(A) TSK fuzzy model generation method based on pseudo metric on fuzzy sets and cluster validation

Abstract

TSK fuzzy model is known to be suitable for approximating a large class of non-linear systems. GK clustering has been widely used to identify the local linear structure of the data set, when building a TSK fuzzy model. Clusters found by GK clustering is used to acquire the antecedents and rules for the TSK fuzzy model. Consequents are acquired using estimation methods like least mean square estimation from the data. There are two problems in this approach: to use GK clustering, the number of clusters has to be given. Unfortunately it is unknown in most cases. Also, the antecedents acquired from clusters are discrete fuzzy sets that are difficult to handle. In this thesis, these two problems are discussed: a pseudo metric on a set of fuzzy sets and a cluster validation index are proposed. It is not easy to measure distance between fuzzy sets since the fuzzy sets carry uncertainty. Distance measures found in the literature are not suitable to measure the distance between discrete fuzzy sets. In this thesis, a pseudo metric, $Dist_V$ is proposed based on satisfaction function. $Dist_V$ can measure the distance between discrete and continuous fuzzy sets. Using $Dist_V$, canonical fuzzy sets with a well known membership function can be found. The canonical fuzzy set for a fuzzy set is the nearest fuzzy set from the fuzzy set in a specific form. The canonical fuzzy sets are used for the antecedents for the TSK fuzzy model. The results of fuzzy clustering are very sensitive to the choice of the number of clusters given. There are a lot of cluster validation indices found for FCM. However, for GK clustering, there are very few of them and the performance is not good. A cluster validation index, $V_{proposed}$, based on the relative degree of sharing is proposed. It shows a very good performance for FCM and GK clustering. $V_{proposed}$ is used to identify the number of clusters in the data that gives the optimal clustering result. Using $Dist_V$ and $V_{propo...