The present study introduces a new framework for the
frequency-domain evaluation of directional influences in jointly stationary
multivariate vector processes. The framework extends to the study of vector
processes the DC/PDC framework, and provides a full multivariate account for
the Geweke framework. As such, it is recommended for the evaluation of causal
relationships between multiple blocks of time series, with typical application
in neurophysiology where multichannel data acquisition technologies allow
to monitor many regions of interest with many recordings per region. The proposed
framework is exploited to define new frequency domain connectivity measures,
which are shown (i) to possess desirable theoretical properties of causality
measures; (ii) to be able to reflect either direct causality or total (i.e.,
direct+indirect) causality from one vector process to another in the multivariate
representation; (iii) to reduce to known causality measures derived from the
Geweke framework in the case of bivariate vector processes, and from the DC/PDC
framework in the case of multivariate scalar processes.
DOWNLOAD:
Zip file with all scripts and functions: blockMVAR.zip
Description of the Toolbox
Functions:
- block_fdMVAR.m
: performs block-based frequency domain connectivity analysis from the parameters
of a strictly causal MVAR model; returns the following coupling measures
block directed coherence, block partial directed coherence, multivariate
direct causality, multivariate total causality.
- block_fdMVAR_diag.m
: similar to block_fdmvar, but forces the input covariance and its inverse
to be diagonal matrices. This is to prevent negative values of the causality
functions on real applications where the model inputs may be correlated
(so the model is not strictly causal as it should be for these analyses).
These functions realize for the block case the so-called "generalized"
formulation (gDTF=DC,gPDC) defined in the Baccalà papers, while the
functions above with non-diagonal covariances realize the so-called "information"
formulation. This modification was proposed in the IJBEM paper, and is used
in the practical analysis of the Biol Cyb paper.
- Geweke_f.m : estimates
f and f_cond, logarithmic frequency domain causality and conditional causality
measures, defined in Geweke (1982) and Geweke (1984) seminal papers.
Scripts:
- example.m : runs the
theoretical example of Faes and Nollo, generating Figure 2 of the paper,
i.e., analytical computation of frequency domain logarithmic and non-logarithmic
causality measures for an illustrative theoretical example.
- example_Fig6.m
: runs the simulated example of Faes and Nollo, generating Figure 6 of the
paper, i.e., estimation of frequency domain logarithmic causality measures
for several realizations of the illustrative theoretical example, and comparison
with estimated known Geweke measures.
The toolbox makes also use of functions taken from the
eMVAR toolbox:
- idMVAR.m : identification
of strictly causal MVAR model: estimates model coefficients, innovations
and innovation covariance from a given time series and a given model order.
The default identification algorithm is the standard least squares method, but several other algorithms may be recalled.
- MVARfilter.m : yields
a single realization of a strictly causal MVAR process of assigned dimensionality
and length, given strictly causal coefficients and residuals.