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Fuzzy Cognitive Map and a Mining Methodology Based on Multi-relational Data Resources

Fuzzy Cognitive Map and a Mining Methodology Based on Multi-relational Data Resources Fuzzy Inf. Eng. (2009) 4: 357-366 DOI 10.1007/s12543-009-0028-7 ORIGINAL ARTICLE Fuzzy Cognitive Map and a Mining Methodology Based on Multi-relational Data Resources Bing-ru Yang· Zhen Peng Received: 26 August 2009/ Revised: 29 October 2009/ Accepted: 19 November 2009/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2009 Abstract Fuzzy Cognitive Map (FCM) is a new kind of intelligent facility, which has many advantages such as intuitive representing knowledge skills and strong in- ference mechanisms based on numeric matrix, etc. In practical application, a major- ity of data is stored into the relational database in the form of Entity-Relationship schema. How to mine FCM directly from multi-relational data resource has be- come a key problem in researching FCM and an important direction and area of data mining. However, traditional approaches for obtaining FCM always rely on experience of domain experts or do not take into account the characteristics of multi- relationship. Based on these, the paper proposes a new model of Two-layer Tree-type FCM (TTFCM) and a new mining methodology based on gradient descent method. Keywords Fuzzy cognitive map· Multi-relational data mining· Entity-relationship schema· Two-layer tree-type FCM· Gradient descent method 1. Introduction FCM [1, 2] is a soft computing method, which combines the elements of fuzzy logic, neural networks and graph theory, stores knowledge in concept nodes and the re- lations between the nodes, and simulates the reasoning process of logic knowledge through the causal link between nodes. FCM models have been widely developed and used in numerous areas of applications such as social, economic and political fields [3,4,5]. However, in order to show these advantages of FCM, the key is to ob- tain corrected FCM. At present, the methods of developing FCM are divided into two main categories. One is manual methods, which exhibit the weaknesses of critical dependence on experts, because the model is manually designed and implemented Bing-ru Yang· Zhen Peng () Information Engineering College, Beijing University of Science and Technology, Beijing 100083, P.R.China email: zhen peng1981@163.com 358 Bing-ru Yang· Zhen Peng (2009) based on experts’ mental understanding. The other is calculation methods, which can automatically or semi-automatically learn FCM from historical data resources, but they do not consider multi-relational properties of these data resources at all. FCM mining methodology from multi-relational data resources is a kind of MRDM [6,7], which is an important direction and area of data mining and can look for pat- terns involving multiple tables (relations) from a relational database. That is to say, it is not necessary to transfer many tables in the relational database into single table so as to effectively avoid the issues of information loss, statistical skew and inefficiency in propositional logic learning. However, a lot of MRDM approaches are based on the database in star schema [8], while a majority of data in practical application is stored in the database in Entity-Relationship (E-R) schema, which is much more complex than star schema. So the paper proposes a TTFCM model and its mining method based on multi- relational data resources in the form of E-R schema. The method first is to make a class node in high level represent each relational table. Then, it is to establish many FCMs in low level by learning algorithms. Finally, One FCM in high level composed of many class nodes and many FCMs in low level form a TTFCM model. The remainders of the paper are organized as follows: Section 2 introduces a fun- damental theory of FCM and traditional learning methods. Section 3 represents a TTFCM modeling for the multi-relational data resources. Finally, a methodology of learning TTFCM based on a gradient descent method is proposed in Section 4. 2. FCM and The Mining Methods 2.1. FCM The topology of FCM is a 4-tuple (C, E, W, A) (Fig.1), in which C = {C , C ,..., C } 1 2 n are a set of nodes representing concepts. E = {< C , C >| C , C ∈ C} are oriented 1 2 1 2 arcs denoting the cause and effect relationship that one concept has on the others, Fig. 1: A simple FCM. W = {W | W is the weight value of the interconnection < C , C >}, W belongs ij ij i j ij to the interval [-1,1]. W in FCM is a square matrix of n× n, shown in Fig. 2. If the Fuzzy Inf. Eng. (2009) 4: 357-366 359 weight value indicates positive causality between concepts C and C , W > 0, then i j ij an increase in the value of concept C will cause an increase in the value of concept C and a decrease in the value of C will lead to a decrease in the value of C . When j i j there is inverse causality between two concepts, W < 0, an increase in the value of ij concept C causes a decrease in the value of the second concept and a decrease in the value of the first concept causes an increase in the value of the concept C . When there is no relationship between the two concepts, W = 0. The strength of weight ij W indicates the degree of influence between concept C and C , ij i j Fig. 2: The weight matrix of FCM. At every simulation step, each node C has a state value A (t), then i i A(t) = {A (t), A (t),..., A (t)}, 1 2 n which is the set of state values of corresponding concepts in C. The value of C at t+1 step can be derived according to A (t + 1). The transformation function commonly used is bivalent function, trivalent function or logistic function. A (t+ 1) = f (A (t)+ A (t)W ), j j i ij i=1,i j f (x) = , −λx 1+ e whereλ> 0 is a parameter determining its steepness, f (x) is a sigmoid function and takes values in the interval [0,1]. In each step, new states of the concepts are derived according to A (t+ 1) and each weight value in the FCM is modified and adjusted to train the FCM. After a number of iterations, FCM may arrive in one of the following states: A: fixed-point attractor; B: limit cycle; C chaotic attractor. When A or B is arrived, FCM system is in a steady state, while C state is instability. 2.2. FCM Mining Methods In general, two approaches to development of FCM are used: manual and computa- tional. Manual methods for development of FCM models have also a major disad- vantage of relying on human knowledge, which implies subjectivity of the developed 360 Bing-ru Yang· Zhen Peng (2009) model and problems with unbiased assessing of its accuracy. Also, in case of large and complex domains, the resulting FCM model requires large amount of concepts and connections that need to be established, adding substantially to the difficulty of a manual development process. These problems have led to the development of computational methods for learn- ing FCM connection matrix, i.e., casual relationships (edges), and their strength (weights) based on historical data. In this way, the expert knowledge is substituted by a set of historical data and a computational procedure that is able to automatically compute the connection matrix. A number of algorithms for learning FCM model structure have been recently proposed. In general two main learning paradigms are used, i.e., Hebbian learning and evolutionary computation. The former includes DHL (Differential Hebbian Learning) [9], BDA (Balanced Differential Algorithm) [10], NHL (Nonlinear Hebbian Learning) [11] and AHL (Ac- tive Hebbian Learning) [12], etc. These methods train FCM from a single sample of one control system to arrive a steady state. The latter is GA (Genetic Strategy) [13], PSO (Particle Swarm Optimization) [14], RCGA (Real Coded Genetic Algorithm) [15] and Parallel Genetic Learning [16], etc. These methods learn FCM from time- series data resources of the single sample to simulate the dynamic behavior of the system. In addition, the paper [17] has proposed a gradient descent method applied in the FCM learning from different samples. All of these learning methods aim to look for the weight matrix of FCM, which can simulate data in single table. But the above models and algorithms proposed are not suitable for the multi-relational data. Besides, FCM models of large-scale complex systems have been proposed, such as aggregation FCM [18], Hierarchical FCM [19] and quotient FCM [20], etc. The com- plex FCMs can be used to model the multi-relational data resource. The researches only have used the FCMs for intelligent inference in steading of discussing the way to learn the FCMs at all. 3. TTFCM 3.1. The Issues Most of MRDM algorithms are researched for multi-relational data in the form of star schema and proposed for improving its performance. And these methods focus on preventing connection operation and reducing the number of scans. However, majority of data in practical application are stored in the relational database in the form of Entity-Relationship schema, which is much more complex than star schema. In the E-R model, the relationships between directly related two tables have two conditions of one-to-one and one-to-many. In addition, there may be many-to-many among any tables as well. Based on the above analysis, if combining the data of many tables into one table, we can get the following conclusions. The combination of tables in accordance with one-to-many and many-to-many causes the explosion of data, in turn leads to the statistical skew of data in one table. Only one-to-one relationship will not arouse the above situations. Fuzzy Inf. Eng. (2009) 4: 357-366 361 Fig. 3: Three relational tables in E-R model. For example, there is a simple relational database, shown in Fig.3, which in- cludes three tables that are respectively Student, SC and Course. The relationships of Student-SC and Course-SC are both one-to-many. Table 1 is an universal ta- ble of three relational tables. The values of the attributes in the Student and the Course are extended from 3 records to 6 records. To the Rule of Age=20 − 25 → 1 1 Address=Haidian, the support is and the confidence is in Fig.3, but the support 3 2 3 3 is and the confidence is in Table1. So the statistical results are inconsistent. 6 5 FCM mining from multi-relational data resources is a kind of MRDM, which does not transfer many tables into a single table. In addition, FCM mining can get the relationship and the strength of the different attributes or attribute sets in the same level and different levels. The aim of the paper is to learn FCM connection matrix by computational method from many samples in E-R model for further knowledge representation and inference. 3.2. TTFCM Model The TTFCM is one 4-tuple, U = (C, E, W, A), in which C is the set of all nodes in 362 Bing-ru Yang· Zhen Peng (2009) Table 1: The universal table of Student-SC-Course. Sno Address Age GPA Grade Credit Type a Haidian 20-25 2-2.5 4 3 1 a Haidian 20-25 2-2.5 2 2 2 a Haidian 20-25 2-2.5 4 2 3 a Xicheng 25-30 2.5-3 3 2 3 a Chaoyang 20-25 3-4 3 2 3 a Chaoyang 20-25 3-4 3 3 1 ... ... ... ... ... ... ... TTFCM, where C = {< C , C ... C >,< C ,...>,...,< C ...>}; a a a b n 1 m these nodes in same symbol of “ <> ” belong to one relation, in which first node is a class node representing all nodes in the same table. E = {< C , C > | C and C i j i j belong to same relation or first nodes in different relationships } denotes the related oriented arc in different nodes of TTFCM. W = {W , W , W ,,W }, 0 a b n where W is an associated matrix consisting of class nodes, the others respectively represent the weight matrix of the relational table. The state A corresponds the states of the nodes in C. A = {< A , A ,... A >,< A ,...>,...,< A ...>}, a a a b n 0 1 m 0 0 A = {Label, Level, [A (t)] | i is the relation ID and j is the attribute ID}, ij ij in which first two parts are the state flags meaning its table label and level, and the final is the value at t-th step. As we can see, TTFCM includes two levels. In the high-level there is only one FCM, obtained from the data in the low-level relations. The low-level contains many FCMs and each FCM corresponds to a table. The creation of the nodes in the high- level follows the principles. (1) If there is a class feature in a table, the feature is the class node in the high-level on behalf of the table. (2) If there is only one feature in a table, the feature is the class node in the high- level and its state value A is got by clustering the data. The low-level FCM i0 does not exist actually. (3) If there is not the above two situations, the class node representing the table is created and its state value A is obtained by clustering the data. i0 Fuzzy Inf. Eng. (2009) 4: 357-366 363 Each low-level FCM includes the feature nodes of the table and one class node that interconnect. In the high-level, the FCM is composed of all class nodes that associate each other by tuple ID propagation. The Fig.4 is the TTFCM of the Fig.3. A high-level FCM in the solid line circle and two low-level FCMs inside the dashed border form one TTFCM. The attribute GPA is the class label of Student, so it is the class node in the high-level FCM. The attribute Grade (Grd) is the only feature in SC and is the high-level class node as well. The Cou represents the Course table. Fig. 4: Two-layer Tree-type FCM model of the relations in Fig. 3. 4. The Methodology of Mining TTFCM based on Multi-relations 4.1. The Algorithm of Learning TTFCM We propose a TTFCM mining algorithm based on gradient descent method. The first process is to establish FCMs in low-level, and then to construct only FCM in high- level. To the i-th relation, the detail procedures of the algorithm are as follows. Step 1 Determine the nodes of FCM in low-level. i. Define the feature attributes and the state flags. Except these fields of class label, primary key and foreign key with the logo role, each field can be seen a node of low- level FCM. These keys only are used to establish the relationship between tables and do not belong to the scope of the attributes. In the state of the node, the Label is the table label and the Level is equal to 1. ii. Determine the class node according to the above three principles. The Label of the class node is the table label and the Level is 2. Step 2 Initialize the state values of the nodes. i. If there is only one feature of the table, the state value needs to be classified. ii. Using database initializing technology (such as discrete, standardization, nor- 364 Bing-ru Yang· Zhen Peng (2009) malization, point scaling methods) makes the initial values in the scope of [0,1]. Step 3 Calculate the correlation matrix Wi following these steps: i. k=0. ii. Set the learning coefficient and the parameter, which are very small positive real numbers. iii. Set the initial value of W (0). iv. Assign initialized values d to A (t)at t = 0. ij ij v. To the k-th record, implement the following steps. a) Calculate A (t+ 1) according to the formula of ij A (t+ 1) = f (A (t)+ A (t)W (t)), ij ij ij ijl j=1, jl where m is the number of attributes including class label. b) Update the W (t+ 1) according to the three formula that are ijl W (t+ 1) = W (t)+ρδ , ijl ijl ijl ∂(E(W )) ρδ = , ijl ∂(W (t)) ijl and E(W ) = (d − A (t+ 1)) . i ij ij j=1 c) Calculate E(W ). d) If E(W ) is small enough or k is equal to K (the total records), go to the Step 4, otherwise go to the Step 3 (vi). vi. k = k+ 1 and go to Step 3 (v). Step 4 Output the finial weight W that is the optimal value. The second process is to learn only FCM in high-level of TTFCM. The detail pro- cedures of the second algorithm are as the follows. Step 1’ k=0. Step 2’ Set the relevant parameters. Step 3’ Start from the node on behalf of one table in one part. Step 4’ Retrieve the initialized k-th class value as the state one of the node. Step 5’ Get the collection of record IDs of the table. Step 6’ Search out the record IDs in the associated tables by tuple ID propagation. Step 7’ Determine the state values of the associated tables by cluster analysis. Step 8’ Go to Step 3’ until the state values of all the nodes in the high-level. Step 9’ Calculate the weight matrix W by gradient descent method that is similar to 0 Fuzzy Inf. Eng. (2009) 4: 357-366 365 the Step 3 (v) of first training algorithm. Step 10’ k = k+ 1, if k < K (K is the number of categories in the one part selected), go to Step 4’, otherwise output the finial weight W as the optimal value. 4.2. The Analysis ++ The experiments of the artificial Student-Score dataset are made by C program- ming. The learning of low-level FCM is a local simulating process avoiding the one- to-many and many-to-many relationships and ensures the information to be valid. In the high-level, the state values of class nodes in the multi-parts are got through online clustering methods so as to avoid the one-to-many and many-to-many relationships and statistical skew. The results are that there are weight values between any two nodes in the same ta- ble or the high-level relation. So the weight matrixes need to be further simplified by setting the threshold value. Finally, the TTFCM obtained can be used for knowledge inference of new data. The inference method firstly is still the FCM inference in the low-level and then the reasoning in the high-level FCM for intelligence-decision or data classification and so on. 5. Conclusion The paper proposes one kind of TTFCM to model the system with the relational data in the form of E-R schema through analyzing the relational database and the multi-relational characteristic. The TTFCM includes one FCM in high-level and many FCMs in low-level forming a two-level tree. The TTFCM can be got from multi-relational data resource through the learning algorithm proposal, which mines respectively low-level FCMs and high-level FCM. The method avoids the statistical skew in the multi-relationship mining and also ensures the information not to be lost. The TTFCM can be effective and widely applied in intelligent inference and decision- making. At the same time, more efficient mining algorithm is studied. Acknowledgments We would like to thank Knowledge Engineering Institute of Beijing University of Sci- ence and Technology and National Natural Science Foundation of China (No. 636750 30, No. 60875029) support. References 1. Kosko B (1986) Fuzzy cognitive maps. International Journal Man-Machine Studies 24: 65-75 2. Aguilar J (2005) A survey about fuzzy cognitive maps papers (Invited Paper). International Journal of Computational Cognition 3(2): 27-33 3. Hafner V V (2000) Cognitive maps for navigation in open environments. Proc. of the 6th Interna- tional Conference on Intelligent Autonomous Systems 4. Tsadiras A K (2003) Using fuzzy cognitive maps for e-commerce strategic planning. Proc. of 9th Panhellenic Conf. on Informatics 366 Bing-ru Yang· Zhen Peng (2009) 5. Ndousse D T, Okuda T (1996) Computational intelligence for distributed fault management in net- works using fuzzy cognitive maps. Proc. of the IEEE International Conference on Communications 6. Dzeroski S, Lavrac N (2001) Relational data mining. Berlin: Springer 7. Dzeroski S (2003) Multi-relational data mining: an introduction. SIGKDD Explorations 5(1): 1-16 8. He J, Liu H Y, Du X Y (2007) Mining of multi-relational association rules. Journal of Software 18(11): 2752-2765 9. Dickerson J A, Kosko B (1994) Virtual worlds as fuzzy cognitive maps. Presence 3(2): 173-189 10. Huerga A V (2002) A balanced differential learning algorithm in fuzzy cognitive maps. Universitat Politecnica de Catalunya(UPC) Spain: Technical Report 11. Papageorgiou E I, Stylios C D, Groumpos P P (2003) Fuzzy cognitive map learning based on nonlin- ear Hebbian rule. Proc. of the Australian Conference on Artificial Intelligence 12. 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): 219-249 13. Koulouriotis D E, Diakoulakis I E, Emiris D M (2001) Learning fuzzy cognitive maps using evolution strategies: a novel schema for modeling and simulating high-level behavior. Proc. of IEEE Congr. on Evolutionary Computation 14. Parsopoulos K E, Papageorgiou E I, Groumpos P P, Vrahatis M N (2003) A first study of fuzzy cognitive maps learning using particle swarm optimization. Proc. of IEEE Congr. on Evolutionary Computation 15. Stach W, Kurgan L, Pedrycz W, Reformar M (2005) Genetic learning of fuzzy cognitive maps. Fuzzy Sets and Systems 15(3): 371-401 16. Stach W (2006) Parallel genetic learning of fuzzy cognitive maps. Final Report for IEEE-CIS Walter Karplus Summer Research Grant 17. Chen Z, Montazemi A R (2007) The methodology of mining cognitive maps based on data resources. Chinese Journal of Computers 30(8): 1446-1454 18. Thomas D, Okuda N T (1996) Computational intelligence for distributed fault management in net- works using fuzzy cognitive maps. Proc. of IEEE ICC 19. 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. Proc. of IEEE Transactions on Biomedical Engineering 20. Zhang G Y, Ma X Y, Yang B R (2007) Decomposition for fuzzy cognitive maps of complex systems. Computer Science 134 (14): 129-134 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Fuzzy Cognitive Map and a Mining Methodology Based on Multi-relational Data Resources

Yang, Bing-ru; Peng, Zhen
Fuzzy Information and Engineering , Volume 1 (4): 10 – Dec 1, 2009
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Abstract

Fuzzy Inf. Eng. (2009) 4: 357-366 DOI 10.1007/s12543-009-0028-7 ORIGINAL ARTICLE Fuzzy Cognitive Map and a Mining Methodology Based on Multi-relational Data Resources Bing-ru Yang· Zhen Peng Received: 26 August 2009/ Revised: 29 October 2009/ Accepted: 19 November 2009/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2009 Abstract Fuzzy Cognitive Map (FCM) is a new kind of intelligent facility, which has many advantages such as intuitive representing knowledge skills and strong in- ference mechanisms based on numeric matrix, etc. In practical application, a major- ity of data is stored into the relational database in the form of Entity-Relationship schema. How to mine FCM directly from multi-relational data resource has be- come a key problem in researching FCM and an important direction and area of data mining. However, traditional approaches for obtaining FCM always rely on experience of domain experts or do not take into account the characteristics of multi- relationship. Based on these, the paper proposes a new model of Two-layer Tree-type FCM (TTFCM) and a new mining methodology based on gradient descent method. Keywords Fuzzy cognitive map· Multi-relational data mining· Entity-relationship schema· Two-layer tree-type FCM· Gradient descent method 1. Introduction FCM [1, 2] is a soft computing method, which combines the elements of fuzzy logic, neural networks and graph theory, stores knowledge in concept nodes and the re- lations between the nodes, and simulates the reasoning process of logic knowledge through the causal link between nodes. FCM models have been widely developed and used in numerous areas of applications such as social, economic and political fields [3,4,5]. However, in order to show these advantages of FCM, the key is to ob- tain corrected FCM. At present, the methods of developing FCM are divided into two main categories. One is manual methods, which exhibit the weaknesses of critical dependence on experts, because the model is manually designed and implemented Bing-ru Yang· Zhen Peng () Information Engineering College, Beijing University of Science and Technology, Beijing 100083, P.R.China email: zhen peng1981@163.com 358 Bing-ru Yang· Zhen Peng (2009) based on experts’ mental understanding. The other is calculation methods, which can automatically or semi-automatically learn FCM from historical data resources, but they do not consider multi-relational properties of these data resources at all. FCM mining methodology from multi-relational data resources is a kind of MRDM [6,7], which is an important direction and area of data mining and can look for pat- terns involving multiple tables (relations) from a relational database. That is to say, it is not necessary to transfer many tables in the relational database into single table so as to effectively avoid the issues of information loss, statistical skew and inefficiency in propositional logic learning. However, a lot of MRDM approaches are based on the database in star schema [8], while a majority of data in practical application is stored in the database in Entity-Relationship (E-R) schema, which is much more complex than star schema. So the paper proposes a TTFCM model and its mining method based on multi- relational data resources in the form of E-R schema. The method first is to make a class node in high level represent each relational table. Then, it is to establish many FCMs in low level by learning algorithms. Finally, One FCM in high level composed of many class nodes and many FCMs in low level form a TTFCM model. The remainders of the paper are organized as follows: Section 2 introduces a fun- damental theory of FCM and traditional learning methods. Section 3 represents a TTFCM modeling for the multi-relational data resources. Finally, a methodology of learning TTFCM based on a gradient descent method is proposed in Section 4. 2. FCM and The Mining Methods 2.1. FCM The topology of FCM is a 4-tuple (C, E, W, A) (Fig.1), in which C = {C , C ,..., C } 1 2 n are a set of nodes representing concepts. E = {< C , C >| C , C ∈ C} are oriented 1 2 1 2 arcs denoting the cause and effect relationship that one concept has on the others, Fig. 1: A simple FCM. W = {W | W is the weight value of the interconnection < C , C >}, W belongs ij ij i j ij to the interval [-1,1]. W in FCM is a square matrix of n× n, shown in Fig. 2. If the Fuzzy Inf. Eng. (2009) 4: 357-366 359 weight value indicates positive causality between concepts C and C , W > 0, then i j ij an increase in the value of concept C will cause an increase in the value of concept C and a decrease in the value of C will lead to a decrease in the value of C . When j i j there is inverse causality between two concepts, W < 0, an increase in the value of ij concept C causes a decrease in the value of the second concept and a decrease in the value of the first concept causes an increase in the value of the concept C . When there is no relationship between the two concepts, W = 0. The strength of weight ij W indicates the degree of influence between concept C and C , ij i j Fig. 2: The weight matrix of FCM. At every simulation step, each node C has a state value A (t), then i i A(t) = {A (t), A (t),..., A (t)}, 1 2 n which is the set of state values of corresponding concepts in C. The value of C at t+1 step can be derived according to A (t + 1). The transformation function commonly used is bivalent function, trivalent function or logistic function. A (t+ 1) = f (A (t)+ A (t)W ), j j i ij i=1,i j f (x) = , −λx 1+ e whereλ> 0 is a parameter determining its steepness, f (x) is a sigmoid function and takes values in the interval [0,1]. In each step, new states of the concepts are derived according to A (t+ 1) and each weight value in the FCM is modified and adjusted to train the FCM. After a number of iterations, FCM may arrive in one of the following states: A: fixed-point attractor; B: limit cycle; C chaotic attractor. When A or B is arrived, FCM system is in a steady state, while C state is instability. 2.2. FCM Mining Methods In general, two approaches to development of FCM are used: manual and computa- tional. Manual methods for development of FCM models have also a major disad- vantage of relying on human knowledge, which implies subjectivity of the developed 360 Bing-ru Yang· Zhen Peng (2009) model and problems with unbiased assessing of its accuracy. Also, in case of large and complex domains, the resulting FCM model requires large amount of concepts and connections that need to be established, adding substantially to the difficulty of a manual development process. These problems have led to the development of computational methods for learn- ing FCM connection matrix, i.e., casual relationships (edges), and their strength (weights) based on historical data. In this way, the expert knowledge is substituted by a set of historical data and a computational procedure that is able to automatically compute the connection matrix. A number of algorithms for learning FCM model structure have been recently proposed. In general two main learning paradigms are used, i.e., Hebbian learning and evolutionary computation. The former includes DHL (Differential Hebbian Learning) [9], BDA (Balanced Differential Algorithm) [10], NHL (Nonlinear Hebbian Learning) [11] and AHL (Ac- tive Hebbian Learning) [12], etc. These methods train FCM from a single sample of one control system to arrive a steady state. The latter is GA (Genetic Strategy) [13], PSO (Particle Swarm Optimization) [14], RCGA (Real Coded Genetic Algorithm) [15] and Parallel Genetic Learning [16], etc. These methods learn FCM from time- series data resources of the single sample to simulate the dynamic behavior of the system. In addition, the paper [17] has proposed a gradient descent method applied in the FCM learning from different samples. All of these learning methods aim to look for the weight matrix of FCM, which can simulate data in single table. But the above models and algorithms proposed are not suitable for the multi-relational data. Besides, FCM models of large-scale complex systems have been proposed, such as aggregation FCM [18], Hierarchical FCM [19] and quotient FCM [20], etc. The com- plex FCMs can be used to model the multi-relational data resource. The researches only have used the FCMs for intelligent inference in steading of discussing the way to learn the FCMs at all. 3. TTFCM 3.1. The Issues Most of MRDM algorithms are researched for multi-relational data in the form of star schema and proposed for improving its performance. And these methods focus on preventing connection operation and reducing the number of scans. However, majority of data in practical application are stored in the relational database in the form of Entity-Relationship schema, which is much more complex than star schema. In the E-R model, the relationships between directly related two tables have two conditions of one-to-one and one-to-many. In addition, there may be many-to-many among any tables as well. Based on the above analysis, if combining the data of many tables into one table, we can get the following conclusions. The combination of tables in accordance with one-to-many and many-to-many causes the explosion of data, in turn leads to the statistical skew of data in one table. Only one-to-one relationship will not arouse the above situations. Fuzzy Inf. Eng. (2009) 4: 357-366 361 Fig. 3: Three relational tables in E-R model. For example, there is a simple relational database, shown in Fig.3, which in- cludes three tables that are respectively Student, SC and Course. The relationships of Student-SC and Course-SC are both one-to-many. Table 1 is an universal ta- ble of three relational tables. The values of the attributes in the Student and the Course are extended from 3 records to 6 records. To the Rule of Age=20 − 25 → 1 1 Address=Haidian, the support is and the confidence is in Fig.3, but the support 3 2 3 3 is and the confidence is in Table1. So the statistical results are inconsistent. 6 5 FCM mining from multi-relational data resources is a kind of MRDM, which does not transfer many tables into a single table. In addition, FCM mining can get the relationship and the strength of the different attributes or attribute sets in the same level and different levels. The aim of the paper is to learn FCM connection matrix by computational method from many samples in E-R model for further knowledge representation and inference. 3.2. TTFCM Model The TTFCM is one 4-tuple, U = (C, E, W, A), in which C is the set of all nodes in 362 Bing-ru Yang· Zhen Peng (2009) Table 1: The universal table of Student-SC-Course. Sno Address Age GPA Grade Credit Type a Haidian 20-25 2-2.5 4 3 1 a Haidian 20-25 2-2.5 2 2 2 a Haidian 20-25 2-2.5 4 2 3 a Xicheng 25-30 2.5-3 3 2 3 a Chaoyang 20-25 3-4 3 2 3 a Chaoyang 20-25 3-4 3 3 1 ... ... ... ... ... ... ... TTFCM, where C = {< C , C ... C >,< C ,...>,...,< C ...>}; a a a b n 1 m these nodes in same symbol of “ <> ” belong to one relation, in which first node is a class node representing all nodes in the same table. E = {< C , C > | C and C i j i j belong to same relation or first nodes in different relationships } denotes the related oriented arc in different nodes of TTFCM. W = {W , W , W ,,W }, 0 a b n where W is an associated matrix consisting of class nodes, the others respectively represent the weight matrix of the relational table. The state A corresponds the states of the nodes in C. A = {< A , A ,... A >,< A ,...>,...,< A ...>}, a a a b n 0 1 m 0 0 A = {Label, Level, [A (t)] | i is the relation ID and j is the attribute ID}, ij ij in which first two parts are the state flags meaning its table label and level, and the final is the value at t-th step. As we can see, TTFCM includes two levels. In the high-level there is only one FCM, obtained from the data in the low-level relations. The low-level contains many FCMs and each FCM corresponds to a table. The creation of the nodes in the high- level follows the principles. (1) If there is a class feature in a table, the feature is the class node in the high-level on behalf of the table. (2) If there is only one feature in a table, the feature is the class node in the high- level and its state value A is got by clustering the data. The low-level FCM i0 does not exist actually. (3) If there is not the above two situations, the class node representing the table is created and its state value A is obtained by clustering the data. i0 Fuzzy Inf. Eng. (2009) 4: 357-366 363 Each low-level FCM includes the feature nodes of the table and one class node that interconnect. In the high-level, the FCM is composed of all class nodes that associate each other by tuple ID propagation. The Fig.4 is the TTFCM of the Fig.3. A high-level FCM in the solid line circle and two low-level FCMs inside the dashed border form one TTFCM. The attribute GPA is the class label of Student, so it is the class node in the high-level FCM. The attribute Grade (Grd) is the only feature in SC and is the high-level class node as well. The Cou represents the Course table. Fig. 4: Two-layer Tree-type FCM model of the relations in Fig. 3. 4. The Methodology of Mining TTFCM based on Multi-relations 4.1. The Algorithm of Learning TTFCM We propose a TTFCM mining algorithm based on gradient descent method. The first process is to establish FCMs in low-level, and then to construct only FCM in high- level. To the i-th relation, the detail procedures of the algorithm are as follows. Step 1 Determine the nodes of FCM in low-level. i. Define the feature attributes and the state flags. Except these fields of class label, primary key and foreign key with the logo role, each field can be seen a node of low- level FCM. These keys only are used to establish the relationship between tables and do not belong to the scope of the attributes. In the state of the node, the Label is the table label and the Level is equal to 1. ii. Determine the class node according to the above three principles. The Label of the class node is the table label and the Level is 2. Step 2 Initialize the state values of the nodes. i. If there is only one feature of the table, the state value needs to be classified. ii. Using database initializing technology (such as discrete, standardization, nor- 364 Bing-ru Yang· Zhen Peng (2009) malization, point scaling methods) makes the initial values in the scope of [0,1]. Step 3 Calculate the correlation matrix Wi following these steps: i. k=0. ii. Set the learning coefficient and the parameter, which are very small positive real numbers. iii. Set the initial value of W (0). iv. Assign initialized values d to A (t)at t = 0. ij ij v. To the k-th record, implement the following steps. a) Calculate A (t+ 1) according to the formula of ij A (t+ 1) = f (A (t)+ A (t)W (t)), ij ij ij ijl j=1, jl where m is the number of attributes including class label. b) Update the W (t+ 1) according to the three formula that are ijl W (t+ 1) = W (t)+ρδ , ijl ijl ijl ∂(E(W )) ρδ = , ijl ∂(W (t)) ijl and E(W ) = (d − A (t+ 1)) . i ij ij j=1 c) Calculate E(W ). d) If E(W ) is small enough or k is equal to K (the total records), go to the Step 4, otherwise go to the Step 3 (vi). vi. k = k+ 1 and go to Step 3 (v). Step 4 Output the finial weight W that is the optimal value. The second process is to learn only FCM in high-level of TTFCM. The detail pro- cedures of the second algorithm are as the follows. Step 1’ k=0. Step 2’ Set the relevant parameters. Step 3’ Start from the node on behalf of one table in one part. Step 4’ Retrieve the initialized k-th class value as the state one of the node. Step 5’ Get the collection of record IDs of the table. Step 6’ Search out the record IDs in the associated tables by tuple ID propagation. Step 7’ Determine the state values of the associated tables by cluster analysis. Step 8’ Go to Step 3’ until the state values of all the nodes in the high-level. Step 9’ Calculate the weight matrix W by gradient descent method that is similar to 0 Fuzzy Inf. Eng. (2009) 4: 357-366 365 the Step 3 (v) of first training algorithm. Step 10’ k = k+ 1, if k < K (K is the number of categories in the one part selected), go to Step 4’, otherwise output the finial weight W as the optimal value. 4.2. The Analysis ++ The experiments of the artificial Student-Score dataset are made by C program- ming. The learning of low-level FCM is a local simulating process avoiding the one- to-many and many-to-many relationships and ensures the information to be valid. In the high-level, the state values of class nodes in the multi-parts are got through online clustering methods so as to avoid the one-to-many and many-to-many relationships and statistical skew. The results are that there are weight values between any two nodes in the same ta- ble or the high-level relation. So the weight matrixes need to be further simplified by setting the threshold value. Finally, the TTFCM obtained can be used for knowledge inference of new data. The inference method firstly is still the FCM inference in the low-level and then the reasoning in the high-level FCM for intelligence-decision or data classification and so on. 5. Conclusion The paper proposes one kind of TTFCM to model the system with the relational data in the form of E-R schema through analyzing the relational database and the multi-relational characteristic. The TTFCM includes one FCM in high-level and many FCMs in low-level forming a two-level tree. The TTFCM can be got from multi-relational data resource through the learning algorithm proposal, which mines respectively low-level FCMs and high-level FCM. The method avoids the statistical skew in the multi-relationship mining and also ensures the information not to be lost. The TTFCM can be effective and widely applied in intelligent inference and decision- making. At the same time, more efficient mining algorithm is studied. Acknowledgments We would like to thank Knowledge Engineering Institute of Beijing University of Sci- ence and Technology and National Natural Science Foundation of China (No. 636750 30, No. 60875029) support. References 1. Kosko B (1986) Fuzzy cognitive maps. International Journal Man-Machine Studies 24: 65-75 2. Aguilar J (2005) A survey about fuzzy cognitive maps papers (Invited Paper). International Journal of Computational Cognition 3(2): 27-33 3. Hafner V V (2000) Cognitive maps for navigation in open environments. 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Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Dec 1, 2009

Keywords: Fuzzy cognitive map; Multi-relational data mining; Entity-relationship schema; Two-layer tree-type FCM; Gradient descent method

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