Convex Group Clustering of Large Geo-referenced Data Sets 1 Convex Group Clustering of Large Geo-referenced Data Sets

Clustering partitions a data set S = fs1;:::;sng < m into groups of nearby points. Distance-based clustering uses op-timisation criteria for deening the quality of the partition. Formulations using representatives (means or medians of groups) have received much more attention than minimisa-tion of the total within group distance (TWGD). However, this non-representative approach has attractive properties while remaining distance-based. While representative approaches produce partitions with non-overlapping clusters, TWGD does not. We investigate the restriction of TWGD to producing convex-hull disjoint groups and show that this problem is NP-complete in the Euclidean case as soon as m 2. Nevertheless we provide eecient algorithms for solving it approximately. 1 Introduction Clustering is a fundamental task in data analysis since it identiies groups in heterogeneous data. Clustering can be seen as a concept formation or class delineation problem. At least the elds of statistics 44, 46], machine intelligence 5, 15, 32] and more recently knowledge discovery and data mining (KDDM) 12, 14, 37, 47] have contributed with algorithms for many clustering approaches. Hierarchical bottom-up approaches form groups by composition or merging items that are close together 10, 29]. However, top-down partition approaches to clustering are also interesting, in particular for spatial data mining 12, 37, 48]. This perspective deenes clustering as partitioning a heterogeneous data set into smaller more homogeneous groups 2, 19, 40]. Clustering typically uses a metric (or distance) to determine the dissimilarity between the items to be clustered. Here we consider the clustering problem in the context of spatial databases, those typically associated with a Geographical Information System (GIS). In spatial settings, the clustering almost invariably makes use of some distance that captures the notion of proximity, as it reeects the essence of