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2.4. MODIFICATION OF FCM ALGORITHM

In order to complete the systematic methodology of fuzzy system identification and modeling, it is necessary to find generic solutions to the above problems. For this purpose, it is proposed that a theoretical basis be found for the first two problems. The proposed approach produces a more efficient strategy than the other solutions, as explained next.

First, in an attempt to derive a generic criterion for assignment of number of clusters from a theoretical point of view, it is proposed (Emami, Türks¸en,and Goldenberg, 1996) that one perform a proper generalization of scatter criteria that are mainly applied as suitable tools for expressing the compactness and separation of the hard clusters (Duda and Hart, 1973). It should be noted that the total mean vector is not weighted in the Sugeno and Yasukawa (1993) index. We propose that the total mean vector, [vee with underline], be weighted with the membership values, uik. It turns out that the difference so generated is significant for large values of weighting exponent (m) in the FCM algorithm.

Secondly, weighting exponent controls the extent of membership sharing between fuzzy clusters in the data set. Therefore, for "m" in the range of (1, [infinity]), the larger "m" generates large overlaps among fuzzy sets. To date, no theoretical basis for an optimal choice of "m" has been suggested. In order to identify the guideline for selection of "m," one should consider the behavior of cluster validity criterion, S, as m varies.

In principle, for a "reliable" performance of the cluster validity criterion, S, the weighting exponent m should be far from both of its limits, i.e., one and infinity (Emami, Türks¸en,and Goldenberg, 1996).

Now, in order to satisfy this principle, the extremes of "m" should be clearly identified. The limit "one" is well defined, but the limit "infinity" for "m" depends on the data at hand. Let us define ST, called fuzzy total scatter matrix, to be the index to indicate the behaviour of "m" as it varies from "one" to "infinity":

It is observed (Emami, Türks¸en,and Goldenberg, 1996) that the trace of ST decreases monotionically from a constant value K to zero, as "m" varies from one to infinity, where K depends only on the data set:

Therefore, for a data set to be clustered in an "efficient" manner, a suitable value of "m" is that which gives a value of ST in a region around K/2. A suitable value of "m" should be determined experimentally in this region around K/2 for different values of "c" that minimize ST.

Third, FCM algorithm generally produce only local minima, as indicated above. Therefore, different initial guesses for cluster centers may lead to different optimum results. This fact affects the cluster validity as well as cluster analysis.

In order to efficiently obtain a preference for initial locations of cluster prototypes, one should start with an Agglomerative Hierarchical Clustering (AHC) algorithm as an introductory procedure to find a suitable guess for the initial locations of cluster prototypes for the FCM algorithm. Among the various AHC algorithms, a specific method to be suggested is Ward's method (1963).

The result of AHC is "c" hard clusters for the data, which is a good start for the fuzzy clustering procedure. By this method, one can choose the initial prototypes without any knowledge of the data a priori. This approach is much more efficient than random searches among different initial guesses.

Finally with the values of "m" and "c" that would be determined with the proposed developments discussed above, one would apply the well-known FCM algorithm in order to find the fuzzy clusters. This will be called the ETG, i.e., Emami, Türks¸en,and Goldenberg, method of clustering.

It should be pointed out that in fuzzy modeling activities in general, the output sample data are assigned into several fuzzy clusters. In order to extend the assigned fuzzy clusters to the entire output space, one more step is required, known as the classification process. In most investigations, this step has not been addressed properly. In the clustering process, we make a suitable clustering for a sample data set X Rh, whereas in the classification procedure, every data point in the entire space Rh is labeled. Therefore, the problem of membership function formation for the entire output space is a classification problem. Since classifier design is usually performed using labeled data, clustering the output sample data is a good tool to design the appropriate classifiers for the entire output space. A fuzzy version of the probabilistic classification method called K-nearest neighbor has been introduced by Keller et al. (1985).

In order to obtain simple membership functions, after the assignment of the data of the entire space to the fuzzy partition, one needs to approximate the classified data by trapezodial functions in such a way that, for each fuzzy cluster, convex points are picked up and a trapezoid is fitted to them (Sugeno and Yasukawa, 1993; Nakanishi, Türks¸en,and Sugeno, 1993).

2.5. INPUT SELECTION

The phase of input selection in fuzzy system identification is to find the most dominant input variables among a finite number of input candidates. A combinatorial approach is proposed by Sugeno and Yasukawa (1993) and implemented by Nakanishi, Türks¸en,and Sugeno (1993), in which all combinations of input candidates are considered and the combinations of input variables that minimize a specific regularity criterion are selected. The main drawback of this approach is that for a system with a large number of input candidates, a large number of combinations must be considered.

This problem of the input selection may be resolved in a simpler and more efficient fashion. It is proposed that the input selection be directly related to the procedure of input membership functions construction. In the proposed approach, those input points that have membership grades close or equal to "one" in each cluster are assigned the same input membership grades, i.e., membership of "one."

Consequently, those input variables that do not have a dominant effect on the output, will have membership grades equal to "one" all across their domain. This is the result of the antecedent aggregation of the fuzzy rules that is performed by t-norm operators. Since "one" is the neutral element for these operators, the ineffective input candidates with membership grades of "one" in their entire range can be canceled from the fuzzy rules. Thus, the remaining variables become selected as input variables.


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