Environmental Engineering Reference
In-Depth Information
10.2.1 Fuzzification: Membership Functions
Membership functions are the core of a fuzzy model, and the most revolutionary
concepts of fuzzy set theory. One of their advantages is that the linguistic expres-
sion of a variable is easily understandable by everyone, whereas numerical values
are meaningful only for experts.
Any model variable is described by a characteristic C,suchas
low
or
high
.
A membership function
m
C
converts the numerical value of the variable v into a
membership grade to the characteristic C. The membership grade ranges from 0 to 1,
and can assume all values in this interval:
m
C
(v)
¼
1, full membership: v displays C completely
m
C
(v)
¼
0, null membership: v does not display C
0
< m
C
(v)
<
1, partial membership, v partially displays C
A straightforward example: consider the variable “water temperature” and its
characteristics
cold
and
hot
. A temperature 0
C could have membership 1 to the
function
cold
and membership 0 to the function
hot
. Conversely, the value 100
C
could have membership 0 to
cold
and membership 1 to
hot
. Membership grades
between 0 and 1 represent intermediate situations. In other approaches, they would
be classified as uncertain values. Fuzzy logic converts uncertainty into enhanced
information. In fact, uncertain values have partial membership to more than one
characteristic. This is due to an important feature of membership functions: they
overlap. A variable value is allowed to have non-null membership for two functions
simultaneously. For instance, a medium temperature value can be partially
cold
and
partially
hot
. The amount of overlap between two functions is related to the amount
of uncertainty included in the model: the more the overlap, the more the uncer-
tainty. Classical “crisp” functions do not tolerate uncertainty, since they have null
overlap (a temperature value can be either
cold
or
hot
). In other terms, whereas in
classical systems there would be a steep threshold between
cold
and
hot
water
temperature, the fuzzy approach offers a gradual transition, which is more similar to
real world conditions.
To develop a fuzzy model, the ecologist is required to define the number of
membership functions for each variable of the model, their shape and their position
along the
x
-axis. These are parameters of the model and indicate the semantic
meaning of the variable characteristics. Membership functions can be linear (trian-
gular and trapezoidal) or nonlinear (bell-shaped, sigmoid, polynomial) (Fig.
10.1
).
Triangular and trapezoidal membership functions indicate that the variable charac-
teristic changes linearly as the variable value changes. Nonlinear functions describe
more complex behaviours. Triangular and trapezoidal functions are easier to define,
whereas more complicated shapes are less intuitive and may require the definition
of more parameters.
However, linearity is not a common behaviour of ecosystems: nonlinear func-
tions such as Gaussian and sigmoid would probably better represent ecological data
and improve model performance.
