# Principal component regression

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In statistics, principal component regression (PCR) is a regression analysis that uses principal component analysis when estimating regression coefficients. It is a procedure used to overcome problems which arise when the exploratory variables are close to being collinear.[1]

In PCR instead of regressing the dependent variable on the independent variables directly, the principal components of the independent variables are used. One typically only uses a subset of the principal components in the regression, making a kind of regularized estimation.

Often the principal components with the highest variance are selected. However, the low-variance principal components may also be important, — in some cases even more important.[2]

## The principle

PCR (principal components regression) is a regression method that can be divided into three steps:

1. The first step is to run a principal components analysis on the table of the explanatory variables,
2. The second step is to run an ordinary least squares regression (linear regression) on the selected components: the factors that are most correlated with the dependent variable will be selected
3. Finally the parameters of the model are computed for the selected explanatory variables.

## References

1. ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 [Amazon-US | Amazon-UK]
2. ^ Ian T. Jolliffe (1982). "A note on the Use of Principal Components in Regression". Journal of the Royal Statistical Society, Series C 31 (3): 300–303. doi:10.2307/2348005. JSTOR 2348005.
• R. Kramer, Chemometric Techniques for Quantitative Analysis, (1998) Marcel-Dekker, ISBN 0-8247-0198-4 [Amazon-US | Amazon-UK].