Dynamic Growth Modeling

1. Dynamic Growth Modeling Paul van Geert University of Groningen Dynamic Growth Modeling

2. 1 Introductory Theoretical Aspects Dynamic Growth Modeling

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

4. Albert Einstein: “Everything should be made as simple as possible, but not simpler...” An example: the vocabulary spurt • Spurt in the lexicon in the second year of life • 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 Dynamic Growth Modeling

5. 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 The quadratic model as explanatory theory 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… • 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 Dynamic Growth Modeling

6. 2 • Dynamic growth models: basic principles Dynamic Growth Modeling

7. 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 Modeling

8. 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 Dynamic Growth Modeling

9. 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 Dynamic Growth Modeling

10. 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 Dynamic Growth Modeling

11. Property A competition competition Property A support Property B support Property A Property B support competition Property B Predator-Prey dynamics Dynamic Growth Modeling

12. Linguistic knowledge concerns Motor system Social knowledge Perceptual system Physical knowledge emotions Pedagogical support External symbol systems • The form of the developmental process is determined by the way the variables interact with each other • Stepwise development (stages) • Temporary regressions Dynamic Growth Modeling

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

14. 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 Pauline Number of Words (1 of 3) Dynamic Growth Modeling

15. Pauline Number of Words (2 of 3) Dynamic Growth Modeling

16. Pauline Number of Words (2 of 3) Dynamic Growth Modeling

17. Pauline Number of Words (2 of 3) Dynamic Growth Modeling

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

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

20. Dynamic Growth Modeling

21. 4 Descriptive curve fitting Dynamic Growth Modeling

22. 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 Dynamic Growth Modeling

23. Example: Peter’s pronomina(1 of 3) Dynamic Growth Modeling

24. Example: Peter’s pronomina(2 of 3) Dynamic Growth Modeling

25. 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 Dynamic Growth Modeling

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

27. Theory-of-Mind: cross-sectional data Dynamic Growth Modeling

28. Theory-of-Mind: cross-sectional data Dynamic Growth Modeling

29. Theory-of-Mind: cross-sectional data Dynamic Growth Modeling

30. 5 • Limits of dynamic growth models (and how they can help to overcome those limits ...) Dynamic Growth Modeling

31. 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 Dynamic Growth Modeling

32. 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 Dynamic Growth Modeling

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

34. Transition marked by unexpected peak (2) Dynamic Growth Modeling

35. Transition marked by jump in extreme range Dynamic Growth Modeling

36. Transition marked by discontinuous membership Dynamic Growth Modeling

37. 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 Dynamic Growth Modeling

38. Emotional expression during interaction • 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? • 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 Dynamic Growth Modeling

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

40. Emotional expression during interaction Individual (dyad) short-term time series Dynamic Growth Modeling

41. Emotional expression during interaction Dynamic Growth Modeling

42. Emotional expression during interaction Dynamic Growth Modeling

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