mvgc_cdf
Sample MVGC thoretical asymptotic cumulative distribution function
Syntax
P = mvgc_cdf(x,X,p,m,N,nx,ny,nz,tstat)
Arguments
See also Common variable names and data structures.
input
x vector of MVGC values
X vector of actual MVGC values
p VAR model order
m number of observations per trial
N number of trials
nx number of target ("to") variables
ny number of source ("from") variables
nz number of conditioning variables (default: 0)
tstat statistic: 'F' or 'chi2' (default: 'F' if nx == 1, else 'chi2')output
P cumulative distribution probabilities evaluated at x
Description
Return theoretical sample MVGC asymptotic cumulative distribution function for actual MVGCs in vector X, evaluated at values in vector x. For p-values, assume null hypothesis H0: X = 0 (see mvgc_pval).
The theoretical distribution is specified by the tstat parameter, which may be 'F' for Granger's F-test (default if nx == 1) or 'chi2' for Geweke's χ2 test (default if nx > 1). For a multivariate predictee (i.e. nx > 1) only the χ2 test is suitable [1].
Note 1: In theory the F-distribution should be preferable for small samples; it has a fatter tail than the corresponding χ2 distribution. However, for a multivariate predictee (nx > 1) it is not appropriate; the usual F-test for a nested linear regression demands the Granger form statistic F = (RSS_reduced - RSS_full) / RSS_full. For a univariate predictee this may be derived from a simple transformation of the Geweke form F = log(det(SIGMA_reduced)) / log(det(SIGMA_full)). However for a multivariate predictee the Granger form cannot be derived from the Geweke form; in fact it corresponds to the "trace" MVGC variant recommended (for different reasons) in [2]; but there are problems with the trace form, in particular lack of invariance to certain transformations of variables and filters [2,3] and lack of an information-theoretic interpretation [1].
Note 2: The non-central F-distribution is currently to be treated as experimental and a warning is issued if it is invoked. Specifically, the problem is: should it be doubly- or singly-noncentral and what should the non-centrality parameter(s) be? Currently we assume singly-noncentral and make an informed guess at the non-centrality parameter. This appears to work well, giving results similar to the χ2 test for medium-sized samples, but warrants further investigation.
References
[1] L. Barnett and A. K. Seth, The MVGC Multivariate Granger Causality Toolbox: A New Approach to Granger-causal Inference, J. Neurosci. Methods 223, 2014 [ preprint ].
[2] C. Ladroue, S. Guo, K. Kendrick and J. Feng, "Beyond element-wise interactions: Identifying complex interactions in biological processes", PLoS ONE 4(9), 2009.
[2] A. B. Barrett, L. Barnett and A. K. Seth, "Multivariate Granger causality and generalized variance", Phys. Rev. E 81(4), 2010.
[3] L. Barnett and A. K. Seth, "Behaviour of Granger causality under filtering: Theoretical invariance and practical application", J. Neurosci. Methods 201(2), 2011.
See also
mvgc_cdfi | mvgc_pval | mvgc_confint | mvgc_cval | mvgc_demo