Dynamic Growth Modeling

Dynamic Growth Modeling 1

Dynamic Growth Modeling

Paul van Geert

University of Groningen


Introductory Theoretical Aspects

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L’ important ....

• Albert Einstein: “Imagination is more important than knowledge” ...

• First comes curiosity, then comes the question, then comes the method• Primacy of theory• Use whatever method(s) that can contribute to

the refinement of the theoretical question

• Historical note• The “Belgians”: Quetelet and Verhulst• Manuel Fawlty Towers

Albert Einstein: “Everything should be made as simple as possible, but not simpler...”


• Ganger and Brent (2004): really?• A spurt requires an S-shaped form of the growth

curve: Logistic equation• 38 longitudinal data sets• In only 5 children the s-shaped function provided a

better fit than the simpler quadratic model• the additional parameter in the S-shaped function did

not result in statistically significant gain in explained variance

An example: the vocabulary spurt • Spurt in the lexicon in the second year of life

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Albert Einstein: “Everything should be made as simple as possible, but not simpler...”


The quadratic model as explanatory theory

• Lt = a + b*t + c*t2

• What is the underlying theory of vocabulary change? • It’s given by the first derivative of the equation• ΔL/ Δt = b + 2ct • the actual learning of words = adding a constant

number of words per unit time (the number b), in addition to adding a number of words, ct, that increases as the child grows older

Is this a reasonable theory of word learning?Is this a reasonable theory of word learning?Word learning depends on age?Word learning depends on age?How does age affect word learning?How does age affect word learning?Because “age” probably stands for something else, Because “age” probably stands for something else, namely the child’s increasing knowledge.namely the child’s increasing knowledge.But the theory does not specify this. But the theory does not specify this. The theory also predicts that a person will either The theory also predicts that a person will either continue to learn ever more words, irrespective of continue to learn ever more words, irrespective of how many words there are in his language, or that how many words there are in his language, or that at some point in time he will start to forget ever at some point in time he will start to forget ever more words…more words…

Word learning depends on the words one already knows and on the words one does not know yet (the number of words in the language)The simplest possible equation expressing this model is the logistic equation


Dynamic growth models: basic principles

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Dynamic Growth Model of Development (1)

• A developing system can be described as a system of variables (or components)

• Variables change according to laws of growth• Auto-catalytic process

“Change (or stability) is its own cause”• Depends on limited resources

Change depends also on other things (the context) But the supply is not unlimited…


Dynamic Growth Model of Development (2)

• We are interested in how phenomena are related• Correlations, explained variance, …

• Dynamic phrasing: how does one thing influence an other? How does one thing make another thing change?

• Dynamic relations are• Supportive• Competitive• Conditional


A one-dimensional growth model

• Example: the lexicon• Learning “now” depends on what one already

knows: a*L• And: Learning now depends on what one does not

know yet: b*(K-L)• Thus: learning now is described by a*L*b*(K-L)• Or, after simplification r*L(1-L/K)• The driving term and the slowing-down term

• The model can be easily extended to any form of resource-dependent growth


Multi-dimensional growth models

• Examples:• Lexicon depends on syntax, and vice versa• Instruction given depends on what the child

already knows, and vice versa…• Language depends on cognition, and vice versa

….

• Coupled growth equations


Property A

Property B

support support

Property A

Property B

competition competition

Property A

Property B

support competition

Predator-Prey dynamics


Motor system

Perceptual system

Linguistic knowledge

Social knowledge

Physical knowledge

Pedagogical support

External symbol systems

concerns

emotions

The form of the developmental process is determined by the way the variables interact with each other

•Stepwise development (stages)Stepwise development (stages)•Temporary regressionsTemporary regressions


Motor system

Perceptual system

Linguistic knowledge

Social knowledge

Physical knowledge

Pedagogical support

External symbol systems

concerns

emotions

Fischer’s developmental theory


Pauline Number of Words (1 of 3)

• Based on a study by Dominique Bassano• Number of words from one-word to multi-

word sentences• 1W-, 2-3W- and 4+W-utterances as fuzzy

indicators of possible underlying generators• Holophrastic, combinatorial, syntactic

• Variability peaks provide an indication of discontinuity or transition



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M1 M1 smooth M23 M23 smooth M422 M422 smooth



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W1 17.4% W23 17.4 W4plus 17.4%



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observed Perc 0.05 Average Perc 0.95W1 17.4% W23 17.4 W4plus 17.4%


Dynamic model building• Use dynamic modeling to investigate

properties of the dynamics• Based on simple relationships between

variables• Supportive• Competitive• conditional


One-word sentences

Holophrastic principle

2&3-word sentences

Combinatorial principle

4&more-word sentencesSyntactic principle

supports

Competes with

supports

Competes with


Dynamic Model and Data of Pauline

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W1 model W23 model W4+ model W1 W23 W4+


Descriptive curve fitting

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Curve fitting…

• Simple curves Linear, quadratic, exponential …

• Transition curves S-shaped curves: logistic, sigmoid,

cumulative Gaussian, … Eventually look very discontinuous…

• Smoothing and denoising curves Loess smoothing, Savitzky-Golay Very flexible


Example: Peter’s pronomina (1 of 3)

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pronomina Linear model Quadratic Model



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pronomina Sigmoid LS Fit Sigmoid Robust Fit



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pronomina Loess 50% Loess 20%

If you want to describe your data by means of a central trend, use Loess* smoothing*(locally weighted least squares regression)Data will be symmetrically distributed around the central trend, without local anomalies


Curve fitting in cross-sectional data

• Theory-of-Mind test: • 324 children between 3 and 11 years• Normal development


Theory-of-Mind: cross-sectional data

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age in months

Tom

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score Model quad2 score Loess



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score boys score girls



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ToM FD1 15% skewness 31


Limits of dynamic growth models (and how they can help to overcome those limits ...)

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Limits• Development is sometimes discontinuous• A developmental level is a range

• Variability and fluctuation

• Fuzziness and ambiguity• One-dimensionality versus multiple states

• Vector-field growth models

• Development through agents• Agent models


Discontinuity and continuity

• a dynamic system can have various attractor states and/or show self-organization

• Which implies that the system will undergo transitions

• Transitions can be continuous or discontinuous, with continuity existing alongside discontinuity

• Discontinuity can be demonstrated by means of so-called catastrophe flags, borrowed from catastrophe theory• Or by means of evidence for some sort of “gap” in the data


Example: Spatial Prepositions

• Marijn van Dijk• 4 sets of data

name ages number of observations gender

Heleen 1;6,4 – 2;5,20 55 FemaleJessica 1;7,12 – 2;6,18 52 FemaleBerend 1;7,14 – 2;7,13 50 Male

Lisa 1;4,12 - 2;4.12 48 Female

• Prepositions used productively in a spatial-referential context


Transition marked by unexpected peak (2)

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Transition marked by jump in extreme range

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Transition marked by discontinuous membership

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Agent models

• Growth models are variable-centered, agent models are agent-centered

• An agent is a collection of variables and relationships between variables

• All agents have the same structure, but different parameters

• Emergent collective behavior and developmental change in the parameters


Emotional expression during interaction

• Henderien Steenbeek• Are there differences in interaction style,

depending on social status? • Method and subjects

• Five- to six-years-olds• Social interaction and emotional expression in

a pretend-play session• Three repeated observations, six week interval

Two “order parameters”: they summarize the behavior of the system

•Action directed towards other person or not•Intensity of emotional expression

What is the time evolution of these order parameters over time?


Realization of concerns

Behaviors of self and other

Emotions of self and other

A dynamic model of social

interaction

determine

determine

Emotional appraisal

dete

rmin

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determine

Strength of concerns

Co-determine

Sets norms to

simulation



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Individual (dyad) short-term time series



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Basic growth equation• In cell for next level type• = preceding cell + RATE * preceding cell +

RATE * ( 1 – preceding cell / K)• Copy to cells below

Documents

Dynamic Growth Modeling