C++ Neural Networks and Fuzzy Logic by Valluru B. Rao M&T Books, IDG Books Worldwide, Inc. ISBN: 1558515526 Pub Date: 06/01/95 |

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A fuzzy number is an ordered pair of numbers **(a, b)** with respect to a reference function **L,** which gives you the membership function. Here **b** has to be a positive number.

Here are the properties of **L,** the reference function:

**1.**It is a function of one variable and is symmetric about**0**. That is,*L*(*x*) =*L*(-*x*).**2.**It has a value of**1**at**x = 0**. In other words,*L*(0) = 1.**3.**It is generally decreasing, when**x**is**0**or a positive number, meaning that its value drops as the value of its argument is increased. For example,*L*(2) <*L*(1). Thus*L*(*x*) has its values less than 1 for positive values of*x*. It does not make sense to have negative numbers as values of**L,**and so you ignore the values of**x**that cause such values for**L**.**4.**The maximum value for the function is**1,**at**x = 0**. It has a sort of a bell-shaped curve.

If **A** is a fuzzy number with the ordered pair (*a*, *b*) with *b* > 0, and if the reference function is **L**, you do the following:

You write the fuzzy number as **A = (a, b) _{L}.** You get the membership of any element

m_{A}(x) =L( (x-a) / b)

Examples of a reference function **L** are:

Example 1: | L(x) = max ( 0, 1- x^{2}) |

You obtain the following shown in Table 16.8.

x | L(x) | |
---|---|---|

-2 | 0 | |

-1 | 0 | |

-0.5 | 0.75 | |

0 | 1 | |

0.5 | 0.75 | |

1 | 0 | |

2 | 0 | |

This function is not *strictly* decreasing though. It remains a constant at 0 for *x* > 1.

Example 2: | L(x) = 1/ (1 + x^{2} ) |

You get the values shown in Table 16.9.

x | L(x) | |
---|---|---|

-7 | 0.02 | |

-2 | 0.2 | |

-1 | 0.5 | |

-0.5 | 0.8 | |

0 | 1 | |

0.5 | 0.8 | |

1 | 0.5 | |

2 | 0.2 | |

7 | 0.02 | |

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.

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

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.

Assume that you have **(n + 1)-tuples** of values of **x _{1}, ... x_{n},** and

Let us give such a model below, for the case with crisp data. Then the fuzziness lies in the coefficients in the model. You use symmetrical fuzzy numbers, **A _{j} = (a_{j}, b_{j})_{L}.** The linear possibility regression model is formulated as:

Y_{j}= A_{0}+ A_{1}X_{j1}+ ... + A_{n}X_{jn}

The value of **Y** from the model is a fuzzy number, since it is a function of the fuzzy coefficients. The fuzzy coefficients **A _{j}**are chosen as those that minimize the width (the base of the triangle) of the fuzzy number

We close this section by observing that linear possibility regression gives triangular fuzzy numbers for **Y,** the dependent variable. It is like doing interval estimation, or getting a regression band. Readers who are seriously interested in this topic should refer to Terano, et al. (see references).

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Copyright © IDG Books Worldwide, Inc.

C++ Neural Networks and Fuzzy Logic

ISBN: 1558515526

EAN: 2147483647

EAN: 2147483647

Year: 1995

Pages: 139

Pages: 139

Authors: Valluru B. Rao, Hayagriva Rao

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