## 8 Discussion and Conclusions  
The simulation study and the analysis on the real dataset provide some very clear conclusions. In lower rank matrices without suspected contamination, classic EK82 can be used without problems with a simple statistic, such as PRESS50 that uses only 50% of the smallest residuals. Thus, being a resistant criterion, it presented good results in scenarios that did not present outliers. According to the results of our analysis, other competitive criteria were \(W_{m50(Max)}\) (which also depends on PRESS50) and \(W_{m(Max)}\). However, when there is any suspicion of contamination, EK82 should be replaced by one of the proposed robust schemes. Thus, if the objective is to choose the rank that produces good predictions with robust lower rank approximations, the EK84 method presented the best performance together with the \(W_{m}\) criterion in most of the situations studied. On the other hand, if the objective is to minimize the critical angle between two principal component subspaces, EK84 (with PRESS or \(W_{m}\)) can be used if the contamination level is high (20%). From our analysis on real data, we can conclude that at lower contamination levels (5 or 10%) EK84 (with PRESS or \(W_{m}\)) should be compared with EK01 (with \(W_{m75}\)), repeating the analysis several times (say, 10 or more) to gain consistent results. This can be achieved, for example, by adding a small number of outliers at random positions or by applying the methods on submatrices obtained from the original study matrix. The methodologies proposed in this paper proved to be efficient in solving the contamination problem without having to use any method of detecting outliers in the original data and without eliminating information, but there are still some aspects that are worth discussing. One of these refers to the simulation study. Similar to any simulation study, more features can be added to make it more complex, but in our case it was enough to show the performance of robust procedures taking the EK82 as the standard method. Our idea of fixing the matrix rank and the level of \"pure noise\" was to detect the strengths and weaknesses of the schemes when the contamination levels were variable, following ideas similar to those used by [PERSON] and [PERSON] [29] and [PERSON] et al. [32]. Future research from the computational point of view can be conducted to determine the robustness of the procedures under other conditions. A common problem in data analysis is performing cross-validation from incomplete matrices. In this case, two options are suggested, the first is to use EK84. This method is based on the rSVD of [PERSON] and [PERSON] [23], and the fit of rank-one approximations leaves out the missing positions and does not need to perform data imputation for this calculation, but if an imputation of the missing data is required, a robust lower rank approximation can be a very useful alternative if contamination is suspected. The second suggestion to get around the problem of missing data in cross-validation without suspected contamination is to apply iterative EK82, in which case there are already algorithms available [3; 33] that can be easily adapted for optimal selection of the rank of a matrix. [PERSON] et al. [3] showed that an iterative algorithm using parity check provides better quality imputations than the expectation-maximization method proposed by [PERSON] et al. [9]. In the dimensions of the matrices that the cross-validation schemes were tested in this study, computational time was not a problem, but if the matrix is of a much larger dimension, there are ways to circumvent this potential problem. For example, [PERSON] and [PERSON] [18] used several random samples from the rows of a multivariate matrix to achieve consistency in the rank selection and reduce computational time. Another alternative is to delete some random positions from the matrix and perform the imputation following one of the previously mentioned schemes.