Abstract
This paper is concerned with the estimation of time-varying networks for high-dimensional nonstationary time series. Two types of dynamic behaviors are considered: structural breaks (i.e., abrupt change points) and smooth changes. To simultaneously handle these two types of time-varying features, a two-step approach is proposed: multiple change point locations are first identified on the basis of comparing the difference between the localized averages on sample covariance matrices, and then graph supports are recovered on the basis of a kernelized time-varying constrained -minimization for inverse matrix estimation (CLIME) estimator on each segment. We derive the rates of convergence for estimating the change points and precision matrices under mild moment and dependence conditions. In particular, we show that this two-step approach is consistent in estimating the change points and the piecewise smooth precision matrix function, under a certain high-dimensional scaling limit. The method is applied to the analysis of network structure of the S&P 500 index between 2003 and 2008.
Highlights
Networks are useful tools to visualize the relational information among a large number of variables
Gaussian graphical models can be represented by the inverse covariance matrix, where a zero entry is associated with a missing edge between two vertices in the graph
Dynamic random networks have been extensively studied from the perspective of large random graphs, such as community detection and edge probability estimation for dynamic stochastic block models (DSBMs) [17,18,19,20,21,22,23,24,25,26,27,28,29,30]
Summary
Networks are useful tools to visualize the relational information among a large number of variables. Theoretical analysis was derived under the fundamental assumption that the observations are either temporally independent, or the temporal dependence has very specific forms, such as Gaussian processes or (linear) vector autoregression (VAR) [14,33,34,37,41,42,43] Such dynamic structures are unduly demanding in view that many time series encountered in real applications have very complex nonlinear spatial-temporal dependency [44,45]. In [53,56,57], the authors considered the theoretical properties of regularized estimation of covariance and precision matrices, based on various dependence measures of high-dimensional time series.
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