Principal Component Regression (PCR) is a statistical technique used in regression analysis that combines the principles of principal component analysis (PCA) with linear regression. It is particularly useful when dealing with multicollinearity, which occurs when independent variables in a regression model are highly correlated, leading to unstable coefficient estimates and reduced interpretability.
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Regress a response on selected principal component scores of centered predictors. If is a singular value decomposition and the first positive-singular-value components are retained, the retained score matrix is and the fitted coefficient in predictor coordinates is . The fitted centered response is : this follows by applying the least-squares normal equations to the orthogonal score columns. Truncating small singular directions limits variance amplification, but directions of large predictor variance need not be those best predicting the response; select using prediction assessment rather than explained predictor variance alone.