Robust triple K-Means

This paper introduces robust dimension-wise and cell-wise extensions of the Triple K-Means algorithm for three-way clustering in the presence of outliers. These methods tackle a tricky problem: clustering units, variables, and occasions all at once, while dealing with the possibility that outliers are lurking in any of the three modes. The dimension-wise Robust Triple K-Means identifies whole units, variables, or occasions that look like outliers and leaves them out entirely. The cell-wise version takes a more precise approach, flagging outliers at the level of individual data cells. Both rely on a trimmed Alternating Least Squares procedure to estimate cluster memberships and centroids. Because the trimmed outliers are kept out of the estimation, they cannot distort the resulting clusters. Furthermore, a data-driven method for selecting the number of outliers is proposed for each method. In both cases, elbow points are used as natural thresholds for outlier detection by leveraging numerical second derivatives. We also discuss the breakdown properties of the algorithms and the methods are evaluated through both a comprehensive simulation study and two real data applications, demonstrating their practical utility and performance in diverse settings.