Measurement based on classes(*)
Thierry Marchant
Ghent University, Belgium


A huge part of the literature about classification, in artificial intelligence and in psychology, is based on models where the objects to be classified are represented by points in an n-dimensional space. An interesting and rich theoretical literature analyses the various classification models based on this spatial representation but almost no paper explores the hypotheses underlying the spatial representation. We adopt in this paper a formalism close to that of measurement theory. We assume the existence of two observable primitives: a set X of objects and a collection of k subsets of X. The k subsets are the classes as observed in an experiment with an agent or subject. No spatial representation is assumed. We then look for axioms or conditions on the primitives ensuring that they can be represented by a given spatial classification model. The models analysed in our paper are the model based on the Euclidean distance w.r.t. a unique prototype for each class and different generalizations of it. We obtain characterizations for some particular and some general models. We improve a theorem of Goldstein, W.M. (1991, Decomposable threshold models. Journal of Mathematical Psychology, 35, 64-79) and we also consider the case of ordered classes.

(*)Denis Bouyssou and Thierry Marchant