Transitions in time : what to look for and how to describe them …

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

2. 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? wobbles, humps and sudden jumps - transitions in time

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

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

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

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

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

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

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

10. 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) wobbles, humps and sudden jumps - discontinuity

11. 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? wobbles, humps and sudden jumps - discontinuity

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

13. 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 wobbles, humps and sudden jumps - continuity

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

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

16. 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 wobbles, humps and sudden jumps - continuity

17. 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… wobbles, humps and sudden jumps - continuity

18. 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 wobbles, humps and sudden jumps - continuity

19. 4 sets of data Spatial Prepositions (1 of 6) • 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 wobbles, humps and sudden jumps - continuity

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

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

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

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

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

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

26. 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 wobbles, humps and sudden jumps - continuity

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

28. 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 Results • The peak is significant in two of the four children wobbles, humps and sudden jumps - continuity

29. 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 wobbles, humps and sudden jumps - continuity

30. 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 wobbles, humps and sudden jumps - continuity

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

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

33. 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 wobbles, humps and sudden jumps - continuity

34. 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 A note on longitudinal data sets wobbles, humps and sudden jumps - continuity

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

36. 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… wobbles, humps and sudden jumps - continuity