Explicit Kernel and Sparsity of Eigen Subspace for the AR(1) Process

Fast implementation of Karhunen-Loeve Transform (KLT) is of great interest to several disciplines, and there were attempts to derive closed-form kernel expressions for certain classes of stochastic processes. Random processes and information sources are described by stochasticsignal models, including autoregressive (AR), moving average (MA), and autoregressivemoving average (ARMA) types. This chapter focuses on the discrete autoregressive order one, AR(1), and the process and derivation of its explicit eigen kernel. It investigates the sparsity of eigen subspace and presents a rate-distortion theory-based sparsing method. The chapter then focuses on eigen subspace of a discrete AR(1) process with closed-form expressions for its eigenvectors and eigenvalues. It provides a comparative performance of the presented method along with the various methods reported in the literature. Finally, the chapter highlights the merit of the method for the AR(1) process as well as for the empirical correlation matrix of stock returns in the NASDAQ-100 index.