Information Technology Reference
In-Depth Information
The possibility of equality of two fuzzy numbers
~
and
~
with membership
functions
()
()
η
x
μ
x
,
, accordingly, is determined by the formula according
to [192]:
~
~
[
]
(
)
() ()
==
In [197, 207] the least squares method is applied to deviations of centers of model
output normal triangular numbers from centers of observable output normal
triangular numbers. The centre of a normal triangular number with membership
function (6.1) is the number
a
. In [207] the optimization problem of a relative
minimum of the sum of fuzziness coefficients of output model normal triangular
numbers is solved. Fuzziness coefficients of a normal triangular number are
numbers
a
,
a
.
However, the methods of hybrid regression analysis as a rule are limited by
consideration of linear regression models and of a slender group of membership
functions (as a rule triangular fuzzy numbers are considered). Moreover, the
hybrid regression analysis must provide a way to model the observed fuzzy data,
such as linguistic descriptions of the type: “good”, “very good”, “excellent”,
which may be
T
- fuzzy numbers
.
In practical problems fuzzy data with tolerance membership functions are often
considered, therefore the problem of their analysis by regression analysis methods
is acute enough. In connection with reviewing of the limited spectrum of
membership functions of input data, a gap in methods of fuzzy regression analysis
occurred which has been partially filled in [194].
The model-building method of hybrid fuzzy least-squares regression in the
form of a system of classical regression equations (for each parameter of
membership functions of initial fuzzy data) described here in, unlike other
methods, can be applied both to unimodal, and to tolerance membership functions
of input data. The method [194] limitations allow constructing regression model
only with definite coefficients. Obviously, it dramatically limits possibilities of
model and makes the problem of developing regression models with fuzzy
coefficients a model of the day.
Methods of the fuzzy information formalization based on COSS described in
Chapter 2 allow representing results of an expert evaluation of qualitative
characteristics in the form of a group of fuzzy numbers explicitly considered in
§2.1. Thus, these fuzzy numbers can be used as input and output data in a fuzzy
regression
Pos
A
B
max
min
η
x
,
μ
x
.
x
model
describing
relations
between
estimated
qualitative
characteristics.
In order to include
T
- fuzzy numbers into a hybrid regression, a need for
developing a new method exists. Therefore, a new linear and nonlinear multiple
hybrid regressions are proposed and developed in this chapter. The developed
methods allow to construct relations among qualitative characteristics and to
predict their meanings.
