= Principal component regression
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= PCR
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Regress a response on selected <principal component scores> of centered predictors. If $X=UDV^T$ is a <singular value decomposition> and the first $q$ positive-singular-value components are retained, the retained score matrix is $U_qD_q$ and the fitted coefficient in predictor coordinates is $V_qD_q^{-1}U_q^Ty$. The fitted centered response is $U_qU_q^Ty$: 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 $q$ using prediction assessment rather than explained predictor variance alone.
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