Generalized fuzzy cognitive maps: a new extension of fuzzy cognitive maps

A fuzzy cognitive maps (FCM) is a cognitive map within the relations between the elements. FCM has been widely used in many applications such as experts system and knowledge engineering. However, classical FCM is inherently short of sufficient capability of representing and aggregating uncertain information. In this paper, generalized FCM (GFCM) is proposed based on genetic algorithm and interval numbers. An application frame of GFCM is detailed. At last, a numerical example about socio-economic system is used to illustrate the effectiveness of the proposed methodology.


Introduction
FCM has received special attentions from the scientific community and done many achievements since it can provide a powerful tool to manipulate knowledge imitating human reasoning and thinking.FCM has used to solve many problems like fuzzy control (Stylios and Groumpos 1999), approximate reasoning (Khan and Quaddus 2004), strategic planning (Konar and Chakraborty 2005), data mining analysis (Yang and Peng 2009), virtual worlds and network models (Dickerson and Kosko 1993), and so on (Gupta andGandhi 2013, 2014;Kandasamy and Indra 2000;Jorge et al. 2011;Yesil et al. 2013;Ganguli 2014;Papageorgiou and Iakovidis 2013;Salmeron and Papageorgiou 2014;Glykas 2013;Na ´poles et al. 2013;Gray et al. 2014;Stylios and Groumpos 2000).It is noted in the real application that Papageorgiou (2011) presents a novel framework for the construction of augmented FCMs based on fuzzy rule-extraction methods for decisions in medical informatics.The study extracted the available knowledge from data in the form of fuzzy rules and inserted them into the FCM, contributing to the development of a dynamic decision support system.FCM has also been investigated for risk analysis of pulmonary infections during patient admission into the hospital (Papageorgiou et al. 2011;Parsopoulos et al. 2004).
Although FCM has achieved success in many fields, there are some limitations inherent in FCM, such as lack of adequate capability to handle uncertain information and lack of enough ability to aggregate the information from different sources.Recently, the uncertain information processing has been heatedly researched in the field of risk analysis (Deng et al. 2011b;Shafiqul Islam et al. 2012) and decision making (Sengupta and Pal 2000;Deng and Chan Bingyi Kang and Hongming Mo have contributed equally to this work. & Yong Deng prof.deng@hotmail.com;ydeng@swu.edu.cn2011;Deng et al. 2011a;Liu et al. 2012;Siraj et al. 2001), environment assessment (Deng et al. 2014), social science (Stakias et al. 2013;Carvalho 2013) and and other fields (Papageorgiou 2013;Kang et al. 2012;Zhang et al. 2013a, b;Chen et al. 2013;Stach et al. 2005;Du et al. 2014;Malik 2013).Some attention has been paid to the first issue by some researchers.For example, Salmeron (2010) proposes an innovative and flexible model based on Grey Systems Theory, called fuzzy grey cognitive maps (FGCM), which can be adapted to a wide range of problems, especially in multiple meaning-based environments.Iakovidis and Papageorgiou (2011) propose an approach based on cognitive maps and intuitionistic fuzzy logic, which is called intuitionistic fuzzy cognitive map (IFCM) to extend the existing FCM by considering the experts hesitancy in the determination of the causal relations between the concepts of a domain.Similarly, after the introduction of neutrosophic logic (similar to intuitionistic fuzzy sets) by Samarandache ( 2002), indeterminacy has been introduced into causal relationships between some of concepts of FCMs.This is a generalization of FCMs and the structure is called neutrosophic cognitive maps (NCMs) (Kandasamy and Smarandache 2003).However, how to extend the ability of FCM to aggregate the information from different sources under uncertain environment is a significant question in the application of FCM and is still an open issue.Hence, a method of aggregating the information from different source base on genetic algorithm and interval number is proposed to enhance the ability of classical FCM to handle the fuzzy information from different sources.The combination of genetic algorithm and FCM base on interval number is shown to be a valuable approach through illustrations.
This paper is organized as follows.In Sect.2, classical FCMs, genetic algorithms and basic operation of interval numbers are introduced.The generalized FCM (GFCM) is proposed in Sect.3. In Sect.4, The application frame of GFCMs is proposed.A numerical example about socioeconomic is used to illustrate the effectiveness of the proposed methodology in Sect. 5.At last, a conclusion is made in Sect.6.

Classical FCMs
Classical fuzzy cognitive map (FCM), an extension of cognitive map, is an illustrative causative representation of the description and models the behavior of any system (Kosko 1986(Kosko , 1996)).Figure 1 (left) illustrates a simple FCM consisting of six concepts C i (i ¼ 1; . ..; 6).The value C i is denoted by A 0 i .Weight x ij 2 ½-1, 1] represents the causal relationship between concept i and concept j, where a negative sign represents inverse causation.This scheme may give rise to the following three types of interactions: x 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; x ij \0, a negative causality, where an increase in the value of the ith concept causes an decrease in the value of the jth concept; x ij ¼ 0, no causal relationship between the ith concept and the jth concept.
The edge matrix of fuzzy cognitive map is denoted as the matrix in Fig. 1 (right) correspondingly.Kosko (1986Kosko ( , 1996) ) proposed a rule to calculate the value of each concept based on the influence of the interconnected concepts, where the content of the following function is normalized in the interval [À1; 1]: where A t j is the normalized (A t j 2 ½0; 1) value (a.k.a activation level) of concept C j at time step t, and f(x) is a threshold function.Generally, a sigmoid function where k [ 0 determines the steepness of f x ð Þ.The coefficient k i 1 express the influence of interconnected concepts in the configuration of the new value of concept A i .For example, in Fig. 1  2 depends on the nature and type of each concept, and may naturally differ from concept to concept.

Genetic algorithms
The simple genetic algorithm over populations defined as multi-sets PðtÞ ¼ ða 0 ðtÞ; a 1 ðtÞ; . ..; a rÀ1 ðtÞÞ consisting of r individual binary lÀtuples a k ðtÞ ¼ ða k;0 ðtÞ; a k;1 ðtÞ; . ..; a k;lÀ1 ðtÞÞ 2 X with fitness values f ða k ðtÞÞ.For the creation of offspring individual in each generation t random genetic operators like crossover v X and mutation l X are applied to parental individuals which are selected according to their fitness values as follows.The population P(0) is initialised appropriately, e.g. by randomly choosing individuals in X (Holland 1975).

Interval number
The distance between two interval numbers Aða 1 ; a 2 Þ and Bðb 1 ; b 2 Þ is defined as (Tran and Duckstein 2002): Let X, Ỹ be two interval numbers, X ¼ ½X L ; X U , Ỹ ¼ ½Y L ; Y U , the operation addition and multiplication of two interval numbers is denoted as 3 Proposed generalized fuzzy cognitive maps based on genetic algorithm and interval number

Generalized fuzzy cognitive maps based on interval number
Generalized fuzzy cognitive map (GFCM), an extension of classical fuzzy cognitive map, is also a directed graph with feedback, consisting of nodes and weighted arcs.Nodes of the graph stands for the concepts that used to described the behavior of the system and they are connected by weighted arcs representing the causal relationships that exist between the concepts.Each concepts Ci is characterized by an interval Ãi that represents its value and it results from the transformation of the fuzzy value of the system's variable.In this way, the representation of the concept is more flexible that the representation of the concept in classical FCM using crisp number.According to the essence of classical cognitive map, experts' opinions are reflected on the estimated of the degree of the cause that is between nodes in the referred concept set, namely weight estimate.Generally, due to the complexity of the relation of concepts and limitation of knowledge and experience of experts, the value of weight estimate should not be simply represented by a crisp number.Hence, the interval number is applied to soften the opinions of experts.
Figure 2 illustrates a simple GFCM consisting of six concepts Ci (i = 1, ..., 6).Weight xij ¼ ½a; b represents the causal relationship between concept i and concept j, where a Â b [ 0; À1\a; b\1.This scheme may give rise to the following three types of interactions: (1) xij ¼ ½a; b [ 0, a positive causality from concept Ci to concept Cj , where 0 a\b 1; (2) xij ¼ ½a; b\0, a negative causality from concept Ci to concept Cj , where À1 a\b 0; With this representing method, the uncertainty of the causal relation between two concepts is more clearly described and handled comparing with the frame of classical FCM.The value Ãi of concept Ci expresses the degree which corresponds to physical value.At each simulation step, the value Ãi of a concept Ci is calculated by computing the influence of the interconnected concepts C0 j s on the specific concept Ci following the calculation rule: where Ãt j is the value of concept Cj at simulation step t, ÃtÀ1 i is the value of concept Ci at simulation step k À 1.The meaning of k i 1 and k i 2 here is the same as meaning of k i 1 and k i 2 in FCM.xij is the weight of the interconnection from concept Ci to concept Cj and f is the nonlinear mapping function which derives from function where X is an interval numbers and k is a parameter determining its steepness.The output of f is also an interval number.It can approximatively handle the uncertain information from concepts and connection weights.The meaning of the f can be illustrated with Fig. 3 (x [ 0).

Aggregation of GFCM using genetic algorithm
This paper integrates the fuzzy numbers by applying the distance based on interval numbers to find a special interval number (½x 1 ; x 2 ; a x 1 x 2 b) whose distance is the optimal value from all other normalized interval numbers with the searching method of Genetic Algorithms.The objective function f ðx 1 ; x 2 Þ is constructed with the summation of all the distance between ½x 1 ; x 2 and each normalized interval number.Hence, the problem is mapped into find the optimal solution of f ðx 1 ; x 2 Þ under the constraint a x 1 \x 2 b with the method of Genetic Algorithms.This process can be mathematically described by Eq. ( 7). min where O i is the interval value of opinion of the ith expert.D 2 ðO; O i Þ is the interval distance between O and O i .a and b is the boundary range of x 1 and x 2 .Now the procedure of aggregating interval numbers (objective factors) is made by GA in detail as follows: (1) Determination of the encoding method The first step is to determine the encoding strategy, in other words, how to represent the data between a and b with the computer language.In this paper, the binary encoding strategy is adopted to represent the chromosome.The length of chromosome depends on the accuracy of encoding.Assume the domain of variable x 1 is [a, b], and the accuracy of encoding prec is the digit after decimal point.The length of a binary string variable L can be calculated as follows: where prec is to represent the precision of the result, max int(x) means the maximum integer which is not more than x.
(2) Determination of the decoding method The decoding of chromosome is to translate the chromosome from binary data to a decimal data in the interval number accordingly.Assume the binary encoding of a chromosome is represented with ðb LÀ1 . ..; b 0 Þ.Firstly, the binary string ðb L ; b LÀ1 . ..; b 0 Þ should be translate to decimal number x 0 , which can be denoted as: Then, the final decoding data in the interval number [a, b] can be calculated as where L is the encoding length of the chromosome.
(3) Construction of the initial population According to Eq. ( 8), the encoding length of chromosome for each variable can be obtained.Hence, the total length of chromosome for variable weight can be accumulated to be composed of the length of a single long chromosome.For the point position of each chromosome, the method of generating population can be denoted as: (4) Determination of the adaptive function and adaptive value Generally, the adaptive function is designed according to the objective function f Ã ðx 1 ; x 2 Þ, and the adaptive function is denoted as Fðx 1 ; x 2 Þ.In order to lay the foundation for calculating the selected probability of each individual behind, the optimalizing direction of adaptive function f Ã ðx 1 ; x 2 Þ should adapt to the incremental direction of adaptive value.Due to the value of objective function f Ã ðx 1 ; x 2 Þ [ 0, the adaptive function Fðx 1 ; x 2 Þ is defined as: In this paper, the objective function f Ã ðx 1 ; x 2 Þ is determined by minSD according to Eq. ( 7).
(5) Determination of the selection criteria In this paper, the proportional selection strategy of adaptive value is introduced, and the the proportion of every individual is defined as selected probability P i .Assume the population whose scale is n as pop ¼ fa 1 ; a 2 ; a 3 ; . ..; a n g, and the adaptive of a i as F i , then the selected probability P i is denoted as: then the accumulative probability Q i of every chromosome is denoted as follows: After a random data r; r 2 ½0; 1 is generated, The selection of chromosome U i for a new population can be selected if (6) Determination of the genetic operators Assume that we have a population pop(1) including four individuals described as follows: After several Roulette Wheel testing, assume chromosome U 2 occupy the most area of the whole circle and chromosome U 3 occupy the least area of the whole circle.According to the selecting criteria, chromosome U 2 is selected to make a reproduction, while chromosome U 3 is fell into disuse.
The crossover operator in this paper adopts the strategy of a single cutting crossover.This method considers the two flanks of the cutting into two substrings, then the right substring should be exchanged with each other to get two new individuals.If the crossover probability P c ¼ 25 %, it means that 25 % of the chromosomes on average exchange each other.
Mutation operator is to change some gene of chromosome with a tiny possibility.If the mutation probability P m ¼ 0:01, it means that 1 % of all the genes are expected to mutate.
(7) Determine the process parameter and terminal condition Through the prior test, the crossover probability is between 0.4 and 0.99, the mutation probability Pmut is 0.0001 and 0.01, the scale of population is between 20 and 100, the terminated condition may be determined by the precise N iterated generation or determined by the minimum bias d, which satisfies where fitness max is the maximum fitness value, and fitness Ã is the objective fitness value.If the judgement condition is not satisfied with the terminal condition, then goto step(4).
A numerical example is used to illustrate the procedure of our method.Three experts, E 1 , E 2 and E 3 , give their linguistic estimation of concept C5 and concept C6 as medium, strong, very strong and their values using interval number are as follows: O 1 ¼ ½0:35; 0:65; O 1 ¼ ½0:5; 0:8; O 1 ¼ ½0:65; 0:9 Assume the importance of each expert is same.First, we suppose that the integrated opinion of the three experts is represented with an interval numbers O ¼ ðx 1 ; x 2 Þ; 0 x 1 \x 2 1.So the distance between O and each expert E i ði ¼ 1; 2; 3Þ can be showed as follows according the distance of the interval numbers.
where 0 x 1 \x 2 1.Hence, the integration of three interval numbers is mapped into solve the minimum of the formula f ðx 1 ; x 2 Þ under the constraint of 0 x 1 \x 2 1.Assuming the method of encoding is binary, the size of population pop_size is 20, the max generation of iteration is 100, the crossover probability pcro is 0.4, the mutation probability pmut is 0.1.The adaptive function is After 100 times repeat, we get the convergent result is [0.5,0.7833], the minimal value the objective function is 0.0381.The trend of objective function can be shown by Fig. 4.

Knowledge acquisition
GFCM can be constructed by a procedure which mainly includes the following three steps: (1) acquisition of the whole concepts related to this issue; (2) refining the concepts; (3) definition of the relation between the concepts (causal or influential relation).Suppose we have m experts, and they are divided into n groups according to their knowledge and background.For each group, the relationship (or the edge weight) between the concepts (or the nodes) is decided by experts exclusively and independently.
The linguistic weight is used so that experts can make a decision more flexible.The values of the linguistic weight are usually very very weak (vvw), very weak (vw), weak (w), medium (m), strong (s), very strong (vs), very very strong (vvs), which is described by Fig. 5.For example in the socio-economic system, three experts are selected to evaluate the influence from population to economic condition, the first expert think that the more scale of the population, the little worse of the economic condition, and the opinion can be shown as -vw.The second expert's opinion is the same as the first one, but more worse, whose opinion is described as -w.The third expert insists that the large scale of the population may improve the economic condition for the labors provided for the economic development are more dominate, the relationship from population to economic is positive, hence, the third opinion can be shown as vw.

Knowledge aggregation
This step allows the aggregation of knowledge acquired from various sources to develop a comprehensive GFCM, which will represent the understanding of the experts about the special issue.The comprehensive GFCM combines partial GFCMs from inner groups and outer groups.The aggregation of knowledge from inner groups is for the opinions of the experts of each group and the aggregation of knowledge from outer groups is for the edges of the partial GFCMs.
In this part, the main problem is how to aggregate the relationship of the each single GFCM and the framework of all the GFCMs.For the relationship between criteria, assume that the weight of all domain experts (whose opinions are described by linguistic language, like Fig. 5) is equal, we aggregate opinions from experts (interval number) by trying to find a general interval number whose total distance from other opinions is minimum.Hence the question of fusion is converted into an optimization problem.GA is good at solving optimization problem than others such as ''ant colony optimization'' for its global search capability.The detailed solution is show in the second part of the proposed methodology.

Training and interpreting GFCM
. ..; n) be an initial vector state in the GFCM and let k i 1 ¼ 1, k j 2 ¼ 1 and the sigmoid function with k ¼ 1 be used as a threshed function.After several times iteration, it can be seen that the FCM reaches an equilibrium state approximately.GFCM is a dynamic system, whether the dynamic is reaches equilibrium, the paper (Boutalis et al. 2009) has provided some inspiration for the interpretation of the condition.
Once the GFCM reaches equilibrium, the activation values provide the 'triggering or firing' strength of those concepts for a given scenario.Generally, when the GFCM reaches equilibrium, the activation levels are transformed back to the corresponding values.These activation levels may be interpreted 'quantitatively' or 'qualitatively'.The interpretation of these concepts will determine the judgement for a given scenario.

numerical example
This section illustrates the application of the proposed method to a socio-economic model.It is constructed with Population, Crime, Economic condition, Poverty, and Unemployment as nodes or concepts.Our purpose is to evaluate the trend of factors changing with any one factor using GFCM.
First, the structure of GFCM should be established using several sources of partial knowledge.All the available experts are divided into three groups (group1, group2, and grou3).and the opinions are provided in Tables 1, 2 and 3.They can be described as Figs.6, 7 and 8 accordingly.
Next, the opinions from different experts and the partial GFCMs are combined together based on genetic algorithm (see Table 4; Fig. 9), then the final combined interval adjacency matric (denoted as Table 5) of the GFCM for socio-economic scenario is established.
For considering the influence of node Population to the socio-economic system, the node of Population is made triggered, and others are non-triggered.Let A 0 ¼ ½ Ã0 1 ; Ã0 2 ; Ã0 3 ; Ã0 4 ; Ã0 5 ¼ ½vvs; 0; 0; 0; 0 ¼ ½½0:8; 1; 0; 0; 0; 0 be  an initial vector state in the GFCM and let k i 1 ¼ 1, k j 2 ¼ 1 and the sigmoid function with k ¼ 1 be used as a threshed function.After six times iteration, it can be seen that the FCM reaches an equilibrium state approximately.
The process of the convergence in this simulation can be as Figs. 10, 11 and 12.The final mean state of the nodes ''Population'', ''Crime'', ''Economic condition'', ''Poverty' ' and ''Unemployment'' is [0.68,0.73,0.58,0.87,0.61].It can be concluded the population is initially triggering, the rate of crime is increasing, the poverty is more serious, and the economic condition (volume of economic) may be improved; the population is decreasing gradually at the same time, and all the nodes reach a new equilibrium state.GFCMs are uncertain-graph structures for representing causal reasoning.GFCM can be considered as the development of FCMs with considering aggregating information from different sources under uncertain environment.It can be widely used in many applications such as decision making and uncertain reasoning.The frame of GFCMs is detailed and a simulation about socio-economic system is also shown to introduce the application of GFCMs.

Fig. 2
Fig. 2 The structural diagram of generalized fuzzy cognitive map

Fig. 4
Fig.4The trend of objective function value in knowledge aggregation

Fig. 9
Fig. 9 Result of GFCM simulations Fig. 10 Results of GFCM simulations (trend of lower bound of each concept value)

Fig. 11 Fig. 12
Fig. 11 Results of GFCM simulations (trend of upper bound of each concept value) the concept C 6 receives inputs from concepts C 1 , C 3 ,

Table 1
Experts' knowledge in group 1 (values of aggregated knowledge are shown in bold)

Table 4
Aggregation of experts' knowledge in groups (values of aggregated knowledge are shown in bold)