Graphical interaction models for multivariate time series 1

Abstract

In this paper we extend the concept of graphical models for multivariate data to multivariate time series. We de ne a partial correlation graph for time series and use the partial spectral coherence between two components given the remaining components to identify the edges of the graph. As an example we consider multivariate autoregressive processes. The method is applied to air pollution data. 1 This work has been supported by a European Union Capital and Mobility Programme (ERB CHRX-CT 940693) AMS 1991 subject classi cations. Primary 62M15 secondary 62F10. Key words and phrases. Graphical models, multivariate time series, partial spectral coherence, spectral estimates, multivariate autoregressive processes, air pollution data.

Graphical modelling for brain connectivity via partial coherence

“…X j ⊥ ⊥X k | X (\jk) means that {X j,t } and {X k,t } are uncorrelated given the other (p − 2) component processes. The following are all equivalent (Dahlhaus, 2000)…”

“…As Eichler (1999) puts it, the inversion method "allows an efficient computation of all frequency domain statistics..." This approach is similarly recommended in Dahlhaus (2000), Dahlhaus et al (1997) and also used in Salvador et al (2005), and it is used here.…”

“…The theory of time series graphical modelling (Dahlhaus, 2000) states that an interaction exists if the partial coherence is non-zero at one or more frequencies. From a practical perspective we might like to distinguish between partial coherencies which are non-zero over multiple frequencies and those which exhibit a spike (perhaps spurious) at one odd frequency.…”

“…Our ultimate objective is to use graphical models to compare brain connectivity in these two groups. If as here, sources of 'driving' are not being investigated, and linear relationships between series are assumed, then partial coherence is a well-established frequency domain approach for constructing graphical models (Dahlhaus, 2000).…”

A Bayes Random Fields Approach for Integrative Large-Scale Regulatory Network Analysis

“…Based on this theory, partial spectral coherence was proposed for frequency domain analysis of time series [27] and it can be obtained by the inverse of the spectral matrix [29]. However, these classical time series analysis techniques are not readily applicable to transciptome data, where the number of data points n far exceed the sample size t, i.e.…”