eMVAR – Extended Multivariate Autoregressive
Modelling Toolbox 
Introduction Multivariate (MV) time series analysis is nowadays extensively
used to investigate the concept of connectivity in dynamical systems through approaches which are ubiquitous,
for instance, in the study of physiological time series. The analysis
of connectivity is not only important to detect coupling, i.e., the presence or absence of interactions, between the
considered processes, but also to identify causality, i.e., the presence of driverresponse relationships. Quantitative
assessment of connectivity is commonly performed representing the
considered MV time series as a realization of a linear MV autoregressive
(MVAR) process, and deriving measures of coupling and causality from
the frequency domain representation of the process. The most popular
of these measures are the Coherence (Coh)^{1}, the Partial Coherence (PCoh)^{2,3}, the Directed Coherence (DC)^{4}, the Partial Directed Coherence (PDC)^{5} and related measures (directed transfer function (DTF)^{6,7}, generalized PDC (gPDC)^{8}). Coh and PCoh are symmetric measures, which can be
decomposed into factors eliciting directionality, these factors being
exactly the DC and the PDC. More specifically, these four measures
describe in the frequency domain the time domain concepts of coupling
(Coh), direct coupling (PCoh), causality (DC), and direct causality
(PDC). Therefore, PCoh and PDC measure direct connectivity between
two processes, while Coh and DC account for both direct and indirect
connections. The parametric model traditionally used to compute Coh,
PCoh, DC and PDC is a strictly causal MVAR model, whereby only lagged effects are modeled, while instantaneous
(i.e., not lagged) effects among the time series are not described
by any model coefficients. Nevertheless, neglecting instantaneous
effects may lead to detection of misleading connectivity patterns.
We have recently shown that causality measures may be adversely affected
by the exclusion of instantaneous effects in the model representation^{9}. To overcome this limitation, we have proposed the utilization
of an extended MVAR model which combines both instantaneous and lagged
effects in order to achieve a full description of the correlation
structure of the observed set of time series^{911}. The extended model allows computation of the same connectivity
measures than the traditional strictly causal one: while Coh and PCoh
are identical, DC and PDC may be evaluated either including or excluding
instantaneous effects in the computation of the causality measure;
in the first case the resulting measures, lDC and lPDC, consider only
lagged effects, in a similar way to DC and PDC but resolving the problems
related to model misspecification; in the second case the resulting
measures, eDC and ePDC, are novel extended measures of causality and
direct causality in which instantaneous causality is accounted for
in combination with the traditionally studied lagged causality^{12}. Note that, in the absence of instantaneous effects,
both lagged and extended measures of causality and causality are equivalent
to the to traditional measures,
because the extended model reduces to the classic strictly causal
model. 
The
eMVAR Toolbox The eMVAR Matlab Toolbox performs both traditional MVAR
analysis and extended MVAR analysis, deriving the corresponding frequency
domain measures of connectivity from the time domain model coefficients.
The toolbox provides also several algorithms for the identification
of the two models from time series data, and is completed with algorithms
for model validation and for the estimation of frequencydomain significance
thresholds. It contains a set of functions realizing model identification
and validation and frequency domain analysis, as well as a set of
scripts illustrating the utilization of the various functions. The
code conforms to methods and notation as described in Faes and Nollo^{12}, which is available from this
link. DOWNLOAD: Zip file with all scripts and functions: eMVAR.zip Note: the “functions” and “external” directories need
to be added to the MATLAB path for proper working Description of the Toolbox Functions:
Scripts:
External
functions: The toolbox makes use of a set of external functions
taken or modified from existing MATLAB toolboxes:

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