Edge Exclusion Tests for Graphical Model Selection: Complex Gaussian Vectors and Time Series

We consider the problem of inferring the conditional independence graph (CIG) of a stationary multivariate Gaussian time series. A <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula>-variate Gaussian time series graphical model associated with an undirected graph with <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula> vertices is defined as the family of time series that obey the conditional independence restrictions implied by the edge set of the graph. In some existing methods, partial coherence has been used as a test statistic for graphical model selection. To test inclusion/exclusion of a given edge in the graph, the test is applied at distinct frequencies, requiring multiple tests and leading to a loss in test power. The nonparametric method of Matsuda uses the Kullback-Leibler divergence measure to define a test statistic which does not need multiple testing to test between two competing models. In this paper we propose a generalized likelihood ratio test (GLRT) based edge exclusion test statistic that also does not need multiple testing. It is computationally significantly faster than the method of Matsuda, and simulations show that we achieve comparable power levels. Our proposed approach is based on a novel formulation of a GLRT based edge exclusion test for <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula>-variate complex Gaussian graphical models (CGGMs); this result is also of independent interest. The computational complexity of the proposed statistic is <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(p^3)$</tex-math></inline-formula> compared to <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(p^5)$</tex-math></inline-formula> for the existing result. We also apply our time series graphical model selection method to a foreign exchange data set consisting of monthly trends of foreign exchange rates of 10 countries.

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