Clustering in ordered dissimilarity data

Abstract

This paper presents a new technique for clustering either object or relational data. First, the data are represented as a matrix D of dissimilarity values. D is reordered to D * using a visual assessment of cluster tendency algorithm. If the data contain clusters, they are suggested by visually apparent dark squares arrayed along the main diagonal of an image I (D * ) of D * . The suggested clusters in the object set underlying the reordered relational data are found by defining an objective function that recognizes this blocky structure in the reordered data. The objective function is optimized when the boundaries in I (D * ) are matched by those in an aligned partition of the objects. The objective function combines measures of contrast and edginess and is optimized by particle swarm optimization. We prove that the set of aligned partitions is exponentially smaller than the set of partitions that needs to be searched if clusters are sought in D. Six numerical examples are given to illustrate various facets of the algorithm. C 2009 Wiley Periodicals, Inc.

Is VAT really single linkage in disguise?

“…The VAT image of D may suggest that the data it represents contains c clusters, but as previously noted, it is CLODD [12] that extracts an aligned c-partition of O * from the image. Proposition 1 shows that when the MST of D is unique, CLODD extracts SL clusters from the VAT image.…”

“…2a possesses five clusters, but the clusters are not identified. Havens et al [12] developed an algorithm called CLODD-CLustering in Ordered Dissimilarity Data that extracts an aligned partition of the objects from a VAT image such as that seen in Fig. 2c. …”

“…A subset of the c-partitions U in M hcn is the set of aligned c-partitions of O [12]. These are defined as the partitions of O that form c contiguous blocks of 1's in U, beginning in the upper left corner, and proceeding down and to the right:…”

“…Researchers have also extended VAT to answer the cluster validity question [4,10,13]. CLustering in Ordered Dissimilarity Data-CLODD is an algorithm that answers the partitioning and validity questions by automatically creating partitions from the VAT image and providing a validity measure of each partition [12]. Reference [29] describes a clustering algorithm that uses VAT to visualize complex networks.…”

“…A wide array of algorithms exists for clustering unlabeled object data O [3,5,9,12,16,26]. Crisp partitions of the unlabeled data are non-intersecting subsets of O such that the union of the subsets cover O.…”