Extending Similarity Measures of Interval Type-2 Fuzzy Sets to General Type-2 Fuzzy Sets

Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures.Comment: International Conference on Fuzzy Systems 2013 (Fuzz-IEEE 2013

A new fuzzy set merging technique using inclusion-based fuzzy clustering

This paper proposes a new method of merging parameterized fuzzy sets based on clustering in the parameters space, taking into account the degree of inclusion of each fuzzy set in the cluster prototypes. The merger method is applied to fuzzy rule base simplification by automatically replacing the fuzzy sets corresponding to a given cluster with that pertaining to cluster prototype. The feasibility and the performance of the proposed method are studied using an application in mobile robot navigation. The results indicate that the proposed merging and rule base simplification approach leads to good navigation performance in the application considered and to fuzzy models that are interpretable by experts. In this paper, we concentrate mainly on fuzzy systems with Gaussian membership functions, but the general approach can also be applied to other parameterized fuzzy sets

An artificial immune systems based predictive modelling approach for the multi-objective elicitation of Mamdani fuzzy rules: a special application to modelling alloys

In this paper, a systematic multi-objective Mamdani fuzzy modeling approach is proposed, which can be viewed as an extended version of the previously proposed Singleton fuzzy modeling paradigm. A set of new back-error propagation (BEP) updating formulas are derived so that they can replace the old set developed in the singleton version. With the substitution, the extension to the multi-objective Mamdani Fuzzy Rule-Based Systems (FRBS) is almost endemic. Due to the carefully chosen output membership functions, the inference and the defuzzification methods, a closed form integral can be deducted for the defuzzification method, which ensures the efficiency of the developed Mamdani FRBS. Some important factors, such as the variable length coding scheme and the rule alignment, are also discussed. Experimental results for a real data set from the steel industry suggest that the proposed approach is capable of eliciting not only accurate but also transparent FRBS with good generalization ability

Fuzzy clustering with volume prototypes and adaptive cluster merging

Two extensions to the objective function-based fuzzy clustering are proposed. First, the (point) prototypes are extended to hypervolumes, whose size can be fixed or can be determined automatically from the data being clustered. It is shown that clustering with hypervolume prototypes can be formulated as the minimization of an objective function. Second, a heuristic cluster merging step is introduced where the similarity among the clusters is assessed during optimization. Starting with an overestimation of the number of clusters in the data, similar clusters are merged in order to obtain a suitable partitioning. An adaptive threshold for merging is proposed. The extensions proposed are applied to Gustafson–Kessel and fuzzy c-means algorithms, and the resulting extended algorithm is given. The properties of the new algorithm are illustrated by various examples