## 5 Simulation Study  
To evaluate the performance of the cross-validation schemes (i.e., EK01, EK13, EK82, and EK84) on contaminated low-rank matrices, one hundred matrices of dimension (\(100\times 8\)) were simulated using the following process: \"Clean\" observations were generated as \(\mathbf{Y}=\mathbf{T}+\mathbf{E}\), where \(\mathbf{T}=\mathbf{A}\mathbf{B}^{T}=\left[t_{ij}\right]\) is a rank four matrix with \(\mathbf{A}\) (\(100\times 4\)), \(\mathbf{B}\) (\(8\times 4\)), and \(\mathbf{E}\) (\(100\times 8\)) is \"pure noise\". The elements of \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{E}\) were iid N(0, 1). Outliers were produced on each \(\mathbf{Y}\) matrix in different percentages (0, 5, 10 and 20%), their positions were chosen randomly and were generated using the normal distribution with mean \(\mu_{j}+100\sigma_{j}^{2}\) and variance \(\sigma_{j}^{2}\). In this case, \(\mu_{j}\) and \(\sigma_{j}^{2}\) represent the mean and variance of the j-th column. Contaminated matrices will be noted by \(\mathbf{Y}_{\text{C}}\).  
The four cross-validation schemes were applied to the \(\mathbf{Y}_{\text{C}}\) matrices to determine their rank \(m\) (each scheme with each statistic considered can provide different ranks). The quality of this choice was evaluated from the predictive point of view, calculating the corresponding (robust) lower rank approximation with \(m\) components (\(\hat{\mathbf{Y}}\)) and comparing it with the original matrix \(\mathbf{Y}\) without contamination. For this, the Procrustes \(M^{2}\) statistic was used [27]. In this case, \(M^{2}=trace\Big{(}\mathbf{Y}\mathbf{Y}^{T}+\hat{\mathbf{Y}}\hat{\mathbf{Y}}^{T}-2\mathbf{Y}\mathbf{Q} \hat{\mathbf{Y}}^{T}\Big{)}\) where \(\mathbf{Q}=\mathbf{V}\mathbf{U}^{T}\) is the rotation matrix calculated from elements of the SVD of the matrix \(\mathbf{Y}^{T}\hat{\mathbf{Y}}=\mathbf{U}\sum\mathbf{V}^{T}\). The \(M^{2}\) statistic measures the difference between two configurations of points, so the (robust) lower rank approximation that minimizes this difference indicates the rank selection method that yields the closest match between clean data and calculated approximations in the presence of outliers.  
Another criterion for comparing the proposed methods, following the work of [PERSON] [8; 28], was the critical angle (\(\theta\)) between two subspaces of principal components. For this, both the SVD of \(\mathbf{Y}=\mathbf{U}\mathbf{D}\mathbf{V}^{T}\) and the (r)SVD of \(\mathbf{Y}_{\text{C}}=\mathbf{W}\mathbf{J}\mathbf{K}^{T}\) were calculated. If the cross-validation shows that the rank is \(m\), the matrices \(\mathbf{V}\) and \(\mathbf{K}\) with \(m\) retained components are compared, that is, \(\mathbf{V}_{(m)}\) and \(\mathbf{K}_{(m)}\). The calculation of the critical angle is defined by \(\theta=cos^{-1}(d)\), where \(d\) is the smallest element of \(\mathbf{G}\) in the SVD of the matrix \(\mathbf{V}_{(m)}^{T}\mathbf{K}_{(m)}=\mathbf{M}\mathbf{G}\mathbf{P}^{T}\). The greater the critical angle, the greater the influence of outliers on the principal coefficients components; therefore, the best cross-validation scheme is the one that provides the lowest value for \(\theta\).  
In each percentage of outliers considered, one hundred values of the Procrustes statistics and one hundred values of critical angles were obtained. From these values, the means of the 90% with the lowest values were calculated. This average is a robust criterion that leaves out some flaws that a method may have, but which, in general, presents a good behavior [20; 29].