Regularized Estimation in Sparse High-Dimensional Time Series Models

• Activity Number: 241
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Sumanta Basu*+ and George Michailidis
• Companies: University of Michigan and University of Michigan
• Keywords: vector autoregression ; time series ; high-dimensional learning ; spectral theory ; covariance estimation ; stochastic regression

Abstract

Many scientific and economic problems require the analysis of high-dimensional time series datasets. However, theoretical studies in high-dimensional statistics to date rely primarily on the assumption of independent and identically distributed (i.i.d.) samples. In this work, we investigated the theoretical properties of l1-regularized estimates in three important statistical problems in the context of high-dimensional time series: (a) stochastic regression with serially correlated errors, (b) transition matrix estimation in vector autoregressive (VAR) models, and (c) covariance matrix estimation from temporal data. For all three problems, we derive non-asymptotic upper bounds on the estimation errors, thus establishing that consistent estimation is possible via l1-regularization for a large class of stationary time series under sparsity constraints. A key technical contribution of the work is to introduce a measure of stability for stationary processes, that provides insight into the effect of dependence on the accuracy of the regularized estimates. Further, we establish some useful deviation bounds for statistics generated from dependent data, which are of independent interest.