Cross-validation in PCA models with the element-wise k-fold (ekf) algorithm: Practical aspects

• PCA cross-validation is discussed for a number of applications.
• Published and original methods are compared.
• The ekf algorithm is only suited for missing data applications.
• Wold and Eastment & Krzanowski provide poor outcomes.
• The error of iterative estimation is theoretically studied.

This is the second paper of a series devoted to provide theoretical and practical results and new algorithms for the selection of the number of Principal Components (PCs) in Principal Component Analysis (PCA) using cross-validation. The study is especially focused on the element-wise k-fold (ekf), which is among the most used algorithms for that purpose. In this paper, a taxonomy of PCA applications is proposed and it is argued that cross-validatory algorithms computing the prediction error in observable variables, like ekf, are only suited for a class of applications. A number of cross-validation methods, several of which are original, are compared in two applications of this class: missing data imputation and compression. The results show that the ekf is especially suited for missing data applications while other traditional cross-validation methods, those by Wold and Eastment and Krzanowski, are not found to provide useful outcomes in any of the two applications. These results are of special value considering that the methods investigated are computed in the main commercial software packets for chemometrics. Finally, the choice of the missing data algorithm within ekf is also investigated.