# Re: Defuzzification

Subject: Re: Defuzzification
WSiler@aol.com
Date: Wed Nov 15 2000 - 21:16:12 MET

In a message dated 11/13/00 5:22:17 AM Central Standard Time,
fractalized@my-deja.com writes:

<< I did Not understand the singleton defuzzification method. I was hoping
that someone here might be able to tell me how to defuzzify a function
using either or both singleton methods.
>>

First, in defuzzification we must have a linguistic variable to defuzzify.
Very simply, a linguistic variable name Speed is a fuzzy set of linguistic
terms describing a numeric quantity (such as Slow, Medium, Fast) with a fuzzy
number or membership function, defined on the real line, assigned to each
linguistic term. For each linguistic term, the membership function value for
each number on the real line represents the confidence, from 0 to 1, that
each number is correctly described by the corresponding linguistic term. For
example, Slow might have a membership function whose value is one from 0 to
10 miles per hour, declining linearly to zero from 10 to 20 mph, being zero
thereafter.

The singleton method uses the simplest possible membership function, with
membership value one at only one number from the real line, and zero
everywhere else. Since only a single value has a non-zero membership value,
such a membership function is called a singleton. For example Slow might have
a singleton at zero; Medium at 30; and Fast at 60.

Defuzzification with singletons is extremely easy. Suppose our rules give
Slow a membership value of 0.1; Medium a membership value of 0.6; and Fast a
membership value of 0.8. The corresponding defuzzified value is:

(0.1 * 0 + 0.6 * 30 + 0.8 * 60) / (0.1 + 0.6 + 0.8) or 44 miles per hour.

Willliam Siler

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