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Causal Dynamic Time Lag (CDT)Applications to Space Weather
Mandar Chandorkar
1Multiscale DynamicsCWI, Amsterdam
2TAU Research UnitINRIA, Paris-Saclay
25 September 2018
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Figure: The Sun-Earth system
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Figure: Effects of Solar Disturbances
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Figure: Yield curves and Recessions in the U.S. Economy
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Granger Causality
1 The cause happens prior to its effect.
2 The cause has unique information about the future values of its effect.
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Figure: By BiObserver - Own work, CC BY-SA 3.0,https://commons.wikimedia.org/w/index.php?curid=33470670
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Nonparametric learning of time-lag
[Zhou and Sornette, 2006] formulated the problem as minimisation of themismatch between two time series.Inputs: X (t),Y (t), two time series.Learn: A mapping φ(t1) = t2 which minimises
ε(tt , t2) = |X (t1)− Y (t2)| (1)
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Figure: An example of energy landscape EX ,Y given by (1) for two noisy timeseries and the corresponding optimal path wandering at the bottom of the valleysimilarly to a river. This optimal path defines the mapping t1→ t2 = φ(t1).
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Causal Dynamic Time-lag Inference (CDT)
Given an input (cause) and output (effect) time series, predict
Magnitude of output signal. (What/How much)
When the effect will be observed in the output signal (When)
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CDT: Formal Definition
Input Signal (Cause)
t ∈ R+
x(t) ∈ X
Output Signal (Effect)
f : X → Rg : X → R+
∆(t) = g [x(t)]
y(t + ∆(t)) = f [x(t)]
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Model Specification
Input Patterns x(t)
Causal Time WindowLower Limit: ` ∈ N ∪ 0Upper Limit: `+ h : h ∈ N
Targets y(t + `), · · · , y(t + `+ h − 1)
Model OutputsPredictions y(t + `), · · · , y(t + `+ h − 1)Time Lag Probabilities p(t + `), · · · , p(t + `+ h − 1)
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Training: Motivation
Balance two incentives
1 Generate accurate predictions for time windowy(t + `), · · · , y(t + `+ h − 1)
2 Learn time lag structure according to some intuition.
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Training Loss
L(y (1:M), y (1:M), p(1:M)) =λ1∑i ,m
1
2M(y
(m)i − y
(m)i )2(1 + p
(m)i )
+
λ2J (y (1:M), y (1:M), p(1:M))
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Training Loss Contd.
The term J (y (1:M), y (1:M), p(1:M)) penalizes the predicted probabilitiesp(1:M), for deviation from some chosen target probability.
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Target Probability
The target probability p for a time window [t + `, t + `+ h − 1] can becharacterized by:
Conjecture: Causal Time Lag
The lagged output y(t + i) which has greater predictability given x(t), is amore likely causal link.
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Target Probability Contd.
1 The target probability distribution for the time lag is,p(m) = softmax((y (m) − y (m))2/T )
2 The term J (y (1:M), y (1:M), p(1:M)) can be computed as the Hellingerdistance between p and p.
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Need for Benchmarks
1 Labelled data sets are small in size.
2 Most real world data sets dont have explicit labels for causal time lag.
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Benchmark Problems
x(t + 1) = (1− β)x(t) +N (0, σ2)
y(t + ∆(t)) = α||x(t)||2
Problem I: Constant Lag
∆(t) = k
Problem II: Constant Velocity ||x(t)||2; Fixed Distance d
∆(t) = d/(α||x(t)||2)
Problem III: Constant Acceleration a; Fixed Distance d
∆(t) = (√α2||x(t)||4 − 2ad − α||x(t)||2)/a
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Data
Figure: Generated Data
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Errors
Figure: Error in Output vs Error in Time Lag
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Predictions
Figure: Test Set Time Series vs Predictions
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Performance
Output
MAE: 8.602Pearson Corr: 0.964Spearman Corr: 0.999
Time Lag
MAE: 0Pearson Corr: N.ASpearman Corr: 1.0
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Data
Figure: Generated Data
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Time Lags
Figure: Time Lags
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Test Data Distribution
Figure: Output vs Time Lags
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Predictions
Figure: Test Set Time Series vs Predictions
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Errors
Figure: Error in Output vs Error in Time Lag
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Performance
Output
MAE: 30.902Pearson Corr: 0.918Spearman Corr: 0.999
Time Lag
MAE: 1.593Pearson Corr: 0.337Spearman Corr: 0.999
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Outline
1 MotivationSpace WeatherFinancial Markets
2 Causality in Time SeriesConceptsExisting Research
3 Problem & ModelDescriptionProposed Solution
4 ApplicationsBenchmarksProblem IProblem IIProblem III
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Data
Figure: Generated Data
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Time Lags
Figure: Time Lags
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Test Data Distribution
Figure: Output vs Time Lags
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Predictions
Figure: Test Set Time Series vs Predictions
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Errors
Figure: Error in Output vs Error in Time Lag
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Performance
Output
MAE: 24.385Pearson Corr: 0.928Spearman Corr: 0.999
Time Lag
MAE: 1.758Pearson Corr: 0.415Spearman Corr: 0.999
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MDG-TAU Team
Website http://mlspaceweather.org
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References I
Zhou, W.-X. and Sornette, D. (2006).Non-parametric determination of real-time lag structure between twotime series: The optimal thermal causal path method withapplications to economic data.Journal of Macroeconomics, 28(1):195 – 224.Nonlinear Macroeconomic Dynamics.
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