PLS method

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

Dunn W J III, S Wold, U Edlund, S Hellberg and J Gasteiger 1984. Multivariate Structure-Activib Relationships Between Data from a Battery of Biological Tests and an Ensemble of Structur Descriptors The PLS Method. Quantitative Structure-Activity Relationships 3 131-137. 

Partial least-squares path modeling with latent variables (PLS), a newer, general method of handling regression problems, is finding wide apphcation in chemometrics. This method allows the relations between many blocks of data ie, data matrices, to be characterized (32—36). Linear and multiple regression techniques can be considered special cases of the PLS method. 

The results obtained by lineal regression (LR) and by pareial least square regression (PLS) methods have been eompared to quantify tlie 0-H signal in anhydrite samples. The PLS quality is eharaeterized by a eoirelation eoeffieient of 0.9942 (eross-validation) using four faetors and a root mean square eiTor of ealibration (RMSEC) of 0.058. Tlie eoirelation eoeffieient of LR metliod obtained was 0.9753. 

We have seen that PCR and RRR form two extremes, with CCA somewhere in between. RRR emphasizes the fit of Y (criterion ii). Thus, in RRR the X-components t, preferably should correlate highly with the original T-variables. Whether X itself can be reconstructed ( back-fitted ) from such components t, is of no concern in RRR. With standard PCR, i.e. top-down PCR, the emphasis is initially more on the X-side (criterion i) than on the T-side. CCA emphasizes the importance of correlation whether the canonical variates t and u account for much variance in each respective data set is immaterial. Ideally, of course, one would like to have the best of all three worlds, i.e. when the major principal components of X (as in PCR) and the major principal components of Y (as in RRR) happen to be very similar to the major canonical variables (as in CCA). Is there a way to combine these three desiderata — summary of X, summary of Y and a strong link between the two — into a single criterion and to use this as a basis for a compromise method The PLS method attempts to do just that. 

Finally, another alternative to continuum regression has been put forward by Wise and de Jong [18]. Their continuum power-PLS (CP-PLS) method modifies the matrix X = USV into X " = i.e. the singular values are raised to a... 

The point being that, as our conclusions indicate, this is one case where the use of latent variables is not the best approach. The fact remains that with data such as this, one wavelength can model the constituent concentration exactly, with zero error - precisely because it can avoid the regions of nonlinearity, which the PCA/PLS methods cannot do. It is not possible to model the constituent better than that, and even if PLS could model it just as well (a point we are not yet convinced of since it has not yet been tried -it should work for a polynomial nonlinearity but this nonlinearity is logarithmic) with one or even two factors, you still wind up with a more complicated model, something that there is no benefit to. 

Since in many applications minor absorption changes have to be detected against strong, interfering background absorptions of the matrix, advanced chemometric data treatment, involving techniques such as wavelet analysis, principle component analysis (PCA), partial least square (PLS) methods and artificial neural networks (ANN), is a prerequisite. 

It is possible to develop the ideas behind PCR and to a lesser extent behind PLS, based on chemical ideas and intuition. Naturally, this is not the only way and both PCR and PLS methods have been developed on pathways that are theoretically oriented. 

In order to understand the difference between the PCR and the PLS methods we first return to PCR. The two central equations of PCR are ... 

Abstract Aspartame (Apt), Acesulfame-K (Ace-K) low-calorie, high-potency artificial sweeteners ate cnnently nsed in beverages and dietary food and drinks. Their increased application in food and drink prodncts has given a new impetus to develop fast and accurate methods for their determination. Absorption spectra of Asp, Caf, Ace-K and BA strongly overlap. Therefore a direct determination of these analytes in quaternary mixture is impossible without a separation step. In order to overcome this difficulty partial least squares (PLS) method has been proposed. 

The PLS method requires a carefully experimental design the standard composition of the calibration set in order to provide good predictions. In this study a calibration... 

B program, PLS-2, uses the partial least squares (PLS) method. This method has been proposed by H. Wold (37) and was discussed by S. Wold (25). In such a problem there are two blocks of data, T and X. It is assumed that T is related to X by latent variables u and t is derived from the X block and u is derived from the Y block. 

The data modeled are from gas chromatograms obtained for Aroclors 1242, 1248, 1254 and 1260. The unknown samples are from the anaysis of used transformer oil obtained from a waste dump in New Jersey. The concentration of individual isomers in selected Aroclor and transformer oil samples are given in Appendix I. The data are organized in a matrix in which the first four data entries for each sample in row 1 of the data array (Table 2, Apendix I) designate the composition of the sample. For standards, these four variables represent the fractional parts of Aroclor 1242, 1248, 1254, or 1260, respectively, that were combined. Results from the analysis of transformer oil (samples 21-23) are of unknown fractional composition and variables 1 through 4 are null entries. In the examples that follow data from samples analyzed (Table 1, Appendix I) were used in part or in total to illustrate the PLS method. 

Because many samples are analzyed in which the analyst is interested in determining which Aroclor mixtures are present, we applied the PLS method to the data obtained from the analysis of... 

The partial least squares (PLS) method has been applied to structure activity problems by Wold al. (38). Recently, Lindberg al. (40) employed this approach to resolve mixtures of humic acid and ligninsulfonate on the basic of fluorescence spectra. 

This example demonstrates that the PLS method gives a stable estimate of the Y-block, even though there are many more X" variables than samples, a condition that removes the possibility of applying multiple regression. Another advantage of the method... 

Dunn, W.J., Wold, S., Edlund, U., Hellbeeg, S., and Gasteigee, J. Multivariate structure-activity relationships between data from a battery of biological tests and an ensemble of stmctural descriptors The PLS method. Quant. Struc-Act. Relat. 1984, 3, 31-137. 

Shen, M., Letiran, A., Xiao, Y., Golbraikh, A., Kohn, H., Tropsha, A. Quantitative structure-activity relationship analysis of functionalized amino acid anticonvulsant agents using k nearest neighbor and simulated annealing PLS methods./. Med. Chem. 2002, 45, 2811-2823. 

The PLS regression method can be extended to accommodate problems where multiple y variables must be predicted. This extension, commonly called PLS2, operates on the same principle as PLS, where the goal is to hnd compressed variables (latent variables) that sequentially describe the most variance in both the x and y data [1]. However, the algorithm for the PLS2 method is slightly different than that of the PLS method, in that one must now account for covariance between different y variables as well as covariance between x variables. 

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. 

In this paper our goal is to introduce the PLS method, to discuss its properties, to compare it with the CMB and TTFA models and to demonstrate its performance on a well known synthetic data set. 

In this paper the PLS method was introduced as a new tool in calculating statistical receptor models. It was compared with the two most popular methods currently applied to aerosol data Chemical Mass Balance Model and Target Transformation Factor Analysis. The characteristics of the PLS solution were discussed and its advantages over the other methods were pointed out. PLS is especially useful, when both the predictor and response variables are measured with noise and there is high correlation in both blocks. It has been proved in several other chemical applications, that its performance is equal to or better than multiple, stepwise, principal component and ridge regression. Our goal was to create a basis for its environmental chemical application. 

Wold, S., Martens, H., Wold, H. The multivariate calibration problem in chemistry solved by the PLS method. Lecture notes in mathematics . Springer Verlag, Heidelberg, in press... 

---