Dynamic Growth Modeling1 Paul van Geert University of Groningen

Dynamic Growth Modeling2 Introductory Theoretical Aspects 1

Dynamic Growth Modeling3 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: “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...”

Dynamic Growth Modeling4 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 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 Albert Einstein: “Everything should be made as simple as possible, but not simpler...”

Dynamic Growth Modeling5 The quadratic model as explanatory theory L t = a + b*t + c*t 2 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 L t = a + b*t + c*t 2 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? Word learning depends on age? How does age affect word learning? Because “age” probably stands for something else, namely the child’s increasing knowledge. But the theory does not specify this. The theory also predicts that a person will either continue to learn ever more words, irrespective of how many words there are in his language, or that at some point in time he will start to forget ever more words… Is this a reasonable theory of word learning? Word learning depends on age? How does age affect word learning? Because “age” probably stands for something else, namely the child’s increasing knowledge. But the theory does not specify this. The theory also predicts that a person will either continue to learn ever more words, irrespective of how many words there are in his language, or that at some point in time he will start to forget ever 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 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 Modeling6 Dynamic growth models: basic principles 2

Dynamic Growth Modeling7 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… 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 Modeling8 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 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

Dynamic Growth Modeling9 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 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

Dynamic Growth Modeling10 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 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

Dynamic Growth Modeling11 Property A Property B support Property A Property B competition Property A Property B support competition Predator-Prey dynamics

Dynamic Growth Modeling12 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 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

Dynamic Growth Modeling13 Motor system Perceptual system Linguistic knowledge Social knowledge Physical knowledge Pedagogical support External symbol systems concerns emotions Fischer’s developmental theory

Dynamic Growth Modeling14 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

Dynamic Growth Modeling15 Pauline Number of Words (2 of 3)

Dynamic Growth Modeling16 Pauline Number of Words (2 of 3)

Dynamic Growth Modeling17 Pauline Number of Words (2 of 3)

Dynamic Growth Modeling18 Dynamic model building Use dynamic modeling to investigate properties of the dynamics Based on simple relationships between variables Supportive Competitive conditional Use dynamic modeling to investigate properties of the dynamics Based on simple relationships between variables Supportive Competitive conditional

Dynamic Growth Modeling19 One-word sentences Holophrastic principle One-word sentences Holophrastic principle 2&3-word sentences Combinatorial principle 2&3-word sentences Combinatorial principle 4&more-word sentences Syntactic principle 4&more-word sentences Syntactic principle supports Competes with supports Competes with

Dynamic Growth Modeling20

Dynamic Growth Modeling21 Descriptive curve fitting 4

Dynamic Growth Modeling22 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 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

Dynamic Growth Modeling23 Example: Peter’s pronomina (1 of 3)

Dynamic Growth Modeling24 Example: Peter’s pronomina (2 of 3)

Dynamic Growth Modeling25 Example: Peter’s pronomina (3 of 3) 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 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

Dynamic Growth Modeling26 Curve fitting in cross-sectional data Theory-of-Mind test: 324 children between 3 and 11 years Normal development Theory-of-Mind test: 324 children between 3 and 11 years Normal development

Dynamic Growth Modeling27 Theory-of-Mind: cross-sectional data

Dynamic Growth Modeling28 Theory-of-Mind: cross-sectional data

Dynamic Growth Modeling29 Theory-of-Mind: cross-sectional data

Dynamic Growth Modeling30 Limits of dynamic growth models (and how they can help to overcome those limits...) 5

Dynamic Growth Modeling31 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 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

Dynamic Growth Modeling32 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 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 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

Dynamic Growth Modeling33 Example: Spatial Prepositions Marijn van Dijk 4 sets of data Prepositions used productively in a spatial- referential context

Dynamic Growth Modeling34 Transition marked by unexpected peak (2)

Dynamic Growth Modeling35 Transition marked by jump in extreme range

Dynamic Growth Modeling36 Transition marked by discontinuous membership

Dynamic Growth Modeling37 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 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

Dynamic Growth Modeling38 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 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? 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?

Dynamic Growth Modeling39 Realization of concerns Behaviors of self and other Emotions of self and other A dynamic model of social interaction determine Emotional appraisal determine Strength of concerns Co- determine Sets norms to simulation

Dynamic Growth Modeling40 Emotional expression during interaction Individual (dyad) short-term time series

Dynamic Growth Modeling41 Emotional expression during interaction

Dynamic Growth Modeling42 Emotional expression during interaction

Dynamic Growth Modeling43 Basic growth equation In cell for next level type = preceding cell + RATE * preceding cell + RATE * ( 1 – preceding cell / K) Copy to cells below In cell for next level type = preceding cell + RATE * preceding cell + RATE * ( 1 – preceding cell / K) Copy to cells below