Definition of Term: Principal Component Analysis

Principal Component Analysis (PCA) is a technique used in multivariate data analysis for dimension reduction. Given a sample of vector random variables, the aim is to find a set of orthogonal coordinate axes, known as principal components, with a small number of principal components describing most of the variation in the data. Projecting the data onto the first principal component maximises the variance of the projected points, compared to all other linear projections. Projecting the data onto the second principal component maximises the variance of the projected points, compared to all other linear projections that are orthogonal to the first principal component. Subsequent principal components are defined likewise.

The principal components are obtained using an eigendecomposition of the variance matrix of the data. The eigenvectors give the principal components, and the eigenvalues divided by their sum give the proportion of variation in the data explained by each associated eigenvector (hence the eigenvector with the largest eigenvalue is the first principal component and so on).

If a simulator output is multivariate, for example a time series, PCA can be used to reduce the dimension of the output before constructing an emulator.