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

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648 K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 FCS beta=0.1, m=2 FCS beta=0.1, m=3 FCS beta=0.1, m=4 1.0 1.0 1.0 membership 0.5 membership 0.5 membership 0.5 0.0 0.0 0.0 0 10 20 30 40 50 x 0 10 20 30 40 50 x 0 10 20 30 x (a) (b) (c) 40 50 4 The phi function of FCS beta=0.1, m=2 4 The phi function of FCS beta=0.1, m=3 4 The phi function of FCS beta=0.1, m=4 2 2 2 phi function 0 -2 -4 phi function 0 -2 -4 phi function 0 -2 -4 -6 -6 -6 -8 0 10 20 30 40 50 x -8 0 10 20 30 40 50 x -8 0 10 20 30 40 50 x (d) (e) (f) Fig. 3. Membership functions (Eq. (30)) and phi functions (Eq. (29)) for a given data point x when m = 2, 3 and 4. n = 800 points consisted of 200 points from each of four clusters. The population mean vectors are l 1 = (3,0,0,0), l 2 = (0, 3,0,0), l 3 = (0, 0,3,0) P and l 4 = (0, 0,0,3). The covariance matrices are i ¼ I 4, i = 1, 2, 3, 4. We use the mean vectors as the initial values for both FCMand FCS <strong>algorithm</strong>s. We compare FCMand FCS with the mean squared error (MSE) criterion. The MSE is calculated by P 4 i¼1 ka i l i k 2 . We implement the FCS <strong>algorithm</strong> with different values combinations of b for (0,0.005, 0.01, 0.05,0.1, 0.2) and m for (1.5,2,2.5,3,3.5). For 100 repeated Normal-4 data sets, the MSE values are shown in Fig. 4(a). The result of b = 0 is equivalent to FCM. As b increases, the volume of the cluster kernel grows. When the <strong>fuzzy</strong> index m is small, the results of FCS (b > 0) and FCM(b = 0) are similar. As m becomes larger, FCMgives larger MSE values than FCS. Thus, the FCS result is more insensitive to the <strong>fuzzy</strong> index m than FCM, especially for b = 0.05, 0.1 and 0.2. In general, the MSE values will increase when m increases. Yu et al. (2004) gave a theoretical upper bound for m for FCM that the grand sample mean x will be a unique optimizer. For each combination of m and b, the range between worst and best MSE values among these 100 repeated MSE values are shown in Fig. 4(b). The cases of m = 3.5 and b = 0,0.05, 0.01 are the examples of the grand sample mean x being the unique optimizer. The range of MSE values for FCS with b = 0.05, 0.1 and 0.2 are still insensitive to the <strong>fuzzy</strong> index m. One may argue that we can process FCMwith a small m value, say 1.5 or 2, to avoid the above defective of FCM. However, FCS are not only insensitive to the <strong>fuzzy</strong> index m than FCM, but also more robust to noise and outliers than FCMwhen m is large. We will show this robust property of FCS in the following examples. Fig. 5 shows a two-cluster data set with unequal sample sizes. The <strong>clustering</strong> results of FCS with m = 2 and b = 0,0.1,0.2 are shown in Fig. 5(a)– (c). Three figures show the similar results. The cluster centers are presented by the solid circle points. Note that, the distance between two cluster centers increases as b increases. This phenomenon can be explained by the update Eq. (17) of FCS.
K.-L. Wu et al. / Pattern Recognition Letters 26 (2005) 639–652 649 MSE 10 5 0 00.005 0.01 0.05 beta 0.1 0.2 (a) 1.5 2.5 m 3.5 range of MSE 25 20 15 10 5 0 00.005 0.01 0.05 0.1 beta 0.2 (b) 1.5 2.5 m 3.5 Fig. 4. MSE values for different combinations of beta and m. When m becomes larger (m = 6), the <strong>clustering</strong> results of FCS with b = 0, 0.1 and 0.2 are shown in Fig. 5(d)–(f). The FCS with cluster kernels (b = 0.1 and 0.2) obtains better performance than FCM(b = 0) which clusters the data set without a cluster kernel. This shows that FCS with a large and suitable m value can detect unequal sample size clusters or is robust to the noise. Fig. 6 shows a two-cluster data set with one outlying point whose coordinate is (100, 0). When m is large (m = 6), the results of FCS with b = 0.1 and 0.2 are more robust to the outlier than FCM (b = 0). These robust properties of FCS can be explained using the FCS update equations. Let ^l ¼ maxfl i1 ; ...; l in g, l 0 ij ¼ l ij=^l, j =1,...,n. We have 80 FCM m=2 80 FCS m=2, beta=0.1 80 FCS m=2, beta=0.2 70 70 70 60 60 60 50 50 50 40 40 40 30 30 30 20 20 20 10 10 10 10 20 30 40 50 60 70 80 90 100 (a) 10 20 30 40 50 60 70 80 90 100 (b) 10 20 30 40 50 60 70 80 90 100 (c) 80 FCM m=6 80 FCS m=6, beta=0.1 80 FCS m=6, beta=0.2 70 70 70 60 60 60 50 50 50 40 40 40 30 30 30 20 20 20 10 10 10 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 (d) (e) (f) Fig. 5. FCMand FCS <strong>clustering</strong> results for unequal sample size data sets.
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