Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 140

It has frequently been reported that the design approaches of fuzzy-logic-based
systems have been found to be very robust when embedded in control and signal-
processing systems. However, the development of fuzzy logic systems, based on
human expert's knowledge, is not an easy attempt, primarily because it is very
difficult to extract the complete and consistent human expert's knowledge correctly
by interviewing him or her.
The objective of this chapter is to develop some suitable fuzzy logic systems
capable of efficiently modelling time series data and forecasting their values.
Because the efficient functioning of fuzzy logic systems depends primarily on
fuzzy rules used for modelling, and because the automated generation of such rules
is rather difficult, various data-driven algorithms for automated rule generation are
presented.

4.2 Fuzzy Sets and Membership Functions

The membership function is the key idea introduced in fuzzy set theory to measure
the degree to which the fuzzy set elements meet the specific properties, i.e. to
measure the degree of belongingness of an element in a specific fuzzy set.
Consequently, the propositions used need not be true or false, but can be to any
degree partially true.
Using a membership function µ , we can define a fuzzy set F on a universe of
discourse U as

> @

P xU : o 0, 1 ,

F

which is nothing but a mapping from the universe of discourse U into the unit
interval [0, 1] and

F P represents the extent (degree/grade) to which x belongs
to fuzzy set F . The concept of membership functions allows any element within the
universe of discourse to have partial membership to a specific fuzzy set and also to
have partial membership to other fuzzy sets. In order to demonstrate the idea of
membership functions, two examples are given, one each for a crisp set and a fuzzy
set.

x where X
is the universe of discourse (domain), then the degree of membership of x in crisp
set C will be 1 and 0 respectively if the element x belongs to C completely (full
member) or it does not belong to it at all. Mathematically, this is stated as

Let C be a crisp set and x be any element of the set C such that ,

^ 1;

i f x C

P x

0; if x C

C

Let us now consider that F be a fuzzy set and x be any element of the fuzzy set F
such that

x where X is the universe of discourse (domain), then the degree of
membership of x in fuzzy set F will be 1 and 0 respectively if the element x
belongs to F completely (full member) or it does not belong to it at all. However, if

,