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Genetic algorithms for decision support in the case of Fuzzy cognitive maps.

1. INTRODUCTION

Axelord (1976) proposed cognitive maps as a formal tool for decision- making, using the matrix representation of the directed graph to represent and study the social scientific knowledge. Based on the cognitive map structure Kosko (1986) proposed Fuzzy Cognitive Maps. Kosko(1988) had applied FCM to neural network and studied the hidden pattern in combined and adaptive knowledge networks. Cognitive Maps were used by Craiger and Coovert 1994) to study the public health scenario of a society, where the edges of the map were assigned a positive or a negative value. Craiger et al(1996) studied modeling organization behaviour with FCM. They had studied FCM as a computer simulation methodology for representing and predicting the behaviour of models of arbitrary complexity. Chong et al (2004) have used FCM in goal oriented decision support systems. A fuzzy cognitive map is a directed graph with feedback. FCM nodes represent concepts like policies, events, inputs, outputs, trends, goals etc. and edges represent causal links among the concepts. Each concept variable is given an initial value based on the belief of the expert(s) about the current state. The FCM is then free to interact until an equilibrium is reached. This is arrived at through a process of forward chaining. Given an initial state of a system represented by a set of values of its constituent concepts, an FCM can stimulate its evolution over time to predict its future behavior.

We may start with a goal and aim to identify the initial state that can trigger a chain of events leading to the state corresponding to the goal. The nonlinear transformation involved in computing successive FCM states and the process of reversing the matrix multiplication make the backward chaining difficult. Hence in this paper we use a methodology based on the use of FCM and Genetic Algorithm for the goal-oriented procedure of a given scenario starting with the desired goal. We apply this procedure for sample study of cancer patients taking treatment in Thanjavur Medical College, Thanjavur, Tamil Nadu. Most of the patients were agricultural labourers. Smoking tobacco, chewing tobacco and using alcohol are common habits among them. We analyse the factors of tobacco and alcohol related cancers using the mentioned methodology.

2. FUZZY COGNITIVE MAP

FCM graph consists of nodes (concepts, agents) and weighted arcs (connection, edge). Nodes on the graph represent concepts describing behavioural characteristics of the system. Concepts can be inputs, outputs, variables, states, events, actions and goals of the system. Signed weighted arcs represent causal relationships that exist among concepts. The edge weight Eij represents the degree of causal relationship between concept i and concept j. This setup gives rise to the following three types of interactions:

[E.sub.ij] > 0, a positive causality, where an increase in the value of the ith concept causes an increase in the value of the jth concept. [E.sub.ij] < 0, a negative causality, where an increase in the value of the ith concept causes a decrease in the value of the jth concept. [E.sub.ij] = 0, no causal relationship between the ith concept and jth concept.

Kosko (1992) proposed a rule to calculate the value of each concept based on the influence of the interconnected concepts which involves vector-matrix multiplication and the forward chaining involves a series of repeated vector-matrix multiplication and transformation. At each time step t, the state vector is calculated as

[C.sub.t] = F([C.sub.t-1] x E)

where [C.sub.t] is the state vector at time step t, [E.sub.ij] is the weight of the arc connecting [C.sub.i] to [C.sub.j] and E is the weight matrix and F(X) = ([f.sub.1], [f.sub.2], ..., [f.sub.n]) is a thresholding function operating on the resulting vector. Generally, a sigmoid function f(x) = 1/(1 + [e.sup.-[lambda]x]) is used for thresholding to constrain the value of f (x) in the interval [0, 1]. After the implementation of thresholding we update the resulting vector by clamping the concepts of interest. This new output vector is treated as the input vector for the next stage.

The vector matrix multiplication operation to derive successive FCM states is iterated until the FCM converges or settles down to one of the following:

(i) A fixed point attractor: The FCM state vector remains unchanged for successive iterations. This stable state is known as the fixed point attractor.

(ii) A Limit Cycle: A sequence of FCM state keeps repeating indefinitely. This sequence is known as limit cycle.

(iii) A chaotic attractor: The FCM state vector keeps changing. Repeating states are never found.

3. GENETIC ALGORITHM

Genetic Algorithms (GA) were first introduced by John Holland (1975) . GA attempts to arrive at optimal solutions through a process similar to biological evolution. This involves following the principles of survival of the fittest, crossbreeding and mutation to generate better solutions from a pool of existing solutions. Genetic algorithms have been found to be capable of finding solutions for a wide variety of problems for which no acceptable algorithmic solutions exist. The GA methodology is particularly used for optimization, a problem solving technique in which one or more very good solutions are searched for in a solution space consisting of a large number of possible solutions. GA reduce the search space by continually evaluating the current generation of candidate solutions. It discards the solutions ranked as poor, and produces a new generation through crossbreeding and mutating those ranked as good. The ranking of candidate solutions is done using some pre-determined measure of fitness.

The GA evolutionary cycle starts with a randomly selected initial population. The changes to the population occur through the processes of selection based on fitness, and alteration using crossover and mutation. The application of selection and alteration leads to a population with a higher proportion of better solutions. The evolutionary cycle continues until an acceptable solution is found in the current population, or some control parameter such as the number of generations is exceeded.

[FIGURE OMITTED]

The steps in the typical genetic algorithm for finding a solution to a problem are listed below:

1. Create an initial solution population of a certain size randomly.

2. Evaluate each solution in the current generation and assign it a fitness value.

3. Select "good" solutions based on fitness value and discard the rest.

4. If acceptable solution(s) are found in the current generation, or maximum number of generations is exceeded, then stop.

5. Alter the solution population using crossover and mutation to create a new generation of solutions.

6. Go to step 2.

The study reported in this paper is to analyse the factors related to tobacco and alcohol related cancers by attempting to optimize the search for the initial starting state vector that will lead to a desired goal state.

4. METHODOLOGY FOR USING FCM WITH GA FOR BACKWARD CHAINING

Here our aim is to identify the initial vector that can cause an FCM to converge to the desired goal vector. Generally we use repeated vector-matrix multiplication when the initial vector is given. Conversely when the desired goal vector is taken as attractor and if we are to find the initial vector avoiding the process involving repeated matrix inverse etc. we use GA based methodology.

i. Start with the desired goal vector G = ([g.sub.1], [g.sub.2]........., [g.sub.n]) which we want to be the attractor.

ii. Define the policy vector P = ([p.sub.1], [p.sub.2].........., [p.sub.n]). This vector specifies for each node, whether it can be a policy node. If the ith node is allowed to be a policy node then pi=1. Otherwise pi is set to 0. When using the FCM we clamp the state vectors as per the information provided by the nodes of the respective policy vector.

iii. When we apply GA, we optimize the following objective function dA (A, G) = min (d) where d [epsilon] D, D = {d (a, G)/a[epsilon]A} and d (a, G) = Euclidean distance of a and G, a is the attractor the FCM converges to, after being fed with an initial vector. A is the set of all attractors, dA defines dissimilarity between the goal vector G and the attractor.

Let P(t) represent the binary coded population generated at time t using the generated initial vectors and policy vectors, attractors are found by applying FCM. Then objective function dA is evaluated. It is minimized by executing the steps based on Goldberg's simple GA as given below.
Begin
t=0
Set mutation rate to a predetermined rate
Set population size to a predetermined size
Initialize P(t) by generating a random population of initial starting
vectors.
Evaluate P(t) using (iii)
While not finished do
t=t+1
Select P(t) from P(t-1)
Rank P(t) according to fitness
Remove the least fit 50% of P(t)
Replace those removed by the process of cross over and mutation of the
best fit 50% of
P(t)
Evaluate P(t) by using (iii)
End while
End.


As a result of above we are left with initial vectors and corresponding policy vectors. There may be more than one possible vector that lead to the state close to the Goal vector. GA is capable of finding many solutions simultaneously.

5. GOAL-ORIENTED WORKING OF FCM WITH GA IN THE CASE OF FACTORS OF ALCOHOL AND TOBACCO RELATED CANCERS

Tobacco addiction ranks as a greatest public health catastrophe of our time. The practice of inhaling cigarette smoke which gained widespread acceptance only during the twentieth century has generated devastating cancer outcomes for our society. Specially lung cancer, previously rare has risen to become the leading cancer killer. A synergistic, multiplicative effect appears to exist between smoking and drinking. Smoking is accepted as the major cause of Cancers of larynx, Pharynx, oral cavity, esophagus and lungs and it is a contributory factor in cancers of the pancreas, bladder, kidney, colen and stomach. Spit Tobacco in its various forms, loose-leaf tobacco, and twist tobacco causes cancers of the oral cavity. ETS (Environmental Tobacco Smoke) also called second hand smoke (combination of smoke released from a burning cigarette between puffs and the smoke exhaled by the smoker) has double the amount of nicotine than main stream smoke and has a higher concentration of carcinogens. People who smoke are likely to drink alcohol and risk of developing cancer is high in people who use tobacco and alcohol. Unemployment, poverty and frustration induce people to those habits. The use of tobacco and alcohol is a social and psychological problem. Agricultural coolies, scavengers, workers are affected by those habits. There is no definite reason for using tobacco and alcohol and we are at the outset justified in using FCM to analyze the problem.Taking in to account of all this factors the nodes considered by an expert are given below.
[C.sub.1] -- Parental use
[C.sub.2] -- Less effective parents
[C.sub.3] -- Smoking tobacco
[C.sub.4] -- Chewing tobacco
[C.sub.5] -- ETS (Environmental tobacco smoke)
[C.sub.6] -- Using alcohol
[C.sub.7] -- Frustration and Low self image
[C.sub.8] -- Mouth cancer
[C.sub.9] -- Throat Cancer
[C.sub.10] -- Lung Cancer
[C.sub.11] -- Stomach Cancer
[C.sub.12] -- Malnutrition
[C.sub.13] -- Unemployment and Poverty


The study is based on an FCM with 13 nodes. The causal weights suggested by the expert are as follows.
[C.sub.1][C.sub.3] = 0.6 [C.sub.1][C.sub.5] = 0.7

[C.sub.3][C.sub.5] = 0.7 [C.sub.3][C.sub.6] = 0.3

[C.sub.3][C.sub.11] = 0.3 [C.sub.3][C.sub.12] = 0.6

[C.sub.5][C.sub.9] = 0.6 [C.sub.5][C.sub.10] = 0.6

[C.sub.6][C.sub.12] = 0.7 [C.sub.7][C.sub.3] = 0.6

[C.sub.13][C.sub.12] = 0.3

[C.sub.1][C.sub.6] = 0.7 [C.sub.2][C.sub.3] = 0.5

[C.sub.3][C.sub.8] = 0.4 [C.sub.3][C.sub.9] = 0.6

[C.sub.4][C.sub.8] = 0.7 [C.sub.4][C.sub.9] = 0.3

[C.sub.6][C.sub.8] = 0.2 [C.sub.6][C.sub.9] = 0.2

[C.sub.7][C.sub.6] = 0.6 [C.sub.12][C.sub.11] = 0.6

[C.sub.2][C.sub.6] = 0.5

[C.sub.3][C.sub.10] = 0.8

[C.sub.4][C.sub.11] = 0.1

[C.sub.6][C.sub.11] = 0.8

[C.sub.13][C.sub.7] = 0.7


The goal of this experiment is to use the workings of the FCM backward chaining technique described in section 4. This experiment models the causal relationships between some life style factors/habits and certain types of cancers. The process of finding the appropriate stimulus vector allows one to investigate the possible causes of a particular type of cancer say throat cancer.

The problem under study may be summarized by the question: "What scenario can lead to throat cancer (represented by node [C.sub.9])"?

(i) The goal vector G is therefore set to (0 0 0 0 0 0 0 0 1 0 0 0 0) where the ninth element represents the maximum state value 1.0 for the concept throat cancer. All of the nodes, except throat cancer, mouth cancer, lung cancer and stomach cancer are potential policy nodes. Accordingly, the policy vector is set as (1 1 1 1 1 1 1 0 0 0 0 1 1).

(ii) The population size is set as 20 and we consider 200 iterations i.e., 200 population sets.

(iii) The stimulus vector generated by genetic algorithm is found to be (.76, .74, .99, .76, 0, 0, .45, .47, .46, .38, .1, .11, .46) with corresponding policy vector (1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0) which leads to a fixed point attractor (.76, .5, .99, .76, .77, .8, .58, .74, .8, .77, .74, .11, .5).

6. CONCLUSION

Thus the level of getting throat cancer is maximum ([C.sub.9] = .8) when the level of smoking tobacco is high ([C.sub.3] = .99). The levels of chewing tobacco ([C.sub.4] = .76) and environment tobacco smoke ([C.sub.5] = .77) being more or less equal in the attractor, it is inferred that both these factors contribute to the incidence of throat cancer in an equal degree. Thus the technique of FCM backward inferencing with Genetic algorithm support is much helpful for the sample study of throat cancer. This procedure can also be extended to other types of tobacco and alcohol related cancers.

REFERENCES

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

(2.) Chong, A., Shamim Khan, M. and Sebastian Khor.2004., "Fuzzy Cognitive Maps with Genetic Algorithm for Goal-oriented decision support", Intl. J. Uncer. Fuzzi. Knowledgebased Sys., World Scientific Publishing Company. (2004).

(3.) Craiger, J.P., and Coovert, M.D.,1994. "Modeling Dynamic Social and Psychological Processes with Fuzzy Cognitive Maps", Proc. of the 3rd IEEE Conference on Fuzzy Systems, 3: 1873-1877 .

(4.) Craiger, J.P., Deborah F.Goodman, Jason Weiss and Adam B.Butler. 1996. "Simulating Organizational Behavior with Fuzzy Cognitive Maps", International Journal of Computational Intelligence and Organizations, 1:120-133.

(5.) George, J. Kleir. and Yuan, B.O., 1995.Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice Hall of India, New Delhi..

(6.) Gold Berg, D.E., 1989.Genetic Algorithms in Search, Optimizations, and Machine Learning, Reading, Addison-Wesley Publishing Company, Massachussetts. (1989).

(7.) Harry Quan, Diane Hershock, Michael Feldman, Duane Sewell and Randal S. Weber, 2004."Cancer of the neck and head", Clinical Oncology [Martin D. Abeloff, James O. Armitage, John E. Niederhuber, Michael B. Kastan and W. Gillies McKenna (eds.)], Elsevier Churchil Livingstone, USA, 3rd edn., 71: 1497-1548. (2004).

(8.) Holland, J. H.,1975. Adaptation in Natural and Artificial Systems. Univ. of Michigan Press, (1975).

(9.) Howard, K. Koh, Christine Kannler and Alan C. Geller,2001. "Cancer Prevention: Preventing Tobacco-Related Cancers", Cancer:Principles and practice of Oncology [Vincent, T. Devita, Jr., Samuel Hellman, and Steven A. Rosenberg (eds.)], A Wollers Kluwer Company, Baltimore, London, 6th Edn., 22: 549-566. (2001).

(10.) Jose, Aguilar. 2005."A survey about Fuzzy Cognitive Maps papers", International Journal of Computational Cognition, 3(2): 27-33.

(11.) Kosko, B., 1986."Fuzzy Cognitive Maps," International Journal of Man-Machine Studies, 34, 65-75 (1986).

(12.) Kosko, B.1988."Hidden Patterns in Combined and Adaptive Knowledge Networks," International Journal of Approximate Reasoning, 2: 377-393 .

(13.) Kosko, B.1992. Neural Networks and Fuzzy Systems: A dynamical systems approach to machine intelligence, New Jersey, USA. Prentice-Hall, Inc., (1992).

(14.) Roy, B., Sessions, Louis, B., Harrison and Arlene A. Forastiere .2001. "Tumors of the Larynx and Hypopharynx", Cancer Principles and practice of Oncology [Vincent, T. Devita, Jr., Samuel Hellman, and Steven A. Rosenberg (eds.)], A Wollers Kluwer Company, Baltimore, London, 6th Edn., 30, Sec.3, pp.861-882. (2001).

(15.) Zadeh, L.A.,1965. "Fuzzy Sets" Information and Control, 8: 139-146 .

R. Nallasamy *, K. Nirmala and K. Kalaiselvi, **

Department of Mathematics, K.N. Govt. Arts College for Women, Thanjavur (T.N.), India

* Department of Mathematics, National Institute of Technology, Tiruchirapalli (T.N.), India

** Additional Director of Medical Education, Directorate of Medical Education, Kilpauk, Chennai (T. N.) India
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Author:Nallasamy, R.; Nirmala, K.; Kalaiselvi, K.
Publication:Bio Science Research Bulletin -Biological Sciences
Date:Jul 1, 2006
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