Fuzzy cognitive map

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Rod Tabers FCM depicting eleven factors of the American drug market

A fuzzy cognitive map (FCM) is a cognitive map within which the relations between the elements (e.g. concepts, events, project resources) of a "mental landscape" can be used to compute the "strength of impact" of these elements. Fuzzy cognitive maps were introduced by Bart Kosko.[1][2] Robert Axelrod introduced cognitive maps as a formal way of representing social scientific knowledge and modeling decision making in social and political systems, then brought in the computation.[3]


Fuzzy cognitive maps are signed fuzzy digraphs. They are visually similar to, but distinct from, Hasse diagrams. Spreadsheets or tables are used to map FCMs into matrices for further computation. FCM is a technique used for causal knowledge acquisition and representation, it supports causal knowledge reasoning process and belong to the neuro-fuzzy system that aim at solving decision making problems, modeling and simulate complex systems.[4] Learning algorithms have been proposed for training and updating FCMs weights mostly based on ideas coming from the field of Artificial Neural Networks.[5] Adaptation and learning methodologies used to adapt the FCM model and adjust its weights. Kosko and Dickerson (Dickerson & Kosko, 1994) suggested the Differential Hebbian Learning (DHL) to train FCM.[6] There have been proposed algorithms based on the initial Hebbian algorithm;[7] others algorithms come from the field of genetic algorithms, swarm intelligence[8] and evolutionary computation.[9] Learning algorithms are used to overcome the shortcomings that the traditional FCM present i.e. decreasing the human intervention by suggested automated FCM candidates; or by activating only the most relevant concepts every execution time; or by making models more transparent and dynamic.[10]

Fuzzy cognitive maps (FCMs) have gained considerable research interest due to their ability in representing structured knowledge and model complex systems in various fields. This growing interest led to the need for enhancement and making more reliable models that can better represent real situations. A first simple application of FCMs is described in a book[11] of William R. Taylor, where the war in Afghanistan and Iraq is analyzed. In Bart Kosko's book Fuzzy Thinking,[12] several Hasse diagrams illustrate the use of FCMs. As an example, one FCM quoted from Rod Taber[13] describes 11 factors of the American cocaine market and the relations between these factors. For computations, Taylor uses pentavalent logic (scalar values out of {-1,-0.5,0,+0.5,+1}). That particular map of Taber uses trivalent logic (scalar values out of {-1,0,+1}). Taber et al. also illustrate the dynamics of map fusion and give a theorem on the convergence of combination in a related article.[14]

While applications in social sciences[11][12][13][15] introduced FCMs to the public, they are used in a much wider range of applications, which all have to deal with creating and using models[16] of uncertainty and complex processes and systems. Examples:

FCMappers is an international online community for the analysis and the visualization of fuzzy cognitive maps.[35] FCMappers offer support for starting with FCM and also provide a Microsoft Excel-based tool that is able to check and analyse FCMs. The output is saved as Pajek file and can be visualized within third party software like Pajek, Visone, etc. They also offer to adapt the software to specific research needs.

Additional FCM software tools, such as Mental Modeler,[36][37] have recently been developed as a decision-support tool for use in social science research, collaborative decision-making, and natural resource planning.

See also[edit]


  1. ^ Bart Kosko (1986). "Fuzzy Cognitive Maps" (PDF). International Journal of Man-Machine Studies. 24: 65–75. doi:10.1016/S0020-7373(86)80040-2.
  2. ^ "Archived copy" (PDF). sipi.usc.edu. Archived from the original (PDF) on 2 September 2006. Retrieved 13 January 2022.{{cite web}}: CS1 maint: archived copy as title (link)
  3. ^ Axelrod, Robert (1976). Structure of Decision: The Cognitive Map of Political Elites. Princeton University Press. ISBN 978-0-691-10050-0.
  4. ^ Salmeron, Jose L. (2012). "Fuzzy Cognitive Maps for Artificial Emotions Forecasting". Applied Soft Computing. 12 (2): 3704–3710. doi:10.1016/j.asoc.2012.01.015.
  5. ^ Salmeron, Jose L.; Froelich, W. (2016). "Dynamic Optimization of Fuzzy Cognitive Maps for Time Series Forecasting". Knowledge-Based Systems. 105: 29–37. doi:10.1016/j.knosys.2016.04.023.
  6. ^ "IEEEBook8.dvi" (PDF). Home.eng.iastate.edu. Retrieved 2017-01-09.
  7. ^ Papageorgiou, E.I.; Stylios, C.D.; Groumpos, P.P. (2004). "Active Hebbian learning algorithm to train fuzzy cognitive maps". International Journal of Approximate Reasoning. 37 (3): 219. doi:10.1016/j.ijar.2004.01.001.
  8. ^ Papageorgiou, Elpiniki I.; Parsopoulos, Konstantinos E.; Stylios, Chrysostomos S.; Groumpos, Petros P.; Vrahatis, Michael N. (2005). "Fuzzy Cognitive Maps Learning Using Particle Swarm Optimization". Journal of Intelligent Information Systems. 25: 95. doi:10.1007/s10844-005-0864-9. S2CID 878213.
  9. ^ Stach, W.; Kurgan, L.; Pedrycz, W.; Reformat, M. (2005). "Evolutionary Development of Fuzzy Cognitive Maps". The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ '05. pp. 619–. doi:10.1109/FUZZY.2005.1452465. ISBN 0-7803-9159-4. S2CID 206671682.
  10. ^ Papageorgiou, Elpiniki I.; Stylios, Chrysostomos; Groumpos, Peter P. (2006). "Unsupervised learning techniques for fine-tuning fuzzy cognitive map causal links". International Journal of Human-Computer Studies. 64 (8): 727. doi:10.1016/j.ijhcs.2006.02.009.
  11. ^ a b William R. Taylor: Lethal American Confusion (How Bush and the Pacifists Each Failed in the War on Terrorism), 2006, ISBN 0-595-40655-6 (FCM application in chapter 14) Archived September 30, 2007, at the Wayback Machine
  12. ^ a b c Bart Kosko: Fuzzy Thinking, 1993/1995, ISBN 0-7868-8021-X (Chapter 12: Adaptive Fuzzy Systems)
  13. ^ a b c Rod Taber: Knowledge Processing with Fuzzy Cognitive Maps, Expert Systems with Applications, vol. 2, no. 1, 83-87, 1991 (Hasse diagram in German Wikipedia)
  14. ^ Taber, Rod; Yager, Ronald R.; Helgason, Cathy M. (2007). "Quantization effects on the equilibrium behavior of combined fuzzy cognitive maps". International Journal of Intelligent Systems. 22 (2): 181. doi:10.1002/int.20185. S2CID 205964356.
  15. ^ Costas Neocleous, Christos Schizas, Costas Yenethlis: Fuzzy Cognitive Models in Studying Political Dynamics - The case of the Cyprus problem Archived September 29, 2007, at the Wayback Machine
  16. ^ Chrysostomos D. Stylios, Voula C. Georgopoulos, Peter P. Groumpos: The Use of Fuzzy Cognitive Maps in Modeling Systems Archived July 20, 2011, at the Wayback Machine
  17. ^ Antonie Jetter: Produktplanung im Fuzzy Front End, 2005, ISBN 3-8350-0144-2
  18. ^ Salmeron, Jose L. (2009). "Supporting decision makers with Fuzzy Cognitive Maps". Research-Technology Management. 52 (3): 53–59. doi:10.1080/08956308.2009.11657569. S2CID 150765164.
  19. ^ Vesa A. Niskanen: Application of Fuzzy Linguistic Cognitive Maps to Prisoner's Dilemma, 2005, ICIC International pp. 139-152, ISSN 1349-4198 Archived September 29, 2007, at the Wayback Machine
  20. ^ Salmeron, Jose L. (2009). "Augmented Fuzzy Cognitive Maps for modelling LMS Critical Success Factors". Knowledge-Based Systems. 22 (4): 53–59. doi:10.1016/j.knosys.2009.01.002.
  21. ^ Georgopoulos, Voula C; Malandraki, Georgia A; Stylios, Chrysostomos D (2003). "A fuzzy cognitive map approach to differential diagnosis of specific language impairment". Artificial Intelligence in Medicine. 29 (3): 261–78. doi:10.1016/S0933-3657(02)00076-3. PMID 14656490.
  22. ^ Papageorgiou, E.I.; Stylios, C.D.; Groumpos, P.P. (2003). "An integrated two-level hierarchical system for decision making in radiation therapy based on fuzzy cognitive maps". IEEE Transactions on Biomedical Engineering. 50 (12): 1326–39. doi:10.1109/TBME.2003.819845. PMID 14656062. S2CID 1434928.
  23. ^ Salmeron, Jose L.; Papageorgiou, E. (2012). "A Fuzzy Grey Cognitive Maps-based Decision Support System for Radiotherapy Treatment Planning". Knowledge-Based Systems. 30 (1): 151–160. doi:10.1016/j.knosys.2012.01.008.
  24. ^ Georgopoulos, Voula C.; Stylios, Chrysostomos D. (2015). "Supervisory Fuzzy Cognitive Map Structure for Triage Assessment and Decision Support in the Emergency Department". Simulation and Modeling Methodologies, Technologies and Applications. Advances in Intelligent Systems and Computing. Vol. 319. pp. 255–69. doi:10.1007/978-3-319-11457-6_18. ISBN 978-3-319-11456-9.
  25. ^ Groumpos, Peter P.; Stylios, Chrysostomos D. (January 2000). "Fuzzy Cognitive Maps in modeling supervisory control systems - IOS Press". Journal of Intelligent & Fuzzy Systems. 8 (1): 83–98. Retrieved 2017-01-09.
  26. ^ Stylios, C.D.; Groumpos, P.P. (2004). "Modeling Complex Systems Using Fuzzy Cognitive Maps". IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans. 34: 155. doi:10.1109/TSMCA.2003.818878. S2CID 10611311.
  27. ^ Salmeron, Jose L.; Gutierrez, E. (2012). "Fuzzy Grey Cognitive Maps in Reliability Engineering". Applied Soft Computing. 12 (12): 3818–3824. doi:10.1016/j.asoc.2012.02.003.
  28. ^ Marc Böhlen: More Robots in Cages,
  29. ^ Benjoe A. Juliano, Wylis Bandler: Tracing Chains-of-Thought (Fuzzy Methods in Cognitive Diagnosis), Physica-Verlag Heidelberg 1996, ISBN 3-7908-0922-5
  30. ^ W. B. Vasantha Kandasamy, Florentin Smarandache: Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps, 2003, ISBN 1-931233-76-4
  31. ^ Rodriguez-Repiso, Luis; Setchi, Rossitza; Salmeron, Jose L. (2007). "Modelling IT projects success with Fuzzy Cognitive Maps". Expert Systems with Applications. 32 (2): 543. doi:10.1016/j.eswa.2006.01.032.
  32. ^ Salmeron, Jose L.; Lopez, C. (2010). "A multicriteria approach for risks assessment in ERP maintenance". Journal of Systems and Software. 83 (10): 1941–1953. doi:10.1016/j.jss.2010.05.073.
  33. ^ Salmeron, Jose L.; Lopez, C. (2012). "Forecasting Risk Impact on ERP Maintenance with Augmented Fuzzy Cognitive Maps". IEEE Transactions on Software Engineering. 38 (2): 439–452. doi:10.1109/TSE.2011.8. S2CID 14713607.
  34. ^ Salmeron, Jose L.; Vidal, R.; Mena, A. (2012). "Ranking Fuzzy Cognitive Maps based scenarios with TOPSIS". Expert Systems with Applications. 39 (3): 2443–2450. doi:10.1016/j.eswa.2011.08.094.
  35. ^ FCMappers - international community for fuzzy cognitive mapping: http://www.FCMappers.net/
  36. ^ Gray, Steven A.; Gray, Stefan; Cox, Linda J.; Henly-Shepard, Sarah (2013). "Mental Modeler: A Fuzzy-Logic Cognitive Mapping Modeling Tool for Adaptive Environmental Management". 2013 46th Hawaii International Conference on System Sciences. pp. 965–. doi:10.1109/HICSS.2013.399. ISBN 978-1-4673-5933-7. S2CID 1413540.
  37. ^ "Fuzzy Logic Cognitive Mapping". Mental Modeler. Retrieved 2017-01-09.