Principal component regression independent matrix

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. 

In the previous chapter, it was commented on that the ordinary least-sqnares approach applied to multivariate data (multivariate linear regression, MLR) suffered from serious uncertainty problems when the independent variables were collinear. Principal components regression (PCR) can solve the collin-earity problem and provide additional benefits of factor-based regression methods, such as noise filtering. Recall that PCR compresses the original X-block e.g. matrix of absorbances) into a new block of scores T, containing fewer variables (the so-called factors, latent variables, or principal components), and then regression is performed between T and the property of... 

Problems with the inversion of the covariance matrix can be overcome by a preceding principal component analysis. Instead of (correlated) features a set of principal component scores is used as independent variables in the regression equation (principal component regression, PCR). Remember that PCA scores are uncorrelated by definition. 

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