Q-Discrimination Analysis

The Q-analysis algorithm is sometimes used as a clustering procedure, sometimes inappropriately. The data for clustering a set of elements in terms of a set of descriptors are assumed to be weighted in a scale. This paper shows the scale requires no structure beyond an ordinal order relation to allow a refinement of the q-connected component partitions by Q-discrimination analysis. This requires a clear understanding of the meaning of the scale values when data are being collected, and gives a precise meaning to the slicing procedure of Q-analysis. The relevant definitions of Q-analysis are presented through examples and the paper is mathematically self-contained. It concludes with two illustrative examples and discusses the questions they raise. Q-discrimination analysis is presented as part of the methodology of Q-analysis, and to the extent that clustering is essentially concerned with set definition its application represents the beginning of a scientific enquiry rather than an end result.