Partial Least Squares extensions

Most of the 2D QSAR methods are based on graph theoretic indices, which have been extensively studied by Randic [29] and Kier and Hall [30,31]. Although these structural indices represent different aspects of molecular structures, their physicochemical meaning is unclear. Successful applications of these topological indices combined with multiple linear regression (MLR) analysis are summarized in Ref. 31. On the other hand, parameters derived from various experiments through chemometric methods have also been used in the study of peptide QSAR, where partial least square (PLS) [32] analysis has been employed [33]. 

Figure 12.8 displays an organization chart of various quantitative methods, in an effort to better understand their similarities and differences. Note that the first discriminator between these methods is the direct versus inverse property. Inverse methods, such as MLR and partial least squares (PLS), have had a great deal of success in PAT over the past few decades. However, direct methods, such as classical least squares (CLS) and extensions thereof, have seen a recent resurgence [46-51]. The criterion used to distinguish between a direct and an inverse method is the general form of the model, as shown 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. 

Another extension is given by Frans et al. This method is adapted to LC-UV. The number of components (5 to 8) is determined with PCA. Lindberg et al. apply a LSO still more close to FA than the method applied by King and King, and they claim a better accuracy for their Partial Least Squares (PLS) method compared to real LSO or FA. 

When compounds are selected according to SMD, this necessitates the adequate description of their structures by means of quantitative variables, "structure descriptors". This description can then be used after the compound selection, synthesis, and biological testing to formulate quantitative models between structural variation and activity variation, so called Quantitative Structure Activity Relationships (QSARs). For extensive reviews, see references 3 and 4. With multiple structure descriptors and multiple biological activity variables (responses), these models are necessarily multivariate (M-QSAR) in their nature, making the Partial Least Squares Projections to Latent Structures (PLS) approach suitable for the data analysis. PLS is a statistical method, which relates a multivariate descriptor data set (X) to a multivariate response data set Y. PLS is well described elsewhere and will not be described any further here [42, 43]. 

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. ... 

Frake and co-workers " extensively evaluated numerous chemometric techniques for the NIRS prediction of mass median particle size determination of lactose monohydrate. Models evaluated in zero order (untreated) and second derivative were MLR, PLS (partial least squares), and ANN (artificial neural network). The researchers concluded that there is more than one way to treat data and achieve a good calibration model. The group also confirms previous observations that derivitization of data does not remove particle size effects (previously thought to contribute to baseline shift). 

Intermediate Least Squares regression (ILS) is an extension of the Partial Least Squares (PLS) algorithm where the optimal variable subset model is calculated as intermediate to PLS and stepwise regression, by two parameters whose values are estimated by cross-validation [Frank, 1987]. The first parameter is the number of optimal latent variables and the second is the number of elements in the weight vector w set to zero. This last parameter (ALIM) controls the number of selected variables by acting on the weight vector of each mth latent variable as the following ... 

The background for the extension of two-way partial least squares regression to multiway data (fV-PLS) was provided by Bro [1996] and further elaborated on by Smilde [1997] and de Jong [1998], An improved model of X was later introduced which, however, maintains the same predictions as the original model [Bro et ol. 2001], Only the three-way version of A-PLS is considered here. It differs from the N-way (N > 3) partial least squares regression algorithm in that it has a closed-form solution for the situation with only one dependent variable. 

Partial Least Squares (PLS) is an extension of PCA where both the x and y data are considered. In PCA only the x data is considered. The goal of the PLS analysis is to build an equation that predicts y values (laboratory data) based on x (spectral) data. The PLS equation or calibration is based on decomposing both the x and y data into a set of scores and loadings, similar to PCA. However, the scores for both the X and y data are not selected based on the direction of maximum variation but are selected in order to maximize the correlation between the scores for both the x and y variables. As with PCA, in the PLS regression development the number of components or factors is an important practical consideration. A more detailed discussion of the PLS algorithm can be found elsewhere [13, 14]. Commercial software can be used to construct and optimize both PCA and PLS calibration models. 

In this section we shall consider the rather general case where for a series of chemical compounds measurements are made in a number of parallel biological tests and where a set of descriptor variables is believed to be related to the biological potencies observed. In order to imderstand the data in their entirety and to deal adequately with the mathematical properties of such data, methods of multivariate statistics are required. A variety of such methods is available as, for example, multivariate regression, canonical correlation, principal component analysis, principal component regression, partial least squares analysis, and factor analysis, which have all been applied to biological or chemical problems (for reviews, see [1-11]). Which method to choose depends on the ultimate objective of an analysis and the property of the data. We have found principal component and factor analysis particularly useful. For this reason and also since many multivariate methods make use of components for factors we will start with these methods in some detail, while the discussion of other approaches will be less extensive. 

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 ... 

In the extension of this least-squares procedure to cover interelement correction terms, other partial differentials must be derived for each term required, that is, of the form... 

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