C++ Neural Networks and Fuzzy Logic: Preface

Conditional Fuzzy Mean and Fuzzy Variance

The conditional B_fuzzy_mean of the takeover price with fuzzy event A as given works out as:

     (1/0.54) x (100 x 0.8 x 0.7 x 0.3 + 85 x 0.4 x 1 x 0.5 + 60 x 0.7 x

0.5 x 0.2) = 70.37

and the conditional B_fuzzy_variance of the takeover price with fuzzy event A, as given, amounts to

1301.76, which is over five times as large as when you did not take the analyst’s fuzzy event as given.

Linear Regression a la Possibilities

When you see the definitions of fuzzy means and fuzzy variances, you may think that regression analysis can

also be dealt with in the realm of fuzzy logic. In this section we discuss what approach is being taken in this

regard.

First, recall what regression analysis usually means. You have a set of x− values and a corresponding set of y

values, constituting a number of sample observations on variables X and Y. In determining a linear regression

of Y on X, you are taking Y as the dependent variable, and X as the independent variable, and the linear

regression of Y on X is a linear equation expressing Y in terms of X. This equation gives you the line ‘closest’

to the sample points (the scatter diagram) in some sense. You determine the coefficients in the equation as

those values that minimize the sum of squares of deviations of the actual y values from the y values from the

line. Once the coefficients are determined, you can use the equation to estimate the value of Y for any given

value of X. People use regression equations for forecasting.

Sometimes you want to consider more than one independent variable, because you feel that there are more

than one variable which collectively can explain the variations in the value of the dependent variable. This is

your multiple regression model. Choosing your independent variables is where you show your modeling

expertise when you want to explain what happens to Y, as X varies.

In any case, you realize that it is an optimization problem as well, since the minimization of the sum of

squares of deviations is involved. Calculus is used to do this for Linear Regression. Use of calculus methods

requires certain continuity properties. When such properties are not present, then some other method has to be

used for the optimization problem.

The problem can be formulated as a linear programming problem, and techniques for solving linear

programming problems can be used. You take this route for solving a linear regression problem with fuzzy

logic.

In a previous section, you learned about possibility distributions. The linear regression problem with fuzzy

logic is referred to as a linear possibility regression problem. The model, following the description of it by

Tarano, Asai, and Sugeno, depends upon a reference function L, and fuzzy numbers in the form of ordered

pairs (a, b). We will present fuzzy numbers in the next section and then return to continue our discussion of

C++ Neural Networks and Fuzzy Logic:Preface

Conditional Fuzzy Mean and Fuzzy Variance

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the linear possibility regression model.

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C++ Neural Networks and Fuzzy Logic:Preface

Conditional Fuzzy Mean and Fuzzy Variance

396

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