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Advances in fuzzy cognitive maps theory
Author’s Accepted Manuscript Advances in Fuzzy Cognitive Maps Theory Wojciech Froelich Jose L. Salmeron
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(16)31498-9 http://dx.doi.org/10.1016/j.neucom.2016.11.058 NEUCOM17840
To appear in: Neurocomputing Accepted date: 19 Cite this article as: Wojciech Froelich and Jose L. Salmeron, Advances in Fuzzy Cognitive Maps Theory, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.11.058 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Editorial Advances in Fuzzy Cognitive Maps Theory Fuzzy cognitive maps (FCMs) are a data model proposed by Bart Kosko in the article “Fuzzy cognitive maps, International Journal of Man-Machine Studies 24 (1986)” as an extension of Axelrod’s cognitive maps. Fuzzy cognitive maps consist of the concepts and linkages among them. The concepts are fuzzy sets; the linkages represent cause-effect relationships among concepts. The linkages of the FCM are labeled by weights that reflect the strengths of the relationships among concepts. The weights are stored in a square adjacency matrix. The influence of multiple concepts on a target concept is aggregated and transformed, usually by a nonlinear transfer function. For this reason, FCMs are nonlinear models. FCMs are used to model information observed in time. The state of every concept is defined at a discrete time step, so this state changes over time. The weights of the FCMs usually don’t change in time, generalizing the relationship existing between concepts over a longer period of time. The main goal of building an FCM around a problem is to be able to predict the outcome by letting the relevant concepts interact with one another. These predictions can be used for finding out whether a decision made by someone is consistent with the entire collection of stated causal assertions. Because they are represented as weighted directed graphs, FCMs can easily be interpreted by humans. FCM graphs can be constructed by experts as a kind of decision support tool reflecting the relationships among concepts observed in the problem domain. In addition, FCMs can be learned from data. In this case, every concept of the FCM is mapped to the data observed in the real world. The relationships and their weights are adjusted using specialized machine learning algorithms. Trained FCMs can be exploited for many different objectives, such as decision support systems, classification, and forecasting among others. After the dynamics, the FCM model reaches one of two states following a number of iterations. In the first state, it settles down to a steady pattern of concepts states, the so-called hidden pattern (also called a fixed-point attractor). Alternatively, it keeps cycling between several fixed states, which is known as a limit cycle. Using a continuous transformation function, there is a third possibility known as a chaotic attractor. This special issue of Neurocomputing presents original articles related to recent advances in fuzzy cognitive maps theory. It aims to promote research and the exchange of ideas related to the theory of FCMs. Submissions came from an open call for papers. All manuscripts were thoroughly reviewed by at least three experts and twelve articles were accepted for publication. A short introduction to the accepted papers is presented below. The first two papers address the problem of the selection of FCM concepts. The problem is related to the initial modeling phase with the use of FCMs. The contribution “Clustering Techniques for Fuzzy Cognitive Map Design for Time Series Modeling” by Homenda and Jastrzebska proposes evaluating fuzzy cognitive maps before training. For this purpose they use cluster validity indexes. The proposed approach is intended to be applied when FCMs are being used as a predictive model for time series forecasting. Results show that fuzzy cognitive maps designed using selected indexes demonstrate superior quality in terms of forecasting errors and interpretability. To a certain extent, a similar problem is addressed in the paper “A Concept Reduction Approach for Fuzzy Cognitive Map Models in Decision Making and Management” by Papageorgiou et al. In this paper a new concept reduction approach for FCMs is proposed. The reduction of the number of an FCM’s concepts is achieved by clustering based on fuzzy tolerance relation. In this way, the resulting FCM is less complex, providing a more transparent and easy-to-use model for policymakers. The results show the advantages of the proposed concept reduction in policymaking. The second and largest group of papers addresses the problem of learning FCMs. In the paper “Continously Self-adjusting Fuzzy Cognitive Map with Semi-autonomous concepts” by Stula et al., an extension to FCMs is proposed. The main contribution of this paper is the self-adjusting FCM, which changes the cause-effect relationships and concept inferences for each considered data point
with the goal of reducing the error between actual data and outcomes produced by the FCM. The authors tested the proposed model on two case studies in which they measured the degree of change to an initial map structure set up by an expert. The experiments showed that the selfadjusted maps produced results that were closer to real data than the maps that were initially set up by the expert. In the paper entitled “Learning FCMs with multi-local and balanced memetic algorithms for forecasting industrial drying processes”, Salmeron et al. propose a new learning approach with a multi-local search in a balanced memetic algorithm. The evolutionary algorithm previously used for learning FCMs is adapted and improved by a multi-local and balanced local search. Moreover, several local search strategies and the balance between local and global searches is checked. The proposed approach is applied to the forecasting of moisture loss in the industrial drying process. The results of the experiments provide evidence that the proposed approach to the exploration of search space is a competitive alternative to other FCM learning algorithms. A new approach to handling incompleteness and natural uncertainty in expert evaluations of the FCM adjacency matrix is proposed by Mls et al. in the paper “Interactive evolutionary optimization of fuzzy cognitive maps”. A modification of interactive evolutionary computing aimed at the improved optimization of FCMs is presented. Experimental results prove that the proposed approach improves the quality of the obtained FCMs. Another extended FCM is introduced in the paper “Multi-stage cognitive map for failures assessment of production processes: An extension in structure and algorithm“ by Rezaee, Yousefi and Babaei. In the proposed model the concepts of the FCM are in various stages associated with each other by causal relationships. The paper also proposes an innovative learning algorithm to train the proposed extended FCMs. The case study on an automotive parts manufacturing unit demonstrates the ability of the proposed approach to assess failures in production processes. FCMs can be used as classifiers. To perform classification using an FCM, a new algorithm for generating thresholds for the discrimination of FCM outcomes is proposed by Froelich in the paper “Towards Improving the Efficiency of the Fuzzy Cognitive Map Classifier”. The thresholds resulting from the proposed algorithm are determined after evolutionary learning of the FCM; they are then applied when classifying new data instances. The application of the proposed algorithm led to improved efficiency of the FCM, making it competitive to the most well-known classifiers. The next paper also addresses the classification problem. In the paper “Wavelet Fuzzy Cognitive Maps”, Wu, Liu, and Chi propose that the wavelet transfer function be applied to FCMs. In addition, a comprehensive analysis of existing transfer functions for the classification problem is made. Finally, according to analysis, a new method involving the selection of transfer functions is proposed. The experimental results demonstrate high effectiveness of the proposed method. In the paper “Medical Diagnosis of Rheumatoid Arthritis Using Data-driven PSO-FCM with Scarce Datasets”, Salmeron, Rahimi, Navali and Sadeghpour propose the use of particle swarm optimization (PSO) and FCMs for the identification and modelling of rheumatoid arthritis. This paper builds on the FCM topology and the influences between the nodes dynamically by the PSO algorithm without human intervention. In addition, the FCM model is built from scarce datasets. The model is exploited for the calculation of the severity of the disease. Accuracy level of the model is satisfactory, with the obtained results closely matching the opinions of medical professionals. The traditional objective of using FCMs for time series forecasting is addressed by Papageorgiou and Poczeta in the paper “A Two-Stage Model for Time Series Prediction based on Fuzzy Cognitive Maps and Neural Networks”. This paper proposes a two-stage forecasting model based on evolutionary FCMs and artificial neural networks (ANNs). In the first stage of learning, an FCM is constructed; the structure optimization genetic algorithm selects the most important concepts of the FCM and the interconnections among them. In the second stage of learning, the obtained FCM defines inputs in an ANN, which is then trained by the back propagation algorithm. Despite the growing interest in FCMs, researchers are facing a lack of tools for the analysis of the developed models. The final two papers of the special issue focus on the analysis of FCMs. In the study presented by Dodurka, Yesil, and Urbas, a new static analysis approach is proposed for enhanced FCMs, which have non-singleton fuzzy numbers in causal relation strength representation.
A new type of analysis is presented for finding the indirect effects and total effects between the concepts of the enhanced FCMs. The results of the proposed causal effect analysis are discussed and compared with those obtained from conventional FCMs. In their study “A Framework for Static and Dynamic Analysis of Multi-Layer Fuzzy Cognitive Maps”, Christoforou and Andreou introduce an integrated analysis framework of multi-layered FCMs. The proposed type of analysis provides information on the model's complexity, which is the tendency to promote or inhibit activation states of concepts as a result of the presence of positive or negative cycles. The framework enables dynamic analysis of what-if scenarios modelled by the FCM. The tool is dedicated to the analysis of real-world problems from the engineering and political decision-making domains.
The guest editors would like to thank all authors for their interesting contributions and all reviewers for their excellent work. We hope that the readers will share our excitement in presenting this special issue of Neurocomputing.
Wojciech Froelich Institute of Computer Science, University of Silesia, Sosnowiec (Poland) e-mail: [email protected] Jose L. Salmeron University Pablo de Olavide, Seville (Spain), the University of Hradec Kralove (Czech Republic), and the Universidad Autónoma de Chile (Chile), e-mail: [email protected].
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(16)31498-9 http://dx.doi.org/10.1016/j.neucom.2016.11.058 NEUCOM17840
To appear in: Neurocomputing Accepted date: 19 Cite this article as: Wojciech Froelich and Jose L. Salmeron, Advances in Fuzzy Cognitive Maps Theory, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.11.058 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Editorial Advances in Fuzzy Cognitive Maps Theory Fuzzy cognitive maps (FCMs) are a data model proposed by Bart Kosko in the article “Fuzzy cognitive maps, International Journal of Man-Machine Studies 24 (1986)” as an extension of Axelrod’s cognitive maps. Fuzzy cognitive maps consist of the concepts and linkages among them. The concepts are fuzzy sets; the linkages represent cause-effect relationships among concepts. The linkages of the FCM are labeled by weights that reflect the strengths of the relationships among concepts. The weights are stored in a square adjacency matrix. The influence of multiple concepts on a target concept is aggregated and transformed, usually by a nonlinear transfer function. For this reason, FCMs are nonlinear models. FCMs are used to model information observed in time. The state of every concept is defined at a discrete time step, so this state changes over time. The weights of the FCMs usually don’t change in time, generalizing the relationship existing between concepts over a longer period of time. The main goal of building an FCM around a problem is to be able to predict the outcome by letting the relevant concepts interact with one another. These predictions can be used for finding out whether a decision made by someone is consistent with the entire collection of stated causal assertions. Because they are represented as weighted directed graphs, FCMs can easily be interpreted by humans. FCM graphs can be constructed by experts as a kind of decision support tool reflecting the relationships among concepts observed in the problem domain. In addition, FCMs can be learned from data. In this case, every concept of the FCM is mapped to the data observed in the real world. The relationships and their weights are adjusted using specialized machine learning algorithms. Trained FCMs can be exploited for many different objectives, such as decision support systems, classification, and forecasting among others. After the dynamics, the FCM model reaches one of two states following a number of iterations. In the first state, it settles down to a steady pattern of concepts states, the so-called hidden pattern (also called a fixed-point attractor). Alternatively, it keeps cycling between several fixed states, which is known as a limit cycle. Using a continuous transformation function, there is a third possibility known as a chaotic attractor. This special issue of Neurocomputing presents original articles related to recent advances in fuzzy cognitive maps theory. It aims to promote research and the exchange of ideas related to the theory of FCMs. Submissions came from an open call for papers. All manuscripts were thoroughly reviewed by at least three experts and twelve articles were accepted for publication. A short introduction to the accepted papers is presented below. The first two papers address the problem of the selection of FCM concepts. The problem is related to the initial modeling phase with the use of FCMs. The contribution “Clustering Techniques for Fuzzy Cognitive Map Design for Time Series Modeling” by Homenda and Jastrzebska proposes evaluating fuzzy cognitive maps before training. For this purpose they use cluster validity indexes. The proposed approach is intended to be applied when FCMs are being used as a predictive model for time series forecasting. Results show that fuzzy cognitive maps designed using selected indexes demonstrate superior quality in terms of forecasting errors and interpretability. To a certain extent, a similar problem is addressed in the paper “A Concept Reduction Approach for Fuzzy Cognitive Map Models in Decision Making and Management” by Papageorgiou et al. In this paper a new concept reduction approach for FCMs is proposed. The reduction of the number of an FCM’s concepts is achieved by clustering based on fuzzy tolerance relation. In this way, the resulting FCM is less complex, providing a more transparent and easy-to-use model for policymakers. The results show the advantages of the proposed concept reduction in policymaking. The second and largest group of papers addresses the problem of learning FCMs. In the paper “Continously Self-adjusting Fuzzy Cognitive Map with Semi-autonomous concepts” by Stula et al., an extension to FCMs is proposed. The main contribution of this paper is the self-adjusting FCM, which changes the cause-effect relationships and concept inferences for each considered data point
with the goal of reducing the error between actual data and outcomes produced by the FCM. The authors tested the proposed model on two case studies in which they measured the degree of change to an initial map structure set up by an expert. The experiments showed that the selfadjusted maps produced results that were closer to real data than the maps that were initially set up by the expert. In the paper entitled “Learning FCMs with multi-local and balanced memetic algorithms for forecasting industrial drying processes”, Salmeron et al. propose a new learning approach with a multi-local search in a balanced memetic algorithm. The evolutionary algorithm previously used for learning FCMs is adapted and improved by a multi-local and balanced local search. Moreover, several local search strategies and the balance between local and global searches is checked. The proposed approach is applied to the forecasting of moisture loss in the industrial drying process. The results of the experiments provide evidence that the proposed approach to the exploration of search space is a competitive alternative to other FCM learning algorithms. A new approach to handling incompleteness and natural uncertainty in expert evaluations of the FCM adjacency matrix is proposed by Mls et al. in the paper “Interactive evolutionary optimization of fuzzy cognitive maps”. A modification of interactive evolutionary computing aimed at the improved optimization of FCMs is presented. Experimental results prove that the proposed approach improves the quality of the obtained FCMs. Another extended FCM is introduced in the paper “Multi-stage cognitive map for failures assessment of production processes: An extension in structure and algorithm“ by Rezaee, Yousefi and Babaei. In the proposed model the concepts of the FCM are in various stages associated with each other by causal relationships. The paper also proposes an innovative learning algorithm to train the proposed extended FCMs. The case study on an automotive parts manufacturing unit demonstrates the ability of the proposed approach to assess failures in production processes. FCMs can be used as classifiers. To perform classification using an FCM, a new algorithm for generating thresholds for the discrimination of FCM outcomes is proposed by Froelich in the paper “Towards Improving the Efficiency of the Fuzzy Cognitive Map Classifier”. The thresholds resulting from the proposed algorithm are determined after evolutionary learning of the FCM; they are then applied when classifying new data instances. The application of the proposed algorithm led to improved efficiency of the FCM, making it competitive to the most well-known classifiers. The next paper also addresses the classification problem. In the paper “Wavelet Fuzzy Cognitive Maps”, Wu, Liu, and Chi propose that the wavelet transfer function be applied to FCMs. In addition, a comprehensive analysis of existing transfer functions for the classification problem is made. Finally, according to analysis, a new method involving the selection of transfer functions is proposed. The experimental results demonstrate high effectiveness of the proposed method. In the paper “Medical Diagnosis of Rheumatoid Arthritis Using Data-driven PSO-FCM with Scarce Datasets”, Salmeron, Rahimi, Navali and Sadeghpour propose the use of particle swarm optimization (PSO) and FCMs for the identification and modelling of rheumatoid arthritis. This paper builds on the FCM topology and the influences between the nodes dynamically by the PSO algorithm without human intervention. In addition, the FCM model is built from scarce datasets. The model is exploited for the calculation of the severity of the disease. Accuracy level of the model is satisfactory, with the obtained results closely matching the opinions of medical professionals. The traditional objective of using FCMs for time series forecasting is addressed by Papageorgiou and Poczeta in the paper “A Two-Stage Model for Time Series Prediction based on Fuzzy Cognitive Maps and Neural Networks”. This paper proposes a two-stage forecasting model based on evolutionary FCMs and artificial neural networks (ANNs). In the first stage of learning, an FCM is constructed; the structure optimization genetic algorithm selects the most important concepts of the FCM and the interconnections among them. In the second stage of learning, the obtained FCM defines inputs in an ANN, which is then trained by the back propagation algorithm. Despite the growing interest in FCMs, researchers are facing a lack of tools for the analysis of the developed models. The final two papers of the special issue focus on the analysis of FCMs. In the study presented by Dodurka, Yesil, and Urbas, a new static analysis approach is proposed for enhanced FCMs, which have non-singleton fuzzy numbers in causal relation strength representation.
A new type of analysis is presented for finding the indirect effects and total effects between the concepts of the enhanced FCMs. The results of the proposed causal effect analysis are discussed and compared with those obtained from conventional FCMs. In their study “A Framework for Static and Dynamic Analysis of Multi-Layer Fuzzy Cognitive Maps”, Christoforou and Andreou introduce an integrated analysis framework of multi-layered FCMs. The proposed type of analysis provides information on the model's complexity, which is the tendency to promote or inhibit activation states of concepts as a result of the presence of positive or negative cycles. The framework enables dynamic analysis of what-if scenarios modelled by the FCM. The tool is dedicated to the analysis of real-world problems from the engineering and political decision-making domains.
The guest editors would like to thank all authors for their interesting contributions and all reviewers for their excellent work. We hope that the readers will share our excitement in presenting this special issue of Neurocomputing.
Wojciech Froelich Institute of Computer Science, University of Silesia, Sosnowiec (Poland) e-mail: [email protected] Jose L. Salmeron University Pablo de Olavide, Seville (Spain), the University of Hradec Kralove (Czech Republic), and the Universidad Autónoma de Chile (Chile), e-mail: [email protected].