Defuzzification methods, as we know, are used to convert defuzzified output put of fuzzy inference engine to crisp value. In previous few articles, we already have discussed maxima methods, Center of Gravity (CoG) method, Center of Sums (CoS) method, Center of Largest Area (CoA) method etc.
In this article, we will see example of various defuzzification techniques.
Example: Find crisp value corresponding to following fuzzy output sets using various defuzzification methods.
To compute the crisp value, we shall place all the function on same axis.
To compute the area of polygon, we need to compute the equation of line enclosing it and integrating it over lines making it.
For discussion, equation of line ab is derived here. Rest can be understand
(y – y1) / (x – x1) = (y2 – y1) / (x2 – x1)
For line ab, (x1, y1) = () and (x2, y2) = ()
(y – 0) / (x – 0) = (1 – 0) / (1.5 – 0)
y / x = 0.67
y = 0.67x
Line ab ranges from 0 to 1.5 on X axis.
In similar way, we can find equation of all the lines. Click here to see detail.
Equation of all the lines are summarized in the table below:
Center of Gravity method:
For computing crisp value using center of gravity method, we shall find the area of overlapped region.
Center of Sums method:
For computing crisp value using center of sums method, we shall find the area of individual triangle rather then overlapped region as in center of gravity method.
Area of triangle is computed as, A=1/2×b×h
A1 = 1/2 × 3 × 1 = 1.5
A2 = 1/2 × 2 × 1 = 1
A3 = 1/2 × 4 × 1 = 2
Putting all these values in the equation of center of sums method,
Center of Largest Area method:
For computing crisp value using center of largest area method, we shall find the centroid of area of largest triangle. From the line equations we have derived,
A3 is the largest area. Lets find the center of it.
Watch on YouTube: Example of Defuzzification Methods
Weighted Area Method:
Weighted area method finds weighted sum of center of area and its membership value.
Center of fuzzy sets 1, 2 and 3 is 1.5, 2 and 3 respectively. And corresponding membership value of them is 1. Crisp value using this methods is computed as,
Maxima method finds the membership value of peak at certain position.
First of Maxima: x* = 1.5
Last of Maxima: x∗ = 3
Middle of Maxima: Not Applicable
Height Method: Not Applicable
Test Your Knowledge:
Example: Find the crisp value for following two fuzzy output functions using various defuzzification methods.
Please post your answer / query / feedback in comment section below !