Clustering analysis has been widely applied in diverse fields such as data mining, access structures, knowledge discovery, software engineering, organization of information systems, and machine learning. The main objective of cluster analysis is to create groups of objects based on the degree of their association (Kaufman & Rousseeuw, 1990; Romesburg, 1990). There are two major categories of clustering algorithms with respect to the output structure: partitional and hierarchical (Romesburg, 1990). K-means is a representative of the partitional algorithms. The output of this algorithm is a flat structure of clusters. The K-means is a very attractive algorithm because of its simplicity and efficiency, which make it one of the favorite choices to handle large datasets. On the flip side, it has a dependency on the initial choice of number of clusters. This choice may not be optimal, as it should be made in the very beginning, when there may not exist an informal expectation of what the number of natural clusters would be. Hierarchical clustering algorithms produce a hierarchical structure often presented graphically as a dendrogram. There are two main types of hierarchical algorithms: agglomerative and divisive. The agglomerative method uses a bottom-up approach, i.e., starts with the individual objects, each considered to be in its own cluster, and then merges the clusters until the desired number of clusters is achieved. The divisive method uses the opposite approach, i.e., starts with all objects in one cluster and divides them into separate clusters. The clusters form a tree with each higher level showing higher degree of dissimilarity. The height of the merging point in the tree represents the similarity distance at which the objects merge in one cluster. The agglomerative algorithms are usually able to generate high-quality clusters but suffer a high computational complexity compared with divisive algorithms. In this paper, we focus on investigating the behavior of agglomerative hierarchical algorithms. We further divide these algorithms into two major categories: group based and single-object based clustering methods. Typical examples for the former category include Unweighted Pair-Group using Arithmetic averages (UPGMA), Centroid Linkage, and WARDS, etc. Single LINKage (SLINK) clustering and Complete LINKage clustering (CLINK) fall into the second category. We choose UPGMA and SLINK as the representatives of each category and the comparison of these two representative techniques could also reflect some similarity and difference between these two sets of clustering methods. The study examines three key issues for clustering analysis: (1) the computation of the degree of association between different objects; (2) the designation of an acceptable criterion to evaluate how good and/or successful a clustering method is; and (3) the adaptability of the clustering method used under different statistical distributions of data including random, skewed, concentrated around certain regions, etc. Two different statistical distributions are used to express how data objects are drawn from a 50-dimensional space. This also differentiates our work from some previous ones, where a limited number of dimensions for data features (typically up to three) are considered (Bouguettaya, 1996; Bouguettaya & LeViet, 1998). In addition, three types of distances are used to compare the resultant clustering trees: Euclidean, Canberra Metric, and Bray-Curtis distances. The results of an exhaustive set of experiments that involve data derived from 50- dimensional space are presented. These experiments indicate a surprisingly high level of similarity between the two clustering techniques under most combinations of parameter settings.
PRIVACY PRESERVING CLUSTERING BASED ON LINEAR APPROXIMATION OF FUNCTION
