Multivariate regression applications

The total residual sum of squares, taken over all elements of E, achieves its minimum when each column Cj separately has minimum sum of squares. The latter occurs if each (univariate) column of Y is fitted by X in the least-squares way. Consequently, the least-squares minimization of E is obtained if each separate dependent variable is fitted by multiple regression on X. In other words the multivariate regression analysis is essentially identical to a set of univariate regressions. Thus, from a methodological point of view nothing new is added and we may refer to Chapter 10 for a more thorough discussion of theory and application of multiple regression. 

A difficulty with Hansch analysis is to decide which parameters and functions of parameters to include in the regression equation. This problem of selection of predictor variables has been discussed in Section 10.3.3. Another problem is due to the high correlations between groups of physicochemical parameters. This is the multicollinearity problem which leads to large variances in the coefficients of the regression equations and, hence, to unreliable predictions (see Section 10.5). It can be remedied by means of multivariate techniques such as principal components regression and partial least squares regression, applications of which are discussed below. 

In the past few years, PLS, a multiblock, multivariate regression model solved by partial least squares found its application in various fields of chemistry (1-7). This method can be viewed as an extension and generalization of other commonly used multivariate statistical techniques, like regression solved by least squares and principal component analysis. PLS has several advantages over the ordinary least squares solution therefore, it becomes more and more popular in solving regression models in chemical problems. 

Artificial neural networks (ANNs) have been widely applied in the electronic tongue literature both for classification and multivariate regression problems almost one-third of the papers on electronic tongues examined for this review show ANN applications (see Fig. 2.10). 

Inductively coupled plasma (ICP) needs careful extraction of the relevant information to obtain satisfactory models and several chemometric studies were found. Indeed, many ICP spectroscopists have applied multivariate regression and other multivariate methods to a number of problems. An excellent review has recently been published on chemometric modelling and applications of inductively coupled plasma optical emission spectrometry (ICP-OES) [79]. 

In Section 3.3.6 we have already performed an example of multivariate regression, another application will be discussed in Section 10.2. In the current example let us again stress the small data set with five objects as last mentioned in Section 5.6.2. 

Not all relationships can be adequately described using the simple linear model, however, and more complex functions, such as quadratic and higher-order polynomial equations, may be required to fit the experimental data. Finally, more than one variable may be measured. For example, multiwavelength calibration procedures are finding increasing applications in analytical spectrometry and multivariate regression analysis forms the basis for many chemometric methods reported in the literature. 

Multivariate regression analysis plays an important role in modem process control analysis, particularly for quantitative UV-visible absorption spectrometry and near-IR reflectance analysis. It is conunon practice with these techniques to monitor absorbance, or reflectance, at several wavelengths and relate these individual measures to the concentration of some analyte. The results from a simple two-wavelength experiment serve to illustrate the details of multivariate regression and its application to multivariate calibration procedures. 

Of the many natural computation techniques, artificial neural networks (ANNs) stand out, particularly their application in carrying out multivariate regression. As they constitute a promising way to cope with complex spectral problems (both molecular and atomic), they are introduced here. After an... 

During the last two or three decades atomic spectroscopists have become used to the application of computers to control their instruments, develop analytical methods, analyse data and, consequently, to apply different statistical methods to explore multivariate correlations between one or more output(s) e.g. concentration of an analyte) and a set of input variables e.g. atomic intensities, absorbances). On the other hand, the huge efforts made by atomic spectroscopists to resolve interferences and optimise the instrumental measuring devices to increase accuracy and precision have led to a point where many of the difficulties that have to be solved nowadays cannot be described by simple univariate linear regression methods (Chapter 1 gives an extensive review of some typical problems shown by several atomic techniques). Sometimes such problems cannot even be addressed by multivariate regression methods based on linear relationships, as is the case for the regression methods described in the previous two chapters. 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

Multivariate calibration has the aim to develop mathematical models (latent variables) for an optimal prediction of a property y from the variables xi,..., jcm. Most used method in chemometrics is partial least squares regression, PLS (Section 4.7). An important application is for instance the development of quantitative structure—property/activity relationships (QSPR/QSAR). 

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