Identifying Dependencies Among Multivariate Time Series

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Identifying Dependencies Among Multivariate Time
Series

• Oscar DE FEO Cristian CARMELISwiss Federal
  Institute of Technology Lausanne

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Outline
0

• ? Framework
• Problem statement
• Available solutions their problems
• ? A Deterministic MV Approach
• Outline working hypothesis
• Estimate connectivity in 3 steps
• ? Tests Results
• Assessment of performances wrt problems
• ? Conclusions Future Work
• What next

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FrameworkProblem statement
?
heterogeneous processes

• Problem
• Given measurements
• multi-site (variate)
• heterogeneous
• Assess interdependencies
• i.e. connectivity graph

• Examples
• Population dynamics
• migration
• Neuroscience
• EEG
• multielectrodes
• Physiology
• heart-breath-brain

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Available SolutionsMethods problems
?

• Old problem ? a lot of methods
• Statistics Information Theory
• Assume random processes
• DFT coh., partial corr.,
• mutual information
• transfer entropy
• Dynamical Systems Theory
• Assume dynamical oscillators
• mutual predictability
• phase synchronization
• mixed state space

Typical problems Symmetry (adirectional) Bivariate
(no marginalization) Assess strong couplings
(DS)
there is space for development
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Proposed Deterministic MV ApproachOutline
working hypothesis
?

• Hypotheses
• Reference model
• heterogeneous network of dynamical oscillators
• weakly coupled

• Working principle
• identify self model for site i
• only use y(i)(t)
• nonlinear model for F (i)
• cross relates the y(j)(t) j ? i
• to the modelling residuals at site i
• linear estimate of K

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Proposed Deterministic MV Approach Estimate
connectivity in 3 steps
?
Step 1 State space reconstruction
Step 2 Self model identification
Step 3 Cross model identification
From y(i)(t) ? R to x(i)(t) ? Rni ? T(i)(t)
From r(i)(t) x(i)(t) - F (i)(x(i)(t-1)) and
x(j)(t) ?j ? i to K (i,j) ?i,j , j ? i r(i)(t)
?j ? i K (i,j)x(j)(t-1) ?
Estimate F (i) x(i)(t1) F (i) (x(i)(t)) ?

• Discrete time modelling
• measures are discrete

• Done with PCA
• noise robustness
• suitable for next step

• Done in two steps
• LS for linear part
• RBF for nonlinear part

• Estimated with LS
• because minimize ?

• Results
• x(i)(t) ?i
• C(i) y(i)(t) C(i) x(i)(t) ?

• Result
• minimal prediction error estimate of F (i) ?i
• compatible with
• small coupling
• small noise

• Result
• coupling matrix C
• cij ?K (i,j)?

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Tests Results IAssessing asymmetry
(directionality) Test bed
?
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Tests Results IAssessing asymmetry
(directionality) Results
?
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Tests Results IIAssessing marginalization
(triangular dependencies) Test bed
?
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Tests Results IIAssessing marginalization
(triangular dependencies) Results
?
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Conclusions Future Work What next
?

• Conclusions
• Method for inferring dependencies among MV
  measurements
• based on deterministic modelling
• assume weak coupling
• Tested on synthetic data proving effective in
• estimating directionality (symmetry problem)
• implicit marginalization (triangular problem)
• Future work
• Address generic couplings
• not necessarily weak
• nonlinear
• Address numeric complexity
• now from divide conquer
• in generic couplings can be a problem