Partial least squares regression, analytical methods

The calibration methods most frequently used to relate the property to be measured to the analytical signals acquired in NIR spectroscopy are MLR,59 60 principal component regression (PCR)61 and partial least-squares regression (PLSR).61 Most of the earliest quantitative applications of NIR spectroscopy were based on MLR because spectra were then recorded on filter instruments, which afforded measurements at a relatively small number of discrete wavelengths only. However, applications involving PCR and PLSR... 

NIR spectroscopy became much more useful when the principle of multiple-wavelength spectroscopy was combined with the deconvolution methods of factor and principal component analysis. In typical applications, partial least squares regression is used to model the relation between composition and the NIR spectra of an appropriately chosen series of calibration samples, and an optimal model is ultimately chosen by a procedure of cross-testing. The performance of the optimal model is then evaluated using the normal analytical performance parameters of accuracy, precision, and linearity. Since its inception, NIR spectroscopy has been viewed primarily as a technique of quantitative analysis and has found major use in the determination of water in many pharmaceutical materials. 

The calibration model referred to a partial least squares regression (PLSR) is a relatively modem technique, developed and popularized in analytical science by Wold. The method differs from PCR by including the dependent variable in the data compression and decomposition operations, i.e. both y and x data are actively used in the data analysis. This action serves to minimize the potential effects of jc variables having large variances but which are irrelevant to the calibration model. The simultaneous use of X and y information makes the method more complex than PCR as two loading vectors are required to provide orthogonality of the factors. 

Bangalore, A. S., Shaffer, R. E., Small, G. W. and Arnold, M. (1996) Genetic algorithm-based method for selecting wavelengths and model size for use with partial least squares regression. Application to near infrared spectroscopy. Analytical Chemistry, 68, 4200-12. 

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. 

If the system is not simple, an inverse calibration method can be employed where it is iKst necessary to obtain the spectra of the pure analytes. The three inverse methods discussed later in this chapter include multiple linear regression (MLR), jirincipal components regression (PCR), and partial least squares (PLS). Wlien using. MLR on data sees found in chemlstiy, variable. sciectson is... 

The method of PLS, also known as Partial Least Squares, is a highly utilized regression tool in the chemometrics toolbox,1 and has been successfully used for many process analytical applications. Like the PCR method, PLS uses the exact same mathematical models for the compression of the X-data and the compression of the Y-data ... 

Finally it is important to note that modern analytical equipment frequently offers opportunities for measuring several or many characteristics of a material more or less simultaneously. This has encouraged the development of multivariate statistics methods, which in principle permit the simultaneous analysis of several components of the material. Partial least squares methods and principal component regression are examples of such techniques that are now finding extensive uses in several areas of analytical science. ... 

The multiple regression method is most often employed to derive predictive QSRR. However, good predictions of GC retention were obtained by means of factorial methods of data analysis. The PLS (partial least squares) treatment of 17 simple descriptors of analytes, such as the number of atoms of each element, of multiple bonds, of functional groups, etc., made predictions of retention of 100 substituted benzenes and pyridines 188],... 

A general requirement for P-matrix analysis is n = rank(R). Unfortcmately, for most practical cases, the rank of R is greater than the number of components, i.e., rank(R) > n, and rank(R) = min(m, p). Thus, P-matrix analysis is associated with the problem of substituting R with an R that produces rank(R ) = n. This is mostly done by orthogonal decomposition methods, such as principal components analysis, partial least squares (PLS), or continuum regression [4]. Dimension requirements of involved matrices for these methods are m > n, and p > n. If the method of least squares is used, additional constraints on matrix dimensions are needed [4]. The approach of P-matrix analysis does not require quantitative concentration information of all constituents. Specifically, calibration samples with known concentrations of analytes under investigation satisfy the calibration needs. The method of PLS will be used in this chapter for P-matrix analysis. 

The utilization of ion-selective membrane electrodes involves their prior calibration. It is necessary to make a regression to obtain good reliability of the analytical information. Because a linear relation between the independent and dependent variables cannot forced, sometimes the nonlinear calibration method is successful. The ion-selective membrane electrode linearization and subsequent calibration use multiple linear regression (MLR) and/or partial least-squares (PLS) when limited calibration data are available.219... 

The improvement in computer technology associated with spectroscopy has led to the expansion of quantitative infrared spectroscopy. The application of statistical methods to the analysis of experimental data is known as chemometrics [5-9]. A detailed description of this subject is beyond the scope of this present text, although several multivariate data analytical methods which are used for the analysis of FTIR spectroscopic data will be outlined here, without detailing the mathematics associated with these methods. The most conunonly used analytical methods in infrared spectroscopy are classical least-squares (CLS), inverse least-squares (ILS), partial least-squares (PLS), and principal component regression (PCR). CLS (also known as K-matrix methods) and PLS (also known as P-matrix methods) are least-squares methods involving matrix operations. These methods can be limited when very complex mixtures are investigated and factor analysis methods, such as PLS and PCR, can be more useful. The factor analysis methods use functions to model the variance in a data set. 

The EDBD variant of BP was used in several chemical kinetic studies. 2-154 This method gave much better estimates of kinetic analytical parameters than either nonlinear regression or principal components regression. EDBD has also been applied to multicomponent kinetic determinations and to the estimation of kinetic compartmental model parameters.i The EDBD method was found to offer increased modeling power for nonlinear multivariate data compared to partial least squares and principal components regression, provided the training set is extensive enough to adequately sample the nonlinear features of the data.i55 Finally, EDBD has been successfully applied to the prediction of retention indices, i ... 

Eor multivariate calibration in analytical chemistry, the partial least squares (PLS) method [19], is very efficient. Here, the relations between a set of predictors and a set (not just one) of response variables are modeled. In multicomponent calibration the known concentrations of / components in n calibration samples are collected to constitute the response matrix Y (n rows, / columns). Digitization of the spectra of calibration samples using p wavelengths yields the predictor matrix X (n rows, p columns). The relations between X and Y are modeled by latent variables for both data sets. These latent variables (PLS components) are constructed to exhaust maximal variance (information) within both data sets on the one hand and to be maximally correlated for the purpose of good prediction on the other hand. From the computational viewpoint, solutions are obtained by a simple iterative procedure. Having established the model for calibration samples. comp>o-nent concentrations for future mixtures can be predicted from their spectra. A survey of multi-component regression is contained in [20],... 

Although colorimetric methods were the earliest to be used for pesticide analysis [203], competitive spectroscopic methodologies for the determination of these pollutants were not developed until the last decade. The spectroscopic determination of several pesticides in mixtures has been the major hindrance, especially when their analytical characteristics are similar and their signals overlap as a result. Multivariate calibration has proved effective with a view to developing models for qualitative and quantitative prediction from spectroscopic data. Thus, partial least squares (PLS) and principal component regression (PCR) have been used as calibration models for the spectrofluorimetric determination of three pesticides (carbendazim, fuberidazole, and thiabendazole) [204]. A three-dimensional excitation-emission matrix fluorescence method has also been used for this purpose (Table 18.3) [205]. 

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