Direction of InformatTime-dependent influences in

Linda So

UniversityCenter for Data AFreiburg InstituteFreiburg Institute

26th M

ion Flow in Networksn noisy multivariate time series

ommerlade

y of Freiburg,Analysis and Modeling

for Advanced Studiesfor Advanced Studies

May 2011

Overview

I t d ti• Introduction

• Renormalized partial dir• Renormalized partial dir

• Challenges• Challenges

• State space modelState space model

• Time-dependent interacp

• Summary

rected coherencerected coherence

ctions

Introduction

C d ff t i• Cause precedes effect i

• Cause is the reason for

Example: the items in your shoppthe items in your shoppthe price you have to pa

i tiin time

effect

ing cart are causal foring cart are causal foray

Granger Causality

Li di t bilit• Linear predictability

if is Grangif is Grang• Restricted to measur

not necessarily true c

ger causal forger causal forement:causality

Autoregressive Processes

V t t i• Vector autoregressive p

• Fourier transformation

process

Renormalized Partial Directed

T di i l t• Two dimensional vector

• Normalization

• Renormalized partial dirp

d Coherence

r

rected coherence (rPDC)( )

Renormalized Partial Directed

T di i l t• Two dimensional vector

• Normalization

• Renormalized partial dirp

d Coherence

r

rected coherence (rPDC)( )

Challenge: Observational Nois

2D AR ith b• 2D AR-process with obs

se

ti l iservational noise

Challenge: Nonstationarity

2D AR ith• 2D AR-process with non t ti i t tinstationary interaction

State Space Model I

R iti VAR f d• Rewriting VAR of order

• Linear state space mod

t VAR f dp to VAR of order one

el

State Space Model II

I l di t ti• Including nonstationary

• State space model for p

ffi i tcoefficients

parameters

Dual Kalman Filter and EM-Al

P t tProcess state space

E-step:pestimate process

M-step:calculate most likelyy

and

gorithm

Dual Kalman Filter and EM-Algorithm

P t t tParameter state space

M-step:pcalculate most likely

E-step:estimate parametersp

Dual Kalman Filter and EM-Al

P t tProcess state space

E-step:pestimate process

M-step:calculate most likelyy

and

gorithm

P t t tParameter state space

M-step:pcalculate most likely

E-step:estimate parametersp

Observational Noise

Nonstationarity

Multivariate example

Summary

E ti ti G• Estimating Granger cau

• Challenges: Obeservati• Challenges: Obeservati

• State space models, p ,Dual Kalman filter & EM

Time-dependent influefrom noisy multivariat

l i fl i PDCusal influences using rPDC

onal noise & nonstationarityonal noise & nonstationarity

M-algorithm

ences can be estimated te time series

The Team

University of Aberdeen:

M. Thiel, B. Platt, A. Plano,

University of Freiburg:

J Timmer B Schelter M KJ. Timmer, B. Schelter, M. K

Thanks to J Wohlmuth for conThanks to J. Wohlmuth for con

Thank you for your attention

G. Riedel

Killmann W MaderKillmann, W. Mader

ntribution to the codentribution to the code.

n!