C++ Neural Networks and Fuzzy Logic: Preface

Linear Possibility Regression Model

Assume that you have (n + 1)−tuples of values of x

1

, ... x

n

, and y. That is, for each i, i ranging from 1 to k,

you have (x

1

, ... , x

n

, y), which are k sample observations on X

1

, ... , X

n

, and Y. The linear possibility

regression model is formulated differently depending upon whether the data collected is crisp or fuzzy.

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 Y

j

.

But A

j

 is also determined by how big the membership of observed y

j

 is to be, in the fuzzy number Y

j

 . This

last observation provides a constraint for the linear programming problem which needs to be solved to find the

linear possibility regression. You select a value d, and ask that m

Y

(y) [ge] d.

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).

Previous Table of Contents Next

Copyright ©

 IDG Books Worldwide, Inc.

C++ Neural Networks and Fuzzy Logic:Preface

Linear Possibility Regression Model

399

---