Biomedical Engineering Reference
In-Depth Information
methodology, and the third presents a tutorial
FDT analysis on the previously described small
example data set.
bers that represent vague concepts and imprecise
terms expressed often in a natural language.
This issue of the expression in a natural lan-
guage is pertinent to the results from analysis when
using FDTs (see later). Here a series of piecewise
linear MFs are used to define linguistic terms to
describe attribute values in a data set. Dombi and
Gera (2005) identifies that fuzzy applications
use piecewise linear MFs because of their easy
handling (see also, Yu and Li, 2001). The general
form of a piecewise linear MF, in the context of the
j
th
linguistic term
Fuzzy Set Theory
In classical set theory, an element (value) either
belongs to a certain set or it does not. It follows, the
definition of a set can be defined by a two-valued
membership function (MF), which takes values 0
or 1, defining membership or non-membership to
the set, respectively. In fuzzy set theory (Zadeh,
1965), a grade of membership exists to charac-
terise the association of a value
x
to a set
S
. The
concomitant MF, defined
μ
S
(
x
), has range [0, 1].
The relevance and impact of the notion of a MF is
clearly stated in Inuiguchi
et al
. (2000, p. 29);
j
T
of a linguistic variable
A
k
, is
shown in Equation 1 with the respective
defining
values
in list form are [α
j,
1
, α
j,
2
, α
j,
3
, α
j,
4
, α
j,
5
]. This
definition is restricted in the sense that the resultant
MF is made up of four 'piecewise' parts only, a
visual example of the kind of structure this type
of MF produces, is given in the fuzzification of a
small example data set next described, including
also the use of the defining values.
The small example data set considered here
consists of four objects,
u
1
,
u
2
,
u
3
and
u
4
, described
by three condition (T1, T2 and T3) and one deci-
sion (C) attribute, see Table 1.
The fuzzification of this data set starts with
the fuzzification of the individual attribute val-
ues. Moreover, each attribute in Table 1 needs
to be viewed in terms of them being a linguistic
variable, which is itself described by a number of
linguistic terms. Here, two linguistic terms are
used to describe a linguistic variable, so two MFs
are used to fuzzify a single attribute, see Figure
1 where the linguistic variable form associated
with the decision attribute C is shown.
In Figure 1, two MFs, m
L
(C) (labelled C
L
) and
m
H
(C) (C
H
), are shown to cover the domain of the
decision attribute C, the concomitant defining
values are, for C
L
: [−∞,−∞, 7, 9, 11] and C
H
: [7,
9, 11, ∞, ∞]. An interpretation could then simply
be the associations of a decision attribute value
to the two linguistic terms denoted by the words,
low (L) and/or high (H). For example, a value
of C = 8, would mean, C
L
= 0.75 and C
H
= 0.25,
k
In fuzzy systems theory, a membership func-
tion plays a basic and significant role since it
characterizes a fuzzy object and all treatments
of those fuzzy objects are made in terms of their
membership functions. The membership function
elicitation is the first and an important stage that
allows us to deal with the fuzzy objects.
It is clear from this statement that the types
of MFs used will have an impact on the analysis
undertaken with them. In this chapter, the use
of MFs is with respect to their association with
defining attributes describing objects as linguistic
variables. Briefly, a linguistic variable is made
up of a number of linguistic terms, which are
each defined by a MF. Garibaldi and John (2003)
consider an important surrounding issue, namely
choosing the type of MFs to define the linguistic
terms. Recent research studies that consider the
issue of the types of MFs used include; Pedrycz
and Vukovich (2002) and Grzegorzewski and
Mrówka (2005). In the latter study, Grzegorzewski
and Mrówka (2005, p. 115) clearly state that;
... the crucial point in fuzzy modeling is to assign
membership functions corresponding to fuzzy num-
Search WWH ::
Custom Search