Cross-validation to select the optimum rank for a reduced-rank approximation to multivariate data

In this paper we consider the Gabriel form of cross-validation (CV) and we investigate how to estimate the optimum rank for lower rank approximations of any dataset that can be written in matrix form, with particular application in multivariate analysis and in the analysis of multienvironment trials. The literature related to the method suggests that it can produce overfitting and poor-quality predictions, characteristics that result in overestimation of the rank. Because of this, it is proposed to change the rank selection criterion, testing thirteen statistics both in the original method and in four proposed extensions that seek to solve the above problems. A comparison is made with two gold standard methods for CV through a simulation study and through the analysis of seventeen real datasets, two of which are general multivariate and fifteen are from experiments with genotype-by-environment interaction. It is concluded that from a predictive point of view, the highest accuracy in estimating the rank is obtained by using a regularized singular value decomposition.