212
Fuzzy Statistical and Modeling Approach to Ecological Assessments
white of true and false. Zadeh referred to this gray
area as "fuzzy", which was an inauspicious choice
of terms for Western science and engineering. But,
once we realize that fuzzy logic and set theory (and
even fuzzy thinking!) are quantitative ways of characterizing intrinsic ambiguity, rather than of substituting imprecision for precision, the shorthand
term fuzzy becomes acceptable in science and engineering. Despite much thought, no better term has
yet emerged, even though fuzzy thinking does not
sound like something intelligent people might do.
Already the application of this theory to the development of fuzzy systems for commercial electronic
products and system controllers in Japan has been
remarkable (e.g., Williams, 1992). Fuzzy mathematics itself is not fuzzy. It is a mathematical theory and method for dealing with fuzzy phenomena
in nature.
Allen and Hoekstra (1992) have described the
conceptual approaches and potential of fuzzy systems analysis for applied ecology. Salski et al.
(1996) and this author (Li, 1996) have also edited
two special issues on fuzzy modeling in ecology
for the international journal Ecological Modelling.
(For people who would like to know more about
fuzzy theory, the easier readable book without too
much mathematical treatment by Kosko, 1993, is a
good starting point.) It would be most valuable for
us to begin to explore the horizons of this new approach for ecological assessment applications.
15.2 Uncertainty and Paradigm
Shift Regarding Uncertainty
When we deal with real-world problems (e.g., ecological risk assessment), we can rarely avoid uncertainty. At the experimental level, uncertainty is
an inseparable companion of any measurement and
observation, resulting from a combination of resolution limits of measuring instruments and inevitable measurement errors. At the cognitive level,
it emerges from the vagueness and ambiguity inherent in natural languages. At the social level, uncertainty results not only from the inevitable incompleteness of shared meanings obtained by
people through social interaction, but it is also created and maintained by people for different purposes. For example, people may have different views
in sustainable development indices on how to incorporate social, societal, resource management, and
scientific perspectives into the assessment process in
order to develop criteria that distinguish good (nominal), marginal, or poor (subnominal) ecological condition for various resources.
Uncertainty is thus fundamental to human beings
at all levels of their interaction with the real world.
The treatment of uncertainty can be time consuming and difficult. Policymakers often tend to ignore
it. Instead, they may consider decision parameters
to be single valued and deterministic or solicit expert opinion and trust it implicitly as the best estimate. It is increasingly recognized, however, that
attitudes toward uncertainty have been undergoing
a significant change in this century. In science, this
change has been manifested by a transition from
the traditional attitude, according to which uncertainty is a plague that should be avoided by all
means, to an alternative attitude, according to
which uncertainty is fundamental to science and its
avoidance is often counterproductive. Two phases
of this transition can be clearly recognized, each
having the characteristics of a paradigm shift in the
sense introduced by Kuhn (1996). When statistical
mechanics was accepted, by and large, by the scientific community as a legitimate area of science
early in this century (Prigogine, 1997), the traditional attitude toward uncertainty was for the first
time revised. Uncertainty became recognized as
useful, or even essential, in certain scientific inquiries. However, this recognition was strongly
qualified: uncertainty was conceived solely in
terms of probability theory.
A mathematical method to quantify uncertainty
was first proposed by Hartley in 1928. This method
was based on classical set theory. Later, Shannon
(1948) proposed a measure for uncertainty based
on probability theory. This measure is called the
entropy measure. Until recently, entropy measure
was the only measure for quantifying uncertainty.
It is generally agreed that an important turning
point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by
Zadeh (1965), even though some ideas presented
in the paper were envisioned by the philosopher
Max Black in 1937. Zadeh (1965) introduced a theory whose objects, fuzzy sets, are sets with imprecise boundaries. Membership in a fuzzy set is not a
matter of affirmation or denial, but rather a matter
of degree. The importance of Zadeh's paper was
that it challenged the adequacy of classical set theory and probability theory as frameworks for expressing uncertainty. By allowing imprecise boundaries, fuzzy sets acquire the capability to express
concepts of natural language that are inherently
vague. They also acquire the capability to bridge,
in whatever crude way, mathematics and empirical
reality (this is really needed for our ecological assessment!).
The concept of a fuzzy set represents a basic
mathematical framework for dealing with vague-
Fuzzy Statistical and Modeling Approach to Ecological Assessments
white of true and false. Zadeh referred to this gray
area as "fuzzy", which was an inauspicious choice
of terms for Western science and engineering. But,
once we realize that fuzzy logic and set theory (and
even fuzzy thinking!) are quantitative ways of characterizing intrinsic ambiguity, rather than of substituting imprecision for precision, the shorthand
term fuzzy becomes acceptable in science and engineering. Despite much thought, no better term has
yet emerged, even though fuzzy thinking does not
sound like something intelligent people might do.
Already the application of this theory to the development of fuzzy systems for commercial electronic
products and system controllers in Japan has been
remarkable (e.g., Williams, 1992). Fuzzy mathematics itself is not fuzzy. It is a mathematical theory and method for dealing with fuzzy phenomena
in nature.
Allen and Hoekstra (1992) have described the
conceptual approaches and potential of fuzzy systems analysis for applied ecology. Salski et al.
(1996) and this author (Li, 1996) have also edited
two special issues on fuzzy modeling in ecology
for the international journal Ecological Modelling.
(For people who would like to know more about
fuzzy theory, the easier readable book without too
much mathematical treatment by Kosko, 1993, is a
good starting point.) It would be most valuable for
us to begin to explore the horizons of this new approach for ecological assessment applications.
15.2 Uncertainty and Paradigm
Shift Regarding Uncertainty
When we deal with real-world problems (e.g., ecological risk assessment), we can rarely avoid uncertainty. At the experimental level, uncertainty is
an inseparable companion of any measurement and
observation, resulting from a combination of resolution limits of measuring instruments and inevitable measurement errors. At the cognitive level,
it emerges from the vagueness and ambiguity inherent in natural languages. At the social level, uncertainty results not only from the inevitable incompleteness of shared meanings obtained by
people through social interaction, but it is also created and maintained by people for different purposes. For example, people may have different views
in sustainable development indices on how to incorporate social, societal, resource management, and
scientific perspectives into the assessment process in
order to develop criteria that distinguish good (nominal), marginal, or poor (subnominal) ecological condition for various resources.
Uncertainty is thus fundamental to human beings
at all levels of their interaction with the real world.
The treatment of uncertainty can be time consuming and difficult. Policymakers often tend to ignore
it. Instead, they may consider decision parameters
to be single valued and deterministic or solicit expert opinion and trust it implicitly as the best estimate. It is increasingly recognized, however, that
attitudes toward uncertainty have been undergoing
a significant change in this century. In science, this
change has been manifested by a transition from
the traditional attitude, according to which uncertainty is a plague that should be avoided by all
means, to an alternative attitude, according to
which uncertainty is fundamental to science and its
avoidance is often counterproductive. Two phases
of this transition can be clearly recognized, each
having the characteristics of a paradigm shift in the
sense introduced by Kuhn (1996). When statistical
mechanics was accepted, by and large, by the scientific community as a legitimate area of science
early in this century (Prigogine, 1997), the traditional attitude toward uncertainty was for the first
time revised. Uncertainty became recognized as
useful, or even essential, in certain scientific inquiries. However, this recognition was strongly
qualified: uncertainty was conceived solely in
terms of probability theory.
A mathematical method to quantify uncertainty
was first proposed by Hartley in 1928. This method
was based on classical set theory. Later, Shannon
(1948) proposed a measure for uncertainty based
on probability theory. This measure is called the
entropy measure. Until recently, entropy measure
was the only measure for quantifying uncertainty.
It is generally agreed that an important turning
point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by
Zadeh (1965), even though some ideas presented
in the paper were envisioned by the philosopher
Max Black in 1937. Zadeh (1965) introduced a theory whose objects, fuzzy sets, are sets with imprecise boundaries. Membership in a fuzzy set is not a
matter of affirmation or denial, but rather a matter
of degree. The importance of Zadeh's paper was
that it challenged the adequacy of classical set theory and probability theory as frameworks for expressing uncertainty. By allowing imprecise boundaries, fuzzy sets acquire the capability to express
concepts of natural language that are inherently
vague. They also acquire the capability to bridge,
in whatever crude way, mathematics and empirical
reality (this is really needed for our ecological assessment!).
The concept of a fuzzy set represents a basic
mathematical framework for dealing with vague-