`stretch' Vs`slice' Methods for Representing Three-way Structure via Matrix Notation

A three-way array must be represented in two-way form if its structure is to be described and manipulated by means of matrix notation. Historically, two methods, here calledàrray stretching' andàrray slicing', have been used. More recently, however, array slicing has often been overlooked, resulting in a loss of mathematical flexibility. `Stretching' involves matricizing (unfolding) the three-way array and applying one's mathematical operations to the resulting two-way matrix; this results in expressions that are often quite useful for parameter estimation but which are relatively long and require practice to interpret properly. `Slicing' involves taking a representative two-way subarray and applying operations to it; this often gives compact and easily understood expressions but requires the introduction of extra matrix names and becomes awkward if the array is not`slicewise regular'. In this paper the advantages of each approach are demonstrated and compared by applying them to a set of models from the Tucker and Parafac families. In addition, we show how slicewise representation can be improved by using (i) angle brackets to eliminate the need for extra diagonal matrices, and (ii) `encapsulated summation' notation to allow representation of array structure that is orderly but not slicewise regular.