C++ Neural Networks and Fuzzy Logic: Preface

Table 16.9 Reference Function L

xL(x)

−70.02

−20.2

−10.5

−0.50.8

01

0.50.8

10.5

20.2

70.02

Let us now determine the membership of 3 in the fuzzy number A = (4, 10)

L

, where L is the function in the

second example above, viz., 1/ (1 + x

2

).

First, you get (x− 4) / 10 = (3 − 4) / 10 = − 0.1. Use this as the argument of the reference function L. 1/ (1 + (−

0.1)

2

 ) gives 0.99. This is expressed as follows:

     m

A

(3) = L( (3−4) / 10) = 1/ (1 + (− 0.1)

2

 ) = 0.99

You can verify the values, m

A

(0) = 0.862, and m

A

(10) = 0.735.

Triangular Fuzzy Number

With the right choice of a reference function, you can get a symmetrical fuzzy number A, such that when you

plot the membership function in A, you get a triangle containing the pairs (x, m

A

(x)), with m

A

(x) > 0. An

example is A = (5, 8)

L

, where L = max(1 − |x|, 0).

The numbers x that have positive values for m

A

(x) are in the interval ( −3, 13 ). Also, m

A

( −3 ) = 0, and

m

A

(13) = 0. However, m

A

(x) has its maximum value at x = 5. Now, if x is less than −3 or greater than 13, the

value of L is zero, and you do not consider such a number for membership in A. So all the elements for which

membership in A is nonzero are in the triangle.

This triangular fuzzy number is shown in Figure 16.1. The height of the triangle is 1, and the width is 16,

twice the number 8, midpoint of the base is at 5. The pair of numbers 5 and 8 are the ones defining the

symmetrical fuzzy number A. The vertical axis gives the membership, so the range for this is from 0 to 1.

Figure 16.1

  Triangular membership function.

C++ Neural Networks and Fuzzy Logic:Preface

Triangular Fuzzy Number

398

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