Partial least squares regression coefficients

Partial least squares regression (PLS). Partial least squares regression applies to the simultaneous analysis of two sets of variables on the same objects. It allows for the modeling of inter- and intra-block relationships from an X-block and Y-block of variables in terms of a lower-dimensional table of latent variables [4]. The main purpose of regression is to build a predictive model enabling the prediction of wanted characteristics (y) from measured spectra (X). In matrix notation we have the linear model with regression coefficients b ... 

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. 

Parent distribution, 3 Partial correlation, 183 Partial least squares regression, 197 Partial regression coefficients, 172 Pattern recognition, supervised, 123 unsupervised, 92 Peak finding, 60 Perceptron, 142 Polynomial interpolation, 48 Polynomial regression, 163... 

Depending on what kind of information is needed, different possibilities exist for using a multilinear partial least squares regression model on new data. If only the prediction of y is wanted, it is possible to obtain a set of regression coefficients that relates X directly to y. This has been shown in detail by Smilde and de Jong for the multilinear PLS1 model [de Jong 1998, Smilde 1997] as follows. The first score vector ti is... 

SE Standard error, R coefficient of determination, SEC standard error of calibration, SEV(C) standard error of cross-validation, PLS terms number of terms used for modified partial least squares regression. 

Partial least squares regression (PLSR) was used again. Leave one out cross validation was carried between the new matrix XRa and concentration matrix Y. After each step of leave one out cross validation, a regression coefficient b was obtained. 

Differences between PIS and PCR Principal component regression and partial least squares use different approaches for choosing the linear combinations of variables for the columns of U. Specifically, PCR only uses the R matrix to determine the linear combinations of variables. The concentrations are used when the regression coefficients are estimated (see Equation 5.32), but not to estimate A potential disadvantage with this approach is that variation in R that is not correlated with the concentrations of interest is used to construct U. Sometiraes the variance that is related to the concentrations is a verv... 

Table 3 (73) compares the retention coefficients for synthetic peptides from various sources. To ensure comparability, the data has been standardized with respect to lysine and assigned a value of 100. The table shows that there are discrepancies between the results obtained using different chromatographic systems. Predictions of retention times should therefore be made using chromatographic systems similar to those used to calculate the retention coefficients for the amino acids. Casal et al. (75a) have made a comparative study of the prediction of the retention behavior of small peptides in several columns by using partial least squares and multiple linear regression analysis. 

A number of variable selection techniques were also suggested for the Partial Least Squares (PLS) regression method [Lindgren et al, 1994]. The different strategies for PLS-based variable selection are usually based on a rotation of the standard solution by a manipulation of the PLS weight vector w or of the regression coefficient vector b of the PLS closed form. 

Numerous software data treatments authorize the elucidation of mixture composition from spectra. One of the best-known methods is the Kalman s least squares filter algorithm, which operates through successive approximations based upon calculations using weighted coefficients (additivity law of absorbances) of the individual spectra of each components contained in the spectral library. Other software for determining the concentration of two or more components within a mixture uses vector quantification mathematics. These are automated methods better known by their initials PLS (partial least square), PCR (principal component regression), or MLS (multiple least squares) (Figure 9.26). 

The coefficients are calculated by multi-linear regression, according to the least squares method. There are a very large number of different programs for doing these calculations. The use of properly structured experimental designs, which are usually quite close to orthogonality, has the result that the more sophisticated methods (partial least squares etc.) are not usually necessary. 

Da Costa Filho in 2009 elaborated a rapid method to determine sucrose in chocolate mass using near infrared spectroscopy. Data were modelled using partial least squares (PLS) and multiple linear regression (MLR), achieving good results (correlation coefficient of 0.998 and 0.997 respectively for the two chemometric techniques). Results showed that NIR can be used as rapid method to determine sucrose in chocolate mass in chocolate factories. 

None of the above approaches optimizes the relationship between NIR absorbances and analyte for a range of sample types. Derivative transformations have been found to be generally useful when stepwise multiple linear regression (SMLR) techniques are used. When multidimensional statistics are employed, e.g., partial least-squares (PLS), principal component regression (PCR), or neural nets, it has been observed in some cases that the untransformed log 1/R data can perform just as well in correlation coefficient and error terms as in any kind of transformation. It is considered that in some cases physical manifestations of the sample contained in the spectra provide valid and useful discriminant data. 

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