RE: JESS: FuzzyJess formula for local centre of gravity

[Orchard, Bob]

 

See the FuzzyJ Toolkit documentation ... chapter on Fuzzy Sets. Note that recently a CenterOfArea defuzzification

was also added.

 

To supplement the documentation I should note that the 'moment' calculation for each of the shapes that

might make up the pieces of a fuzzy set are:

 

Given: x1, x2 are the left and right x values of the shape

          y1, y2 are the membership (y) values of the left and right edges of the shape

 

        /* rectangle of zero height or zero width? */
        moment is 0

        area is 0

 

       /* rectangle */
       moment is 0.5 * (x1 + x2) 

       area is ( x2 - x1 ) * y1;

       /* triangle, where height is y2 */
       moment is  (2.0/3.0) * ( x2 - x1 ) + x1;
       area  is 0.5 * ( x2 - x1 ) * y2;

       /* triangle, where height is y1 */
       moment is (1.0/3.0) * ( x2 - x1 ) + x1;
       area is 0.5 * (x2 - x1 ) * y1;

       /* trapezoid */
       moment is  ( (2.0/3.0) * (x2-x1) * (y2+0.5*y1))/(y1+y2) + x1;
       area is  0.5 * ( x2 - x1 ) * ( y1 + y2 );

 

Bob Orchard
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-----Original Message-----
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]On Behalf Of yuping he
Sent: Monday, June 26, 2006 7:47 PM
To: jess-users@sandia.gov
Subject: JESS: FuzzyJess formular for local centre of gravity

Dr. Friedman-Hill:

 

I am trying to test Fuzzy Jess inside my rules and had one question - what is the formular to calculate the local centre of gravity?

 

Thanks.

Yuping

 

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momentDefuzzify
Moment defuzzification defuzzifies the FuzzySet of the FuzzyValue returning a floating point (double value) that represents the fuzzy set. It calculates the first moment of the area of a fuzzy set about the y axis. The set is subdivided into different shapes by partitioning vertically at each point in the set, resulting in rectangles, triangles, and trapezoids. The centre of gravity (moment) and area of  each subdivision is calculated using the appropriate formulas for each shape. The first moment of area of the whole set is then:

where x_i´ is the local centre of gravity, A _i is the local area of the shape underneath line segment (p_i-1, p_i), and n is the total number of points. As an example,

For each shaded subsection in the diagram above, the area and centre of gravity is calculated according to the shape identified (i.e., triangle, rectangle or trapezoid). The centre of gravity of the whole set is then determined: