Constructing a Fuzzy Clustering Model Based on its Data Distribution
4. Experimental Result
Gaussian functions for the j’th input variable in the i’th rule. They also signify the premise parameters of Equation (1).
After partitioning a given data set into clusters, Equations (17), (18), (19) and (20) can be used to derive the linear equation (15) for each of clusters. Equation (15) includes information for the consequent parameters of a given rule. For the premise parameters, Equation (21) has information used to define the range of input variables in every cluster.
Fuzzy rules are therefore created and the initial TS fuzzy model is constructed for the given data set.
In order to achieve a better fuzzy model, the ϕ1ij, ϕ2ij and σ1ij, σ2ij as well as the
i
aj parameters of every rule in the initial TS fuzzy model can be fine-tuned using a Gradient Descent Algorithm to lower the MSE value.
4. Experimental Result
This illustration used by Sugeno and Yasukawa [9] has two inputs and one output.
The nonlinear system is described in Equation (23).
y =(1+1/x1−2 +1/x2−1.5)2 (23)
Figure 10: y =(1+1/x1−2 +1/x2−1.5)2 Figure 11: The decrease of MSE
The x1 and x2 represent two inputs and y is the output. The graph in Figure 10 shows the relationship between the inputs and output. There are 50 training data. Using the method of multiple linear regression proposed in this paper, the 50 training data are seperated into five clusters resulting five rules for the initial TS fuzzy model and the value of 0.0713 for MSE.
This is better than the result of Sugero and Yasukawa [9] in terms of MSE and number of rules, even after their parameters were fine-tuned. After fine-tuning the parameters in this intial model with gradient descent algorithm, the final MSE is reduced to be 0.0528. Figure 11 shows the change on MSE throughout the fine-tuning process of 1000 iterations. It can be seen that the system has pretty much settled after 250 iterations. Figure 12 is the rule base of the established model after fine-tuning. Table (2) compares the result of the proposed method with those suggested from other papers. Table (2) indicates the proposed method in this paper works better than the others.
R1：If x1 is mf1 and x2 is mf1 Then y = -1.4416x1 -1.6610x2 + 8.9607 R2：If x1 is mf1 and x2 is mf1
Then y = -1.246x1 –0.3338x2 + 5.9043 R3：If x1 is mf2 and x2 is mf1
Then y = 44.3466x1 -1.0878x2 –121.17 R4：If x1 is mf2 and x2 is mf2 Then y = -0.0995x1 –0.1639x2 + 2.515 R5：If x1 is mf3 and x2 is mf1
Then y = 0.0636x1 -1.084x2 –4.5143
Figure 12: The TS fuzzy model rule base after fine-tuning
Although the method suggested by Sugeno and Yasukawa in 1993 obtained an MSE of 0.0197, it requires
xi
y
∂
∂ ，i=1,2….k, and linguistic statements in every rule, which makes
the calculations much more complex. The paper by M. Delgado and Antonio F. used five methods to create a model of the system in Equation (23). The best MSE of the five methods is 0.231 [5].
Table 1: Comparson of MSE values
model Rule number
MSE
Sugeno and Yasukawa [9] 6 0.079 Gomez-Skarmeta and Martin[5] 5 0.231
Our Model 5 0.0528
5. Conclusion
The algorithm proposed in this paper can swiftly establsih an initial, but relatively good, TS fuzzy model for a system that can be further fine-tuned by Gradient Descent Algorithm. The two experiements indicated the performance of our proposed method has better performance than other methods.
The contribution of this paper can be summarized as:
1. Suggesting a quick and simple method to establish a TS model. Compared with the recursive algorithm of FCM, the partitioning algorithm suggested in this paper can establish an intitial model within O(c) amount of time, where c is the number of partitioned clusters.
2. It is not necessary to know the number of clusters beforehand. When FCM creates a fuzzy model, it must first know how many clusters are desired. The proposed method partitions a given data set according to the distribution of the training data. There is no need to know how many clusters are required.
3. The performance is good on both MSE and number of rules. This means the constructed model is more close to the actual behaviour of the system and the structure of the constructed model is simplier, resulting on accuracy and execution speed.
Acknowledgements
This research was supported by NSC-91-213-E-011-055
References
1. J. Zhao, V. Wertz and R. Gorez, “A Fuzzy Clustering Method for the Identification of Fuzzy Models for Dynamic Systems”, IEEE International Symposium on Intelligent Control, 16-18 Aug. 1994 pp172-177.
2. J. Bezdek, “A Convergence Theorem for the Fuzzy ISODATA Clustering Algorithms”, IEEE Trans. on PAMI, PAMI-2 (1980) pp1-8.
3. N. Zahid, M Limouri and A. Eassaid “A New Cluster–Validity for Fuzzy Clustering”, Pattern Recognition, 32 (1999), pp1089-1097.
4. G. Zouridakis, B. Jansen and N. Boutros, “A Fuzzy Clustering Approach to EP Estimation”, IEEE Trans. on Biomedical Engineering, vol 44, no 8, Aug. 1977, pp673-680.
5. M. Gomez-Skarmeta and F. Martin “A Fuzzy Clustering-Based Rapid Prototyping for Fuzzy Rule-Based Modeling”, IEEE Trans. on Fuzzy Systems, vol 5, no 2, May 1997, pp223-233.
6. T. Takagi and M. Sugeno, ”Structure Identification of Fuzzy Model”, Fuzzy Sets and
Systems, 28, 1988, pp15-33.
7. E. Kim and M. Park and S. Ji, “A New Approch to Fuzzy Modeling”, IEEE Trans. on Fuzzy Systems, vol 5, no 3, Aug. 1997, pp328-337.
8. L. Wang, Adaptive Fuzzy System and Control：Design and Stability Analysis, Prentice-Hall, 1994.
9. M. Sugeno and T. Yasukawa, ”A Fuzzy-Logic-Based Approch to Qualitative Modeling”, IEEE Trans. on Fuzzy Systems, vol 1, Feb. 1993, pp7-31.
10. L. Fu, “Rule Generation From Neural Network”, IEEE Trans. on Systems, Man and Cybernetics, vol 24, no 8, Aug. 1994, pp1114-1124.
11. S. Horikawa, T. Furuhashi and Y, Uchikawa, ”On Fuzzy Modeling Using Fuzzy Neural Networks with the Back-Propagation Algorithm”, IEEE Trans. on Neural Networks, vol 3, May 1992, pp801-806.
12. Y. Lin and G. Cunningham, “A New Approach to Fuzzy-Neural System Modeling”, IEEE Trans. on Fuzzy Systems, vol 3, no 2, May 1995, pp190-197.
13. C. Juang and C. Lin, “An On Line Self-Constructing Neural Fuzzy Inference Network and Its Applications”, IEEE Trans. on Fuzzy Systems, vol 6, no 1, Feb. 1998, pp12-32.
14. T. Takagi and M. Sugeno，”Fuzzy Identification of System and Its Applications to Modeling and Control”, IEEE Trans. on System, Man and Cybernetics, vol SMC-15, no 1, Jan. 1985.
15. Walpole and Myers, Probablity and Statistics, Prentice Hall, 1998.
16. J. Jang, C. Sun and E. Mizutani, Neuro-Fuzzy and Soft Computing, Prentice-Hall, 1997, pp74-78