An Efficient Method to Derive Explicit KLT Kernel for First-Order Autoregressive Discrete Process

Abstract

Signal dependent Karhunen-Loève transform (KLT), also called factor analysis or principal component analysis (PCA), has been of great interest in applied mathematics and various engineering disciplines due to optimal performance. However, implementation of KLT has always been the main concern. Therefore, fixed transforms like discrete Fourier (DFT) and discrete cosine (DCT) with efficient algorithms have been successfully used as good approximations to KLT for popular applications spanning from source coding to digital communications. In this paper, we propose a simple method to derive explicit KLT kernel, or to perform PCA, in closed-form for first-order autoregressive, AR (1), discrete process. It is a widely used approximation to many real world signals. The merit of the proposed technique is shown. The novel method introduced in this paper is expected to make real-time and data-intensive applications of KLT, and PCA, more feasible.Index Terms-Covariance analysis, eigenanalysis, explicit Karhunen-Loève transform (KLT) kernel, factor analysis, first-order autoregressive process, principal component analysis (PCA), signal dependent transform.

A fast derivation of Karhunen-Loeve transform kernel for first-order autoregressive discrete process

“…In order to derive an explicit expression for the roots of the transcendental equation that are required in the definition of the discrete KLT kernel given in (12), we need to calculate the first N/2 positive roots of two transcendental equations as given [11,12] tan ω…”

“…Recently, an efficient method to derive explicit KLT kernel for AR(1) process was developed [12]. This paper investigates the computational performance of the new kernel derivation method for large dimensions that may benefit big data applications, and compares with D&Q under the same test conditions.…”

“…Namely, a. correlation measurements of raw data to populate covariance matrix; b. derivation of KLT matrix through eigendecomposition of covariance matrix; c. implementing KLT transform for a given data vector. This paper investigates computational cost of the fast KLT kernel generation method derived in [12] for step b here.…”

“…II. A fast method proposed in [12] to derive explicit KLT kernel for AR(1) discrete process, FD-AR(1), is highlighted in Sec. III.…”

“…We will summarize below a recently introduced fast eigenanalysis method for AR(1) discrete process [12]. Then, we will investigate its computational requirements and compare with the currently available techniques.…”