Fast principal component analysis of large data sets

Abstract

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.