Biomedical Engineering Reference
In-Depth Information
Chapter 8
Estimation of Causal Networks:
Source-Space Causality Analysis
8.1 Introduction
This chapter reviews the methodology for estimating causal relationships among
cortical activities in MEG/EEG source space analysis. The source space causality
analysis computes some types of causality measures using the estimated time series of
the source activities of interest. Commonly-used measures are the Granger causality
related measures, which are based on the MVAR modeling of the voxel time series.
We first describe these Granger causality based measures, including the original def-
inition in the temporal domain [
1
,
2
], Geweke's extension to the spectral domain
[
3
,
4
], and related measures such as the directed transfer function (DTF) [
5
] and the
partial directed coherence (PDC) [
6
]. The transfer entropy [
7
], which is a representa-
tive non-parametric measure, is also discussed, with a proof of its equivalence to the
Granger causality under the Gaussianity assumption. The last section of this chapter
describes methods of estimating the MVAR coefficients. Here we present a con-
ventional least-square-based algorithm and a method based on the sparse Bayesian
inference for this MVAR estimation problem. The sparse Bayesian algorithm out-
performs the conventional method when the sensor data contains a large amount of
interferences.
8.2 Multivariate Vector Autoregressive (MVAR) Process
8.2.1 MVAR Modeling of Time Series
The Granger-causality measures make use of multivariate vector auto regressive
(MVAR) process of the source time series. We first explain the modeling of time
series using the MVAR process. A general approach to the source-space causality
analysis first chooses a relatively small number of voxels corresponding to the source
activities of interest, and the causal relationships among time series of the selected
voxels are estimated.
