Principal components regression cross-validation

M. Stone and R.J. Brooks, Continuum regression cross-validated sequentially constructed prediction embracing ordinary least sqaures, partial least squares, and principal component regression. J. Roy. Stat. Soc. B52 (1990) 237-269. 

For partial least-squares (PLS) or principal component regression (PCR), the infrared spectra were transferred to a DEC VAX 11/750 computer via the NIC-COM software package from Nicolet. This package also provided utility routines used to put the spectra into files compatible with the PLS and PCR software. The PLS and PCR program with cross-validation was provided by David Haaland of Sandia National Laboratory. A detailed description of the program and the procedures used in it has been given (5). 

For a well-behaved sensor array, only a small subset k of n available PCs is sufficient to characterize the matrix. Once again, Principal Component Regression (PCR) is a data reduction tool. The robustness of the selection of k can be tested by cross-validation in which case data subsets are randomly selected and the error matrix H xn is calculated. 

There is an approach in QSRR in which principal components extracted from analysis of large tables of structural descriptors of analytes are regressed against the retention data in a multiple regression, i.e., principal component regression (PCR). Also, the partial least square (PLS) approach with cross-validation 29 finds application in QSRR. Recommendations for reporting the results of PC A have been published 130). 

Likewise, a PARAFAC model for X can be assumed and y can subsequently be regressed on the proper PARAFAC components [Bro 1997, Gel adi etui. 1998], Obviously, the number of PARAFAC components or the number of components in the Tucker3 model has to be found. This can be done, e.g., by cross-validation (see Chapter 7). The procedure of Equations (4.21)-(4.23) resembles principal component regression and can be generalized to higher-way X and two-way Y. 

M. Stone and R. J. Brooks, Continuum Regression Cross-validated SequentiaUy-consIructedPrediction Embracing Ordinary Least Squares, Partial Least Squares, and Principal Component Regression, J. R. Stat. Soc. B., 52 337-369 (1990). 

Principal component analysis (PCA) and principal component regression (PCR) were used to analyze the data [39,40]. PCR was used to construct calibration models to predict Ang II dose from spectra of the aortas. A cross-validation routine was used with NIR spectra to assess the statistical significance of the prediction of Ang II dose and collagen/elastin in mice aortas. The accuracy of the PCR method in predicting Ang II dose from NIR spectra was determined by the F test and the standard error of performance (SEP) calculated from the validation samples. 

A crucial decision in PLS is the choice of the number of principal components used for the regression. A good approach to solve this problem is the application of cross-validation (see Section 4.4). 

With respect to the appHed regression methodologies, RR is similar to PCR in that the independent variables are transformed to their principal components (PCs). However, while PCR utilizes only a subset of the PCs, RR retains them all but downweights them based on their eigenvalues. With PLS, a subset of the PCs is also used, but the PCs are selected by considering both the independent and dependent variables. For each model developed, the cross-validated R was obtained using the leave-one-out (LOO) approach and can be calculated as follows ... 

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