Fast Principal Component Analysis Research Articles

Fast principal component analysis of large data sets based on information extraction

AbstractPrincipal component analysis (PCA) and principal component regression (PCR) are routinely used for calibration of measurement devices and for data evaluation. However, their use is hindered in some applications, e.g. hyperspectral imaging, by excessive data sets that imply unacceptable calculation time. This paper discusses a fast PCA achieved by a combination of data compression based on a wavelet transformation and a spectrum selection method prior to the PCA itself. The spectrum selection step can also be applied without previous data compression. The calculation speed increase is investigated based on original and compressed data sets, both simulated and measured. Two different data sets are used for assessment of the new approach. One set contains 65 536 synthetically generated spectra at four different noise levels with 256 measurement points each. Compared with the conventional PCA approach, these examples can be accelerated 20 times. Evaluation errors of the fast method were calculated and found to be comparable with those of the conventional approach. Four experimental spectra sets of similar size are also investigated. The novel method outperforms PCA in speed by factors of up to 12, depending on the data set. The principal components obtained by the novel algorithm show the same ability to model the measured spectra as the conventional time‐consuming method. The acceleration factors also depend on the possible compression; in particular, if only a small compression is feasible, the acceleration lies purely with the novel spectrum selection step proposed in this paper. Copyright © 2002 John Wiley & Sons, Ltd.

Time Series Segmentation Using a Novel Adaptive Eigendecomposition Algorithm

In this paper we present a new technique for time series segmentation built around a fast principal component analysis (PCA) algorithm that is on-line and stable. The traditional Generalized Likelihood Ratio Test (GLRT) has been used to solve the segmentation problem, but this has enormous limitations in terms of complexity and speed. Newer methods use gated experts and mixture models to detect transitions in time series. These techniques perform better than GLRT, but most of them require extensive training of relatively large neural networks. The segmentation method discussed in this paper is based on a novel idea that involves solving the generalized eigendecomposition of two consecutive windowed time series and can be formulated as a two-step PCA. Thus, the performance of our segmentation technique mainly depends on the efficiency of the PCA algorithm. Most of the existing techniques for PCA are based on gradient search procedures that are slow and they also suffer from convergence problems. The PCA algorithm presented in this paper is both online, and is proven to converge faster than the current methods.

Fast principal component analysis of large data sets

Principal component analysis (PCA) and principal component regression (PCR) are widespread algorithms for calibration of spectrometers and evaluation of unknown measurement spectra. In many measurement tasks, the amount of calibration data is increasing nowadays due to new devices like hyperspectral imagers. Core of PCA is the singular value decomposition (SVD) of the matrix containing the calibration spectra. SVD of large calibration sets is computational, very expensive and often gets unreasonable due to excessive calculation times. With hyperspectral imaging as application in mind, an algorithm is proposed for compressing calibration spectra based on a wavelet transformation before performing the SVD. Considering only relevant wavelet coefficients can accelerate the SVD. After determining the relevant principal components (PCs) from this shrunken calibration matrix in the wavelet domain, they are expanded again by insertion of zeros at the right positions. Denoised PCs are then obtained by the inverse wavelet transform into the wavelength domain. An additional computation speed increase is described for “landscape” matrices by transposing the matrix before performing the SVD. In the Results section, both PCA approaches are demonstrated to result in comparable PCs. This is done by means of synthetically generated spectra as well as by experimental FTIR-data. By this algorithm, the PCA of the discussed examples could be accelerated up to a factor of 52. Additionally, concentrations of synthetic spectra are evaluated by means of the PCs obtained by the different PCA algorithms. Both PC sets, the conventional and the one based on the new technique, result in equivalent concentration values.

Wavelet packet transform applied to a set of signals: A new approach to the best-basis selection

A new approach to best-basis selection for a set of signals, enabling significant and uniform data compression in the time-frequency domain, is proposed. The high degree of data compression can be used as the first step of fast approximate principal component analysis (PCA) of data sets with high rank (∼ 10 4). The performance of the proposed approach is tested for two data sets and compared with Wickerhauser's approach of fast approximate PCA.