Principal component regression algorithms

Tablet hardness is a property that, when measured, destroys the sample. The destructive nature of the test, coupled with the variability of the test itself does not contribute to an incentive to test a large number of samples. Morisseau and Rhodes99 correlated the diffuse reflectance NIR spectra of tablets pressed at different pressures and subsequently tested the tablet hardness with an Erweka Hardness Tester. The tablet hardness, as predicted by the NIR method, was at least as precise as the laboratory test method. Kirsch and Drennen100 evaluated NIR as a method to determine potency and tablet hardness of Cimetidine tablets over a range of 1-20% potency and 107-kPa compaction pressure. Hardness at different potency levels was used to build calibration models using PCA/ principal component regression and a new spectral best-fit algorithm. Both methods provided acceptable predictions of tablet hardness.

An evaluation of the performance of these algorithms in predictions using inside model space and outside model space was conducted. In principal component regression, principal axes highly correlated with sample constituents of interest are considered to be inside model space, while axes typically attributed to spectral noise are termed outside model space. 

Depezynski, U., Frost, V.J. and Molt, K. (2000) Genetic algorithms applied to the selection of factors in principal component regression. Anal. Chim. Aaa, 420, 217-227. 

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). 

Principal component regression (PCR) is the algorithm by which PCA is used for quantitative analysis and involves a two-step process. The first is to decompose a calibration data set with PCA to calculate all the significant principal components, and the second step is to regress the concentrations against the scores to produce the component calibration coefficients. Generally, the ILS model is preferred, as it does not require knowledge of the complete composition of all the spectra. Therefore, if we use the ILS model from Eq. 9.16 but rewrite it for scores, S, instead of absorbances. A, we have... 

Kohonen network Conceptual clustering Principal Component Analysis (PCA) Decision trees Partial Least Squares (PLS) Multiple Linear Regression (MLR) Counter-propagation networks Back-propagation networks Genetic algorithms (GA)... 

The NIPALS algorithm extracts one factor (a principal component) at a time from the mean-centred data matrix. Each factor is obtained iteratively by repeated regression of response (absorbance) data, F , on the scores (principal components) Z to obtain improved loadings (eigenvectors) V, and of F on V to obtain improved Z. 

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