On the convergence of sigmoidal fuzzy grey cognitive maps

Harmati, István Á. ; Kóczy, László T.

International Journal of Applied Mathematics and Computer Science, Tome 29 (2019) no. 3, pp. 453-466.

Résumé

Fuzzy cognitive maps (FCMs) are recurrent neural networks applied for modelling complex systems using weighted causal relations. In FCM-based decision-making, the inference about the modelled system is provided by the behaviour of an iteration. Fuzzy grey cognitive maps (FGCMs) are extensions of fuzzy cognitive maps, applying uncertain weights between the concepts. This uncertainty is expressed by the so-called grey numbers. Similarly as in FCMs, the inference is determined by an iteration process which may converge to an equilibrium point, but limit cycles or chaotic behaviour may also turn up. In this paper, based on the grey connections between the concepts and the parameters of the sigmoid threshold function, we give sufficient conditions for the existence and uniqueness of fixed points of sigmoid FGCMs.

Détail

Keywords: fuzzy cognitive map, grey system theory, fuzzy grey cognitive map, fixed point
Mots-clés : mapa kognitywna rozmyta, teoria szarych systemów, rozmyta mapa poznawcza, punkt stały

Bibliographie

[1] Axelrod, R. (1976). Structure of Decision: The Cognitive Maps of Political Elites, Princeton University Press, Princeton, NJ.

[2] Bartczuk, Ł., Przybył, A. and Cpałka, K. (2016). A new approach to nonlinear modelling of dynamic systems based on fuzzy rules, International Journal of Applied Mathematics and Computer Science 26(3): 603–621, DOI: 10.1515/amcs-2016-0042.

[3] Boutalis, Y., Kottas, T.L. and Christodoulou, M. (2009). Adaptive estimation of fuzzy cognitive maps with proven stability and parameter convergence, IEEE Transactions on Fuzzy Systems 17(4): 874–889.

[4] Buruzs, A., Hatwágner, M.F. and Kóczy, L.T. (2015). Expert-based method of integrated waste management systems for developing fuzzy cognitive map, in Q. Zhu and A. Azar (Eds), Complex System Modelling and Control Through Intelligent Soft Computations, Springer, Cham, pp. 111–137.

[5] Busemeyer, J.R. (2001). Dynamic decision making, in N.J. Smelser and P.B. Baltes (Eds), International Encyclopedia of the Social Behavioral Sciences, Elsevier, New York, NY pp. 3903–3908.

[6] Carlsson, C. and Fullér, R. (2011). Possibility for Decision: A Possibilistic Approach to Real Life Decisions, Studies in Fuzziness and Soft Computing Series, Vol. 270/2011, Springer Publishing Company, Berlin/Heidelberg.

[7] Carvalho, J.P. (2013). On the semantics and the use of fuzzy cognitive maps and dynamic cognitive maps in social sciences, Fuzzy Sets and Systems 214: 6–19.

[8] Felix, G., Nápoles, G., Falcon, R., Froelich, W., Vanhoof, K. and Bello, R. (2017). A review on methods and software for fuzzy cognitive maps, Artificial Intelligence Review 2017: 1–31.

[9] Ferreira, F.A., Ferreira, J.J., Fernandes, C.I., Meidutė-Kavaliauskienė, I. and Jalali, M.S. (2017). Enhancing knowledge and strategic planning of bank customer loyalty using fuzzy cognitive maps, Technological and Economic Development of Economy 23(6): 860–876.

[10] Harmati, I. Á ., Hatwágner, M.F. and Kóczy, L.T. (2018). On the existence and uniqueness of fixed points of fuzzy cognitive maps, in J. Medina et al. (Eds), Information Processing and Management of Uncertainty in Knowledge-Based Systems: Theory and Foundations, Springer International Publishing, Cham, pp. 490–500.

[11] Harmati, I.Á . and Kóczy, L.T. (2018). On the convergence of fuzzy grey cognitive maps, in P. Kulczycki et al. (Eds), Contemporary Computational Science, AGH-UCT Press, Cracow, p. 139.

[12] Harmati, I.Á . and Kóczy, L.T. (2019 ). On the convergence of fuzzy grey cognitive maps, in P. Kulczycki et al. (Eds), Information Technology, Systems Research and Computational Physics, Advances in Intelligent Systems and Computing, Springer, Cham, pp. 74–84.

[13] Knight, C.J., Lloyd, D.J. and Penn, A.S. (2014). Linear and sigmoidal fuzzy cognitive maps: An analysis of fixed points, Applied Soft Computing 15: 193–202.

[14] Kosko, B. (1986). Fuzzy cognitive maps, International Journal of Man-Machine Studies 24(1): 65–75.

[15] Liu, S. and Lin, Y. (2006). Grey Information: Theory and Practical Applications, Springer Science Business Media, London.

[16] Lorenz, S., Martinez-Fernández, V., Alonso, C., Mosselman, E., de Jalón, D.G., del Tánago, M.G., Belletti, B., Hendriks, D. and Wolter, C. (2016). Fuzzy cognitive mapping for predicting hydromorphological responses to multiple pressures in rivers, Journal of Applied Ecology 53(2): 559–566.

[17] Nápoles, G., Papageorgiou, E., Bello, R. and Vanhoof, K. (2016). On the convergence of sigmoid fuzzy cognitive maps, Information Sciences 349–350: 154–171.

[18] Nápoles, G., Papageorgiou, E., Bello, R. and Vanhoof, K. (2017). Learning and convergence of fuzzy cognitive maps used in pattern recognition, Neural Processing Letters 45(2): 431–444.

[19] Papageorgiou, E.I. and Salmeron, J.L. (2012). Learning fuzzy grey cognitive maps using nonlinear Hebbian-based approach, International Journal of Approximate Reasoning 53(1): 54–65.

[20] Papageorgiou, E.I. and Salmeron, J.L. (2013). A review of fuzzy cognitive maps research during the last decade, IEEE Transactions on Fuzzy Systems 21(1): 66–79.

[21] Papageorgiou, E.I. and Salmeron, J.L. (2014). Methods and algorithms for fuzzy cognitive map-based modeling, in E. Papageorgiou (Ed.), Fuzzy Cognitive Maps for Applied Sciences and Engineering, Springer, Berlin/Heidelberg, pp. 1–29.

[22] Salmeron, J.L. (2010). Modelling grey uncertainty with fuzzy grey cognitive maps, Expert Systems with Applications 37(12): 7581–7588.

[23] Salmeron, J.L. and Gutierrez, E. (2012). Fuzzy grey cognitive maps in reliability engineering, Applied Soft Computing 12(12): 3818–3824.

[24] Salmeron, J.L. and Papageorgiou, E.I. (2012). A fuzzy grey cognitive maps-based decision support system for radiotherapy treatment planning, Knowledge-Based Systems 30: 151–160.

[25] Smoczek, J. (2013). Evolutionary optimization of interval mathematics-based design of a TSK fuzzy controller for anti-sway crane control, International Journal of Applied Mathematics and Computer Science 23(4): 749–759, DOI: 10.2478/amcs-2013-0056.

[26] Stylios, C.D. and Groumpos, P.P. (2004). Modeling complex systems using fuzzy cognitive maps, IEEE Transactions on Systems, Man, and Cybernetics A: Systems and Humans 34(1): 155–162.

[27] Tsadiras, A.K. (2008). Comparing the inference capabilities of binary, trivalent and sigmoid fuzzy cognitive maps, Information Sciences 178(20): 3880–3894.

[28] Vidhya, R. and Hepzibah, R.I. (2017). A comparative study on interval arithmetic operations with intuitionistic fuzzy numbers for solving an intuitionistic fuzzy multi-objective linear programming problem, International Journal of Applied Mathematics and Computer Science 27(3): 563–573, DOI: 10.1515/amcs-2017-0040.

[29] Zanon, L.G. and Carpinetti, L.C.R. (2018). Fuzzy cognitive maps and grey systems theory in the supply chain management context: A literature review and a research proposal, 2018 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Rio de Janerio, Brazil, pp. 1554–1561.

[30] Ziv, G., Watson, E., Young, D., Howard, D.C., Larcom, S.T. and Tanentzap, A.J. (2018). The potential impact of Brexit on the energy, water and food nexus in the UK: A fuzzy cognitive mapping approach, Applied Energy 210: 487–498.