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Blind Signal Separation &
Multivariate Non-Linear
Dependency Measures
SYstemic Risk TOmography:
Signals, Measurements, Transmission Channels, and Policy Interventions
Peter Martey Addo (Speaker) (Centre National De La Recherche Scientifique (CNRS), Universite Paris1 Pantheon-Sorbonne ) Hayette Gatfaoui (NEOMA Business School) Philippe de Peretti (Universite Paris1 Pantheon-Sorbonne) CSRA research meeting – December, 15 2014
Outline: a package of (2) methodologies ©
1 Part I: Causal dependencies in multivariate time seriesTime series graphs & information theoretic measure
2 Part II: Tracking dependency in large multivariate financial systems.Coupling & decoupling information
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 2 / 20
Part I: Causal dependencies in multivariate time series Time series graphs & information theoretic measure
Part I: Multivariate Dependency Measures
“The kiss of information theory that captures systemic risk tomography” (jointwith Philippe de Peretti)
Detection and quantification of causal dependencies in multivariate timeseries
How do we create a formal measure of systemic risk that adequately capturescomplex linkages in the financial system? ©
Granger causality & transfer entropy ??
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 3 / 20
Introduction
Drawbacks of Granger causality & transfer entropy §
Transfer entropy is the information-theoretic analogue of Granger causality
It reduces to Granger causality for vector auto-regressive processes
It is advantageous for the analysis of non-linear signals where the modelassumption of Granger causality might not hold.
The transfer entropy is not uniquely determined by the interaction of the twocomponents alone and depends on misleading effects such as autodependencyand interaction with other process.
It requires arbitrary truncation during estimation, it usually requires moresamples for accurate estimation
Can lead to false interpretation since it is not lag-specific.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 4 / 20
Introduction
Our Offer ©
A novel information theoretic model-free approach to understanding systemicrisk
To detect and quantify causal dependencies from multivariate time series.
It gives similar scores to equally noisy dependencies.
It is uniquely determined by the interaction of the two components alone andin a way autonomous of their interaction with the remaining process.
Excludes the misleading influence of autodependency within a process
Lag-specific. Enhance better interpretation.
Only requires that the multivariate time series be stationary.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 5 / 20
Overview of approach
Methodology
Let X be a multivariate time series with a set of subprocesses V at each timet ∈ Z and directional links be defined in E . Then
G = (V × Z,E )
is the time series graph of X, where the set of nodes in the graph are made up ofV .
Like graphical models (Lauritzen 1996), TSG’s are based on the concept ofconditional independence.
Note that the time-dependence in the time series is used to define directionallinks in the graph.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 6 / 20
Overview of approach
Definition (Lag-specific directed link)
We say that the nodes At−τ ∈ G and Bt ∈ G are connected by a lag-specificdirected link “At −→τ Bt” pointing forward in time if and only if τ > 0 and
IA link−→B
(τ) ≡ I(At−τ ;Bt
∣∣ X−\{At−τ})> 0. (1)
Thus, At−τ and Bt are connected if they are not independent conditionally on thepast of the whole process excluding {At−τ} (denoted by the symbol \) which
implies a lag-specific causality with respect to X. I(·; ·| ·
)denotes conditional
mutual information.
If A 6= B then “At −→τ Bt” represents a coupling at lag τ .
An autodependency at lag τ corresponds to A = B.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 7 / 20
Overview of approach
Definition (Undirected contemporaneous link)
The nodes At ∈ G and Bt ∈ G are connected by an undirected contemporaneouslink “A− B” if and only if
IA
link−B≡ I(At ;Bt
∣∣ X−t+1\{At ,Bt})> 0. (2)
Definition (Notion of parents and neighbors of subprocesses.)
Given the nodes At ∈ G and Bt ∈ G, the parents PBt and neighbors NBt of nodeBt are defined as
PBt ≡ {At−τ : A ∈ X, τ > 0,At−τ −→ Bt} (3)
NBt ≡ {At : A ∈ X,At − Bt} (4)
Definition (Causal Markov Condition)
Any node Bt ∈ G in the time series graph is conditionally independent of X−t \PBt
given its parents PBt .
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 8 / 20
Overview of approach
Definition (Momentary Information Transfer (MIT) Links)
For two subprocesses A, B of a stationary multivariate discrete time process Xwith parents PAt and PBt in the associated time series graph and τ > 0, thegeneral information theoretic measure between nodes At−τ and Bt is given by
IAMIT−→B
(τ) ≡ I(At−τ ;Bt
∣∣ PBt\{At−τ},PAt−τ
)> 0
= H(Bt∣∣ PBt\{At−τ},PAt−τ
)− H
(Bt∣∣ PBt
)and contemporaneous MIT defined by
IA
MIT− B≡ I(At ;Bt
∣∣ PBt ,PAt ,NAt\{Bt},NBt\{At},PNAt \{Bt},PNBt \{At}
). (5)
where H(X ) is Shannon’s entropy and H(X|Y) denotes conditional Shannon’sentropy.
The parents of all subprocesses in X together with the contemporaneous linksforms the time series graph.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 9 / 20
Overview of approach
Definition (Coupling Strength)
The partial correlation MIT measure, denoted ρMIT , associated with equation (5),for the strength of coupling mechanism between At−τ and Bt is given by
ρAMIT−→B
(τ) ≡ ρ(At−τ ;Bt
∣∣ PBt\{At−τ},PAt−τ
). (6)
The measure ρMIT quantifies how much the variability in A at the exact lag τdirectly influences B, irrespective of the past of At−τ and Bt .
ρMIT is the cross-correlation of the residual after At−τ and Bt have beenregressed on both the parents of At−τ and Bt .
The contemporaneous MIT in the linear case is equivalent to the partialcorrelation of the errors after regressing each process on its parents.
Unlike classical statistics, interactions in the framework of information theoryare viewed as transfers of information and thus this approach is model-free.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 10 / 20
Overview of approach
Estimation
Summary
In the first Step, PC algorithm (Spirtes et al [SSCR 1991]) is used toestimate the parents of each process, i.e., as a variable selection method.
Unlike graphical models, only undirected links are inferred and the second stepof PC algorithm is omitted.
This first step determines the existence or absence of a link, which also provideuseful information on the causality between lagged components of themultivariate process.
In the second step, MIT is used and all possible links are tested again.
In this step, the problem of serial dependencies is drastically reduced usingMIT (Runge et al[Physical Review E, 2012]).
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 11 / 20
Overview of approach
Simulation Exercise
Consider a simulated 1000 points of a stationary multivariate autoregressiveprocess made up of four subprocesses {Xt ,Yt ,Zt ,Wt}
′defined by
Xt = aXt−1 + cZt−4 + εx (7)
Yt = kXt−1 + hYt−1 + εy (8)
Zt = dYt−2 + bZt−1 + fWt−1 + εz (9)
Wt = eYt−3 + gWt−1 + εw (10)
and the innovation covariance matrix given by Σε =
1 0 d 00 1 0 dd 0 1 00 d 0 1
, where
a = 0.6, b = 0.4, c = 0.3, d = −0.3, e = −0.6, f = 0.2, g = 0.4, k = 0.3, andh = 0.6.
Notice that the lagged causal chain for this process is X −→1 Y −→2 Z withfeedback Z −→4 X , and Y −→3 W −→1 Z , plus contemporaneous links X − Zand Y −W .
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 12 / 20
Overview of approach
X
Y
Z
W
12
3
41
0.0 0.2 0.4 0.6 0.8 1.0
Auto-MIT1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0
Cross-MIT
0.60.40.20.00.20.40.6
X
→ X → Y → Z → W
0.60.40.20.00.20.40.6
Y
0.60.40.20.00.20.40.6
Z
0 1 2 3 4 5 60.60.40.20.00.20.40.6
W
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6
ρMIT
lag τ [units]
Figure : MIT plot for the simulated process. The plot of significant lags for simulatedprocess associated with the MIT plot.
The results indicates a lagged causal chain as X −→1 Y −→2 Z with feedbackZ −→4 X , and Y −→3 W −→1 Z , plus contemporaneous links X − Z andY −W .
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 13 / 20
Overview of approach
Research-in-progress
Develop nonlinear equivalent measure of the coupling strength.
Empirical application: Causal dependencies in financial institutions
Further Application: Sovereign CDS, Credit Risk etc
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 14 / 20
Part II: Tracking dependency in large multivariate financial systems. Coupling & decoupling information
Part II: Blind Signal Separation
Tracking dependency in large multivariate financial systems.
Track dependency in large multivariate financial systems.
Study the time-varying information coupling and decoupling
Deduce measures of excess dependency, frailty of the market or multivariatefinancial risk
Deduce Early Warning Indicators.
Build dependency measures for multivariate systems that exhibit :
Rapid changing dynamics, i.e. exhibit a high degree of non-linearity
Non-Gaussianity
Non-stationarity.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 15 / 20
Part II: Tracking dependency in large multivariate financial systems. Coupling & decoupling information
Independent Component Analysis (ICA) ©
Assume that the dynamics of the multivariate signal is driven by latentindependent non-gaussian signals
Let {Xt}Tt=1 be a multivariate process with d ∈ Z+ components such that
Xt = ASt (11)
where Xt is the observed process, St are the unobserved independent signals.St signals, as well as the unmixing matrix W :
St =WXt
Thus W = A−1 will contain all relevant information about the dependencystructure of the system, revealed by off-diagonal elements.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 16 / 20
Part II: Tracking dependency in large multivariate financial systems. Coupling & decoupling information
The Idea is to build information coupling measures entirely based on W.
Here we have full information decoupling if W is the identity matrix, andinformation coupling between some components if some elements in rows arenon-zero.
Then given this measures, use rolling windows to study how the couplinginformation evolves over time. Uses it to develep new risk measures or earlywarnings.
The mixing process A might change with time - thus a dynamic mixingprocess can arise due to regime changes or structural changes.
Note that mixing of the latent signals does not need to be linear.
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 17 / 20
Part II: Tracking dependency in large multivariate financial systems. Coupling & decoupling information
Research continues
“If you’re not prepared to be wrong, you’ll never come up with anything original.”(Sir Ken Robinson at TED 2006)
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 18 / 20
Part II: Tracking dependency in large multivariate financial systems. Coupling & decoupling information
EU’s Seventh Framework Programme (SYRTO Project)
This project has received funding from the European Union’s Seventh FrameworkProgramme (FP7-SSH/2007-2013) for research, technological development anddemonstration under grant agreement no320270 (SYRTO)
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 19 / 20
Part II: Tracking dependency in large multivariate financial systems. Coupling & decoupling information
Thank you for your attention ©
Peter Martey Addo (joint with Hayette Gatfaoui & Philippe de Peretti) Centre National De La Recherche Scientifique Universite Paris1 Pantheon-Sorbonne NEOMA Business School The Consortium for Systemic Risk Analytics (CSRA) Massachusetts Institute of Technology (MIT ) Cambridge, MassachusettsInformation theory, Econometrics and Systemic risk December 15, 2014 20 / 20
This project has received funding from the European Union’s
Seventh Framework Programme for research, technological
development and demonstration under grant agreement n° 320270
www.syrtoproject.eu
This document reflects only the author’s views.
The European Union is not liable for any use that may be made of the information contained therein.