Partial least squares basis

B. Waiczack and D.L. Massart, Application of radial basis functions-partial least squares to non-linear pattern recognition problems diagnosis of process faults. Anal. Chim. Acta, 331 (1996) 187-193. 

PLS (partial least squares) multiple regression technique is used to estimate contributions of various polluting sources in ambient aerosol composition. The characteristics and performance of the PLS method are compared to those of chemical mass balance regression model (CMB) and target transformation factor analysis model (TTFA). Results on the Quail Roost Data, a synthetic data set generated as a basis to compare various receptor models, is reported. PLS proves to be especially useful when the elemental compositions of both the polluting sources and the aerosol samples are measured with noise and there is a high correlation in both blocks. 

Fourier transform infrared (FTIR) spectroscopy of coal low-temperature ashes was applied to the determination of coal mineralogy and the prediction of ash properties during coal combustion. Analytical methods commonly applied to the mineralogy of coal are critically surveyed. Conventional least-squares analysis of spectra was used to determine coal mineralogy on the basis of forty-two reference mineral spectra. The method described showed several limitations. However, partial least-squares and principal component regression calibrations with the FTIR data permitted prediction of all eight ASTM ash fusion temperatures to within 50 to 78 F and four major elemental oxide concentrations to within 0.74 to 1.79 wt % of the ASTM ash (standard errors of prediction). Factor analysis based methods offer considerable potential in mineral-ogical and ash property applications. 

The method which satisfies these conditions is partial least squares (PLS) regression analysis, a relatively recent statistical technique (18, 19). The basis of tiie PLS method is that given k objects, characterised by i descriptor variables, which form the X-matrix, and j response variables which form the Y-matrix, it is possible to relate the two blocks (or data matrices) by means of the respective latent variables u and 1 in such a way that the two data sets are linearly dependent ... 

Partial least squares regression (PLS) [WOLD et al., 1984] is a generalized method of least squares regression. This method uses latent variables i, 2,. .., i.e. matrix U, for separately modeling the objects in the matrix of dependent data Y, and t, t2,. .., i.e. matrix T, for separately modeling the objects in the matrix of independent data X. These latent variables U and T are the basis of the regression model. The starting points are the centered matrices X and Y ... 

The advent of personal computers greatly facilitated the application of spectroscopic methods for both quantitative and qualitative analysis. It is no longer necessary to be a spectroscopic expert to use the methods for chemical analyses. Presently, the methodologies are easy and fast and take advantage of all or most of the spectral data. In order to understand the basis for most of the current processing methods, we will address two important techniques principal component analysis (PCA) and partial least squares (PLS). When used for quantitative analysis, PCA is referred to as principal component regression (PCR). We will discuss the two general techniques of PCR and PLS separately, but we also will show the relationship between the two. 

The numbers of partial least squares (PLS) components were higher in CoMSIA than in CoMFA. This difference probably resulted from the significantly higher number of lattice points showing steadily varying field values (e.g., inside the molecules). The optimal numbers of components were selected on the basis of lowest Spress. 

From the total sample set (48 samples), 45 samples were used as calibration samples. The three samples excluded from the calibration set were selected on the basis of a representative variation of their active ingredient concentrations, and finally used as unknown test samples to predict the content of their active ingredients. Partial least squares (PLS) models for each active ingredient were developed with the Unscrambler Software (version 9.6 CAMO Software AS, Oslo, Norway) from the MSC-pretreated median spectra of all pixels of each of the 45 calibration sample images. Based on these calibration models, the predictions of the active ingredient content for each pixel of the imaging data of the three test samples and their evaluation as histograms, contour plots and RGB plots was performed with Matlab v. 7.0.4 software (see below). 

The similarity between a Tucker 1 model of a three-way array X and a PCA model of a two-way array X has already been mentioned in Section 4.1. This similarity is the basis for the most popular three-way regression model, the Tucker 1-PLS model. In the literature this regression method has also been called three-way partial least squares regression [Wold et al. 1987]. This name should be avoided for the Tucker 1 version specifically, because it gives rise to much confusion as to what is multi-way methodology. Even though the Tuckerl version of three-way partial least squares regression is indeed one way to implement a three-way method, it is actually the model that assumes the least three-way structure in the data. 

A different approach to mathematical analysis of the solid-state C NMR spectra of celluloses was introduced by the group at the Swedish Forest Products Laboratory (STFI). They took advantage of statistical multivariate data analysis techniques that had been adapted for use with spectroscopic methods. Principal component analyses (PCA) were used to derive a suitable set of subspectra from the CP/MAS spectra of a set of well-characterized cellulosic samples. The relative amounts of the I and I/3 forms and the crystallinity index for these well-characterized samples were defined in terms of the integrals of specific features in the spectra. These were then used to derive the subspectra of the principal components, which in turn were used as the basis for a partial least squares analysis of the experimental spectra. Once the subspectra of the principal components are validated by relating their features to the known measures of variability, they become the basis for analysis of the spectra of other cellulosic samples that were not included in the initial analysis. Here again the interested reader can refer to the original publications or the overview presented earlier. ... 

Seven parameters of physicochemical properties, such as acid number, color, density, refractive index, moisture and volatility, saponification value and PV, were measured for quality and adnlter-ated soybean, as well as quality and rancid rapeseed oils. The chemometric methods were then applied for qualitative and quantitative discrimination and prediction of the oils by methods snch as exploratory principal component analysis (PCA), partial least squares (PLS), radial basis function-artificial neural networks (RBF-ANN), and multi-criteria decision making methods (MCDM), PROMETHEE and GAIA.260... 

Paracetamol, propiphenazone and caffeine First to fourth derivative spectra of components were subjected to chemometric analysis (principal component regression, PCR partial least squares with one dependent variable, PLS-1 three dependent variables, PLS2) and adopted for multicomponent analysis. The third derivative spectra of aU ingredients became a basis of quantification method. 39... 

Harding, Popelier, and co-workers [285,286] have employed a variety of quantum chemical approaches in their estimation of the pK s ol oxyacids. In a study of 228 carboxylic acids they used what they call quantum chemical topology to find pK estimates. They tested several different methods, including partial least squares (PLS), support vector machines (SVMs), and radial basis function neural networks (RBFNNs) with Hartree-Fock and density functional calculations, concluding that the SVM models with HF/6-31G calculations were most efficient [285]. Foi a data set of 171 phenols they found that the C-0 bond length provided an effective descriptor for pK estimation [286]. 

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