This was part of Expressing and Exploiting Structure in Modeling, Theory, and Computation with Gaussian Processes

## Graphical models for nonstationary time series

Suhasini Subba Rao, Texas A&M University
Tuesday, August 30, 2022

Abstract: We propose NonStGM, a general nonparametric graphical modeling framework for studying dynamic associations among the components of a nonstationary multivariate time series. It builds on the framework of Gaussian Graphical Models (GGM) and stationary time series Graphical models (StGM), and complements existing works on parametric graphical models based on change point vector autoregressions (VAR). Analogous to StGM, the proposed framework captures conditional noncorrelations (both intertemporal and contemporaneous) in the form of an undirected graph. In addition, to describe the more nuanced nonstationary relationships among the components of the time series, we introduce the new notion of conditional nonstationarity/stationarity and incorporate it within the graph architecture. This allows one to distinguish between direct and indirect nonstationary relationships among system components, and can be used to search for small subnetworks that serve as the source'' of nonstationarity in a large system. Together, the two concepts of conditional noncorrelation and nonstationarity/stationarity provide a parsimonious description of the dependence structure of the time series. In GGM, the graphical model structure is encoded in the sparsity pattern of the inverse covariance matrix. Analogously, we explicitly connect conditional noncorrelation and stationarity between and within components of the multivariate time series to zero and Toeplitz embeddings of an infinite-dimensional inverse covariance operator. In order to learn the graph, we move to the Fourier domain. We show that in the Fourier domain, conditional stationarity and noncorrelation relationships in the inverse covariance operator are encoded with a specific sparsity structure of its integral kernel operator. Within the local stationary framework we show that these sparsity patterns can be recovered from finite-length time series by node-wise regression of discrete Fourier Transforms (DFT) across different Fourier frequencies. We illustrate the features of our general framework under the special case of time-varying Vector Autoregressive models. We demonstrate the feasibility of learning NonStGM structure from data using simulation studies.