Accelerating the analyses of 3‐way and 4‐way PARAFAC models utilizing multi‐dimensional wavelet compression

Abstract

Parallel factor analysis (PARAFAC) is one of the most popular methods for evaluating multi-way data sets, such as those typically acquired by hyphenated measurement techniques. One of the reasons for PARAFAC popularity is the ability to extract directly interpretable chemometric models with little a priori information and the capability to handle unknown interferents and missing values. However, PARAFAC requires long computation times that often prohibit sufficiently fast analyses for applications such as online sensing. An additional challenge faced by PARAFAC users is the handling and storage of very large, high-dimensional data sets. Accelerating computations and reducing storage requirements in multi-way analyses are the topics of this manuscript. This study introduces a data pre-processing method based on multi-dimensional wavelet transforms (WTs), which enables highly efficient data compression applied prior to data evaluation. Because multidimensional WTs are linear, the intrinsic underlying linear data construction is preserved in the wavelet domain. In almost all studied examples, computation times for analyzing the much smaller, compressed data sets could be reduced so much that the additional effort for wavelet compression was more than recompensated. For 3-way and 4-way synthetic and experimental data sets, acceleration factors up to 50 have been achieved; these data sets could be compressed down to a few per cent of the original size. Despite the high compression, accurate and interpretable models were derived, which are in good agreement with conventionally determined PARAFAC models. This study also found that the wavelet type used for compression is an important factor determining acceleration factors, data compression ratios and model quality.

Trilinear analysis of images obtained with a hyperspectral imaging confocal microscope

“…A significant challenge in dealing with HSI-CM produced data is its huge volume. Utilizing compression prior to PARAFAC-ALS can significantly reduce the computational workload [25][26][27]. One easy-to-program and highly effective method of compression is to co-add (bin) or average spectra from adjacent pixels in the image space.…”

“…S/N increase should approach the square root of the compression factor for spatially uniform neighborhoods. Pixel binning also compares favorably with alternative spatial compression methods such as wavelet compression [25,28]. For instance, since variances add, simple pixel binning preserves the fundamental noise structure of the original data and uncorrelated noise remains uncorrelated.…”

“…For instance, since variances add, simple pixel binning preserves the fundamental noise structure of the original data and uncorrelated noise remains uncorrelated. Nonnegativity constraints can also be applied when analyzing binned data since no negative coefficients are introduced by the procedure, unlike typical wavelet methods [25]. Finally, since binning is a linear function, its properties carry across other linear functions such as principal component analysis and PARAFAC.…”

Accelerating kernel principal component analysis (KPCA) by utilizing two‐dimensional wavelet compression: applications to spectroscopic imaging

“…K is built using measurement spectra that have previously been mapped from the 'input space' X into the 'feature space' H (see Equations (24), (25) and related discussion). As outlined in the appendix, K has dimensions M Â M with M being the number of (here) spectra.…”

Introducing multi‐dimensional ‘hybrid wavelets’ for enhanced evaluation of hyperspectral image cubes and multi‐way data sets

“…Two-dimensional (2D) WTs [15,23] have been utilized in image analyses and the compression of 2-way data sets [24][25][26]. Recently the extension to three and four dimensions has been presented for analyses of hyperspectral image cubes [27] as well as for 3-way and 4-way [28] parallel factor analysis (PARAFAC) [29][30][31][32][33][34][35][36][37]. The general concept of WTs in arbitrary dimensionality is presented in reference [38].…”