A novel fuzzy clustering algorithm based on a fuzzy scatter matrix ...

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644 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 will move away from x with a corresponding weighted term g i . Since g i is a monotone increasing function of b, the cluster centers obtained by FCS with a large b value will be more far away from x than the centers obtained with a small b value. The proposed FCS <strong>clustering</strong> <strong>algorithm</strong> is summarized as follows: FCS Algorithm 1. (see also Yang et al. (2003)) Set the iteration counter ‘ = 0 and choose the initial values a ð0Þ i , i =1,...,c. Given b, e >0 Step 1. Find g ð‘þ1Þ i using (15) Step 2. Find l ð‘þ1Þ ij using (13) Step 3. Find a ð‘þ1Þ i using (14) Increment ‘; until max i ka ð‘þ1Þ i a ð‘Þ i k < e. We now use a simple example to illustrate the FCS properties. This is a sampling data set drawn from a one-dimensional normal mixture (1/ 3)N(0, 1) + (1/3)N(4, 1) + (1/3)N(10, 1) with three populations of means 0, 4 and 10. The histogram of this data set is shown in Fig. 2(a). Fig. 2(b) shows the FCMmembership function that belongs to the mean 0 cluster. Fig. 2(c)–(f) are the FCS membership functions that belong to the mean 0 cluster with the values 0.9, 0.5, 0.1 and 0.05 of the parameter b. The cluster kernel volumes always decrease when b is decreasing. The volume of the cluster kernel can also be presented using the range with the value of one of the cluster membership functions. In our 3-cluster example, Fig. 2(b)–(f) draw only the membership function of the mean 0 cluster. Thus, the range with the membership value one presents the volume of the mean 0 cluster kernel, and the range with the membership value zero presents the volumes of the means 4 and 10 cluster kernels. The crisp membership values and <strong>fuzzy</strong> membership values co-exist in FCS. When b = 0, the FCS membership values will be equivalent to FCMwith no existing cluster kernel. The three cluster centers obtained using FCM and FCS are shown in Table 1. When b decreases, the cluster centers obtained by FCS will then be closer to the grand mean x and also closer to the cluster centers obtained by the FCM. These results coincide with the property shown in Eq. (15). Histogram FCM FCS beta=0.9 15 1.0 1.0 Frequency 10 5 membership 0.5 membership 0.5 0 0.0 0.0 0 5 10 15 0 5 10 0 5 10 data set data set data set (a) (b) (c) FCS beta=0.5 FCS beta=0.1 FCS beta=0.05 1.0 1.0 1.0 membership 0.5 membership 0.5 membership 0.5 0.0 0.0 0.0 0 5 10 0 5 10 0 5 10 data set data set data set (d) (e) (f) Fig. 2. Membership functions of FCM and FCS.
K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 645 Table 1 Cluster centers obtained by FCS Beta Cluster centers 0.900 0.366 3.667 12.319 0.500 0.281 3.687 11.221 0.200 0.164 3.714 10.557 0.100 0.111 3.726 10.356 0.050 0.083 3.733 10.256 0 (FCM) 0.052 3.740 10.156 Although the FCMis a popular <strong>clustering</strong> <strong>algorithm</strong>, the cluster centers obtained by it will be closer to the grand mean x when the data set heavily overlaps. According to the FCS properties, the cluster centers obtained by it can be more accurate than the FCMin some situations. This will be illustrated in Section 5. In the next section, more details about the FCS parameters are presented. 4. Optimality tests and parameter selection of FCS In general cases, when the data set is clustered into c (c > 1) subsets, each subset is often expected to have a different prototype (or cluster center) than the others. However, the grand sample center x of the data set is always a fixed point in the FCS <strong>algorithm</strong>. The FCS output will be x with a great probability if x is one stable solution in the FCS <strong>algorithm</strong>. To avoid such cases, we hope that x is not an attracted point in the FCS. How do we judge if x is attracted or a stable point of FCS? The Hessian matrix of the FCS objective function (12) must be studied. In order to simplify the calculations, substituting (13) into (12) yields (17) J ¼ Xn j¼1 X c i¼1 ðkx j a i k 2 g i ka i xk 2 Þ 1 m 1! 1 m : ð17Þ It can be proved that J = min l J FCS . Therefore, it is enough to judge whether or not x is attracted or is a stable point in FCS using the Hessian matrix of (17). Let us set qðx j ; a i Þ¼kx j a i k 2 g i ka i xk 2 ; S j ¼ Xc i¼1 m qðx j ; a i Þ 1 1 ; l ij ¼ qðx m j; a i Þ 1 1 S j and h ij ¼ l m ij ½ð1 g iÞa i ðx j g i xÞŠ: We know that ð1Þ oJ oa i ¼ 2 Xn ð2Þ j¼1 o 2 J ¼ 4m oa i oa k m 1 þ 2d ik l m ij ½ð1 g iÞa i ðx j g i xÞŠ: X n j¼1 X n j¼1 ðS j Þ m l m ij ð1 1 h ij ðh kj Þ t ! g iÞ I ss 4m m 1 d X n ik ½qðx j ; a i ÞŠ 1 h ij ½ð1 j¼1 8i; 8j g i Þa i ðx j g i xÞŠ t : ð18Þ Therefore, the second-order term of TaylorÕs series expansion of (17) cab be expressed as follows: o 2 ! J u oa i oa a ¼ 4m 2 X n X ðS j Þ m 1 c l m ij k m 1 ut a i ½ð1 g i Þa i ðx j g i xÞŠ j¼1 i¼1 ! þ2 Xc ð1 g i Þ Xn l m 4m X c X n ij ut a i u ai u t a m 1 i l m ð1 g i Þa i ðx j g i xÞð1 g i Þa i ðx j g i xÞ t ij u qðx i¼1 j¼1 i¼1 j¼1 j ;a i Þ ai : ð19Þ u t a
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