1. wobbles, humps and sudden jumps1 Transitions in time: what to look for and how to describe them …

2. wobbles, humps and sudden jumps - transitions in time2 Measuring transitions-in-time (1 of 2) Transition = transitory change from one set of constraints to another What are the empirical indicators of a transition? What methods can be used to find and characterize a transition? Transition = transitory change from one set of constraints to another What are the empirical indicators of a transition? What methods can be used to find and characterize a transition?

3. wobbles, humps and sudden jumps - transitions in time3 Measuring transitions-in-time (2 of 2) time continuity discontinuity

4. wobbles, humps and sudden jumps - transitions in time4 Transitions-in-time and anomaly time continuity discontinuity Anomalies Transition from one set of constraints to another causes Extremes, sudden change, mixtures, regression, slowing down, …

5. wobbles, humps and sudden jumps5 Methods for finding transitions-in-time Direct fitting of transition models Discontinuous models Continuous models Looking for qualitative indicators Catastrophe flags Qualitative indicators Direct fitting of transition models Discontinuous models Continuous models Looking for qualitative indicators Catastrophe flags Qualitative indicators

6. wobbles, humps and sudden jumps6 Transitions, discontinuity and catastrophe theory

7. wobbles, humps and sudden jumps - discontinuity7 Control parameter Performance Discontinuity: cusp catastrophe (1 of 3)

8. wobbles, humps and sudden jumps - discontinuity8 (2 of 3)

9. wobbles, humps and sudden jumps - discontinuity9 (3 of 3) Inaccessible region

10. wobbles, humps and sudden jumps - discontinuity10 Cusp catastrophe research Empirical indicators: 8 catastrophe “flags” Sudden jump, anomalous variance, inaccessible region, … Applied to Conservation (van der Maas and Molenaar) Reaching and grasping (Wimmers & Savelsbergh) Function words (Ruhland & VG) Analogous reasoning (van der Maas, Hosenfeld,..) Balance Scale task (van der Maas) Empirical indicators: 8 catastrophe “flags” Sudden jump, anomalous variance, inaccessible region, … Applied to Conservation (van der Maas and Molenaar) Reaching and grasping (Wimmers & Savelsbergh) Function words (Ruhland & VG) Analogous reasoning (van der Maas, Hosenfeld,..) Balance Scale task (van der Maas)

11. wobbles, humps and sudden jumps - discontinuity11 Cusp catastrophe research: problems Based on two control parameters Only few of the 8 flags are found Some require experimental manipulation What if the states of the control parameters are fuzzy (ranges)? Is this the only definition of discontinuity ? Based on two control parameters Only few of the 8 flags are found Some require experimental manipulation What if the states of the control parameters are fuzzy (ranges)? Is this the only definition of discontinuity ?

12. wobbles, humps and sudden jumps12 Transitions, continuity and curve fitting

13. wobbles, humps and sudden jumps - continuity13 Continuous models Simple curves Linear, quadratic, exponential … Not a transition Transition curves S-shaped curves: logistic, sigmoid, cumulative Gaussian, … Eventually look very discontinuous… Smoothing and denoising curves Loess smoothing Very flexible Simple curves Linear, quadratic, exponential … Not a transition Transition curves S-shaped curves: logistic, sigmoid, cumulative Gaussian, … Eventually look very discontinuous… Smoothing and denoising curves Loess smoothing Very flexible

14. wobbles, humps and sudden jumps - continuity14 Example: Peter’s pronomina (1 of 3)

15. wobbles, humps and sudden jumps - continuity15 Example: Peter’s pronomina (2 of 3)

16. wobbles, humps and sudden jumps - continuity16 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

17. wobbles, humps and sudden jumps - continuity17 A critical note on curve fitting We fit a continuous model through the data and assume it approximates the real, underlying curve Observed data = curve plus error OK if the underlying phenomenon is indeed a point source and noise is added from an external source However, if we deal with behavior, the real thing is the range A curve isn’t but a “geographical” marking point, no underlying reality The Greenwich meridian… We fit a continuous model through the data and assume it approximates the real, underlying curve Observed data = curve plus error OK if the underlying phenomenon is indeed a point source and noise is added from an external source However, if we deal with behavior, the real thing is the range A curve isn’t but a “geographical” marking point, no underlying reality The Greenwich meridian…

18. wobbles, humps and sudden jumps - continuity18 Indicators of transitions in ranges Spatial prepositions Is there a discontinuity? Number of words in early sentences Is variability an indicator of a transition? Cross-sectional Scores on a theory-of-mind test An anomaly in cross-sectional data? Stability of Sociometric ratings of children Is there a categorical distinction between stable and unstable ratings Spatial prepositions Is there a discontinuity? Number of words in early sentences Is variability an indicator of a transition? Cross-sectional Scores on a theory-of-mind test An anomaly in cross-sectional data? Stability of Sociometric ratings of children Is there a categorical distinction between stable and unstable ratings

19. wobbles, humps and sudden jumps - continuity19 Spatial Prepositions (1 of 6) 4 sets of data Prepositions used productively in a spatial- referential context Why language? Categorical nature: preposition or not Relatively easy to observe and interpret High sampling frequency possible Prepositions used productively in a spatial- referential context Why language? Categorical nature: preposition or not Relatively easy to observe and interpret High sampling frequency possible

20. wobbles, humps and sudden jumps - continuity20 Spatial Prepositions (2 of 6)

21. wobbles, humps and sudden jumps - continuity21 Spatial Prepositions (3 of 6)

22. wobbles, humps and sudden jumps - continuity22 Spatial Prepositions (4 of 6)

23. wobbles, humps and sudden jumps - continuity23 Spatial Prepositions (5 of 6)

24. wobbles, humps and sudden jumps - continuity24 Spatial Prepositions (6 of 6) Hypothesis: a discontinuous transition Alternative hypothesis: continuous increase in level and variability Simple linear model provides best description

25. wobbles, humps and sudden jumps - continuity25 Discontinuity in linear model (1 of 2)

26. wobbles, humps and sudden jumps - continuity26 Discontinuity in linear model (2 of 2) What is the probability that a linear increase in level and variability produces maximal gaps as big as or bigger than the maximal gap observed in the data? Method Simulate datasets based on the linear model of level and variability Calculate the maximal gap for every simulated set Count the number of times the simulated gap is as big as or bigger than the observed one Divide this number by the number of simulations: p-value What is the probability that a linear increase in level and variability produces maximal gaps as big as or bigger than the maximal gap observed in the data? Method Simulate datasets based on the linear model of level and variability Calculate the maximal gap for every simulated set Count the number of times the simulated gap is as big as or bigger than the observed one Divide this number by the number of simulations: p-value

27. wobbles, humps and sudden jumps - continuity27 Transition marked by unexpected peak (1 of 2)

28. wobbles, humps and sudden jumps - continuity28 Transition marked by unexpected peak (2 of 2) What is the probability that a linear increase in level and variability produces peaks as big as or bigger than the maximal peak observed in the data? Method Simulate datasets based on the linear model of level and variability Calculate the peak for every simulated set Count the number of times the simulated peak is as big as or bigger than the observed one Divide this number by the number of simulations: p-value What is the probability that a linear increase in level and variability produces peaks as big as or bigger than the maximal peak observed in the data? Method Simulate datasets based on the linear model of level and variability Calculate the peak for every simulated set Count the number of times the simulated peak is as big as or bigger than the observed one Divide this number by the number of simulations: p-value Results The peak is significant in two of the four children Results The peak is significant in two of the four children

29. wobbles, humps and sudden jumps - continuity29 Transition marked by jump in maximum Method Apply progressive maximum to time series Keep maximum of an expanding time window (focusing on extremes) Results All samples significant “Eyeball” estimation matches maximum level criterion Discussion Transition marked by a discontinuous jump in the maximal level of production See Fischer Method Apply progressive maximum to time series Keep maximum of an expanding time window (focusing on extremes) Results All samples significant “Eyeball” estimation matches maximum level criterion Discussion Transition marked by a discontinuous jump in the maximal level of production See Fischer

30. wobbles, humps and sudden jumps - continuity30 Transition marked by jump in extreme range Method Add regressive maximum to time series Start at end and keep minimum of time window expanding towards the beginning (focusing on extremes in maximum and minimum) Results All samples significant “Eyeball” estimation exactly matches range criterion Discussion Transitions are expressed through the extremes Method Add regressive maximum to time series Start at end and keep minimum of time window expanding towards the beginning (focusing on extremes in maximum and minimum) Results All samples significant “Eyeball” estimation exactly matches range criterion Discussion Transitions are expressed through the extremes

31. wobbles, humps and sudden jumps - continuity31 Pauline Number of Words (1 of 3) Number of words from one-word to multi- word sentences Mean-length-of-utterance = continuous development Variability provides an indication of discontinuity or transition

32. wobbles, humps and sudden jumps - continuity32 Pauline Number of Words (2 of 3)

33. wobbles, humps and sudden jumps - continuity33 Pauline Number of Words (3 of 3) Method Use the smoothed curves as an estimation of the probability that an M1, M23 or M4-22 sentence will be produced and simulate sets of 60 sentences over 46 simulated observations. Calculate difference between simulated sentences and model; calculate total variability and retain highest peak Repeat 1000 times Results Simulation reconstructs average variability, but not the observed variability peak Discussion Increased variability at the transition from combinatorial to grammatical sentences Method Use the smoothed curves as an estimation of the probability that an M1, M23 or M4-22 sentence will be produced and simulate sets of 60 sentences over 46 simulated observations. Calculate difference between simulated sentences and model; calculate total variability and retain highest peak Repeat 1000 times Results Simulation reconstructs average variability, but not the observed variability peak Discussion Increased variability at the transition from combinatorial to grammatical sentences

34. wobbles, humps and sudden jumps - continuity34 A note on longitudinal data sets Time-series data from language are not representative: most time-series sets are smaller! Size of the data set, nature of the missing data, conditional dependencies and violations of “normality” are characteristic of the data Permutation, resampling and monte-carlo techniques are good alternatives to standard statistical tests

35. wobbles, humps and sudden jumps - continuity35 An example from cross-sectional data Scores on a Theory-of-Mind test 233 children from 3 to 11 years old Normally developing children Scores on a Theory-of-Mind test 233 children from 3 to 11 years old Normally developing children

36. wobbles, humps and sudden jumps - continuity36 An example from cross-sectional data Method Loess smoothed curve (40% window) Compared with quadratic model 200 datasets simulated based on quadratic model and model of variance All sets smoothed with same Loess procedure Look for a piece of the curve that’s as anomalous as the anomaly in the real data Results Anomaly cannot be reconstructed by quadratic model Discussion Could still be an artifact of the subject sampling… Method Loess smoothed curve (40% window) Compared with quadratic model 200 datasets simulated based on quadratic model and model of variance All sets smoothed with same Loess procedure Look for a piece of the curve that’s as anomalous as the anomaly in the real data Results Anomaly cannot be reconstructed by quadratic model Discussion Could still be an artifact of the subject sampling…