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 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.

 

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:

  • fdMVAR.m : performs frequency domain connectivity analysis from the parameters of a strictly causal MVAR model; returns causality and coupling measures (i.e., DC4, DTF7, PDC5, gPDC8, Coh1, PCoh3), as well as spectral and transfer function matrices1.
  • fdMVAR0.m : performs frequency domain connectivity analysis from the parameters of an extended MVAR model12; returns frequency domain spectral functions, as well as frequency domain extended and lagged measures of causality and coupling (eDC, lDC, ePDC, lPDC, Coh, PCoh).
  • 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 method12, but several other algorithms may be recalled13.
  • idMVAR0prior.m : identification of extended MVAR model through the approach using prior information to estimate the instantaneous model: estimates strictly causal and extended model coefficients, innovations and innovation covariance.
  • idMVAR0ng.m : identification of extended MVAR model through the approach using non-gaussianity of the innovations to estimate the instantaneous model: estimates strictly causal and extended model coefficients, innovations and innovation covariance.
  • mos_idMVAR.m : model order selection for strictly causal MVAR model, using Akaike Information Criterion14 or Minimum description Length criterion15.
  • test_whiteness.m : test for whiteness of strictly causal model residuals; it is the multivariate Ljung-Box portmanteau test16
  • test_independence.m : test for independence of strictly causal model residuals; may be Spearman rho or Kendall tau test.
  • test_gaussianity.m : test for gaussianity of strictly causal residuals; it is the Jarque-Bera test for non-normality16.
  • surrVFT.m : generation of FT surrogates for testing the significance of the coherence17.
  • surrVCFTf.m : generation of causal FT surrogates in which full causality from one series to another is destroyed; they can be used for testing the significance of the DC18. This implementation works for a strictly causal MVAR model.
  • surrVCFTd.m : generation of causal FT surrogates in which only direct causality from one series to another is destroyed; they can be used for testing the significance of the PDC18. This implementation works for a strictly causal MVAR model.
  • surrVCFTf0.m : generation of causal FT surrogates in which full causality from one series to another is destroyed; they can be used for testing the significance of eDC and lDC18. This implementation works for an extended MVAR model.
  • surrVCFTd0.m : generation of causal FT surrogates in which only direct causality from one series to another is destroyed; they can be used for testing the significance of the ePDC and lPDC18. This implementation works for an extended MVAR model.
  • InstModelfilter.m : realizes strictly causal residuals of assigned dimensionality, given either their instantaneous covariance or their dependence on extended residuals (i.e., given instantaneous effects and instantaneous covariance of extended residuals); in the second case, non-gaussianity may be allowed for the extended residuals.
  • MVARfilter.m : yields a single realization of a strictly causal MVAR process of assigned dimensionality and length, given strictly causal coefficients and residuals.
  • choldiag.m : performs diagonal Cholesky decomposition; useful to derive extended input covariance and mixing matrix (related to instantaneous effects matrix) from the strictly causal input covariance.
  • diag_coeff_rev.m : moves from extended MVAR representation to strictly causal MVAR representation (useful in Section 1 only when coeffs are given in the extended representation)
  • simuMVARcoeff.m : generates the theoretical coefficients of some illustrative MVAR processes9,12,19.

 

Scripts:

  • example1.m : runs the theoretical example 1 in Faes and Nollo12, generating Figs 2,3,4, i.e., spectral functions and decompositions for an illustrative strictly causal MVAR process.
  • example1_realization.m : performs a full realization of the theoretical example 1 in Faes and Nollo12 (and of any other theoretical MVAR process in simuMVARcoeff.m). The script computes the theoretical strictly causal model describing the given process as well as the theoretical frequency domain functions, generates a finite length realization of the process, estimates the strictly causal model and spectral functions from the realization, performs tests on the residuals, as well as full significance testing through surrogates.
  • example2.m : runs the theoretical example 2 in Faes and Nollo12, generating Figs 6,7,8, i.e., spectral, coupling and causality functions for an illustrative extended MVAR process.
  • example2_realization_prior.m : performs a full realization of the theoretical example 2 in Faes and Nollo12 (and of any other theoretical MVAR process in simuMVARcoeff.m). The script computes the theoretical extended model describing the given process as well as the theoretical frequency domain functions, generates a finite length realization of the process, estimates the extended model and spectral functions from the realization, performs tests on the residuals, as well as full significance testing through surrogates. Identification is based on the method using prior information to estimate the instantaneous model.
  • example2_realization_ng.m : performs a full realization of the theoretical example 2 in Faes and Nollo12 (and of any other theoretical MVAR process in simuMVARcoeff.m). The script computes the theoretical extended model describing the given process as well as the theoretical frequency domain functions, generates a finite length realization of the process, estimates the extended model and spectral functions from the realization, performs tests on the residuals, as well as full significance testing through surrogates. Identification is based on the method using non-gaussianity of the innovations to estimate the instantaneous model.

 

External functions:

The toolbox makes use of a set of external functions taken or modified from existing MATLAB toolboxes:

  • the BioSig Toolbox (http://biosig.sourceforge.net/), with two functions used for providing various methods for strictly causal MVAR estimation (mvar.m, covm.m);
  • the FastICA toolbox (http://research.ics.tkk.fi/ica/fastica), with five functions used to perform Independent Component Analysis in the identification of the instantaneous model based on non-gaussianity of the innovations (fastica.m, remmean.m, pcamat.m, whiten.m, fpica.m);
  • the LiNGAM Toolbox (http://www.cs.helsinki.fi/group/neuroinf/lingam), with seven functions used to integrate the identification of the extended MVAR model based on non-gaussianity of the innovations (permnozeribrutal.m, permnozerihungarian.m, permslowertriagbrutal.m, hungarian.m, iperm.m, sltprune.m, slttestperm.m).

 

 

References

 

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       (8)    Baccala L, Sameshima K, Takahashi DY. Generalized partial directed coherence.Proceedings of the 2007 15th International Conference on Digital Signal Processing. Cardiff, UK: 2007.

       (9)    Faes L, Nollo G. Extended causal modelling to assess Partial Directed Coherence in multiple time series with significant instantaneous interactions. Biol Cybern 2010; 103(5):387-400.

     (10)    Erla S, Faes L, Tranquillini E, Orrico D, Nollo G. Multivariate autoregressive model with instantaneous effects to improve brain connectivity estimation. Int J Bioelectromag 2009; 11(2):74-79.

     (11)    Faes L, Nollo G. Assessing frequency domain causality in cardiovascular time series with instantaneous interactions. Methods Inf Med 2010; 49(5):453-457.

     (12)    Faes L, Nollo G. Multivariate frequency domain analysis of causal interactions in physiological time series. In: Laskovski AN, editor. Biomedical Engineering, Trends in Electronics, Communications and Software. Rijeka, Croatia: InTech, 2011: 403-428.

     (13)    Schlogl A. A comparison of multivariate autoregressive estimators. Sign Proc 2006; 86:2426-2429.

     (14)    Akaike H. A new look at the statistical model identification. IEEE Trans Autom Contr 1974; 19:716-723.

     (15)    Rissanen J. An universal prior for the integers and estimation by minimum description length. Ann Stat 1983; 11:417-431.

     (16)    Lutkepohl H. Introduction to multiple time series analysis. Berlin Heidelberg New York: Springer, 1993.

     (17)    Faes L, Pinna GD, Porta A, Maestri R, Nollo G. Surrogate data analysis for assessing the significance of the coherence function. IEEE Trans Biomed Eng 2004; 51(7):1156-1166.

     (18)    Faes L, Porta A, Nollo G. Testing Frequency Domain Causality in Multivariate Time Series. IEEE Trans Biomed Eng 2010; 57(8):1897-1906.

     (19)    Faes L, Erla S, Tranquillini E, Orrico D, Nollo G. An identifiable model to assess frequency-domain granger causality in the presence of significant instantaneous interactions. Conf Proc IEEE Eng Med Biol Soc 2010; 1:1699-1702.

 

 

Last edited March 2, 2011