EXTENDED FUZZY COGNITIVE MAPS
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1 EXTENDED FUZZY COGNITIVE MAPS Masafumi Hagiwara Department of Psychology Stanford University Stanford, CA 930 U.S.A. Abstract Fuzzy Cognitive Maps (FCMs) have been proposed to represent causal reasoning by using numeric processing. They graphically represent uncertain causal reasoning. In the resonant states, there emerges a limit cycle or a hidden pattern, which is a FCM inference. However, there are some shortcomings concerned with knowledge representation in the conventional FCMs. In this paper, we propose Extended Fuzzy Cognitive Maps (E-FCMs) to represent causal relationships more naturally. The features of the E-FCMs are: there are nonlinear membership functions, conditional weights, and time delay weights. Computer simulation results indicate the effectiveness of the E-FCMs. 1. Introduction Knowledge representation and inference have been very important and difficult problems in Artificial Intelligence (AI). Traditional AI systems such as expert systems can store and process knowledge; generally they are based on symbolic, not numeric. In symbolic processing, language strings represent thought or short-term memory, rules and relations between language strings correspond to long-term memory, and programming corresponds learning. In traditional AI systems, it is easy to represent structured knowledge as rules. However, symbolic processing is not good at dealing with mathematical analysis in the traditional senses of engineering and the physical sciences; it is hard to directly apply the tools of numerical mathematics. Neural networks have a numerical framework with theorems and algorithms. They have demonstrated their superior performance in some applications (Rumelhart, McClelland, et al., 1986, Gorman & Sejnowski, 1988,Waibel, Hanazawa, Hinton, Shikano & Lang, 1989). However there are still some drawbacks: they cannot directly encode structured knowledge, and generally learning requires many cycles. Fuzzy Cognitive Maps (FCMs) have been proposed to represent causal reasoning (Kosko, 198, Taber & Siegel, 198, Zhang & Chen, 1988). FCMs are useful not only to symbolic representation of semantic networks, but also to distributed representation of cognitive processes. An addition, it is quite appealing that a biologically observed excitatory (positive) and inhibitory (negative) associations between neurons are represented in this logical architecture (Zhang and Chen, 1988). The features of the FCMs are: 1) They store domain knowledge in the nodes and the directional connections. ) They graphically represent uncertain causal reasoning. 3) Their matrix representation allow causal inferences to be made as feedback associative memory recollections. Since FCMs are dynamical systems, their resonant states are limit cycles. The limit cycle or hidden pattern is a FCM inference (Kosko, 1991). However, there are three important drawbacks in the conventional FCMs: 1) Connections in FCMs are just numeric ones: relationship of two events should be linear. ) Lack of a concept of time; practically each causal has different time delay. 3) They cannot deal with co-occurrence of multiple causes such as expressed by "and" conditions /9 $ IEEE 9
2 In this paper, we propose Extended Fuzzy Cognitive Maps (E-FCMs) to overcome the above drawbacks. The features of the E-FCM are as follows: 1) Weights having nonlinear membership functions. ) Conditional weights. 3) Time delay weights. In, conventional FCMs are briefly explained. In 3, E-FCMs are proposed. And then computer simulation results are shown in.. Conventional Fuzzy Cognitive Maps (FCM) In this section, the structure and behavior of Fuzzy Cognitive Maps (FCMs) are briefly explained. Generally FCMs are representated by fuzzy signed digraph with feedback. A FCM graphically represents uncertain causal reasoning. FCM causal connection weights are numbers in [- 1, 11 to allow degrees of causality to be represented. Fig. 1 shows an example of a simple FCM for director of public health (Zhang & Chen, 1988). The FCM has the following causal connection matrix E. c, c c3 c, cs c, c r Fig. 1 An example of a simple Fuz~y Cognitive Map. 96
3 In addition, any set of FCMs can be naturally combined. Each expert can draw a different size of FCM causal concepts. There is no restriction on the number of experts or on the number of concepts. To combine arbitrary FCM connection matrices E,, E,, * * e, E,, first, these matrices are augmented. Each augmented matrix Fi has n rows and n columns, where n is the total number of distinct concept used by the experts. The combined matrix can be expressed as (Kosko, 198), R F =x W, r=l F, where wi is a nonnegative credibility weight of each expert. 3. Proposed Extended Fuzzy Cognitive Maps (E-FCM) In this section, first,three drawbacks of the conventional FCMs are mentioned, and then proposed Extended Fuzzy Cognitive Maps (E-FCM) are explained. As for the conventional FCMs, these are three important drawbacks. 1) Connections in FCMs are just numeric ones; relationship of two events should be linear such as shown in Fig.(a). ) Lack of a concept of time; practically each causal has different time delay. 3) They cannot deal with co-occurrence of multiple causes such as (Gupta, 1991, Kaufmann & Gupta, 1983), if X,, and X, -1 and... and Xn, then Y. (3 1 As for l), according to the connection wi.<, If (the number of people in a city increases) then (modernization of the city advances) with the proportion of 0.6. () However, it is not true in a sense "with the proportion of 0.6'; the most populous city is not necessarily the most modernized. In general the relationship (0.6 in this case) is not a linear but a nonlinear, such as shown in Fig.(b)..- 3 cd e.,u.- n,k - d c.c 8 - C d % E, " 9
4 As for ), it is natural that the time delay in the connection ws6 [if (sanitation facilities are improved) then (the number of diseases per lo00 residents decreases)] is longer than that in the connections w,[if (migration in a city increases) then (the number of people in a city)j. As for 3), if the government of the country controls migration, connection w3 changes; w3 should be conditional such as: if (modernization of a city advances) and (the government restricts migration) () then (migration into the country increases) with the proportion x. Considering the above discussions, we propose Extended Fuzzy Cognitive Maps (E-FCMs). In E-FCMs, total input to node Cj at time t can be expressed as, where, C,(t) is a causal concept at time t, w,(.) is a weight function from concept C,(t) to concept C,(r), and defuvlj is a time delay from causal concept C,(t) to concept C,(t). Similar to the conventional FCMs, output of each causal node CJ(t) is a nonlinear function that transforms the path-weighted activation flowing into it into a value in 10, 11. This nonlinear function is in general a bounded monotonous increasing transformation, such as a sigmoid or S-shaped functions. Fig.3 An e\ample of it simple Extended Furq Cognitile Map. 98
5 . Computer simulations Here, computer simulation results are shown to demonstrate the superiority of E-FCMs. The same example as in Fig. 1 (Zhang and Chen, 1988) was used for the conventional FCM. As for the proposed E-FCMs, the map shown in Fig.3 was used. We modified the conventional FCM just parts for simplicity, which are shown by dotted circles in Fig.3. a) Nonlinear weights: C, -->C, (~13) and C, -->C, (w p) Conditional weight: C,, C, -->C, (w3,=0./s). If (the government controls migration, C,) then {if (migration into the country increases) then (modernization of the city advances) with proportion 0./ (not 0.)) () y) Time delays: there is one time delay in the weights C, -->C, (ws6). There are, of course, many parts to be changed in Fig. I, however, we changed just the above parts to make clear the effects. Table I shows each node activation as a function of time for the case of the conventional FCM. Table shows that for the case of the proposed E-FCMs. Table (a) is a result using only a), Table (b) is a result using a) and p), and Table (c) is a result using a), p). and y). It can be observed many things from these tables. For example, the modernization (C,) in the conventional FCM is saturated from time, on the other hand as for the E-FCMs, when there are two nonlinear weights (wi3 and wl,), it is saturated from time 8 (Table (a)). When there are nonlinear weights and a conditional weight, it is from 10 (Table (b)). And the case of Table (c), it is from 11. Another conspicuous thing is about "bacteria per area (C,)". For the conventional FCM, the value of C, converges to 0.08, not 0. However, for the proposed E-FCMs, they converge to 0. From above results and discussion, it is clear that nonlinear weights, conditional weights, and time delay in weights greatly influence the behavior of FCM. -. Tinic in a tit\ into a cit). Lation C CS C6 c Garbage Sanitation n of peop~c per Bacteria per area facilities 100oresideiice per area I I.ooo I.ooo , ooo ooo I.000 I.ooo h 8 9 IO Tablc 1 hch ntdc actit ation as ii function of time for the case of the con\entional FCM 99
6 Cl c C3 C Cs Ch c C8 'I ime # of people Rligration hlodemi- Garbage Sanitation # of people per Bacteria ('ontrol bq in a cit) into a city Lation pe r area racilities IOOOresidence per area government 0 I r,. I ime. 0 I # of people hligration Modemi- Garbage Sanitation #of people per Bacteria ('ontrol by in a city into a citj lation per area facilities 1000residcnce per area government : OOO X into a city dation pe r area facilities IOOOresidenee per area government I so I
7 . Conclusions In this paper, we have proposed Extended Fuzzy Cognitive Maps (E-FCMs) to represent causal relationship more naturally. The features of the E-FCMs are as follows: 1) Weights having nonlinear membership functions. ) Conditional weights. 3) Time delay weights. Computer simulation results have demonstrated the effectiveness of the proposed E-FCMs. The proposed methods will greatly improve the behavior of FCM. The example used in this paper was a simple one, however, the proposed E-FCMs are useful to many fields such as cognitive science, social science, computer vision, and so on. Acknowledgement The auther wish to thank Prof. David E. Rumelhart for his valuable comments to improve this paper. REFERENCES Gorman, R.P. & Sejnowski, T.J. (1988). Analysis of hidden units in a layered network trained to classify sonar targets. Neurul Networks, 1, -89. Gupta, M.M., and Qi, J. (1991). On fuzzy neuron models. Proceeding of Internutionul Joint Conference on Neural Networks,, Kaufmann, A., and Gupta, M.M. (1988). Fuzzy mathematical models in engineering and management science. North-Holland. Kosko, B. ( 198). Adaptive inference in fuzzy knowledge networks. Proceeding of the First International Conference on Neural Networks,, Kosko, B. (1991). Neural networks and fuzzy system. Prentice Hall. Rumelhart, D.E., McClelland, J.L. & the PDP Research Group (1986). Parallel Distributed Processing, MIT Press. Taber, W.R., and Siegel, M.A. (198). Estimation of expert weights using fuzzy cognitive maps. Proceeding of the First Internutionul Conference on Neural h'etworkb,, Waibel, A., Hanazawa, T., Hinton, G., Shikano, K., and Lang, K., (1989). Phoneme recognition using time-delay neural networks. IEEE Truns. on Acoustics, Speech, and Signal Processing, 3, Zhang, W., and Chen, S. (1988). A logical architecture for cognitive maps. Proceeding of the - nd International Conference on Neural Networks, 1,
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