2.03 - Principal Component Analysis

Principal Component Analysis (PCA) is a multivariate exploratory analysis method, useful to separate systematic variation from noise. It allows to define a space of reduced dimensions that preserves the relevant information of the original data and allows visualization of objects (scores) and variables (loadings). PCA requires multivariate data, meaning many variables measured on many objects. Data, vectors and matrices are defined and a short summary of necessary linear algebra is given. Purely mathematical almost identical definitions of PCA and Singular Value Decomposition (SVD) are shown, but in chemometrics, PCA always has a residual and a number of meaningful components, the rank. This leads to a discussion of numerical and visual diagnostics for finding the rank and checking the residual. The visualization of scores and loadings is introduced by means of two small examples. Data preprocessing is also given consideration.