transitions in time : what to look for and how to describe them …
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Transitions in time : what to look for and how to describe them …. 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?. - PowerPoint PPT PresentationTRANSCRIPT
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?
wobbles, humps and sudden jumps - transitions in time 3
Measuring transitions-in-time (2 of 2)Measuring transitions-in-time (2 of 2)
time
continuity
discontinuity
wobbles, humps and sudden jumps - transitions in time 4
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, …
wobbles, humps and sudden jumps 5
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
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Transitions, discontinuity and catastrophe theory
Transitions, discontinuity and catastrophe theory
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Control parameter
Co
ntro
l param
eter
Perfo
rman
ce
Discontinuity: cusp catastrophe (1 of 3)Discontinuity: cusp catastrophe (1 of 3)
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(2 of 3) (2 of 3)
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(3 of 3) (3 of 3)
Inaccessibleregion
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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)
wobbles, humps and sudden jumps - discontinuity 11
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?
wobbles, humps and sudden jumps 12
Transitions, continuity and curve fitting
Transitions, continuity and curve fitting
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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
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Example: Peter’s pronomina (1 of 3)Example: Peter’s pronomina (1 of 3)
-70
-20
30
80
130
180
230
280
330
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430
75 85 95 105 115 125 135
pronomina Linear model Quadratic Model
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Example: Peter’s pronomina (2 of 3)Example: Peter’s pronomina (2 of 3)
-70
-20
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75 85 95 105 115 125 135
pronomina Sigmoid LS Fit Sigmoid Robust Fit
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Example: Peter’s pronomina (3 of 3)Example: Peter’s pronomina (3 of 3)
-70
-20
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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
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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…
wobbles, humps and sudden jumps - continuity 18
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
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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
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Spatial Prepositions (2 of 6)Spatial Prepositions (2 of 6)
0
5
10
15
20
25
30
35
40
-180 -130 -80 -30 20 70 120 170 220
age
freq
uen
cy
lisa
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Spatial Prepositions (3 of 6)Spatial Prepositions (3 of 6)
0
5
10
15
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30
-270 -220 -170 -120 -70 -20 30 80
age
freq
uen
cy
heleen
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Spatial Prepositions (4 of 6)Spatial Prepositions (4 of 6)
0
5
10
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-40 10 60 110 160 210 260 310
age
freq
uen
cy
berend
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Spatial Prepositions (5 of 6)Spatial Prepositions (5 of 6)
0
5
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-100 -50 0 50 100 150 200
age
freq
uen
cy
jessica
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Spatial Prepositions (6 of 6)Spatial Prepositions (6 of 6)
• 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
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Discontinuity in linear model (1 of 2)Discontinuity in linear model (1 of 2)
0
5
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15
20
25
30
-270 -220 -170 -120 -70 -20 30 80
age
freq
uenc
y
data
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Discontinuity in linear model (2 of 2)Discontinuity in linear model (2 of 2)
0
5
10
15
20
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-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
wobbles, humps and sudden jumps - continuity 27
Transition marked by unexpected peak (1
of 2)
Transition marked by unexpected peak (1
of 2)
0
5
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15
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-180 -130 -80 -30 20 70 120 170
age
freq
uenc
y
data Lisa linear model
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Transition marked by unexpected peak (2
of 2)
Transition marked by unexpected peak (2
of 2)
0
5
10
15
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-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
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Transition marked by jump in maximumTransition marked by jump in maximum
0
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-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
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Transition marked by jump in extreme rangeTransition marked by jump in extreme range
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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
wobbles, humps and sudden jumps - continuity 31
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
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Pauline Number of Words (2 of 3)Pauline Number of Words (2 of 3)
-10
0
10
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30
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14 19 24 29 34
age
freq
uen
cy
M1 M1 smooth M23 M23 smooth M422 M422 smooth
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Pauline Number of Words (3 of 3)Pauline Number of Words (3 of 3)
-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
wobbles, humps and sudden jumps - continuity 34
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
wobbles, humps and sudden jumps - continuity 35
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
wobbles, humps and sudden jumps - continuity 36
An example from cross-sectional dataAn example from cross-sectional data
20
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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…