NONLINEAR AND DYNAMIC EXTENSIONS FOR FUZZY COGNITIVE MAPS (FCM) TOOLS

Transcription

1 OLIEAR AD DYAMIC EXTESIOS FOR FUZZY COGITIVE MAPS (FCM) TOOLS Raimundas Jasinevicius, Vytautas Petrauskas Kaunas University of Technology, Centre of Computer Literacy at the Department of Informatics, Studentu A, Kaunas, Lithuania, Abstract. Widely spread fuzzy cognitive maps (FCM) approach to the imprecise, ambiguous and uncertainly described situations analysis has delivered a lot of new tools serving as a support for decision makers. This paper is devoted to summarize some well-known theoretical FCMs extensions used in the certain tools including those which were proposed and implemented by the authors of this presentation. So, the descriptions and examples of nonlinear and dynamic FCM-based modelling tools are presented in which nonlinear behaviour as well as a dynamic delay on FCM edges and a phenomenon of nodes inertia can be demonstrated. Keywords: Fuzzy cognitive map, ordinary, nonlinear, dynamics, tools. Introduction The cognitive maps approach to decision processes analysis was started by R. Axelrod at Princeton University ([2]). But it is widely recognized that only after famous L.A. Zadeh s papers ([29]-[3]) the contemporary avalanche of fuzzy sets applications has burst. Fuzzy control systems (FCS) and fuzzy cognitive maps (FCM) are the best confirmation of this tendency. The background for fuzzy thinking and for fuzzy cognitive maps applications to decision making processes was preliminary mainly developed at the USC (University of Southern California) by B. Kosko ([8]-[20]) and later extended by J.P. Carvalho, J.A. Tome, M.S. Kahn, G. Xirogiannis ([6], [7], [6], [27], [28]) and many, many others. Following the world wide experience in soft computing (for example, [7], [22], [25]) as well as requirements, emphasized by different decision makers, looking for efficient computerized advisers in various cases of very sophisticated and sensitive situations like financial risk management, medical diagnostics, politics and international relations, environmental protection, terrorism and security, pattern recognition and so on ([], [3]-[5], [8], [2], [23]), we have summarized main theoretical features, properties and limitations, used, implemented and/or inherent for FCM-principles-based decision making support tools. The main ideas, captured from the references mentioned above and some theoretical ones developed by the authors of this presentation, were implemented in different decision makers support tools at the Centre of Computer Literacy (CCL) of the Department of Informatics at the Kaunas University of Technology ([9]-[5], [24]). This paper aims to deliver the most popular theoretical FCMs extensions used in the certain tools and to demonstrate some examples of their implementation. 2 Description of an Ordinary Fuzzy Cognitive Map (OFCM) The most thorough description of the structure, mathematics and phenomenon such as an ordinary fuzzy cognitive map (OFCM) is presented in [20]. There the structure is called a simple FCM. It has bivalent nodes (i =, 2,, j, k,,) representing fuzzy events, which can be evaluated by concept values in {0, }. The nodes are connected by causal edges. The strength of any particular edge, connecting Ci and Ck is represented by wik which takes trivalent value in {-, 0, }. The positive value corresponds to the phenomenon when causal event, actor, goal, trend or concept Ci stimulates the causal event C k, the negative one corresponds to the effect of suppression and zero demonstrates total indifference. So the connection matrix E lists the causal links between nodes as follows [20] (Figure ): Figure. The connection matrix E - -

2 In the i-th row the edges w ik (k =,, ) are collected from causal concept to causal concept C k as well as in the k-th column the edges wik (i =,, ) from the concept C k to the concept are listed. A causal dynamism and conceptual interaction is expressed by the FCM when the concept value Ci on the certain step n of behaviour is subjected to the nonlinear transformation. Such a system can show all possible aspects of interaction between causal concepts (i =,, ) including hidden fixed point, limit cycle or chaotic attractors, corresponding to the real life to be modelled by a simple FCM. Unfortunately the real life is much more complicated. Usually neither concepts are bivalent nor edges are trivalent. In general, are real numbers from the interval [0, ] and values of edges e ik as well are from interval [-, +]. So, an Ordinary Fuzzy Cognitive Map (OFCM) is described by the following formula: = w ij C j [] n [ n ] () C [] = Ψ ( [] n ) i n i which corresponds to the structure shown in Figure 2. Here Ψ i (*) are different nonlinearities for all i =,,, corresponding to the physical meaning of the i-th entity. C [n-] w i C j [n-] Ψ j (*) Ψ i (*) w ij [n], 2,, i,, C [n-] w i Figure 2. Structure of the OFCM A simplified fragment of a tool s screen in case of a OFCM-based modelling is presented in Figure 3. Here one can see an example of the ATO enlargement simulation, performed concerning the security of Baltic States (all comments are written in Lithuanian) [4]. Figure 3. A simplified fragment of a screen of the OFCM-based modelling tool 3 Description of a onlinear Fuzzy Cognitive Map (FCM) Much more sophisticated relations and interactions between entities can be represented and modelled taking into account the fact that sometimes edges are not linear. An entity of the nonlinear FCM (FCM) is described according to the following formulae: - 2 -

3 α i[] n = w ij ϕij ( C j [ n ] ) [] n = Ψ ( [] n ) i (2) The structure of an entity i is shown in the Figure 4. C [n-] φ i (*) w i Ф(*) j C j [n-] φ ij (*) w ij Ф(*) i [n], 2,, i,, C [n-] φ i (*) w i Figure 4. Structure of the FCM Here Ф(*) is a linear function with saturation and φ ij (*) are different for all i =,, j,,. A simplified fragment of a screen of the FCM-based modelling tool is presented in Figure 5. Here are seen some samples of nonlinearities to be selected for an edge with the weight 0.7 between two nodes (Mazgas 0 and Mazgas ). All comments here are made also in the one of the oldest European Union languages Lithuanian. Figure 5. A simplified fragment of a screen in case of the FCM-based modelling tool 4 Description of a Dynamics, Delay and Inertia Phenomenon in FCM-based Modelling Tools Dynamics of process performance in the OFCM is involved: a) by using so-called delay operation (DOP) on each edge of the FCM, and b) by changing (slowing down and modelling the phenomenon of inertia) the frequency of calculation steps (steps operation SOP) in each node of the FCM under consideration. So, the involvement of DOP (case a)) for an OFCM can be achieved by introducing the number of delay steps d ij in (): - 3 -

4 [] n [ n ] = w ij C j d ij [] n = Ψ ( [] n ) i (3) The DOP in a case of FCM is shown in the (4), which is derived from the (2): [] n ϕ ( [ n ] α i = w ij ij C j d ij ) [] n = Ψ ( [] n ) i (4) An involvement of SOP (case b)) for the OFCM as well as for the FCM can be achieved from the (3) and (4) according to the (5) and (6), where k i means slowing down ratio and m i the new step number for the i- th node: m i k i = w ij C j k i m i j = [ ] [ ] ( ) d ij [ m ] = Ψ ( [ m ]) i k i i i k i α i [ m i k i ] = w ij ϕij ( C j [ k i ( m i ) d ij ]) j = [ m ] = Φ( [ m ]) i k i i k i (5) (6) In both cases m i = ] n/k i [ =, 2, 3, - are strictly integers.] 5 Acknowledgements Authors are eager to express their gratitude to young scientists who took part in tools development processes (A. Liutkevicius, V. Bivainis, V. Ratkelis,) and to acknowledge the synergetic influence on theses considerations made by the COST program Action IC0702 Combining Soft Computing Techniques and Statistical Methods to Improve Data Analysis Solutions under coordination of prof. Christian Borgelt from European Centre for Soft Computing (Mieres, Spain). 6 Conclusions. This paper has represented a systematic approach to the FCM extensions towards expert fuzzy knowledge management. The descriptions of four possible cases of FCM-based tools are delivered. 2. Such a background is laid for industrial software tools implementation for decision making and risk management processes as well as for analysis of international relations. 3. Further extension of a practical approach lays in a broader time dimension inclusion, while theoretical one in causal dynamics and stability research. References [] Aguilar J. A Survey about Fuzzy Cognitive Maps Papers (Invited Paper), International Journal of Computational Cognition, v. 3, r. 2, June 2005, p.p [2] Axelrod R. Structure of decision: the cognitive maps of political elites. Princeton,.J.: Princeton University Press, 976. [3] Beaton S. Maritime Security & Maritime Domain Awareness. InfraGard 2005 ational Conference, Hosted by the InfraGard ational Members Alliance and the FBI, August 9, ,C [4] Berner, E. S. (ed.). Clinical Decision Support Systems: Theory and Practice. Springer-Verlag, ew York. 999 [5] Brabazon A., O eil. Biologically Inspired Algorithms for Financial Modelling, Springer [6] Carvalho J. P., Tome J. A. Fuzzy mechanisms for causal reasoning. Proc. Eighth Internat. Fuzzy Systems Association World Congress, IFSA 99 Taiwan, 999, p.p

5 [7] Carvalho J. P., Tome J. A. Interpolated linguistic terms. Proc. 23-rd Internat. Conf. of the orth American Fuzzy Information Processing Society, AFIPS2004 Banff, Canada, 2004,.p.p [8] Goward D. A. Maritime Domain Awareness the Key to Maritime Security. IAC Luncheon, US Coast Guard Maritime Domain Awareness, 23 May 2006, ( [9] Jasinevicius R., Petrauskas V. The new tools for systems analysis // Informacinės technologijos ir valdymas = Information technology and control / Kauno technologijos universitetas. ISS X. 2003, o 2(27). p [0] Jasinevicius R., Petrauskas V. Dynamic SWOT Analysis as a Tool for System Experts. Engineering Economics/ Kaunas university of technology. Kaunas: Technologija, ISS , 2006, o 5(50), p.p [] Jasinevicius R., Petrauskas V. Fuzzy expert maps: the new approach // WCCI 2008 Proceedings: 2008 IEEE World Congress on Computational Intelligence, June -6, 2008, Hong Kong: 2008 IEEE International Conference on Fuzzy Systems IEEE International Joint Conference on eural etworks IEEE Congress on Evolutionary Computation. Piscataway: IEEE, ISB p [2] Jasinevicius R., Petrauskas V. Dynamic SWOT analysis as a tool for environmentalists // Environmental research, engineering and management. ISS , o (43). [3] Jasinevicius R., Petrauskas V. Fuzzy expert maps for risk management systems // US/EU-Baltic 2008 International Symposium: Ocean Observations, Ecosystem-based Management & Forecasting, May , Tallin, Estonia. Piscataway: IEEE, ISB [4] Jasinevicius R. Fuzzy inference tools for decision makers // ISAGA 2008 : the 39th Conference International Simulation and Gaming Association: Games: Virtual Worlds and Reality : 7- July 2008, Kaunas, Lithuania : conference book. Kaunas: Technologija, ISB p. 28. [5] Jasinevicius R., Petrauskas V. Rule-based extensions of fuzzy cognitive maps for decision support systems // Information Technologies' 2008: proceedings of the 4th International Conference on Information and Software Technologies, IT 2008, Kaunas, Lithuania, April 24-25, 2008 / Kaunas University of Technology. ISS p [6] Kahn M. S., Quaddus M. Group Decision Support using Fuzzy Cognitive Maps for Causal Reasoning. Group Decision and egotiation Journal, vol.3, o 5, p.p [7] Konar A. Computational Intelligence: Principles, Techniques and Applications, Springer [8] Kosko B. Fuzzy cognitive maps. International Journal of Man-Machine Studies, 24, 986, p.p [9] Kosko B. Fuzzy thinking: the new science of fuzzy logic. Flamingo, London, 994. [20] Kosko B. Fuzzy engineering. Prentice-Hall,.J., 997. [2] Li H., Chen Ph., Huang H-P. Fuzzy neural intelligent systems: mathematical foundations and the applications in engineering, RCA Press LLC, 200. [22] Lin C.-T., Lee S. G. eural fuzzy systems, Prentice Hall, 996. [23] Maringer D. Heuristic Optimization for Portfolio Management, IEEE Computational Intelligence, v.3 r. 4, ov p.p [24] Mohr T. S. Software Design for a Fuzzy Cognitive Map Modelling Tool, Master s Project Rensselaer Polytechnic Institute, 997, 9p.. [25] Passino P. M, Jurkovich S. Fuzzy control, Addison-Wesley, 998. [26] Venayagamoortthy G. K. A Successful Interdisciplinary Course on Computational Intelligence. IEEE Computational Intelligence Magazine, v.4, umber, p.p [27] Xirogiannis G., Stefanou J., Glykas M. A fuzzy cognitive map approach to support urban design. Journal of Expert Systems with Applications, 26(2), p.p [28] Xirogiannis G., Glykas M., Staikouras Ch. Fuzzy Cognitive Maps as a Back End to Knowledge-based Systems in Geographically Dispersed Financial Organizations. Knowledge and Process Management, vol. (2), 2004, p.p [29] Zadeh L. A. Fuzzy sets. Information and control, 8, 965, p. p [30] Zadeh L. A. Fuzzy algorithms. Information and control, 2, 968, p. p [3] Zadeh L. A. The concept of a linguistic variable and its application to approximate reasoning. Information sciences, 8, 975, p. p