Quotient FCMs-a decomposition theory for fuzzy cognitive maps

In this paper, we introduce a decomposition theory for fuzzy cognitive maps (FCM). First, we partition the set of vertices of an FCM into blocks according to an equivalence relation, and by regarding these blocks as vertices we construct a quotient FCM. Second, each block induces a natural sectional FCM of the original FCM, which inherits the topological structure as well as the inference from the original FCM. In this way, we decompose the original FCM into a quotient FCM and some sectional FCM. As a result, the analysis of the original FCM is reduced to the analysis of the quotient and sectional FCM, which are often much smaller in size and complexity. Such a reduction is important in analyzing large-scale FCM. We also propose a causal algebra in the quotient FCM, which indicates that the effect that one vertex influences another in the quotient depends on the weights and states of the vertices along directed paths from the former to the latter. To illustrate the process involved, we apply our decomposition theory to university management networks. Finally, we discuss possible approaches to partitioning an FCM and major concerns in constructing quotient FCM. The results represented in this paper provide an effective framework for calculating and simplifying causal inference patterns in complicated real-world applications.     

Fuzzy cognitive maps

Fuzzy cognitive maps (FCMs) are fuzzy-graph structures for representing causal reasoning. Their fuzziness allows hazy degrees of causality between hazy causal objects (concepts). Their graph structure allows systematic causal propagation, in particular forward and backward chaining, and it allows knowledge bases to be grown by connecting different FCMs. FCMs are especially applicable to soft knowledge domains and several example FCMs are given. Causality is represented as a fuzzy relation on causal concepts. A fuzzy causal algebra for governing causal propagation on FCMs is developed. FCM matrix representation and matrix operations are presented in the Appendix.     

Unsupervised learning techniques for fine-tuning fuzzy cognitive map causal links

Fuzzy Cognitive Maps (FCMs) constitute an attractive knowledge-based methodology, combining the robust properties of fuzzy logic and neural networks. FCMs represent causal knowledge as a signed directed graph with feedback and provide an intuitive framework which incorporates the experts’ knowledge. FCMs handle available information and knowledge from an abstract point of view. They develop behavioural model of the system exploiting the experience and knowledge of experts. The construction of FCMs is based mainly on experts who determine the structure of FCM, i.e. concepts and weighted interconnections among concepts. But this methodology may not be a sufficient model of the system because the human factor is not always reliable. Thus the FCM model of the system may requires restructuring which is achieved through adjustment the weights of FCM interconnections using specific learning algorithms for FCMs. In this article, two unsupervised learning algorithms are presented and compared for training FCMs; how they define, select or fine-tuning weights of the causal interconnections among concepts. The implementation and results of these unsupervised learning techniques for an industrial process control problem are discussed. The simulations results of training the process system verify the effectiveness, validity and advantageous characteristics of those learning techniques for FCMs.     

Dynamical cognitive network - an extension of fuzzy cognitive map

We present the dynamic cognitive network (DCN) which is an extension of the fuzzy cognitive map (FCM). Each concept in the DCNs can have its own value set, depending on how precisely it needs to be described in the network. This enables the DCN to describe the strength of causes and the degree of effects that are crucial to conducting meaningful inferences. The arcs in the DCN define dynamic, causal relationships between concepts. Structurally, DNCs are scalable and more flexible as compared to FCMs. A DCN can be as simple as a cognitive map and FCM, or as complex as a nonlinear dynamic system. To demonstrate the potential applications of DCNs, we present some simulation results. This paper represents our first attempt to develop a dynamic fuzzy inference system using causal relationships. There are many interesting and challenging theoretical and practical issues in DCNs open to further research     

Comparing the inference capabilities of binary, trivalent and sigmoid fuzzy cognitive maps

In this paper, we compare the inference capabilities of three different types of fuzzy cognitive maps (FCMs). A fuzzy cognitive map is a recurrent artificial neural network that creates models as collections of concepts/neurons and the various causal relations that exist between these concepts/neurons. In the paper, a variety of industry/engineering FCM applications is presented. The three different types of FCMs that we study and compare are the binary, the trivalent and the sigmoid FCM, each of them using the corresponding transfer function for their neurons/concepts. Predictions are made by viewing dynamically the consequences of the various imposed scenarios. The prediction making capabilities are examined and presented. Conclusions are drawn concerning the use of the three types of FCMs for making predictions. Guidance is given, in order FCM users to choose the most suitable type of FCM, according to (a) the nature of the problem, (b) the required representation capabilities of the problem and (c) the level of inference required by the case.     

A novel semi-quantitative Fuzzy Cognitive Map model for complex systems for addressing challenging participatory real life problems

Fuzzy Cognitive Maps (FCM) are a promising approach for socio-ecological systems modelling. FCMs represent problem knowledge extracted from different stakeholders in the form of connected factors/variables with imprecise cause-effect relationships and many feedback loops. These typically large maps are condensed and aggregated to obtain a summary view of the system. However, representation, condensation and aggregation of previous FCM models are qualitative due to lack of appropriate quantitative methods. This study tackles these drawbacks by developing a semi-quantitative FCM model consisting of robust methods for adequately and accurately representing and manipulating imprecise data describing a complex problem involving stakeholders for pragmatic decision making. The model starts with collecting qualitative imprecise data from relevant stakeholders. These data are then transformed into stakeholder perceptions/FCMs with different causal relationship formats (linguistic or numeric) which the proposed model then represents in a unified format using a 2-tuple fuzzy linguistic representation model which allows combining imprecise linguistic and numeric values with different granularity and/or semantic without loss of information. The proposed model then condenses large FCMs using a semi-quantitative method that allows multi-level condensation. In each level of condensation, groups of similar variables are subjectively condensed and the corresponding imprecise connections are computationally condensed using robust calculations involving credibility weights assigned to variables (variables’ importance). The model then uses a quantitative fuzzy method to aggregate perceptions/FCMs into a stakeholder group or social perception/FCM based on the 2-tuple model and credibility weights assigned to FCMs (stakeholders’ importance). Thereafter, the structure of produced FCMs is analysed using graph theory indices to examine differences in perceptions between stakeholders or groups. Finally, the model applies various what-if policy scenario simulations on group FCMs using a dynamical systems approach with neural networks and analyses scenario outcomes to provide appropriate recommendations to decision makers. An example application illustrates method’s effectiveness and usefulness.     

Quantization effects on the equilibrium behavior of combined fuzzy cognitive maps

Fuzzy cognitive maps (FCMs) allow experts to express their knowledge by drawing weighted causal digraphs. Experts can pool or fuse their knowledge by adding the underlying FCM causal matrices. This naturally extends the ordered‐weighted‐averaging (OWA) technique to averaging dynamical systems and can create complex dynamical systems from several simpler ones. Edge quantization allows experts to state their knowledge in the simpler terms of causal increase (1), decrease (?1), or absence (0). We model the expert FCMs as a sequence of random fields to study the small‐sample effects of quantizing both the causal edges and the fuzzy‐set concept nodes. The averaged quantized random matrices exhibit large‐sample convergence to the population means of the unquantized matrices in accordance with the Strong Law of Large Numbers. But the small‐sample averages can show substantial diversity of equilibrium attractors (fixed points or limit cycles). We use statistical tests—chi‐square tests, Spearman's rank coefficient, the Kolmogorov–Smirnov test, and the fuzzy equality of limit cycle histograms—to show that this small‐sample equilibrium diversity increases as the node multivalence or fuzzy‐set quantization increases. The appendix presents a new probabilistic convergence theorem that shows that edge quantization or thresholding does not affect FCM combination for large expert sample sizes: the sample mean of quantized expert causal edge values converges with probability one to the population mean causal edge values. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 181–202, 2007.     

Application of fuzzy cognitive maps for cotton yield management in precision farming

The management of cotton yield behavior in agricultural areas is a very important task because it influences and specifies the cotton yield production. An efficient knowledge-based approach utilizing the method of fuzzy cognitive maps (FCMs) for characterizing cotton yield behavior is presented in this research work. FCM is a modelling approach based on exploiting knowledge and experience. The novelty of the method is based on the use of the soft computing method of fuzzy cognitive maps to handle experts’ knowledge and on the unsupervised learning algorithm for FCMs to assess measurement data and update initial knowledge.The advent of precision farming generates data which, because of their type and complexity, are not efficiently analyzed by traditional methods. The FCM technique has been proved from the literature efficient and flexible to handle experts’ knowledge and through the appropriate learning algorithms can update the initial knowledge. The FCM model developed consists of nodes linked by directed edges, where the nodes represent the main factors in cotton crop production such as texture, organic matter, pH, K, P, Mg, N, Ca, Na and cotton yield, and the directed edges show the cause–effect (weighted) relationships between the soil properties and cotton field.The proposed method was evaluated for 360 cases measured for three subsequent years (2001, 2003 and 2006) in a 5 ha experimental cotton yield. The proposed FCM model enhanced by the unsupervised nonlinear Hebbian learning algorithm, was achieved a success of 75.55%, 68.86% and 71.32%, respectively for the years referred, in estimating/predicting the yield between two possible categories (“low” and “high”). The main advantage of this approach is the sufficient interpretability and transparency of the proposed FCM model, which make it a convenient consulting tool in describing cotton yield behavior.     

Fuzzy cognitive map modelling educational software adoption

Educational software adoption across UK secondary schools is seen as unsatisfactory. Based on stakeholders’ perceptions, this paper uses fuzzy cognitive maps (FCMs) to model this adoption context. It discusses the development of the FCM model, using a mixed-methods approach and drawing on participants from three UK secondary schools. The study presents three phases involved in the development of the model, where individual FCMs were developed in phase one and then the individual FCMs were aggregated in phase two. In phase three, further interrelationships identified from the empirical data were assigned weightings and added, resulting in the final FCM model. Following a static analysis of the model, the resulting FCM offers a visual medium of factors key to the adoption of educational software as perceived by relevant stakeholders. As a holistic model it provides insight into the context of educational software adoption in schools, which can be used to guide both educational decision-makers in where to focus their efforts and software developers in terms of more focused and appropriate software development efforts.     

An FCM–FAHP approach for managing readiness-relevant activities for ERP implementation

Any organization which plans to introduce a new enterprise resource planning (ERP) system will carry out a range of activities to improve its readiness for the new system. This paper develops a new approach for managing these interrelated activities using fuzzy cognitive maps (FCMs) and the fuzzy analytical hierarchy process (FAHP). This approach enables the organization to (1) identify the readiness-relevant activities, (2) determine how these activities influence each other, (3) assess how these activities will contribute to the overall readiness and (4) prioritize these activities according to their causal interrelationships to allocate management effort for the overall readiness improvement. The approach first uses FCMs and a fuzzy connection matrix to represent all possible causal relationships between activities. It then uses FAHP to determine the contribution weights and uses FCM inference to include the effects of feedback between the activities. Based on the contribution and interrelationships between activities, a management matrix is developed to categorize them into four management zones for effective allocation of limited management efforts. An empirical study is conducted to demonstrate how the approach works.     

A hybrid fuzzy cognitive model based on weighted OWA operators and single‐antecedent rules

Conventional fuzzy cognitive maps (FCMs) can only represent monotonic or symmetric causal relationships and cannot simulate the AND/OR combinations of the antecedent nodes. The rule‐based fuzzy cognitive maps (RBFCMs) usually suffer from the well‐known combinatorial rule explosion problem. A hybrid fuzzy cognitive model based on weighted OWA operators and single‐antecedent rules is proposed to eliminate the drawbacks of the existing FCM models. Hybrid fuzzy cognitive maps (HFCMs) represent the causal relationships with single‐antecedent fuzzy rules and handle the various AND/OR relationships among the antecedent nodes with weighted OWA aggregation operators. Compared with conventional FCMs, HFCMs have more powerful cognitive capability. Compared with RBFCMs, HFCMs reduce the scale and complexity of the rule bases significantly and have better representation and inference performance. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1189–1196, 2007.     

Fuzzy cognitive map modelling educational software adoption

Educational software adoption across UK secondary schools is seen as unsatisfactory. Based on stakeholders’ perceptions, this paper uses fuzzy cognitive maps (FCMs) to model this adoption context. It discusses the development of the FCM model, using a mixed-methods approach and drawing on participants from three UK secondary schools. The study presents three phases involved in the development of the model, where individual FCMs were developed in phase one and then the individual FCMs were aggregated in phase two. In phase three, further interrelationships identified from the empirical data were assigned weightings and added, resulting in the final FCM model. Following a static analysis of the model, the resulting FCM offers a visual medium of factors key to the adoption of educational software as perceived by relevant stakeholders. As a holistic model it provides insight into the context of educational software adoption in schools, which can be used to guide both educational decision-makers in where to focus their efforts and software developers in terms of more focused and appropriate software development efforts.     

A fuzzy cognitive map approach for effect-based operations: An illustrative case

This paper proposes the use of fuzzy cognitive maps (FCMs) as a technique for supporting the decision-making process in effect-based planning. The goal is to determine alternative courses of action to realize the aims of an operation, and choose the best option among them. With adequate consideration of the problem features and the constraints governing the method used, an FCM is developed to model effect-based operations (EBOs). In this study, certain features that do not exist in the classical FCM method were added to our FCM concept value calculation algorithm; these include influence possibility, influence duration, dynamic influence value-changing, and influence permanence. The model developed was applied to an illustrative scenario involving military planning, and we comment on the usefulness of the proposed methodology.     

On causal inference in fuzzy cognitive maps

Fuzzy cognitive maps (FCM) is a powerful paradigm for representing human knowledge and causal inference. This paper formally analyzes the causal inference mechanism of FCM. We focus on binary concept states. It is known that given initial conditions, FCM is able to reach only certain states in its state space. We prove that the problem of finding whether a state is reachable in the FCM is nondeterministic polynomial (NP) hard, that we can divide fuzzy cognitive maps containing circles into basic FCM modules. The inference patterns in these basic modules can be studied individually in a hierarchical fashion. This paper also presents a recursive formula for computing FCM's inference patterns in terms of key vertices. The theoretical results presented in this paper provide a feasible and effective framework for the analysis and design of fuzzy cognitive maps in real-world large-scale applications     

Intelligent modeling of e-business maturity

E-business has a significant impact on managers and academics. Despite the rhetoric surrounding e-business strategy formulation mechanisms, which support reasoning of the effect of strategic change activities to the maturity of the e-business models, are still emerging. This paper describes an attempt to build and operate such a reasoning mechanism as a novel supplement to e-business strategy formulation exercises. This new approach proposes the utilization of the fuzzy causal characteristics of Fuzzy Cognitive Maps (FCMs) as the underlying methodology in order to generate a hierarchical and dynamic network of interconnected maturity indicators. By using FCMs, this research aims at simulating complex strategic models with imprecise relationships while quantifying the impact of strategic changes to the overall e-business efficiency. This research establishes generic adaptive domains – maps in order to implement the integration of hierarchical FCMs into e-business strategy formulation activities. Finally, this paper discusses experiments with the proposed mechanism and comments on its usability.     

Fuzzy cognitive map software tool for treatment management of uncomplicated urinary tract infection

Uncomplicated urinary tract infection (uUTI) is a bacterial infection that affects individuals with normal urinary tracts from both structural and functional perspective. The appropriate antibiotics and treatment suggestions to individuals suffer of uUTI is an important and complex task that demands a special attention. How to decrease the unsafely use of antibiotics and their consumption is an important issue in medical treatment. Aiming to model medical decision making for uUTI treatment, an innovative and flexible approach called fuzzy cognitive maps (FCMs) is proposed to handle with uncertainty and missing information. The FCM is a promising technique for modeling knowledge and/or medical guidelines/treatment suggestions and reasoning with it. A software tool, namely FCM-uUTI DSS, is investigated in this work to produce a decision support module for uUTI treatment management. The software tool was tested (evaluated) in a number of 38 patient cases, showing its functionality and demonstrating that the use of the FCMs as dynamic models is reliable and good. The results have shown that the suggested FCM-uUTI tool gives a front-end decision on antibiotics’ suggestion for uUTI treatment and are considered as helpful references for physicians and patients. Due to its easy graphical representation and simulation process the proposed FCM formalization could be used to make the medical knowledge widely available through computer consultation systems.     

RuleML representation and simulation of Fuzzy Cognitive Maps

Fuzzy Cognitive Map (FCM) technique is a combination of Fuzzy Logic and Artificial Neural Networks that is extensively used by experts and scientists of a diversity of disciplines, for strategic planning, decision making and predictions. A standardized representation of FCMs accompanied by a system that would assist decision makers to simulate their own developed Fuzzy Cognitive Maps would be highly appreciated by them, and would help the dissemination of FCMs. In this paper, (a) a RuleML representation of FCM is proposed and (b) a system is designed and implemented in Prolog programming language to assist experts to simulate their own FCMs. This system returns results in valid RuleML syntax, making them readily available to other cooperative systems. The representation capabilities and the design choices of the implemented system are discussed and a variety of examples are given to demonstrate the use of the system.     

基于结合空间信息的FCM聚类的分水岭图像分割

提出了一种分水岭变换和结合空间信息的FCM聚类相结合的图像分割方法。方法采用基于图论的结合区域特征信息和空间信息的距离度量,以分水岭变换得到的图像分割小区域为节点构建一个连通加权图,通过计算图上不同节点之间的最短路径来度量不同区域之间的相似程度,从而实现过分割小区域的合并。该方法综合考虑了区域的特征之间的差异和空间位置的差异,与传统的FCM聚类方法在特征空间进行聚类相比,具有较强的噪声抑制能力。图像分割的实验结果证明了该算法的可行性和有效性。