On Relationships between Uncentred and Column-centred Principal Component Analysis

Principal component analysis (PCA) can be seen as a singular value decomposition (SVD) of a column-centred data matrix. In a number of applications, no pre-processing of the data is carried out, and it is the uncentred data matrix that is subjected to an SVD, in what is often called an uncentred PCA. This paper explores the relationships between the results from both the standard, column-centred, PCA, and its uncentred counterpart. In particular, it obtains both exact results and bounds relating the eigenvalues and eigenvectors of the covariation matrices, as well as the principal components, in both types of analysis. These relationships highlight how the eigenvalues of both the covariance matrix and the matrix of non-central second moments contain much information that is highly informative for a comparative assessment of PCA and its uncentred variant. The relations and the examples also suggest that the results of both types of PCA have more in common than might be supposed.