Fuzziness of a membership function

WSiler (wsiler@aol.com)
Wed, 26 Aug 1998 20:28:40 +0200 (MET DST)

Someone wanted a measure of the fuzziness of a membership function. A simple
measure for a discrete fuzzy set, derived from information theory, is:

Fuzziness = sum over i(-mu(i)logmu(i) - (1 - mu(i))log(1 - mu(i))) / log(2)

For a continuous fuzzy set, this would be:

Fuzziness = integral(-mu(x)log(mu(x)) - (1 - mu(x)log(1 - mu(x)))dx

If the membership function is a singleton, this gives ambiguity and fuzziness
zero.

Of course, the depends on the fact that the product mu(x) log(mu(x)) gpes to
zero as x approaches zero.

If we have more than one discrete fuzzy set member, as in Slow, Medium and
Large, a useful measure of the ambiguity among these members is:

ambiguity = exp(sum(-mu(i) log(mu(i)))

If only one member of the fuzzy set has membership one, and the rest have
membership zero, this gives zero ambiguity.

William Siler

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