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A good source of references

to clustering outside of the data mining field is the article by Arabie and Hu-

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bert [376]. A paper by Kleinberg [401] provides a discussion of some of the

trade-offs that clustering algorithms make and proves that it is impossible to

for a clustering algorithm to simultaneously possess three simple properties’

The K-means algorithm has a long history, but is still the subject of current

research. The original K-means algorithm was proposed by MacQueen [403]. The ISODATA algorithm by Ball and Hall [377] was an early, but sophisticated

version of K-means that employed various pre- and postprocessing techniques

to improve on the basic algorithm. The K-means algorithm and many of its

variations are described in detail in the books by Anderberg PTal and Jain

and Dubes [396]. The bisecting K-means algorithm discussed in this chapter

was described in a paper by Steinbach et al. [4L4], and an implementation

of this and other clustering approaches is freely available for academic use in

the CLUTO (ClUstering TOolkit) package created by Karypis [382]. Bolev

[380] has created a divisive partitioning clustering algorithm (PDDP) based

on finding the first principal direction (component) of the data, and Savaresi

and Boley [411] have explored its relationship to bisecting K-means. Recent

variations of K-means are a new incremental version of K-means (Dhilion et al.

[383]), X-means (Pelleg and Moore 1408]), and K-harmonic means (Zhang et aI

[416]). Hamerly and Elkan [392] discuss some clustering algorithms that pro-

duce better results than K-means. While some of the previously mentioned

approaches address the initialization problem of K-means in some manller)

8 .6



556 Chapter 8 Cluster Analysis: Basic Concepts and Algorithms

other approaches to improving K-means initialization can also be found in the

work of Bradley and Fayyad [381]. Dhillon and Modha [384] present a gen-

eralization of K-means, called spherical K-means, that works with commonly used similarity functions. A general framework for K-means clustering that

uses dissimilarity functions based on Bregman divergences was constructed by

Banerjee et al. 1378]. Hierarchical clustering techniques also have a long history. Much of the

initial activity was in the area of taxonomy and is covered in books by Jardine

and Sibson [398] and Sneath and Sokal 1412]. General-purpose discussions of

hierarchical clustering are also available in most of the clustering books men-

tioned above. Agglomerative hierarchical clustering is the focus of most work

in the area ofhierarchical clustering, but divisive approaches have also received

some attention. For example, Zahn l4l5l describes a divisive hierarchical tech- nique that uses the minimum spanning tree of a graph. While both divisive

and agglomerative approaches typically take the view that merging (splitting) decisions are final, there has been some work by Fisher [389] and Karypis et

al. [399] to overcome these limitations. Ester et al. proposed DBSCAN 1387], which was later generalized to the

GDBSCAN algorithm by Sander et al. 1410] in order to handle more general

types of data and distance measures) such as polygons whose closeness is mea-

sured by the degree of intersection. An incremental version of DBSCAN was

developed by Kriegel et al. [386]. One interesting outgrowth of DBSCAN is

OPTICS (Ordering Points To Identify the Clustering Structure) (Ankerst et

al. [375]), which allows the visualization of cluster structure and can also be used for hierarchical clustering.

An authoritative discussion of cluster validity, which strongly influenced the discussion in this chapter, is provided in Chapter 4 of Jain and Dubes’

clustering book [396]. More recent reviews of cluster validity are those of

Halkidi et al. [390, 391] and Milligan [ 0a]. Silhouette coefficients are described in Kaufman and Rousseeuw’s clustering book [400]. The source of the cohesion and separation measures in Table 8.6 is a paper by Zhao and Karypis [417], which also contains a discussion of entropy, purity, and the hierarchical F-

measure. The original source of the hierarchical F-measure is an article by Larsen and Aone [402].

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