Cross-validation (CV) is a common approach for determining the optimal number of components in a principal component analysis model. To guarantee the independence between model testing and calibration, the observationwise k-fold operation is commonly implemented in each cross-validation step. This operation renders the CV algorithm computationally intensive, and it is the main limitation to apply CV on very large data sets. In this paper, we carry out an empirical and theoretical investigation of the use of this operation in the element-wise k-fold (ekf) algorithm, the state-of-the-art CV algorithm. We show that when very large data sets need to be cross-validated and the computational time is a matter of concern, the observation-wise k-fold operation can be skipped. The theoretical properties of the resulting modiﬁed algorithm, referred to as column-wise k-fold (ckf) algorithm, are derived. Also, its performance is evaluated with several artiﬁcial and real data sets. We suggest the ckf algorithm to be a valid alternative to the standard ekf to reduce the computational time needed to cross-validate a data set.
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