Factorizing Three-Way Ordinal Data Using Triadic Formal Concepts

The paper presents a new approach to factor analysis of three-way ordinal data, i.e. data described by a 3-dimensional matrix I with values in an ordered scale. The matrix describes a relationship between objects, attributes, and conditions. The problem consists in finding factors for I, i.e. finding a decomposition of I into three matrices, an object-factor matrix A, an attribute-factor matrix B, and a condition-factor matrix C, with the number of factors as small as possible. The difference from the decomposition-based methods of analysis of three-way data consists in the composition operator and the constraint on A, B, and C to be matrices with values in an ordered scale. We prove that optimal decompositions are achieved by using triadic concepts of I, developed within formal concept analysis, and provide results on natural transformations between the space of attributes and conditions and the space of factors. We present an illustrative example demonstrating the usefulness of finding factors and a greedy algorithm for computing decompositions.

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