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    A Fuzzy Clustering Validity Index Induced by Triple Center Relation
    (Ieee-Inst Electrical Electronics Engineers Inc, 2023) Tang, Yiming; Huang, Jiajia; Pedrycz, Witold; Li, Bing; Ren, Fuji
    The existing clustering validity indexes (CVIs) show some difficulties to produce the correct cluster number when some cluster centers are close to each other, and the separation processing mechanism appears simple. The results are imperfect in case of noisy data sets. For this reason, in this study, we come up with a novel CVI for fuzzy clustering, referred to as the triple center relation (TCR) index. The originality of this index is twofold. On the one hand, a new fuzzy cardinality is built on the strength of the maximum membership degree, and a novel compactness formula is constructed by combining it with the within-class weighted squared error sum. On the other hand, starting from the minimum distance between different cluster centers, the mean distance as well as the sample variance of cluster centers in the statistical sense are further integrated. These three factors are combined by means of product to form a triple characterization of the relationship between cluster centers, and hence a 3-D expression pattern of separability is formed. Subsequently, the TCR index is put forward by combining the compactness formula with the separability expression pattern. By virtue of the degenerate structure of hard clustering, we show an important property of the TCR index. Finally, based on the fuzzy C-means (FCMs) clustering algorithm, experimental studies were conducted on 36 data sets (incorporating artificial and UCI data sets, images, the Olivetti face database). For comparative purposes, 10 CVIs were also considered. It has been found that the proposed TCR index performs best in finding the correct cluster number, and has excellent stability.
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    Universal Quintuple Implicational Algorithm: A Unified Granular Computing Framework
    (Ieee-Inst Electrical Electronics Engineers Inc, 2024) Tang, Yiming; Chen, Jingjing; Pedrycz, Witold; Ren, Fuji; Zhang, Li
    In the field of fuzzy inference, the universal triple I algorithm integrated the CRI (Compositional Rule of Inference) algorithm with the triple I algorithm. Later the triple I algorithm was generalized to the QIP (quintuple implication principle) algorithm. Whether the QIP algorithm and the CRI algorithm can be unified has become an interesting question. Therefore, in this study, a fuzzy inference scheme referred to as the universal quintuple implicational (UQI) algorithm is proposed. First, we establish a unified granular computing framework with the UQI algorithm, which is a generalization of the QIP algorithm, the CRI algorithm as well as the universal triple I algorithm. The optimal UQI solutions derived from the fundamental principle of determining inference results are obtained for the FMP (fuzzy modus ponens) problem, in which some specific solutions are also given. Second, the reversible property of the UQI algorithm is verified for FMP, while aiming at the metric derived from the biresiduum operation, the robustness of the UQI algorithm is validated. Third, under the environment of multiple rules, two general cases of FITA (First-Inference-Then-Aggregate) and FATI (First-Aggregate-Then-Inference) are constructed for the UQI algorithm. The corresponding equivalence relation between continuity and interpolation is analyzed. Fourth, the fuzzy system is established based on the UQI algorithm, the singleton fuzzier as well as the centroid defuzzier. Its response ability is analyzed and it is shown that such fuzzy system is a universal approximator. Lastly, we compare the results of the UQI algorithm with the QIP algorithm by five examples for FMP. It is found that the UQI algorithm is able to acquire more and better forms of the fuzzy inference in contrast with the QIP algorithm.