Center of Largest Area (CoA) method for defuzzification

Center of Largest Area (CoA) method is simple, computationally effective and widely used defuzzification.

If the fuzzy set has two sub-regions, then the center of gravity of the sub-region with the largest area can be used to calculate the defuzzified value.

x= (∫ μCm(x)  .  x′  dx) / ( ∫ μCm(x)  dx )

Here, Cm is the region with the largest area, x′ is the center of gravity of Cm

Example: Center of Largest Area (CoA) method

Find the crisp value corresponding to given fuzzy output functions

Solution:

To compute the crisp value for given fuzzy output sets, we shall place them on one axis and should find the area of nonoverlapping regions. The region with the highest value is used to compute the crisp value.

The aggregated fuzzy output functions are shown below. First, we will compute the area of region A1.

To find the area of region A1, we need to find the line equations for the ab and bc lines

line ab:

(y – y1) / (x – x1) = (y2 – y1) / (x2 – x1)

for line ab, (x1, y1) = (0, 0), and (x2, y2) = (1.5, 0.5)

⇒ (y – 0) / (x – 0) = (0.5 – 0) / (1.5 – 0)

⇒ y = 0.33x

The line ranges from 0 to 1.5 on the X axis.

line bc:

for line bc, (x1, y1) = (1.5, 0.5), and (x2, y2) = (x, 0).

We don’t know what is the x coordinate of point c. We shall compute the equation of line cd also. The x coordinate of both the line would be the same at the intersection point

for line bc:

(y – 0.5) / (x – 1.5) = (0 – 0.5) / (3 – 1.5)

⇒ y = 0.33x + 1

for line cd:

line cd:

y = 0.67x – 0.67

By comparing equations of line bc and line cd,

0.33x + 1 = 0.67x – 0.67

x = 1.67 ≈ 1.7

so line bc ranges from 1.5 to 1.7 on the X axis.

The area enclosed by line ab and bc:

A1 = 0.466

To find the area of region A2, we need to find the line equations for the cd and de lines

line cd:

(y – y1) / (x – x1) = (y2 – y1) / (x2 – x1)

for line cd, (x1, y1) = (1, 0), and (x2, y2) = (2.5, 1)

⇒ (y – 0) / (x – 1) = (1 – 0) / (2.5 – 1)

⇒ y = 0.67x -0.67   [1.7, 2.5]

The line ranges from 1.7 to 2.5 on the X axis.

line de:

for line de, (x1, y1) = (2.5, 1), and (x2, y2) = (4, 0).

(y – 1) / (x – 2.5) = (0 – 1) / (4 – 2.5)

⇒ y = -0.67x + 2.67

line ef:

y = 0.2x – 0.6

By comparing equations of line de and line ef, we can find the x coordinate of point e

-0.67x + 2.67 = 0.2x – 0.6

x = 3.8

so line de ranges from 2.5 to 3.8 on the X axis.

The area enclosed by line cd and de:

A2 = 1.32

To find the area of region A3, we need to find the line equations for ef and fg lines

line ef:

(y – y1) / (x – x1) = (y2 – y1) / (x2 – x1)

for line cd, (x1, y1) = (0, 3), and (x2, y2) = (4, 0.2)

⇒ (y – 0) / (x – 3) = (0.2 – 0) / (4 – 3)

⇒ y = 0.2x – 0.6

The line ranges from 3.8 to 4 on the X axis.

line fg:

for line de, (x1, y1) = (4, 0.2), and (x2, y2) = (5, 0).

(y – 0.2) / (x – 4) = (0 – 0.2) / (5 – 4)

⇒ y = -0.2x + 1

so line fg ranges from 4 to 5 on the X axis.

The area enclosed by line ef and fg:

A3 = 0.136

Thus, the area enclosed by lines are summarized in the following table:

As we can see, region 2 has the largest area. So center of A2 will be used to find the crisp value.

Watch on YouTube: Center of Largest Area method

All the defuzzification methods are summarized below: