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Knowledge discovery and semantic learning in the framework of axiomatic fuzzy set theory

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Axiomatic fuzzy set (AFS) theory facilitates a way on how to transform data into fuzzy sets (membership functions) and implement their fuzzy logic operations, which provides a flexible and powerful tool for representing human knowledge and emulate human recognition process. In recent years, AFS theory has received increasing interest. In this survey, we report the current developments of theoretical research and practical advances in the AFS theory. We first review some notion and foundations of the theory with an illustrative example, then, we focus on the various extensions of AFS theory for knowledge discovery, including clustering, classification, rough sets, formal concept analysis, and other learning tasks. Due to its unique characteristics of semantic representation, AFS theory has been applied in multiple domains, such as business intelligence, computer vision, financial analysis, and clinical data analysis. This survey provides a comprehensive view of these advances in AFS theory and its potential perspectives. This article is categorized under: Technologies > Computational Intelligence
The comparison among natural, disgust, and surprise on CK+, with the semantic concept is that: “the angle between corners of mouth and the down point of mouth is larger”; with the semantic concept is that: “the distance between the point of inner canthus point and the point of inner eyebrow is smaller” or “the height of eyes is smaller”; with the semantic concept is that: “the area of mouth is larger and the angle of right corner of the mouth is larger” or “the height of mouth is larger and the angle of right corner of the mouth is larger” (Z. Li, Zhang, Duan, Wang, et al., )
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Comparisons of “large eyes cluster ” and “small eyes cluster ,” with the semantic interpretation: “the eyes in this cluster have large perimeter, large center length and large area,” with the semantic interpretation: “the eyes in this cluster have small area” (Ren et al., )
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The feature extraction of face image on f1 = “right eye” (Ren et al., )
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Membership degree of ,, and ζ:“iris‐setosa”(Liu et al., )
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Membership degrees of , , and (Liu et al., )
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(a) the number of testing samples distributing different scopes of confidence degree; (b) The distribution of the error rate with different confidence degree (Liu et al., )
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Membership degree of the fuzzy descriptions with the semantic “short petal length and narrow petal width” or “not mid petal width and narrow petal width”; with the semantic “mid petal length and mid petal width”; with the semantic “long sepal length and long petal length” or “wide petal width” for iris data (Liu & Pedrycz, )
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The number of clusters, clustering accuracy, and the fuzzy cluster validity index Iα of the algorithm applied to iris data of the threshold α (Liu & Pedrycz, )
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Membership functions of fuzzy descriptions , with the semantics “narrow petal width , short petal length and not mid petal length”, , with the semantics “narrow petal width and not mid petal width”, , with the semantics “short petal width and mid petal width”, , with the semantics “long sepal length and long petal length” (Liu & Pedrycz, )
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The procedure of axiomatic fuzzy set clustering algorithm
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Classification boundaries generated by axiomatic fuzzy set membership function (Liu et al., )
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Classification boundaries generated by triangular membership function (Liu et al., )
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The framework of axiomatic fuzzy set theory and its applications
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