linear model.
Given input values XI,, xz,,
...,
x,,, the prediction of output y, is calculated bywhere p denotes the number of rules, A;(xj.) the membership grade of xj, to the fuzzy set A;, and yf the prediction by the rule
R'.
The most important feature of a fuzzy model is that it behaves as a nonlinear model though it consists of a set of linear equations. The main tasks in the fuzzy modeling
are: (i) division of the d a t a space into fuzzy subspaces, (ii) identification of membership functions, and (iii) statistical modeling with selection of explanatory variables. Here, the data space can be identified with the (q
+
1)-dimensional Euclidean space Rq+' in the present case. We use the subset of input-output measurement data corresponding to the subspace t o build the linear model in a rule.A fuzzy subspace corresponds t o the product set defined by, for example, Af
x
A; xx
A:x
A t , where At is the fuzzy set related to the output y. Hence, the number of fuzzy subspaces is the same as that of rules. Note that every fuzzy set A: should not be empty, but can be identical to its support set; which means that some variables can disappear from premises of some rules. Needless to say, some coefficients in the linear model can be zero. We omit t o explain these points in this paper, (see Nakamori and Sawaragi, 1989).The above mentioned tasks are mutually dependent, and very difficult if we would follow the traditional analytical approach. One can introduce some criteria in carrying out those tasks. But, the final result heavily depends on the capability and experience of the individual modeler. This is the reason why we have developed an interactive and intelligent environment in model building.
The fuzzy modeling is a nice idea to obtain nonlinear relationships by acquiring knowl- edge and judgment of domain experts. However, we sometimes find fuzzy subspaces in which we can hardly obtain linear models because of the nature of data. To cope with such a case, we propose a hybrid fuzzy modeling which develops a pattern model or linear model for each explained variable in each subspace. Moreover, we build pattern models in the subspaces with no data, which may occur in the future.
7 Conclusion
We proposed the Shinayakana Systems Approach in systems analysis. In designing and implementing the systems for modeling and decision support, it requires three I's, that is, the support system should be designed in an interactive, intelligent and interdisciplinary fashion. As a realization of the Shinayakana Systems Approach, we introduced a big experiment of developing an integrated support system for environmental planning. The developed system has user-friendly human-computer interfaces t o elicit intuitive or inner, personal knowledge or idears about the problem under study.
Acknowledgments
We wish t o thank Dr. Morita and Dr. Kainuma, the National Institute for Environ- mental Studies, the Environment Agency of Japan, for their cooperation in developing the system for environmental planning.
References
Beer, S. (1985). Diagnosing the Systems for Organizations. John Wiley.
Checkland, P.B. (1981). Systems Thinking, Systems Practice. John Wiley.
Checkland, P.B. (1983). OR and the Systems Movement: Mappings and Conflicts. J.
Opl. Res. Soc., Vo1.34, No.8, pp.661-675.
Fedorowicz, J., and G.B. Williams (1 986). Representing Modeling Knowledge in an Intelligent Decision Support System, Decision Support Systems, Vo1.2, pp.3-14.
Gordon, T. J., and 0 . Helmer (1964). Report on a Long-Range Forecasting Study. RAND Corp.
Gruver, W.A., A. Ford, and P.C. Gardiner (1984). Public Policy Analysis Using Three Science Techniques. IEEE Trans. on Syst. Man Cybern., Vol.SMC-14, No.2, pp.355-361.
Jackson, M.C., and P. Key (1984). Towards a System of Systems Methodologies. J. Opl.
Res. Soc., Vo1.35, No.6, pp.473-486. Support, Part I: Theoretical and Methodological Backgrounds. WP-88-03, Interna- tional Institute for Applied Systems Analysis, Laxenburg, Austria.
Nakamori, Y., (1989). Development and Application of an Interactive Modeling Support System. Automatica, Vo1.25, No.2, pp.185-206.
Nakamori, Y., and Y. Sawaragi (1989). Fuzzy Modeling and Simulation for an Environ- mental System. Proc. IFAC/IFORS/IMACS Symp. on Large Scale Systems 89, Theory and Application, pp.538-543, Berlin, GDR.
Nishioka, S., and M. Naito (1984). Information System for Environmental Quality As- sessment. Int. Symp. on Regional Information Systems (IRIS), Japan Association for Planning Administration, Amagi, Japan.
Norberg, A.L., and G.P. Johnson (1979). Structural and Underst anding: Some Obser- vations on Current Activities in the Field of Structural Modeling. Technological Forecasting and Social Change, Vo1.14, pp.277-289.
Sage, A.P. (1 981). Behavioral and Organizational Considerations in the Design of In- formation Systems and Processes for Planning and Decision Support. IEEE Trans.
on Syst. Man Cybern., Vol.SMC-11, No.9, pp.640-678.
Sage A.P., and C.C. White (1984). 111, ARIADNE: A Knowledge-Based Interactive System for Planning and Decision Support. IEEE Trans. on Syst. Man Cybern., Vol.SMC-14, No.1, pp.35-47.
Stanciulescu, F. (1986). Principles of Modeling and Simulation of Large-Scale and Com- plex Systems, Applications in Ecology. Syst. Anal. Model. Simul., Vo1.3, No.5, pp.409-423.
Sugeno, M., and G.T. Kang (1988). Structure Identification of Fuzzy Model. Fuzzy Sets and Systems, Vo1.28, pp.15-33.
Sugiyama, K., S. Tagawa, and M. Toda (1981). Methods for Visual Understanding of Hierarchical System Structure. IEEE Trans. on Syst. Man Cybern., Vol.SMC-11, No.2, pp.109-125.
Sutherland, J.W. (1986). Assessing the Artificial Intelligence Contribution to Decision Technology. IEEE Trans. on Syst. Man Cybern., Vol.SMC-16, No.1, pp.3-20.
Ulrich, W. (1981). A Critique of Pure Cybernetic Reason: the Chilean Experience with Cybernetics. J. Appl. Sys. Anal., No.8, pp.33-59.
U'lrich, W. (1983). Critical Heuristics of Social Planning. Haupt, Bern.
Wang, M.S., and J.F. Courtney, Jr. (1984). A Conceptural Architecture for Generalized Decision Support System Software. IEEE Trans. on Syst. Man Cybern., Vol.SMC- 14, No.5, pp.701-711.
Warfield, J.N. (1974). Toward Interpretation of Complex Structural Modeling. IEEE Trans. on Syst. Man Cybern., Vol.SMC-4, No.5, pp.405-407.
Warfield, J.N. (1 976). Societal Systems: Planning, Policy and Complexity. John Wiley, New York.
Warfield, J.N. (1977). Crossing Theory and Hierarchy Mapping. IEEE Trans. on Syst.
Man Cybern., Vol.SMC-7, No.7, pp.505-523.
Zadeh, L.A. (1973). Outline of a New Approach to the Analysis of Complex Systems and Decision Processes. IEEE Trans. Syst. Man Cybern., Vol.SMC-3, No. 1, pp.28-44.
Zadeh, L.A. (1975). The Concept of a Linguistic Variable and Its Application to Ap- proximate Reasoning 1-111. Inform. Sci., Vols.8-9, pp.199-249, pp.301-357, pp.43-80.