Despite its popularity, it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, unlike the situation with other statistical problems as regression or classification, for some of the clustering methodologies it is difficult to specify both, the target object that they seek after from a population point of view, and the population goal to which the data-based clustering algorithms try to get close. This paper aims to provide some insight into the theoretical foundations of clustering by focusing on two main objectives: to provide an explicit formulation for the ideal population goal of the modal clustering methodology, which understands clusters as regions of high density; and to present two new risk functions, applicable to any clustering methodology, to evaluate the performance of a data-based clustering algorithm with respect to the ideal population goal. In particular, it is shown that only mild conditions on a sequence of density estimators are needed to ensure that the sequence of modal clusterings that they induce is consistent.
View on arXiv