# Dynamic Systems methods

Urban and Civil

Nov 16, 2013 (4 years and 7 months ago)

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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

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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

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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

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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

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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

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Application of Dynamics to Social Sciences

dynamic

model

of
complex

systems

that are
non
-
ergodic

Explain
complex system

Explain
ergodic

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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

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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

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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

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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

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What does
development

mean?
(3)

developmental viewpoint: development is
the overarching term

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

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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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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)

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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

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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

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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

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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

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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

-

and long
-
term dynamics

Serve as criteria for existing theories

Applying dynamic systems to
development ….

4

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4 areas of discussion

Theory formation

Existing theories

Theories that await application to development

Model building

Empirical design

Statistical methods

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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)

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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

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Existing theories
(3)

theory of developmental dynamics

Classical developmentalists (Van Geert
1998)

based on the fundamental developmental
mechanisms in classical theories

highly developed in macro
-
social theory

beginning application to development

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theories that await application to
development

symbolic dynamics, categorical
dynamics

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

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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

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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

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Empirical design
(2)

mixed time
-
serial designs

combination of time
-
serial, longitudinal and
cross
-
sectional

Statistical Methods …

6

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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

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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

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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

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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

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categorical analysis of state spaces
(1)

State space grids

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

Who: Lewis,
Hollenstein

example:

See Hollenstein’s workshop

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categorical analysis of state spaces
(2)

Karnaugh maps

Why: describing transitions through Boolean
logic

Who: Dumas et al, Schiepek, Tschacher

example:

mother
-
child interaction

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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)

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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

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methods that await application to
development

fuzzy logic and categorical methods

why: behavioral data are categorical, but fuzzy