Financial Signal Processing and Machine Learning

Chapter 5Explicit Kernel and Sparsity of Eigen Subspace for the AR(1) Process

Mustafa U. Torun, Onur Yilmaz and Ali N. Akansu

New Jersey Institute of Technology, USA

5.1 Introduction

Karhunen–Loeve Transform (KLT), also called Eigen decomposition or principal component analysis (PCA), is the optimal orthogonal subspace method (block transform) that maps wide-sense stationary (WSS) stochastic signals with correlations into nonstationary and pairwise uncorrelated transform coefficients. The coefficient with the highest variance corresponds to the most covariability among the observed signals, hence the most meaningful information (Akansu and Haddad, 1992). Therefore, the coefficients with large variances are kept and the ones with low variances corresponding to noise are discarded in noise-filtering and compression applications (Jolliffe, 2002). KLT basis functions are the eigenvectors of the given signal covariance matrix that define the corresponding unique eigen subspace. Therefore, it is a signal-dependent transform as opposed to some other popular transforms like discrete Fourier transform (DFT) and discrete cosine transform (DCT). DFT and DCT have their kernels that are independent of signal statistics. They are called fixed transforms, and their good performance with efficient implementations make them desirable choices for various applications (Akansu and Haddad, 1992). Fast implementation of KLT is of great interest to several disciplines, and there were attempts to ...