## 3 Proposed Methodology  
The choice of the optimal rank for a lower rank approximation using the EK method basically depends on two aspects: the predictions of the elements of \(\mathbf{Y}\) and the criterion to determine if a component is important or not. It is known that the quality of predictions using the standard SVD decreases in the presence of outliers in the data matrix [20]. For that reason, our first attempt at circumventing this problem was to create a robust version of the EK schema. For this, the usual standardization of \(\mathbf{Y}\) was replaced by a robust standardization and, in the calculation of the elements of the predictor described in (4), the SVDs of \(\mathbf{Y}^{(-i)}\) and \(\mathbf{Y}_{(-j)}\) were replaced by an rSVD (details for calculating it are described in the next section). Although this approach produced good-quality predictions in small test matrices (for instance, \(20\times 5\)), the main disadvantage was computational because the implementation for larger matrices became too heavy due to the inclusion of rSVDs. In this way, the problem of outliers was circumvented, but the computational speed of the EK method was lost. This could later be a problem in the future for large arrays. Thus, we now have a more complex problem: how do we improve the predictions of the EK scheme in the presence of outliers, without eliminating them while at the same time trying to preserve the computational speed of the method as much as possible? To resolve this issue, we decided to use the work of [PERSON] et al. [21] and our proposal was to conduct the cross-validation in two stages: (i) calculate an rSVD on the original matrix \(\mathbf{Y}\) and obtain a robust lower rank approximation \(\mathbf{Y}_{RLRA}\) with the maximum possible number of components, that is, with the smallest number between \(n\) and \(p\). With this \(\mathbf{Y}_{RLRA}\), we can obtain a good approximation of the original values, and in the case of outliers we can obtain a robust approximation without eliminating any information and following the maximum-data precept. (ii) The EK method is then applied to \(\mathbf{Y}_{RLRA}\), and the optimal rank is determined. Such a combination of methodologies (rSVD + EK) was not found in our literature review so becomes a possible way to perform a robust cross-validation on contaminated lower rank matrices that is is computationally efficient as the size of the study matrix increases. Determining the rank depends on the chosen criterion; therefore, in addition to \(W_{m}\), we could use several criteria based on trimmed PRESS [22]. The following is a list of the possible criteria we considered:  
1. PRESS: According to Equation (5), the optimal rank is the one that minimizes the statistic. [PERSON] et al. [9] found that, in some cases, this criterion may be more effective than \(W_{m}\). 2. PRESS75: A resistant PRESS statistic is constructed by averaging the 75% of the smallest squared errors \(\left(\hat{y}_{ij}^{(m)}-y_{ij}\right)^{2}\). The optimal rank is the one that minimizes PRESS75. 3. PRESS50: A resistant PRESS statistic is constructed by averaging the 50% of the smallest squared errors \(\left(\hat{y}_{ij}^{(m)}-y_{ij}\right)^{2}\). The optimal rank is the one that minimizes PRESS50. 4. \(W_{m}\): According to Equation (6), the optimal rank is the total number of important components. 5. \(W_{m(Max)}\): The optimal rank is the number of the largest important component using \(W_{m}\). [PERSON] [8] found that, in some data sets, the \(W_{m}\) statistic does not always show a monotonically decreasing behaviour. For example, if \(W_{1}=26.22\), \(W_{2}=5.95\), \(W_{3}=0.19\) and \(W_{4}=1.03\), the optimal rank is 4, even though component 3 is not important. 6. \(W_{m75}\): In Equation (6), PRESS is replaced by PRESS75 and the optimal rank is the total number of important components. 7. \(W_{m75(Max)}\): The optimal rank corresponds to the number of the largest important components using \(W_{m75}\). 8.