Factor analysis | Regression analysis

Principal component regression

In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). More specifically, PCR is used for estimating the unknown regression coefficients in a standard linear regression model. In PCR, instead of regressing the dependent variable on the explanatory variables directly, the principal components of the explanatory variables are used as regressors. One typically uses only a subset of all the principal components for regression, making PCR a kind of regularized procedure and also a type of shrinkage estimator. Often the principal components with higher variances (the ones based on eigenvectors corresponding to the higher eigenvalues of the sample variance-covariance matrix of the explanatory variables) are selected as regressors. However, for the purpose of predicting the outcome, the principal components with low variances may also be important, in some cases even more important. One major use of PCR lies in overcoming the multicollinearity problem which arises when two or more of the explanatory variables are close to being collinear. PCR can aptly deal with such situations by excluding some of the low-variance principal components in the regression step. In addition, by usually regressing on only a subset of all the principal components, PCR can result in dimension reduction through substantially lowering the effective number of parameters characterizing the underlying model. This can be particularly useful in settings with high-dimensional covariates. Also, through appropriate selection of the principal components to be used for regression, PCR can lead to efficient prediction of the outcome based on the assumed model. (Wikipedia).

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From playlist Linear Regression.

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Dimensionality reduction | Total sum of squares | Symmetric function | Regression analysis | Coordinate system | Statistics | Gauss–Markov theorem | Mallows's Cp | High-dimensional statistics | Estimator | Canonical correlation | Principal component analysis | Linear independence | Map (mathematics) | Covariance matrix | Positive-definite kernel | Transformation matrix | Singular value decomposition | Variance inflation factor | Linearity | Bias of an estimator | Regularization (mathematics) | Partial least squares regression | Variance | Dimension (vector space) | Estimation | Constrained optimization | Ordinary least squares | Linear regression | Multicollinearity | Centering matrix | Orthonormality | Linear combination | Deming regression | Ridge regression | Dependent and independent variables | Eigenvalues and eigenvectors | Orthogonal matrix | Linear form | Efficient estimator | Reproducing kernel Hilbert space | Cross-validation (statistics) | Mean squared error | Rank (linear algebra) | Eigendecomposition of a matrix | Linear approximation