Recursive Principal Components Analysis Using Eigenvector Matrix Perturbation

Abstract

<p/> <p>Principal components analysis is an important and well-studied subject in statistics and signal processing. The literature has an abundance of algorithms for solving this problem, where most of these algorithms could be grouped into one of the following three approaches: adaptation based on Hebbian updates and deflation, optimization of a second-order statistical criterion (like reconstruction error or output variance), and fixed point update rules with deflation. In this paper, we take a completely different approach that avoids deflation and the optimization of a cost function using gradients. The proposed method updates the eigenvector and eigenvalue matrices simultaneously with every new sample such that the estimates approximately track their true values as would be calculated from the current sample estimate of the data covariance matrix. The performance of this algorithm is compared with that of traditional methods like Sanger's rule and APEX, as well as a structurally similar matrix perturbation-based method.</p

Fault detection and isolation for dynamic processes using recursive principal component analysis (PCA) based on filtering of signals

“…Then, apply PCA on X I to generate the LPC and compute r I by Eqn (9). The vector r I will be used for isolating this detected fault.…”

“…The faults considered include constant-bias, high-frequency noise originating from sensor measurement and errors resulting from input disturbance or change in the process gain. To facilitate the on-line computation of LPC, a recursive algorithm [9] based on rank-one matrix update of the covariance is incorporated with the PCA. Finally, a distillation column of Wood and Berry [10] is used to illustrate the proposed method.…”

Independent Component Analysis Applied to Voice Activity Detection

“…In order to recursively evaluate the LRT over the set of uncorrelated signals in the current frame l we use a result in [9]. In the frame l + 1 the PCA components for the MO-window l − m, .…”

“…Using a matrix perturbation analysis approach of a matrix in the form (Λ + αα T ), we can obtain a recursion for the eigenvalues and eigenvectors as [9]:…”

“…This approach is also assumed in the Fourier domain [7]. We use the latter algorithm in conjunction with the recursive PCA algorithm presented in [9] to build an efficient "on line" VAD, assessing its performance in an HMM-based speech recognition system.…”