15.4 Applications of the Fuzzy Modeling Approach to Ecological Assessment
215
Inclusion of fuzzy sets is defined by the inequality
of their membership functions:
VxES.
Union and intersection of two fuzzy sets are defined by the maximum and minimum, respectively.
Union:
AUB: Il-A U B(X)
= max[Il-A(X), Il-B(X)].
Intersection: A n B: Il-A n B(X)
= min[Il-A(X), Il-B(X)].
The complement of a fuzzy set A, which is denoted
by AC, is defined as follows:
Complement: AC: Il-AC(X) = 1 - Il-A(X).
For example, let S = {I, 2, 3,4, 5} again and assume that fuzzy sets A and B are given by
A = 0/1 + 0.5/2 + 0.8/3 + 114 + 0.2/5,
B = 0.9/1 + 0.4/2 + 0.3/3 + 0.1/4 + 015.
Then
A U B = 0.9/1 + 0.5/2 + 0.8/3 + 114 + 0.2/5,
A n B = 0/1 + 0.412 + 0.3/3 + 0.1/4 + 015,
AC = 111 + 0.5/2 + 0.2/3 + 0/4 + 0.8/5.
Many extensions of the above min-max definition to the fuzzy set operations can be found in
Zimmermann (1991) and many other texts. These
fuzzy set operations can be used to aggregate different ecological indicators during ecological risk
analysis. Applications of fuzzy mathematical
methods in ecology have been published by many
authors (e.g., Li, 1986, 1996; Li and Zhou, 1987;
Banyikwa et al., 1990; Bardossy et al., 1990a;
Keesman and van Straten, 1990; Lindsey et aI.,
1992; McBratney et al., 1992; Bardossy, 1994; Feoli and Zuccarello, 1994; Salski et al., 1996).
15.4 Applications of the Fuzzy
Modeling Approach to
Ecological Assessment
Because fuzzy sets are a generalization of a classical set theory, the embedding of conventional models into a larger setting endows fuzzy models with
greater flexibility to capture various aspects of incompleteness or imperfection in whatever information and data are available about a real process.
For example, we developed a semiarid grazing
ecosystem simulation model combining fuzzy imprecision with probabilistic uncertainty to characterize climate-plant-herbivore interactions (Wu et
al., 1996); Cheng et al. (1996) used fuzzy systems
analysis and conventional dynamic programming
to optimize biological control of a greenhouse spider mite--cucumber system; Singer and Singer
(1993) presented a fuzzy set formulation of the phenomenological equations of nonequilibrium chemical kinetics, which could be easily adopted to deal
with nonequilibrium ecological "reactions" of multiple species; and Levary (1990) established fuzzylogic-based system dynamics methodology for
modeling, simulating, and analyzing real-life systems with imprecise and vague variables or events.
In general, we can use fuzzy models in any of
the following circumstances: (1) the modeled
process is too complicated and its exact description
is not known; (2) many factors are not quantified,
or are impossible to quantify, and they are characterized only verbally; (3) more exact specifications
cannot be employed and, hence, our research would
be gratuitous; (4) the exact description is too time
consuming (the use of natural language is advantageous because it follows human thinking in the
best way, and the necessary time for fuzzy model
construction is very small); (5) the exact model
construction might need some experiments (this
might not, however, be possible either due to economic reasons or due to the nature of the process
itself); and (6) the fuzzy model is more objective
than verbal description based on a purely intuitive
approach.
The general methodology for the fuzzy model
construction can be summarized as follows:
1. Description of the problem
2. Factor identification, definition, and classification as dependent or independent
3. Analysis of all factors from the viewpoint of
their dependence
4. Determination of the factors that will be
processed as linguistic variables and determination of the scales of linguistic values
5. Determination of the universes, that is, determination of the kind of elements (numbers, intervals) and their extent (see the above example
on membership functions of vegetation cover)
6. Transformation of linguistic values into fuzzy
sets, that is, the estimation of membership functions
7. Formulation of the fuzzy model using conditional statements and/or its mathematical elaboration
This process should be understood to be iterative.
It means that we must repeat some steps with modifications to reach satisfactory results.
215
Inclusion of fuzzy sets is defined by the inequality
of their membership functions:
VxES.
Union and intersection of two fuzzy sets are defined by the maximum and minimum, respectively.
Union:
AUB: Il-A U B(X)
= max[Il-A(X), Il-B(X)].
Intersection: A n B: Il-A n B(X)
= min[Il-A(X), Il-B(X)].
The complement of a fuzzy set A, which is denoted
by AC, is defined as follows:
Complement: AC: Il-AC(X) = 1 - Il-A(X).
For example, let S = {I, 2, 3,4, 5} again and assume that fuzzy sets A and B are given by
A = 0/1 + 0.5/2 + 0.8/3 + 114 + 0.2/5,
B = 0.9/1 + 0.4/2 + 0.3/3 + 0.1/4 + 015.
Then
A U B = 0.9/1 + 0.5/2 + 0.8/3 + 114 + 0.2/5,
A n B = 0/1 + 0.412 + 0.3/3 + 0.1/4 + 015,
AC = 111 + 0.5/2 + 0.2/3 + 0/4 + 0.8/5.
Many extensions of the above min-max definition to the fuzzy set operations can be found in
Zimmermann (1991) and many other texts. These
fuzzy set operations can be used to aggregate different ecological indicators during ecological risk
analysis. Applications of fuzzy mathematical
methods in ecology have been published by many
authors (e.g., Li, 1986, 1996; Li and Zhou, 1987;
Banyikwa et al., 1990; Bardossy et al., 1990a;
Keesman and van Straten, 1990; Lindsey et aI.,
1992; McBratney et al., 1992; Bardossy, 1994; Feoli and Zuccarello, 1994; Salski et al., 1996).
15.4 Applications of the Fuzzy
Modeling Approach to
Ecological Assessment
Because fuzzy sets are a generalization of a classical set theory, the embedding of conventional models into a larger setting endows fuzzy models with
greater flexibility to capture various aspects of incompleteness or imperfection in whatever information and data are available about a real process.
For example, we developed a semiarid grazing
ecosystem simulation model combining fuzzy imprecision with probabilistic uncertainty to characterize climate-plant-herbivore interactions (Wu et
al., 1996); Cheng et al. (1996) used fuzzy systems
analysis and conventional dynamic programming
to optimize biological control of a greenhouse spider mite--cucumber system; Singer and Singer
(1993) presented a fuzzy set formulation of the phenomenological equations of nonequilibrium chemical kinetics, which could be easily adopted to deal
with nonequilibrium ecological "reactions" of multiple species; and Levary (1990) established fuzzylogic-based system dynamics methodology for
modeling, simulating, and analyzing real-life systems with imprecise and vague variables or events.
In general, we can use fuzzy models in any of
the following circumstances: (1) the modeled
process is too complicated and its exact description
is not known; (2) many factors are not quantified,
or are impossible to quantify, and they are characterized only verbally; (3) more exact specifications
cannot be employed and, hence, our research would
be gratuitous; (4) the exact description is too time
consuming (the use of natural language is advantageous because it follows human thinking in the
best way, and the necessary time for fuzzy model
construction is very small); (5) the exact model
construction might need some experiments (this
might not, however, be possible either due to economic reasons or due to the nature of the process
itself); and (6) the fuzzy model is more objective
than verbal description based on a purely intuitive
approach.
The general methodology for the fuzzy model
construction can be summarized as follows:
1. Description of the problem
2. Factor identification, definition, and classification as dependent or independent
3. Analysis of all factors from the viewpoint of
their dependence
4. Determination of the factors that will be
processed as linguistic variables and determination of the scales of linguistic values
5. Determination of the universes, that is, determination of the kind of elements (numbers, intervals) and their extent (see the above example
on membership functions of vegetation cover)
6. Transformation of linguistic values into fuzzy
sets, that is, the estimation of membership functions
7. Formulation of the fuzzy model using conditional statements and/or its mathematical elaboration
This process should be understood to be iterative.
It means that we must repeat some steps with modifications to reach satisfactory results.