On balance of uncertainty in shadowed sets
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A shadowed set, $S$, facilitates crisp decision-making with a fuzzy set $F$. It is constructed with the aid of different optimization-based principles. Among these principles, the requirement of uncertainty balance guarantees preservation of the uncertainty of $F$ in $S$. In order to gain further insight on uncertainty balance, some essential mathematical properties which characterize uncertainty-balance-based objective function, $J(\alpha)$, are studied. These properties provide theoretical explanation for interpreting and analyzing $J(\alpha)$ and its ensuing optimum partition threshold $\alpha$. Two senses of uncertainty balance are discussed in this paper. Their combined efficiency in enhancing clustering results is illustrated with the aid of synthetic data set used in shadowed $C$-means clustering. Finally a need for five-region shadowed sets, $S_5$, is pointed out. A closed-form formula for determining its optimum thresholds is proposed and exemplified on typical fuzzy set and synthetic dataset.
Abstract
A shadowed set, $S$, facilitates crisp decision-making with a fuzzy set $F$. It is constructed with the aid of different optimization-based principles. Among these principles, the requirement of uncertainty balance guarantees preservation of the uncertainty of $F$ in $S$. In order to gain further insight on uncertainty balance, some essential mathematical properties which characterize uncertainty-balance-based objective function, $J(\alpha)$, are studied. These properties provide theoretical explanation for interpreting and analyzing $J(\alpha)$ and its ensuing optimum partition threshold $\alpha$. Two senses of uncertainty balance are discussed in this paper. Their combined efficiency in enhancing clustering results is illustrated with the aid of synthetic data set used in shadowed $C$-means clustering. Finally a need for five-region shadowed sets, $S_5$, is pointed out. A closed-form formula for determining its optimum thresholds is proposed and exemplified on typical fuzzy set and synthetic dataset.
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