What Clusters Are Generated by Normal Mixtures?

Model based cluster analysis is often carried out by estimation of the parameters of a normal mixture. But mixture components do not necessarily reeect the idea of a \cluster". I discuss how to formalize the concept of \clusters" w.r.t. probability distributions on the real line by means of xed point clusters, i.e., sets that do not contain any outlier and with respect to which the rest of the real line consists of outliers. The concept is applied to some normal mixtures. 1 Theoretical clusters of distributions There are lots of diierent goals of cluster analysis (CA) and therefore there are lots of diierent meanings and formal deenitions of the term \cluster". Most of them have in common that a cluster should be a group of entities which is \homogeneous" in some manner, i.e., the entities are similar to each other, and which is in some manner \separated" from the rest. In CA based on probability models, it is assumed that the entities (usually from IR p ; here I consider only IR) are generated by some probability distribution. The clusters of the data are determined by using estimators of the features of the distribution (i.e., parameters or regions of high density). That is, the found clusters of data can be viewed as estimators of certain properties of the distribution as well, which I call \theoretical clusters". In order to explain the very meaning of the data clusters, it is necessary to think about the theoretical clusters generated by the considered model. Often a CA is based on a normal mixture model of the form (1) where N(b; 2) denotes the normal distribution with mean b and variance 2 , the (b i ; 2 i); i = 1; : : : ; s; being pairwise distinct. Most of the following considerations apply also to normal xed partition models. See Bock (1996) for literature concerning the use of these kinds of models in CA. How to deene the theoretical clusters of such normal mixtures? To my knowledge there are three approaches up to now used in the literature more or less implicitly (see Bock (1996) for references):