Component Analysis of a Three-way Dataset of TV-ratings


Speaker


Abstract

This talk is about component analysis of three-way arrays, where a three-way array of size IxJxK can be seen as K matrices of size IxJ. For a matrix, component analysis can be done via principal component analysis. This boils down to finding a best rank-R approximation via the truncated singular value decomposition. For a three-way array an analogous approach exists, which is called Candecomp/Parafac. However, a best rank-R approximation may not exist for three-way arrays when R>1. When a best rank-R approximation does not exist, trying to compute it results in rank-1 components that diverge to linear dependency and infinite magnitude while the rank-R approximation itself converges to an optimal boundary point of the rank-R set. This situation can be avoided by imposing constraints such as orthogonality or nonnegativity. A different and more versatile approach is to include specific interaction terms in the rank-R decomposition. We demonstrate this procedure for a three-way dataset containing 15 TV shows that are scored on 16 rating scales by 30 persons. Trying to compute a best rank-3 approximation results in two diverging rank-1 terms, while fitting a model with one additional interaction term yields an interpretable solution.