basic mechanism
(2)


it is a fundamentally interactional notion


the dynamics occurs through the interaction
with an environment or context


this environment is of many kinds: biological,
spatial, cultural, ...


a badly missing component: the utility
-
driven nature of human action and the
motor of action and development


biology and economics emphasize the utility
-
driven nature of action

Dynamic systems of Development

19

The drive for developmental change
(1)


appears as a factor among many others:
motivation for instance


relates to control theory, happiness/pleasure
theories; appraisal theory of emotion, self
-
actualization (Maslow), drives (Freud) ….


but is far from the fundamental dynamic factor
that features in biology (fitness
-
maximization)
or economics (utility
-
drive)


you need to understand the dynamics of the utility
function in order to understand the dynamics of the
long
-
term process, e.g. biological evolution, economic
processes and trade, ...


Dynamic systems of Development

20

The drive for developmental change
(2)


the dynamics of the utility function is
essential for understanding the
short
-
term dynamics

of change


the short
-
term dynamics of development
involves the dynamics of action


see the model of interaction dynamics S and
VG

Dynamic systems of Development

21

Dynamic systems approach to
development
(1)


you can define an organism as a
manifold, a space of variables


specify its changing position on a
developmental ruler


Properties worth studying


discontinuity next to continuity


construction of novelty next to transmission and
appropriation


Fuzziness and ambiguity


intra
-
, inter
-

and contextual variability


Dynamic systems of Development

22

Dynamic systems approach to
development
(2)


A Dynamic Systems theory of
development incorporates


the basic recursive developmental mechanism


Interaction and transaction


Action drives, evaluation and control


Short
-
term dynamics of action


Long
-
term dynamics of development


Link between short
-

and long
-
term dynamics


Serve as criteria for existing theories


Applying dynamic systems to
development ….

4

Dynamic systems of Development

24

4 areas of discussion


Theory formation


Existing theories


Theories that await application to development


Model building


Empirical design


Statistical methods


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25

Existing theories
(1)


qualitative use of complex dynamic
systems concepts


Lewis, Fogel, Granic, Dishion, …


theory of embedded
-
embodied action


Thelen and Smith (Spencer, Schoner, ...)


short
-
term interactional aspect of development:
organism
-
environment interaction


development as change in the dynamic field
(Schoner)


related theories: ecological psychology (Gibson)


Dynamic systems of Development

26

Existing theories
(2)


Connectionism


Really a DST approach? Or are they
supplementary?


Focuses on the organismic
-
brain component


Dynamic growth theory


Van Geert,
Fischer, Case,language
development


emphasizes long
-
term dynamics: growth as an
auto
-
catalytic process under limited resources


related theories: biological theory of ecology


Dynamic systems of Development

27

Existing theories
(3)


theory of developmental dynamics


Classical developmentalists (Van Geert
1998)


based on the fundamental developmental
mechanisms in classical theories


theory of dyadic agents


highly developed in macro
-
social theory


beginning application to development


Dynamic systems of Development

28

theories that await application to
development


symbolic dynamics, categorical
dynamics


their major advantage: they link categorical with
quantitative descriptions and modeling, a link
that is badly needed in developmental (and
clinical) psychology


fuzzy control system dynamics


control theory and theory of agents


Model building …

5

Dynamic systems of Development

30

model building


connectionist modeling


differential and difference models of
growth phenomena


Growth models


Interaction models (“The Mathematics of
Marriage”, Gottman et al.


agent models


cognitive simulation and AI
-
models
(Anderson, ACT)


Empirical Design …

6

Dynamic systems of Development

32

Empirical design
(1)


High
-
frequency, time
-
serial N=1 studies


samples result as collections of time serial studies


the time
-
serial study should capture the characteristic
dynamics at the time scale at issue (which longitudinal
studies nromally not do)


Experiments as perturbations


experimental studies in psychology involve specific
perturbations of an ongoing process


the experimental manipulation must be studied time
-
serially, as a perturbation that is assimilated by the
process or to which the process accommodates


Dynamic systems of Development

33

Empirical design
(2)


mixed time
-
serial designs


combination of time
-
serial, longitudinal and
cross
-
sectional


Statistical Methods …

6

Dynamic systems of Development

35

Overview of methods


Standard statistical methods


Statistical methods for non
-
linear time series


Standard optimization techniques for curve
-
fitting


Analysis of categorical state spaces


State space grids


Karnaugh maps


Finite state diagrams, Markov chains and t
-
patterns


Monte Carlo tools for statistical simulation


methods that await application to development

Dynamic systems of Development

36

standard statistical methods


Why: analysis of sample predictions
based on dynamic models


“Who”: standard statistical packages


Example:


dynamic model of dyadic play in children of
different sociometric statuses


principal component analysis of dyadic conflict
trajectories


Dynamic systems of Development

37

statistical methods for non
-
linear time
series


Why: statistical description and analysis
of time series


Who: Molenaar,
Hamaker
, et al.


Example:


non
-
linear time
-
serial factor analysis


See Hamaker’s workshop


Dynamic systems of Development

38

standard optimization techniques for
curve
-
fitting


Why: fitting dynamic models in the form
of differential equations or maps
(difference equations) to data


“Who”: standard fitting techniques


Example:


fitting growth models to data, qualitative and
quantitative fitting


See Van Geert’s Workshop


Dynamic systems of Development

39

categorical analysis of state spaces
(1)


State space grids


Why: describe transitions among categorical
states and finding categorical attractor states


Who: Lewis,
Hollenstein


example:


dyadic interactions among adolescents


See Hollenstein’s workshop

Dynamic systems of Development

40

categorical analysis of state spaces
(2)


Karnaugh maps


Why: describing transitions through Boolean
logic


Who: Dumas et al, Schiepek, Tschacher


example:


mother
-
child interaction


Dynamic systems of Development

41

finite state diagrams, Markov chains and t
-
patterns


Why: finding patterns in time series


who: Magnusson; Markov chains,


example:


time patterns in teacher
-
child interactions


See Van Geert’s workshop (if possible)


Dynamic systems of Development

42

Monte Carlo tools for statistical simulation


Why: applicable to non
-
standard data


“Who”:


often used in biology, non
-
standard problems,
small sample problems etc.


Manly; Todman and Dugard


example:


significant peaks in variability of langauge
production (time series)


Significance testing of dynamic model of dyadic
play

Dynamic systems of Development

43

methods that await application to
development


fuzzy logic and categorical methods


why: behavioral data are categorical, but fuzzy


Who: Zadeh, Ragin, Smithson, Verkuilen


example:


Emergence of linguistic categories in young children


analysis of computer use in toddlers