Least-squares Bilinear Clustering of Three-way Data

__Abstract__ A least-squares bilinear clustering framework for modelling three-way data, where each observation consists of an ordinary two-way matrix, is introduced. The method combines bilinear decompositions of the two-way matrices into overall means, row margins, column margins and row-column interactions with clustering along the third way. Different clusterings are defined for each part of the decomposition, so that up to four different classifications are defined jointly. The computational burden is greatly reduced by the orthogonality of the bilinear model, such that the joint clustering problem reduces to separate ones which can be handled independently. Three of these sub-problems are specific cases of $k$-means clustering; a special algorithm is formulated for the row-column interactions, which are displayed in clusterwise biplots. The method is illustrated via two empirical examples and interpreting the interaction biplots are discussed.