A Population Background for Nonparametric Density-Based Clustering
arXiv:1408.1381 · doi:10.1214/15-STS526
Abstract
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, whereas for other statistical problems the theoretical population goal is clearly defined (as in regression or classification), for some of the clustering methodologies it is difficult to specify the population goal to which the data-based clustering algorithms should 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 loss functions, applicable in fact 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.
Published at http://dx.doi.org/10.1214/15-STS526 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:1212.1384
References in corpus (3)
Cited by in corpus (8)
- Fully adaptive density-based clustering
- Nonparametric Clustering of Functional Data Using Pseudo-Densities
- Nonparametric clustering for image segmentation
- The emptiness inside: Finding gaps, valleys, and lacunae with geometric data analysis
- Bump hunting through density curvature features
- Skeleton Clustering: Dimension-Free Density-based Clustering
- Bayesian taut splines for estimating the number of modes
- Mode-based estimation of the center of symmetry