Apr 30, 2008 - techniques. Fuzzy sets as a theoretical basis for clustering algorithms were first suggested by ... Bezdek all presume a common algebraic framework for fuzzy c-partitions of A', the topic ...... Washington, D.C.: Thompson Book.
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Cluster Validity with Fuzzy Sets James C. Bezdek† a
Department of Mathematics State University College Oneonta, New York Available online: 30 Apr 2008
To cite this article: James C. Bezdek† (1973): Cluster Validity with Fuzzy Sets, Journal of Cybernetics, 3:3, 58-73 To link to this article: http://dx.doi.org/10.1080/01969727308546047
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Journal of Cybernetics, 1974, 3, 3, pp. 58-73
Cluster Validity with Fuzzy Sets
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James C. Bezdekf Department of Mathematics State University College Oneonta, New York Abstract Given a finite, unlabelled set of real vectors X, one often presumes the existence of (c) subsets (clusters) in X, the members of which somehow bear more similarity to each other than to members of adjoining clusters. In this paper, we use membership function matrices associated with fuzzy c-partitions of X, together with their values in the Euclidean (matrix) norm, to formulate an a posteriori method for evaluating algorithmically suggested clusterings of X. Several numerical examples are offered in support of the proposed technique.
1. Introduction Algorithms for discovering a preassigned number (c), 2 < £ • < / ; , of clusters in a finite set of real, ^-dimensional vectors X = {.v,,.. ..v n } have proliferated with expanding computing capabilities. The central issues in this paper are the selection of (c)-how many clusters shall we search for?, and how much reliance can we place on algorithmically suggested substructure in A"? Uncertainties concerning substructure (or the lack of it) in X recommends the use of fuzzy sets as a "hedge" against nonstatistical ambiguities in the data: indeed, Zadch [22] devised the notion in an effort to quantify analysis in problem areas where vagueness seemed to preclude the use of conventional (hard) mathematical techniques. Fuzzy sets as a theoretical basis for clustering algorithms were first suggested by Bellman, Kalaba, and Zadeh [ 3 ] . Subsequently, the papers of Wee [ 2 0 ] , Flake and Turner [ 9 ] , Gitman and Levine [ I I ] , Ruspini [17, 18], and Dunn [7, 8] have concerned themselves with various theories of fuzzy clustering. The papers of Adey [ 1 ] , Larsen et al. [ 1 4 ] , and Bezdek [4] address some promising applications of fuzzy clustering methods in EEC analysis and numerical taxonomy. The works of Ruspini, Dunn, and Bezdek all presume a common algebraic framework for fuzzy c-partitions of A', the topic to which we now turn. 2. Partition Spaces Let R be the reals, Rs the usual s-dimensional Euclidean Hilbert space over /?, equipped with standard inner product, induced norm, and norm metric: s
= xy' = y
Xiy,
(la)
/=!
2 (.Y/)2 =
db)
1=1
'Requests for reprints should be sent to Dr. James C. Bezdek, Department of Mathematics, State University College, Oneonta, New York 13820.
58 Copyright © 1974 by Scripta Publishing Co
CLUSTER VALIDITY WITH FUZZY SETS
x, y) = Hx - yll = ( ^ (xi ~ v,)2j
59
(lc)
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V (row) vectors .v = (,v1?. . .xs), y = O'i . . .ys) e Rs, where superscript (/) here denotes the transpose operation. Suppose X = {.v, .. . xn}CRs is given. For 2 < c < / ; , any (c) "hard" (as opposed to fuzzy) subsets of A', say {YJ: 1 < / < c } , such that A' = U Yj
(2a)
/= 1
,• $ =£Yj
v /* j VKKf
(2c)
are called a nondcgcnerate hard c-partition of X. The class of all such r-tuples of hard subsets of X is denoted by Pc. Let Wj-.X-* {0, l} be the characteristic function of Y',. 1 < / < c . Thus, u',-(.vfc) = wjk = 1 iff x^eYj, and is 0 otherwise. For each P = {Yj,. . . Yc }ePc, there is unique 0 = {if,,.. .wc) so that c
1 = V nv
(3a)
0 = nv A ny
V / ¥= j
(3b)
0 ¥= nv
V1 Fc (if) > 1 - f} (2y/n -• e)
(20)
Proof. 0 < V " = IIIt'll = \\W-U+ U\\ < lilt' - U\\ + IIMl < e + \\U\\, thus (y/Ji - e) < lltf||=* (s/ii- 0.750 only for c = 2, 5 = 4,5,6,7. Figure 2 exhibits the hard 2-partitions closest to FP of fuzzy ISODATA. At 6 = 3, F(c) first falls below 0.750, and it is here that J2 first produces the undesirable dichotomy of X favored by / , for 5 < 5.5. This indicates the plausibility of our assertions above. ^ ^ In Fig. 2, values of 0(2,W) for W the visually correct 2-partitions of X are listed adjoining their corresponding partition coefficients F(2). X contains CWS clusters only for. 6 = 6,7, but even with 0(2,J?') considerably less than 1, inference of (c) at 6 = 4,5 yields the most appealing interpretation of_clusters in X. Only when 5 = 3, at which the optimum of J2 is unsatisfactory, does F(c) fall below our tolerance limit. Further inspection of Table 5 reveals that as 6 increases, F(2)-and only />"(2)-increases sharply, while F(c) remains uniformly lower and nearly constant independent of (c) and 5. The diagonal entries of parameter matrices S2(U) at each value of 5, along with («,•/'»)> the actual subsample proportions in each pattern are
0.207 0.152 0.143 0.140 0.1390.138 0.535 0.686 0.737 0.766 0.786 0.862
TABLES
Example
4: ?( c )
c
* =3
S=4
5=5
S=6
«=7
2
0.742 0.656 0.624 0.611 0.564 0.531
0 .837 0 .674 0 .636 0 .625 0 .580 0 ,533
0.880 0.687 0.643 0.632 0.589 0.538
0 .907 0 .694 0 .648 0 .600 0 .594 0 .543
0.925 0.699 0.652 0.605 0.598 0.543
3 4 5 6 7
72
JAMES C.BEZDEK
Pattern
Interface Distance
X
(5)
Partition Coefficient
Separation Index
0.742
0.531
0.837
0.707
0.880
0.885
0.907
1.060
0.925
•1.240
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