Cross-validatory choice in principal component analysis; some sampling results

Abstract

Sampling behaviour is investigated of the statistic W in the cross-vaiidatory scheme for selectionof number of components in a principal component analysis, as proposed by Eastment and Krzanowski [1982, Technometrics, 24, 73–77]. Two possible levels of variability are isolated, conditional and unconditional. The former is shown to be appropriate when assessing results from analysis of real data, and a previously given example is assessed in this way. As a by-product of this analysis, reasons for earlier erratic behaviour of W are suggested and the previous tentative proposals about the cut-off value of W are confirmed. The two possible sources of variability inW lead to a hierarchical design of the main Monte Carlo experiment. Results on sampling behaviour of W are given, and compared with traditional eigenvalue-based methods of choosing components. A possible interpretation of W is put forward.

Keywords:

• Cross-validation
• Monte-Carlo methods
• Random orthogonal matrices
• Singular value decomposition
• Wishart variates