Regression analysis principal components

Roy, K. and Ghosh, G. (2004c) QSTR with extended topochemical atom indices. 4. Modeling of the acute toxicity of phenylsulfonyl carboxylates to Vibrio fischeri using principal component factor analysis and principal component regression analysis. QSAR Comb. Sci., 23, 526-535. 

PCA is not only used as a method on its own but also as part of other mathematical techniques such as SIMCA classification (see section on parametric classification methods), principal component regression analysis (PCRA) and partial least-squares modelling with latent variables (PLS). Instead of original descriptor variables (x-variables), PCs extracted from a matrix of x-variables (descriptor matrix X) are used in PCRA and PLS as independent variables in a regression model. These PCs are called latent variables in this context. 

On the other hand, Gao et al. [50], using principal component regression analysis, derived a QSPR model for the prediction of solubility of a set of 930 diverse compounds, including pharmaceuticals, pollutants, nutrients, herbicides and pesticides. The model, which involves 24 molecular descriptors, predicts the log(S) with a squared correlation coefficient of 0.92 and rms error of 0.53 log units. 

Leonard and Roy [ 194] recently reported QSAR 70-73 on the HIV protease inhibitory data of 1,2,5,6-tetra-o-benzyl-D-mannitols (62) studied by Bouzide et al. [195]. Several statistical techniques such as stepwise regression, multiple linear regression with factor analysis as the data preprocessing step (FA-MLR), principal component regression analysis (PCRA) and partial least square (PLS) analysis were appHed to identify the structural and physicochemical requirements for HIV protease inhibitory activity. 

Because of the enormous number of x variables that are generated in the field calculations, regression analysis cannot be applied. In the very beginning of 3D QSAR studies, principal components were derived from the X block (i.e., the table of field values) and then correlated with the biological activity values. In 1986, Svante Wold proposed to use PLS analysis. PLS analysis resembles principal component regression analysis in its derivation of vectors from the Y and the X blocks. However, there is a fundamental difference in PLS analysis, the orientation of the so-called u and t vectors does not exactly correspond to the orientation of the principal components. They are slightly skewed within their confidence hyperboxes, in order to achieve a maximum intercorrelation (Figure 8). 

To gain insight into chemometric methods such as correlation analysis, Multiple Linear Regression Analysis, Principal Component Analysis, Principal Component Regression, and Partial Least Squares regression/Projection to Latent Structures... 

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

Factor spaces are a mystery no more We now understand that eigenvectors simply provide us with an optimal way to reduce the dimensionality of our spectra without degrading them. We ve seen that, in the process, our data are unchanged except for the beneficial removal of some noise. Now, we are ready to use this technique on our realistic simulated data. PCA will serve as a pre-processing step prior to ILS. The combination of Principal Component Analysis with ILS is called Principal Component Regression, or PCR. 

Because of peak overlappings in the first- and second-derivative spectra, conventional spectrophotometry cannot be applied satisfactorily for quantitative analysis, and the interpretation cannot be resolved by the zero-crossing technique. A chemometric approach improves precision and predictability, e.g., by the application of classical least sqnares (CLS), principal component regression (PCR), partial least squares (PLS), and iterative target transformation factor analysis (ITTFA), appropriate interpretations were found from the direct and first- and second-derivative absorption spectra. When five colorant combinations of sixteen mixtures of colorants from commercial food products were evaluated, the results were compared by the application of different chemometric approaches. The ITTFA analysis offered better precision than CLS, PCR, and PLS, and calibrations based on first-derivative data provided some advantages for all four methods. ... 

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 ... 

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. 

The method of PCA can be used in QSAR as a preliminary step to Hansch analysis in order to determine the relevant parameters that must be entered into the equation. Principal components are by definition uncorrelated and, hence, do not pose the problem of multicollinearity. Instead of defining a Hansch model in terms of the original physicochemical parameters, it is often more appropriate to use principal components regression (PCR) which has been discussed in Section 35.6. An alternative approach is by means of partial least squares (PLS) regression, which will be more amply covered below (Section 37.4). 

N.B Vogt, Polynomial principal component regression an approach to analysis and interpretation of complex mixture relationships in multivariate environmental data, Chemom. Intell. Lab. Syst, 7, 119-130 (1989). 

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. 

Haaland and coworkers (5) discussed other problems with classical least-squares (CLS) and its performance relative to partial least-squares (PLS) and factor analysis (in the form of principal component regression). One of the disadvantages of CLS is that interferences from overlapping spectra are not handled well, and all the components in a sample must be included for a good analysis. For a material such as coal LTA, this is a significant limitation. 

Experience in this laboratory has shown that even with careful attention to detail, determination of coal mineralogy by classical least-squares analysis of FTIR data may have several limitations. Factor analysis and related techniques have the potential to remove or lessen some of these limitations. Calibration models based on partial least-squares or principal component regression may allow prediction of useful properties or empirical behavior directly from FTIR spectra of low-temperature ashes. Wider application of these techniques to coal mineralogical studies is recommended. 

Principal Component (PC) In this book, the tenn principal component is used as a generic term to indicate a factor or dimension when using SIMCA, principal components analysis, or principal components regression. Using this terminology, there are scores and loadings associated with a given PC. (See also Factor.)... 

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