15.6 References
f Then, by the extension principle in fuzzy mathematics,
/LB(Y) =
max
XiEX;. WiEZ
i= 1,2, ... , n
y = !(x1, ... ,xn)
{min[/LA1 (XI), ... ,
/LAn (Xn), /Lw1 (WI),
... , /Lw n (Wn)]}
where /LAi and /LWi are the membership functions
of fuzzy number Ai and Wi, respectively, i ;= 1,2,
... , n. Dong and Wong (1987) provided a way to
calculate them.
Many other fuzzy modeling techniques can be
used for ecological assessment, for example, fuzzy
approach to expert data analysis (Khurgin and
Polyakov, 1986; Bardossy et aI., 1993), fuzzy spatial analysis (Leung, 1985), fuzzy urban design assessment model (Kumar and Goel, 1994), fuzzy
kriging (Diamond, 1989; Bardossy et aI., 1990b),
fuzzy rule-based GIS technology (Saint-Joan and
Desachy, 1995), and fuzzy cellular automata (Baldwin et aI., 1993).
15.5 Conclusion
Integrated ecological assessment at multiple scales
is complex and multidimensional. Uncertainty is a
major issue in all applications of risk assessment,
but it presents a particular problem for ecological
risk assessment due to the inherent variability of
biological and ecological systems. Errors in the information used for assessments can arise from errors in measurements (e.g., toxicology index, environmental concentrations); extrapolation from
species to species or from one set of conditions to
another and in other ways in which expert judgment is used; and the assumptions and arbitrary
thresholds that are made at some stages. There is
nothing to be gained from ignoring the potential inaccuracies that these errors may cause or the substantial risks of misjudging the scale of problems if
they are not examined, even imperfectly. Increasing awareness of ecological issues has emphasized
the need for improved ecological risk assessment
methodology. This chapter briefly introduced a
fuzzy statistical and modeling approach as a potential tool for integrated ecological risk assessment. The significance of this approach is that it
can generalize conventional assessment models
and, more importantly, it provides new perspectives, a higher degree of flexibility, and more appropriate approximations to reality. Although the
phenomena analyzed are vague, the analysis,
though not necessarily quantitative, is precise.
While the present scope and depth of the fuzzy eco219
logical assessment modeling is limited, with more
researchers taking an active role in its development
and application, its increased impact on complex
ecological assessment and constructing ecological
indicators is a strong possibility.
15.6 References
Allen, T. F. H.; Hoekstra, T. W. 1992. Toward a unified
ecology. New York: Columbia University Press.
Ayoubi, B. A; Beninel, F.; Le Calve, G. 1991. Comparison of some techniques of multidimensional scaling in soil fauna ecology. Appl. Stochastic Models
Data Analysis 7:263-271.
Baldwin, J. F.; Martin, T. P.; Zhou, Y. 1993. Fuzzy cellular automata. In: Bouchon-Meunier, B., ed. Uncertainty in intelligent systems. Amsterdam: Elsevier Science: 235-245.
Bandemer, H.; Nather, W. 1992. Fuzzy data analysis.
Dordrecht: Kluwer.
Banyikwa, F. F.; Feoli, E.; Zuccarello, V. 1990. Fuzzy
set ordination and classification of Serengeti short
grasslands, Tanzania. J. Vegetation Sci. 1 :97-104.
Bardossy, A. 1994. Downscaling from GCMs to local
climate through stochastic linkages. In: Paoli, G., ed.
Climate change, uncertainty, and decision-making,
lnst. for Risk Res.: 33-46.
Bardossy, A; Bogardi, I.; Duckstein, L. 1990a. Fuzzy
regression in hydrology. Water Resources Res. 26:
1497-1508.
Bardossy, A.; Bogardi, I.; Kelly, W. E. 1990b. Kriging
with imprecise (fuzzy) variograms. Math. Geol. 22:
63-79.
Bardossy, A.; Duckstein, L.; Bogardi, I. 1993. Combination of fuzzy numbers representing expert opinions.
Fuzzy Sets Systems 57:173-181.
Black, M. 1937. Vagueness. Phil. Sci. 4:427-455.
Chen, S.; Jiang, A; Domroes, M. 1992. Studies on the
impact of winter climate on rubber and wheat cultivation in the mountains of southern China, applying a
fuzzy cluster analysis. Int. J. Biometeorology 36:
159-164.
Cheng, Z.; Hom, D. J.; Lindquist, R. K.; Taylor, R. A
J. 1996. Fuzzy analysis for a greenhouse spider mite
management system. Ecol. Modelling 90: 111-121.
Diamond, P. 1989. Fuzzy kriging. Fuzzy Sets Systems
33:315-332.
Dong, W. M.; Wong, F. S. 1987. Fuzzy weighted averages and implementation of the extension principle.
Fuzzy Sets Systems 21:183-199.
Dubois, D.; Prade, H. 1988. Possibility theory. New
York: Plenum Press.
EI-Shishiny, H.; Ghabbour, S. I. 1991. A fuzzy interpretation for an ecological site classification case. Appl.
Stochastic Models Data Analysis 7:257-261.
Feoli, E.; Zuccarello, V. 1994. Naivete of fuzzy system
spaces in vegetation dynamics? Coenoses 9:25-32.
f Then, by the extension principle in fuzzy mathematics,
/LB(Y) =
max
XiEX;. WiEZ
i= 1,2, ... , n
y = !(x1, ... ,xn)
{min[/LA1 (XI), ... ,
/LAn (Xn), /Lw1 (WI),
... , /Lw n (Wn)]}
where /LAi and /LWi are the membership functions
of fuzzy number Ai and Wi, respectively, i ;= 1,2,
... , n. Dong and Wong (1987) provided a way to
calculate them.
Many other fuzzy modeling techniques can be
used for ecological assessment, for example, fuzzy
approach to expert data analysis (Khurgin and
Polyakov, 1986; Bardossy et aI., 1993), fuzzy spatial analysis (Leung, 1985), fuzzy urban design assessment model (Kumar and Goel, 1994), fuzzy
kriging (Diamond, 1989; Bardossy et aI., 1990b),
fuzzy rule-based GIS technology (Saint-Joan and
Desachy, 1995), and fuzzy cellular automata (Baldwin et aI., 1993).
15.5 Conclusion
Integrated ecological assessment at multiple scales
is complex and multidimensional. Uncertainty is a
major issue in all applications of risk assessment,
but it presents a particular problem for ecological
risk assessment due to the inherent variability of
biological and ecological systems. Errors in the information used for assessments can arise from errors in measurements (e.g., toxicology index, environmental concentrations); extrapolation from
species to species or from one set of conditions to
another and in other ways in which expert judgment is used; and the assumptions and arbitrary
thresholds that are made at some stages. There is
nothing to be gained from ignoring the potential inaccuracies that these errors may cause or the substantial risks of misjudging the scale of problems if
they are not examined, even imperfectly. Increasing awareness of ecological issues has emphasized
the need for improved ecological risk assessment
methodology. This chapter briefly introduced a
fuzzy statistical and modeling approach as a potential tool for integrated ecological risk assessment. The significance of this approach is that it
can generalize conventional assessment models
and, more importantly, it provides new perspectives, a higher degree of flexibility, and more appropriate approximations to reality. Although the
phenomena analyzed are vague, the analysis,
though not necessarily quantitative, is precise.
While the present scope and depth of the fuzzy eco219
logical assessment modeling is limited, with more
researchers taking an active role in its development
and application, its increased impact on complex
ecological assessment and constructing ecological
indicators is a strong possibility.
15.6 References
Allen, T. F. H.; Hoekstra, T. W. 1992. Toward a unified
ecology. New York: Columbia University Press.
Ayoubi, B. A; Beninel, F.; Le Calve, G. 1991. Comparison of some techniques of multidimensional scaling in soil fauna ecology. Appl. Stochastic Models
Data Analysis 7:263-271.
Baldwin, J. F.; Martin, T. P.; Zhou, Y. 1993. Fuzzy cellular automata. In: Bouchon-Meunier, B., ed. Uncertainty in intelligent systems. Amsterdam: Elsevier Science: 235-245.
Bandemer, H.; Nather, W. 1992. Fuzzy data analysis.
Dordrecht: Kluwer.
Banyikwa, F. F.; Feoli, E.; Zuccarello, V. 1990. Fuzzy
set ordination and classification of Serengeti short
grasslands, Tanzania. J. Vegetation Sci. 1 :97-104.
Bardossy, A. 1994. Downscaling from GCMs to local
climate through stochastic linkages. In: Paoli, G., ed.
Climate change, uncertainty, and decision-making,
lnst. for Risk Res.: 33-46.
Bardossy, A; Bogardi, I.; Duckstein, L. 1990a. Fuzzy
regression in hydrology. Water Resources Res. 26:
1497-1508.
Bardossy, A.; Bogardi, I.; Kelly, W. E. 1990b. Kriging
with imprecise (fuzzy) variograms. Math. Geol. 22:
63-79.
Bardossy, A.; Duckstein, L.; Bogardi, I. 1993. Combination of fuzzy numbers representing expert opinions.
Fuzzy Sets Systems 57:173-181.
Black, M. 1937. Vagueness. Phil. Sci. 4:427-455.
Chen, S.; Jiang, A; Domroes, M. 1992. Studies on the
impact of winter climate on rubber and wheat cultivation in the mountains of southern China, applying a
fuzzy cluster analysis. Int. J. Biometeorology 36:
159-164.
Cheng, Z.; Hom, D. J.; Lindquist, R. K.; Taylor, R. A
J. 1996. Fuzzy analysis for a greenhouse spider mite
management system. Ecol. Modelling 90: 111-121.
Diamond, P. 1989. Fuzzy kriging. Fuzzy Sets Systems
33:315-332.
Dong, W. M.; Wong, F. S. 1987. Fuzzy weighted averages and implementation of the extension principle.
Fuzzy Sets Systems 21:183-199.
Dubois, D.; Prade, H. 1988. Possibility theory. New
York: Plenum Press.
EI-Shishiny, H.; Ghabbour, S. I. 1991. A fuzzy interpretation for an ecological site classification case. Appl.
Stochastic Models Data Analysis 7:257-261.
Feoli, E.; Zuccarello, V. 1994. Naivete of fuzzy system
spaces in vegetation dynamics? Coenoses 9:25-32.