Why Fuzzy Cognitive Maps Are Efficient
Keywords:
fuzzy cognitive maps, neural networks, seven plus minus two law.Abstract
In many practical situations, the relation between the experts’ degrees of confidence in different related statements is well described by Fuzzy Cognitive Maps (FCM). This empirical success is somewhat puzzling, since from the mathematical viewpoint, each FCM relation corresponds to a simplified one-neuron neural network, and it is well known that to adequately describe relations, we need multiple neurons. In this paper, we show that the empirical success of FCM can be explained if we take into account that human’s subjective opinions follow Miller’s seven plus minus two law.References
C.M. Bishop (2006), Pattern Recognition and Machine Learning, Springer, New York.
E. Bourgani, C.D. Stylios, G. Manis, V. Georgopoulos (2013), A study on Fuzzy Cognitive Map Structures for Medical Decision Support Systems, Proc. of the 8th Conf. of the European Society for Fuzzy Logic and Technology EUSFLAT'2013, Milano, Italy, September 11-13, 744-751. http://dx.doi.org/10.2991/eusflat.2013.111
Y. Boutalis, T. Kottas, M. Christodoulou M. (2009), Adaptive Estimation of Fuzzy Cognitive Maps with Proven Stability and Parameter Convergence, IEEE Trans. on Fuzzy Systems, 17(4): 874-889. http://dx.doi.org/10.1109/TFUZZ.2009.2017519
B. Chokr, V. Kreinovich (1994), How Far Are We from the Complete Knowledge: Complexity of Knowledge Acquisition in Dempster-Shafer Approach, In: R.R. Yager, J. Kacprzyk, M. Pedrizzi (Eds.), Advances in the Dempster-Shafer Theory of Evidence, Wiley, New York, 555-576.
E.T. Jaynes, G.L. Bretthorst (2003), Probability Theory: The Logic of Science, Cambridge University Press, Cambridge, UK. http://dx.doi.org/10.1017/CBO9780511790423
G. Klir, B. Yuan (1995), Fuzzy Sets and Fuzzy Logic, Prentice Hall, Upper Saddle River, New Jersey.
C. Knight, D. Lloyd, A. Penn (2014), Linear and Sigmoidal Fuzzy Cognitive Maps: An Analysis of Fixed Point, Applied Soft Computing, 15: 193-202. http://dx.doi.org/10.1016/j.asoc.2013.10.030
B. Kosko (1986), Fuzzy Cognitive Maps, International J. of Man-Machine Studies, 7: 65-75. http://dx.doi.org/10.1016/S0020-7373(86)80040-2
V. Kreinovich, A. Bernat (1994), Parallel Algorithms for Interval Computations: An Introduction, Interval Computations, 1994(3):3, 6-62.
V. Kreinovich, C. Quintana (1991), Neural Networks: What Non-Linearity to Choose?, Proc. of the 4th University of New Brunswick Artificial Intelligence Workshop, Fredericton, New Brunswick, Canada, 627-637.
W. Lu, J. Yang, X. Liu (2014), Numerical Prediction of Time Series Based on FCMs with Information Granules, International J. of Computers Communications & Control, 9(3): 313-324. http://dx.doi.org/10.15837/ijccc.2014.3.210
Y. Miao, C.Y. Miao, X.H. Tao, Z.Q. Shen, Z.Q. Liu (2010), Transformation of Cognitive Maps, IEEE Trans. on Fuzzy Systems, 18(1): 114-124. http://dx.doi.org/10.1109/TFUZZ.2009.2037218
G.A. Miller (1956), The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information", Psychological Review, 63(2): 81-97. http://dx.doi.org/10.1037/h0043158
H.T. Nguyen, V. Kreinovich, Applications of Continuous Mathematics to Computer Science, Kluwer, Dordrecht.
H.T. Nguyen and E.A. Walker (2006), A First Course in Fuzzy Logic, Chapman and Hall/CRC, Boca Raton, Florida.
E. Papageorgiou, C. Stylios (2008), Fuzzy Cognitive Maps, In: W. Pedrycz, A. Skowron, V. Kreinovich, Handbook of Granular computing, John Wiley & Sons, 755-776. http://dx.doi.org/10.1002/9780470724163.ch34
E.I. Papageorgiou, C. Stylios, P.P. Groumpos (2006), Introducing Interval Analysis in Fuzzy cog- nitive Map Framework, In: G. Antoniou et al. (eds), Proc. of the 4th Hellenic Conf. on Artificial Intelligence SETN'2006, Heraklion, Crete, May 18-20, 2006, Springer Lecture Notes in Artificial Intelligence, 3955: 571-575.
W. Pedrycz (2010), The Design of Cognitive Maps: A Study in Synergy of Granular Computing and Evolutionary Optimization, Expert Systems with Applications, 37(10): 7288-7294. http://dx.doi.org/10.1016/j.eswa.2010.03.006
W. Pedrycz, W. Homenda, From Fuzzy Cognitive Maps to Granular Cognitive Maps, IEEE Trans. on Fuzzy Systems, 22(4): 859-869. http://dx.doi.org/10.1109/TFUZZ.2013.2277730
Y.G. Petalas, E.I Papageorgiou, K.E. Parsopoulos, P.P. Groumpos, M.N. Vrahatis (2005), Interval Cognitve Maps, Proc. of Intl. Conf. of Numerical Analysis and Applied Mathematics ICNAAM'05, Rhodes, Greece, September 16-20, 2005, 1120-1123.
S.K. Reed (2010), Cognition: Theories and Application, Wadsworth Cengage Learning, Belmont, California.
J.T. Rickard, J. Aisbett, R.R. Yager (2015), A New Fuzzy Cognitive Map Structure Based on the Weighted Power Mean, IEEE Trans. on Fuzzy Systems. http://dx.doi.org/10.1109/TFUZZ.2015.2407906
O. Sirisaengtaksin, V. Kreinovich, H.T. Nguyen (1995), Sigmoid Neurons Are the Safest Against Additive Errors, Proc. of the First Intl. Conf. on Neural, Parallel, and Scientific Computations, Atlanta, Georgia, May 28-31, 1995, 1: 419-423.
H.J. Song, C.Y. Miao, Z.Q. Shen, W. Roel, D.H. Maja, C. Francky (2010), Design of Fuzzy Cognitive Maps Using Neural Networks for Predicting Chaotic Time Series, Neural Networks, 23: 1264-1275. http://dx.doi.org/10.1016/j.neunet.2010.08.003
H. Song, C. Mia, R.Wuyts, Z. Shen, M. Hondt, F. Catthoor (2011), An Extension to Fuzzy Cognitive Maps for Classification and Prediction, IEEE Trans. on Fuzzy Systems, 19(1): 116-135. http://dx.doi.org/10.1109/TFUZZ.2010.2087383
W. Stach, L. Kurgan, W. Pedrycz (2008), Numerical and Linguistic Prediction of Time Series With the Use of Fuzzy Cognitive Maps, IEEE Trans. on Fuzzy Systems, 16(1): 61-72. http://dx.doi.org/10.1109/TFUZZ.2007.902020
C.D. Stylios, P.P. Groumpos (2000), Fuzzy Cognitive Maps in Modeling Supervisory Control Sys- tems, Journal of Intelligent & Fuzzy Systems, 8(2): 83-98.
C. Stylios, P. Groumpos (2004), Modeling Complex Systems Using Fuzzy Cognitive Maps, IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 34(1): 155-162. http://dx.doi.org/10.1109/TSMCA.2003.818878
R. Taber (1991) Knowledge Processing with Fuzzy Cognitive Maps, Expert Systems with Applica- tions, 2(1): 83-87. http://dx.doi.org/10.1016/0957-4174(91)90136-3
R. Trejo, V. Kreinovich, I.R. Goodman (2002), J. Martinez, R. Gonzalez, A Realistic (Non- Associative) Logic And a Possible Explanations of 7±2 Law, International Journal of Approximate Reasoning, 29: 235-266. http://dx.doi.org/10.1016/S0888-613X(01)00065-2
L.A. Zadeh (1965), Fuzzy Sets, Information and Control 8: 338-353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X
Published
Issue
Section
License
ONLINE OPEN ACCES: Acces to full text of each article and each issue are allowed for free in respect of Attribution-NonCommercial 4.0 International (CC BY-NC 4.0.
You are free to:
-Share: copy and redistribute the material in any medium or format;
-Adapt: remix, transform, and build upon the material.
The licensor cannot revoke these freedoms as long as you follow the license terms.
DISCLAIMER: The author(s) of each article appearing in International Journal of Computers Communications & Control is/are solely responsible for the content thereof; the publication of an article shall not constitute or be deemed to constitute any representation by the Editors or Agora University Press that the data presented therein are original, correct or sufficient to support the conclusions reached or that the experiment design or methodology is adequate.
