Returns-based style analysis

In mathematics, vague sets are an extension of fuzzy sets. In a fuzzy set, each object is assigned a single value in the interval [0,1] reflecting its grade of membership. Unfortunately, this single value does not allow a separation of evidence for membership and evidence against membership. Gau et al. proposed the notion of vague sets, where each object is characterized by two different membership functions: a true membership function and a false membership function. This kind of reasoning is also called interval membership, as opposed to point membership in the context of fuzzy sets.

Mathematical definition

A vague set $V$ is characterized by

• its true membership function $f_{v}(x)$
• its false membership function $t_{v}(x)$
• with $0\leq t_{v}(x)+f_{v}(x)\leq 1$

The grade of membership for x is not a crisp value anymore, but can be located in $[t_{v}(x),1-f_{v}(x)]$. This interval can be interpreted as an extension to the fuzzy membership function. The vague set degrades to a fuzzy set, if $1-f_{v}(x)=t_{v}(x)$ for all x. The uncertainty of x is the difference between the upper and lower bounds of the membership interval; it can be computed as $(1-f_{v}(x))-t_{v}(x)$.

References

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External links

• Vague Sets