By Anil K. Jain

ISBN-10: 013022278X

ISBN-13: 9780130222787

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C (for example, select c objects randomly as mi , i = 1, . . , c). LVQC2. For t = 1, 2, . . , repeat LVQC3–LVQC5 until convergence (or until the maximum number of iterations is attained). LVQC3. Select randomly x(t) from X. LVQC4. Let ml (t) = arg min x(t) − mi (t) . 1≤i≤c 30 Basic Methods for c-Means Clustering LVQC5. Update m1 (t), . . , mc (t): ml (t + 1) = ml (t) + α(t)[x(t) − ml (t)], mi (t + 1) = mi (t), i = l. Object represented by x(t) is allocated to Gl . End LVQC. In this algorithm, the parameter α(t) satisﬁes ∞ ∞ α(t) = ∞, t=1 α2 (t) < ∞, t = 1, 2, · · · t=1 For example, α(t) = Const/t satisﬁes these conditions.

3. 3 Covariance Matrices within Clusters Inclusion of yet another variable is important and indeed has been studied using diﬀerent algorithms. That is, the use of ‘covariance matrices’ within clusters. 4 where we ﬁnd two groups, one of which is circular while the other is elongated. 5 which fails to separate the two groups. All methods of crisp and fuzzy c-means as well as FCMA in the last section fails to separate these groups. The reason of the failure is that the cluster allocation rule is basically the nearest neighbor allocation, and hence there is no intrinsic rule to recognize the long group to be a cluster.

23) is related to the crisp A question arises how the fuzzy solution U one. We have the next proposition. 2. 4), on the condition that the nearest center to any xk is unique. In other words, for all xk , there exists unique vi such that i = arg min D(xk , v ). 1≤ ≤c Proof. Note 1 −1= uki 1 m−1 D(xk , vi ) D(xk , vj ) j=i . Assume vi is nearest to xk . Then all terms in the right hand side are less than unity. Hence the right hand side tends to zero as m → 1. Assume vi is not nearest to xk . Then a term in the right hand side exceeds unity.

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