# Techniques for Robust Imputation in Incomplete Two-Way Tables
## 2 Material and Methods
### Robust Singular Value Decomposition  
The presence of outliers in a data set can reduce the effectiveness of least squares techniques [12], which in this case is the standard SVD. To avoid this behavior, robust lower-rank approximations or equivalent robust SVD could be used. Fortunately, the literature provides several alternatives that can be considered. Recently, [14] presented a review of four options for producing robust SVD. The first is an approximation of alternating regressions based on a mixture of M estimators and trimmed least squares estimators. For more details on these types of regressions, see [15]. The second consists of a sub-sampling method whose objective is to find a sub-matrix of rows without outliers with high probability and from this sub-matrix to estimate the effects of rows and columns by means of a robust function. For more details see [16]. The third option is to use iterative reweighted least square algorithms to obtain robust estimators in principal component analysis (PCA). A complete description of this approach and extensions can be found in [17] and a computational implementation is available in the R RobRSVD package [13]. The last option is an optimization with restrictions in which the data matrix is assumed to be a sum of three matrices, one of low rank, a sparse matrix to represent the outliers, and a random noise matrix. Algorithms for such optimization and additional details can be found in [14]. Other very useful work on robust SVD has been carried out by [18], who used alternating robust fittings with trimmed least squares to find the SVD and apply it in microarray data analysis; [PERSON] and [PERSON] [19] who used alternating M regressions to obtain robust low-rank approximations; and [PERSON] et al. [20; 21], who used the SVD proposed by [PERSON] et al. [22] for robust singular spectral analysis and for a robust version of AMMI models. The computational implementation of this SVD can be found in the R pcaMethods package ([13]; pcaMethods available from Bioconductor; [[https://bioconductor.org/packages/release/bioc/html/pcaMethods.html](https://bioconductor.org/packages/release/bioc/html/pcaMethods.html)]([https://bioconductor.org/packages/release/bioc/html/pcaMethods.html](https://bioconductor.org/packages/release/bioc/html/pcaMethods.html)), accessed on 25 June 2021). Given this literature review and taking our objectives into account, we delimit the research and choose an option to produce a robust SVD that can use the SVD88 method. To focus specifically on matrices with genotype-by-environment interaction, we adopt the suggestion of [20]. Thus, we use the [PERSON] et al. [22] SVD which is based on an alternating L1 regression algorithm (rSVD01), and the generalization proposed by [23] using weighted least absolute deviation regression (rSVD10). The frequent use of alternating regressions in the literature cited above obliged us to also consider the classic reference of [PERSON] [24], whose main objective was to produce resistant low-rank matrices (rSVD84). We thus initially focused on comparing the performances of rSVD01, rSVD10, and rSVD84 when they are introduced into the SVD88 imputation system. However, their individual performance was also considered, i.e., without introducing them into the imputation system. This is because the SVD from alternating regressions does not need any data imputation as it uses only the observed information. Moreover, a robust low-rank approximation can provide estimates of the missing positions of an incomplete matrix [18] without any additional calculations. The three SVDs were tested, and in the case of rSVD01 and rSVD10, we also tested replacing L1 regressions with M regressions using the rlm function of R ([13]; RobRSVD available from CRAN; see [[https://cran.r-project.org/web/packages/RobRSVD/index.html](https://cran.r-project.org/web/packages/RobRSVD/index.html)]([https://cran.r-project.org/web/packages/RobRSVD/index.html](https://cran.r-project.org/web/packages/RobRSVD/index.html)), accessed on 25 June 2021).