# Re: Can a fuzzy model be made from an incomplete set of variables?

From: WSiler@aol.com
Date: Thu Aug 23 2001 - 07:16:36 MET DST

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In a message dated 8/22/2001 1:23:26 PM Eastern Daylight Time,
aquila.em.hx.deere.com@hx.deere.com writes:

> , I am trying to use NeuroFuzzy technique to map a number of input =
> variables to an output variable. The data are from field measurements = and
> are very vague. In addition, some of the input variables are not =
> measurable although they are the less important variables. I am wondering
>

The answer is Yes, but not with the conventional fuzzy rules. You need to use
the method of William Combs. Briefly, the method is this.

Suppose you have two input variables, fuzzified into two discrete fuzzy sets,
say Size with members Small and Big, and Speed with members Slow and Fast,
and one output discrete fuzzy set with members Little and Big..

Ordinarily you would write four rules, like this:

IF (x1 is Small and x2 is Slow) THEN y is Big
IF (x1 is Big and x2 is Slow) THEN y is Little
IF (x1 is Small and x2 is Fast) THEN y is Big
IF (x1 is Big and x2 is Fast) THEN y is Little

This is what Combs calls the Intersection Rule Configuration, or IRC. There
are two problems with this. First, the number of rules grows exponentially
with the number of input variables. (With 20 input variables, you will have a
huge number of rules.) Second, if say x2 is missing you are dead; no rules
will fire, and you will get no answer.

Combs proposes the Union Rule Configuration, or URC. In the Combs method,
yhou rules would be:

IF (x1 is Small) THEN y is Big
IF (x2 is Slow) THEN y is Little
IF (x1 is Small) THEN y is Big
IF (x2 is Fast) THEN y is Little
IF (x1 is Small) THEN y is Big
IF (x2 is Slow) THEN y is Little
IF (x1 is Small) THEN y is Big
IF (x2 is Fast) THEN y is Little

As each rule fires, we will get different values for the grade of membership
of Big or Little, say Big1, Big2, Big3, Big4 and Little1, Little2, Little3
and Little4. These are aggregated by simply taking the average value: the
final value for Big's greade of membership would be

Big = (Big1+Big2+Big3+Big4)/4, and
Little = (Little1+Little2+Little3+Little4)/4.

Although in this case the Combs URC method has twice the number or rule as
the IRC, the number of rules increases LINEARLY with the number of input
variables, so handling 20 input variables still produces a manageable number
of rules. Further, missing data for some input variables will not stop you
from getting an answer, although the answer might not be quite as reliable as
if no input data were missing.

I have checked the method out on a number of problems, and it works, although
the answers from the URC and IRC can be slightly different.

William Siler

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<HTML><FONT FACE=arial,helvetica><FONT SIZE=2>In a message dated 8/22/2001 1:23:26 PM Eastern Daylight Time,
<BR>aquila.em.hx.deere.com@hx.deere.com writes:
<BR>
<BR>
<BR><BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">, I am trying to use NeuroFuzzy technique to map a number of input =
<BR>variables to an output variable. The data are from field measurements = and
<BR>are very vague. In addition, some of the input variables are not =
<BR>measurable although they are the less important variables. I am wondering
<BR>if a fuzzy model can be made from an incomplete set of = variables.=20 </BLOCKQUOTE>
<BR>
<BR>The answer is Yes, but not with the conventional fuzzy rules. You need to use
<BR>the method of &nbsp;William Combs. Briefly, the method is this.
<BR>
<BR>Suppose you have two input variables, fuzzified into two discrete fuzzy sets,
<BR>say Size with members Small and Big, and Speed with members Slow and Fast,
<BR>and one output discrete fuzzy set with members Little and Big..
<BR>
<BR>Ordinarily you would write four rules, like this:
<BR>
<BR>IF (x1 is Small and x2 is Slow) THEN y is Big
<BR>IF (x1 is Big and x2 is Slow) THEN y is Little
<BR>IF (x1 is Small and x2 is Fast) THEN y is Big
<BR>IF (x1 is Big and x2 is Fast) THEN y is Little
<BR>
<BR>This is what Combs calls the Intersection Rule Configuration, or IRC. There
<BR>are two problems with this. First, the number of rules grows exponentially
<BR>with the number of input variables. (With 20 input variables, you will have a
<BR>huge number of rules.) Second, if say x2 is missing you are dead; no rules
<BR>will fire, and you will get no answer.
<BR>
<BR>Combs proposes the Union Rule Configuration, or URC. In the Combs method,
<BR>yhou rules would be:
<BR>
<BR>IF (x1 is Small) THEN y is Big
<BR>IF (x2 is Slow) THEN y is Little
<BR>IF (x1 is Small) THEN y is Big
<BR>IF (x2 is Fast) THEN y is Little
<BR>IF (x1 is Small) THEN y is Big
<BR>IF (x2 is Slow) THEN y is Little
<BR>IF (x1 is Small) THEN y is Big
<BR>IF (x2 is Fast) THEN y is Little
<BR>
<BR>As each rule fires, we will get different values for the grade of membership
<BR>of Big or Little, say Big1, Big2, Big3, Big4 and Little1, Little2, Little3
<BR>and Little4. These are aggregated by simply taking the average value: the
<BR>final value for Big's greade of membership would be
<BR>
<BR>Big = (Big1+Big2+Big3+Big4)/4, and
<BR>Little = (Little1+Little2+Little3+Little4)/4.
<BR>
<BR>Although in this case the Combs URC method has twice the number or rule as
<BR>the IRC, the number of rules increases LINEARLY with the number of input
<BR>variables, so handling 20 input variables still produces a manageable number
<BR>of rules. Further, missing data for some input variables will not stop you
<BR>from getting an answer, although the answer might not be quite as reliable as
<BR>if no input data were missing.
<BR>
<BR>I have checked the method out on a number of problems, and it works, although
<BR>the answers from the URC and IRC can be slightly different.
<BR>
<BR>William Siler
<BR></FONT></HTML>

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