eMVAR – Extended Multivariate Autoregressive Modelling Toolbox
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 driver-response 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 representation9. 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 series9-11. 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 mis-specification; 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 causality12. 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 frequency-domain 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 Nollo12, which is available from this link.
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
The toolbox makes use of a set of external functions taken or modified from existing MATLAB toolboxes:
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Last edited March 2, 2011