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We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary, multivariate long-range dependent (LRD) Gaussian time series. In a time series graph, each component of the vector series is represented by a distinct node, and associations between components are represented by edges between the corresponding nodes. In a recent work on graphical modeling of short-range dependent (SRD) Gaussian time series, the problem was cast as one of multi-attribute graph estimation for random vectors where a vector is associated with each node of the graph. At each node, the associated random vector consists of a time series component and its delayed copies. A theoretical analysis based on short-range dependence has been given in Tugnait (2022 ICASSP). In this paper we analyze this approach for LRD Gaussian time series and provide consistency results regarding convergence in the Frobenius norm of the inverse covariance matrix associated with the multi-attribute graph.
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This content will become publicly available on November 6, 2027
Conditionally specified stochastic processes for graphical modeling of stationary multivariate time series
Graphical models are ubiquitous for summarizing conditional relations in multivariate data. In many applications involving multivariate time series, it is of interest to learn an interaction graph that treats each individual time series as nodes of the graph, with the presence of an edge between two nodes signifying conditional dependence given the others. Typically, the partial covariance is used as a measure of conditional dependence. However, in many applications, the outcomes may not be Gaussian and/or could be a mixture of different outcomes. For such time series using the partial covariance as a measure of conditional dependence may be restrictive. In this article, we propose a broad class of time series models which are specifically designed to succinctly encode process-wide conditional independence in its parameters. For each univariate component in the time series, we model its conditional distribution with a distribution from the exponential family. We develop a notion of process-wide compatibility under which such conditional specifications can be stitched together to form a well-defined strictly stationary multivariate time series. We call this construction a conditionally exponential stationary graphical model (CEStGM). A central quantity underlying CEStGM is a positive kernel which we call the interaction kernel. Spectral properties of such positive kernel operators constitute a core technical foundation of this work. We establish process-wide local and global Markov properties of CEStGM exploiting a Hammersley-Clifford type decomposition of the interaction kernel. Further, we study various probabilistic properties of CEStGM and show that it is geometrically mixing. An approximate Gibbs sampler is also developed to simulate sample paths of CEStGM.
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- Award ID(s):
- 2210726
- PAR ID:
- 10657847
- Publisher / Repository:
- Arxiv
- Date Published:
- Journal Name:
- Under submission
- ISSN:
- i do not know yet
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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