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

wobbles, humps and sudden jumps 1

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

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

wobbles, humps and sudden jumps - transitions in time 2

Measuring transitions-in-time (1 of 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?

• 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?


Measuring transitions-in-time (2 of 2)Measuring transitions-in-time (2 of 2)

time

continuity

discontinuity


Transitions-in-time and anomalyTransitions-in-time and anomaly

time

continuity

discontinuity

AnomaliesAnomalies

Transition from one set of constraints to another causes

Transition from one set of constraints to another causes

Extremes, sudden change, mixtures, regression, slowing down, …

Extremes, sudden change, mixtures, regression, slowing down, …


Methods for finding transitions-in-timeMethods 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


Transitions, discontinuity and catastrophe theory

Transitions, discontinuity and catastrophe theory

wobbles, humps and sudden jumps - discontinuity 7

Control parameter

Co

ntro

l param

eter

Perfo

rman

ce

Discontinuity: cusp catastrophe (1 of 3)Discontinuity: cusp catastrophe (1 of 3)


(2 of 3) (2 of 3)


(3 of 3) (3 of 3)

Inaccessibleregion


Cusp catastrophe researchCusp 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)


Cusp catastrophe research: problemsCusp 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?


Transitions, continuity and curve fitting

Transitions, continuity and curve fitting

wobbles, humps and sudden jumps - continuity 13

Continuous modelsContinuous 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


Example: Peter’s pronomina (1 of 3)Example: Peter’s pronomina (1 of 3)

-70

-20

30

80

130

180

230

280

330

380

430

75 85 95 105 115 125 135

pronomina Linear model Quadratic Model



-70

-20

30

80

130

180

230

280

330

380

75 85 95 105 115 125 135

pronomina Sigmoid LS Fit Sigmoid Robust Fit



-70

-20

30

80

130

180

230

280

330

380

75 85 95 105 115 125 135

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

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


A critical note on curve fittingA 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…


Indicators of transitions in rangesIndicators 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


Spatial Prepositions (1 of 6)Spatial Prepositions (1 of 6)

• 4 sets of data• 4 sets of data

name agesnumber of

observationsgender

Heleen 1;6,4 – 2;5,20 55 Female

Jessica 1;7,12 – 2;6,18 52 Female

Berend 1;7,14 – 2;7,13 50 Male

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

name agesnumber of

observationsgender

Heleen 1;6,4 – 2;5,20 55 Female

Jessica 1;7,12 – 2;6,18 52 Female

Berend 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

• 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



0

5

10

15

20

25

30

35

40

-180 -130 -80 -30 20 70 120 170 220

age

freq

uen

cy

lisa



0

5

10

15

20

25

30

-270 -220 -170 -120 -70 -20 30 80

age

freq

uen

cy

heleen



0

5

10

15

20

25

30

35

40

-40 10 60 110 160 210 260 310

age

freq

uen

cy

berend



0

5

10

15

20

25

30

35

40

-100 -50 0 50 100 150 200

age

freq

uen

cy

jessica



• Hypothesis: a discontinuous transition

• Alternative hypothesis: continuous increase in level and variability• Simple linear model provides best description

• Hypothesis: a discontinuous transition

• Alternative hypothesis: continuous increase in level and variability• Simple linear model provides best description


Discontinuity in linear model (1 of 2)Discontinuity in linear model (1 of 2)

0

5

10

15

20

25

30

-270 -220 -170 -120 -70 -20 30 80

age

freq

uenc

y

data


Discontinuity in linear model (2 of 2)Discontinuity in linear model (2 of 2)

0

5

10

15

20

25

30

-270 -220 -170 -120 -70 -20 30 80

age

freq

uenc

y

data

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


Transition marked by unexpected peak (1

of 2)


of 2)

0

5

10

15

20

25

30

35

40

-180 -130 -80 -30 20 70 120 170

age

freq

uenc

y

data Lisa linear model



of 2)


of 2)

0

5

10

15

20

25

30

-270 -220 -170 -120 -70 -20 30 80

age

freq

uenc

y

data

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

p-value peak

berend 0.264

heleen 0.406

jessica 0.0004

lisa 0.008

meta analysis

0.0004


Transition marked by jump in maximumTransition marked by jump in maximum

0

5

10

15

20

25

30

35

40

-100 -50 0 50 100 150 200

age

freq

uenc

y

data progmax

MethodApply progressive maximum to time seriesKeep maximum of an expanding time window (focusing on extremes) ResultsAll samples significant“Eyeball” estimation matches maximum level criterionDiscussionTransition marked by a discontinuous jump in the maximal level of productionSee Fischer

MethodApply progressive maximum to time seriesKeep maximum of an expanding time window (focusing on extremes) ResultsAll samples significant“Eyeball” estimation matches maximum level criterionDiscussionTransition marked by a discontinuous jump in the maximal level of productionSee Fischer

estimated position

p-value distance

Berend 0 0.029Heleen 5 0.01Jessica 0 0.05

Lisa 0 0.02


Transition marked by jump in extreme rangeTransition marked by jump in extreme range

0

5

10

15

20

25

30

35

40

-100 -50 0 50 100 150 200

age

freq

uenc

y

data progmax regmin

MethodAdd regressive maximum to time seriesStart at end and keep minimum of time window expanding towards the beginning (focusing on extremes in maximum and minimum) ResultsAll samples significant“Eyeball” estimation exactly matches range criterion DiscussionTransitions are expressed through the extremes

MethodAdd regressive maximum to time seriesStart at end and keep minimum of time window expanding towards the beginning (focusing on extremes in maximum and minimum) ResultsAll samples significant“Eyeball” estimation exactly matches range criterion DiscussionTransitions are expressed through the extremes

estimated position

p-value distance

Berend 0 0.04Heleen 0 0.16Jessica 0 0.07

Lisa 0 0.03


Pauline Number of Words (1 of 3)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

• Number of words from one-word to multi-word sentences

• Mean-length-of-utterance = continuous development

• Variability provides an indication of discontinuity or transition



-10

0

10

20

30

40

50

60

14 19 24 29 34

age

freq

uen

cy

M1 M1 smooth M23 M23 smooth M422 M422 smooth



-0.2

0

0.2

0.4

0.6

0.8

1

1.2

14 19 24 29 34

age

freq

uenc

y

M1 M23 M422 total var smoothed rescaled

MethodUse 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 peakRepeat 1000 timesResultsSimulation reconstructs average variability, but not the observed variability peak DiscussionIncreased variability at the transition from combinatorial to grammatical sentences

MethodUse 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 peakRepeat 1000 timesResultsSimulation reconstructs average variability, but not the observed variability peak DiscussionIncreased variability at the transition from combinatorial to grammatical sentences


A note on longitudinal data setsA 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

• 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


An example from cross-sectional dataAn 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


An example from cross-sectional dataAn example from cross-sectional data

20

30

40

50

60

70

80

90

100

35 55 75 95 115 135

age in months

To

m s

co

re

score Model quad2 score Loess

MethodLoess smoothed curve (40% window)Compared with quadratic model200 datasets simulated based on quadratic model and model of varianceAll sets smoothed with same Loess procedureLook for a piece of the curve that’s as anomalous as the anomaly in the real dataResultsAnomaly cannot be reconstructed by quadratic model DiscussionCould still be an artifact of the subject sampling…

MethodLoess smoothed curve (40% window)Compared with quadratic model200 datasets simulated based on quadratic model and model of varianceAll sets smoothed with same Loess procedureLook for a piece of the curve that’s as anomalous as the anomaly in the real dataResultsAnomaly cannot be reconstructed by quadratic model DiscussionCould still be an artifact of the subject sampling…

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

Documents