Cluster Analysis Using Rough Clustering and k-Means Clustering

Cluster Analysis Using Rough Clustering and k-Means Clustering

Kevin E. Voges (University of Canterbury, New Zealand)
DOI: 10.4018/978-1-60566-026-4.ch091
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Abstract

Cluster analysis is a fundamental data reduction technique used in the physical and social sciences. It is of potential interest to managers in Information Science, as it can be used to identify user needs though segmenting users such as Web site visitors. In addition, the theory of Rough sets is the subject of intense interest in computational intelligence research. The extension of this theory into rough clustering provides an important and potentially useful addition to the range of cluster analysis techniques available to the manager. Cluster analysis is defined as the grouping of “individuals or objects into clusters so that objects in the same cluster are more similar to one another than they are to objects in other clusters” (Hair, Black, Babin, Anderson, & Tatham, 2006). There are a number of comprehensive introductions to cluster analysis (Abonyi & Feil, 2007; Arabie, Hubert, & De Soete, 1994; Cramer, 2003; Everitt, Landau, & Leese, 2001; Gan, Ma, & Wu, 2007; Härdle & Hlávka, 2007). Techniques are often classified as hierarchical or nonhierarchical (Hair et al., 2006), and the most commonly used nonhierarchical technique is the k-means approach developed by MacQueen (1967). Recently, techniques based on developments in computational intelligence have also been used as clustering algorithms. For example, the theory of fuzzy sets developed by Zadeh (1965), which introduced the concept of partial set membership, has been applied to clustering (Abonyi & Feil, 2007; Dumitrescu, Lazzerini, & Jain, 2000). Another technique receiving considerable attention is the theory of rough sets (Pawlak, 1982), which has led to clustering algorithms referred to as rough clustering (do Prado, Engel, & Filho, 2002; Kumar, Krishna, Bapi, & De, 2007; Parmar, Wu, & Blackhurst, 2007; Voges, Pope, & Brown, 2002). This article provides brief introductions to k-means cluster analysis, rough sets theory, and rough clustering, and compares k-means clustering and rough clustering. It shows that rough clustering provides a more flexible solution to the clustering problem, and can be conceptualized as extracting concepts from the data, rather than strictly delineated subgroupings (Pawlak, 1991). Traditional clustering methods generate extensional descriptions of groups (i.e., which objects are members of each cluster), whereas clustering techniques based on rough sets theory generate intentional descriptions (i.e., what are the main characteristics of each cluster) (do Prado et al., 2002). These different goals suggest that both k-means clustering and rough clustering have their place in the data analyst’s and the information manager’s toolbox.
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Background

k-Means Cluster Analysis

In the k-means approach, the number of clusters (k) in each partition of the data set is decided prior to the analysis, and data points are randomly selected as the initial estimates of the cluster centers (referred to as centroids). The remaining data points are assigned to the closest centroid on the basis of the distance between them, usually using a Euclidean distance measure. The aim is to obtain maximal homogeneity within clusters (i.e., members of the same cluster are most similar to each other) and maximal heterogeneity between clusters (i.e., members of different clusters are most dissimilar to each other).

K-means cluster analysis has been shown to be quite robust (Punj & Stewart, 1983). Despite this, the approach suffers from many of the problems associated with all traditional multivariate statistical analysis methods. These methods were developed for use with variables that are normally distributed and have an equal variance-covariance matrix in all groups. In most realistic data sets, neither of these conditions necessarily holds.

Key Terms in this Chapter

Rough Set: The concept of rough, or approximation, sets was introduced by Pawlak and is based on the single assumption that information is associated with every object in an information system. This information is expressed through attributes that describe the objects; objects that cannot be distinguished on the basis of a selected attribute are referred to as indiscernible. A rough set is defined by two sets, the lower approximation and the upper approximation.

K-Means Clustering: A cluster analysis technique in which clusters are formed by randomly selecting k data points as initial seeds or centroids, and the remaining data points are assigned to the closest cluster on the basis of the distance between the data point and the cluster centroid.

Cluster Analysis: A data analysis technique involving the grouping of objects into sub-groups or clusters so that objects in the same cluster are more similar to one another than they are to objects in other clusters.

Market Segmentation: A central concept in marketing theory and practice; involves identifying homogeneous sub-groups of buyers within a heterogeneous market. It is most commonly conducted using cluster analysis of the measured demographic or psychographic characteristics of consumers. Forming groups that are homogenous with respect to these measured characteristics segments the market.

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