Interrogating the structure of fuzzy cognitive maps

 Causal algebra in fuzzy cognitive maps (FCMs) plays a critical role in the analysis and design of FCMs. Improving causal algebra in FCMs to model complicated situations has been one of the major research topics in this area. In this paper we propose a dynamic causal algebra in FCMs which can improve FCMs' inference and representation capability. The dynamic causal algebra shows that the indirect, strongest, weakest and total effects a vertex influences another in the FCM not only depend on the weights along all directed paths between the two vertices but also the states of the vertices on the directed paths. Therefore, these effects are nonlinear dynamic processes determined by initial conditions and propagated in the FCM to reach a static or cyclic pattern. We test our theory with a simple example.     

Linear and sigmoidal fuzzy cognitive maps: An analysis of fixed points

Fuzzy cognitive mapping is commonly used as a participatory modelling technique whereby stakeholders create a semi-quantitative model of a system of interest. This model is often turned into an iterative map, which should (ideally) have a unique stable fixed point. Several methods of doing this have been used in the literature but little attention has been paid to differences in output such different approaches produce, or whether there is indeed a unique stable fixed point. In this paper, we seek to highlight and address some of these issues. In particular we state conditions under which the ordering of the variables at stable fixed points of the linear fuzzy cognitive map (iterated to) is unique. Also, we state a condition (and an explicit bound on a parameter) under which a sigmoidal fuzzy cognitive map is guaranteed to have a unique fixed point, which is stable. These generic results suggest ways to refine the methodology of fuzzy cognitive mapping. We highlight how they were used in an ongoing case study of the shift towards a bio-based economy in the Humber region of the UK.     

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     

Learning of fuzzy cognitive maps using density estimate

Fuzzy cognitive maps (FCMs) are convenient and widely used architectures for modeling dynamic systems, which are characterized by a great deal of flexibility and adaptability. Several recent works in this area concern strategies for the development of FCMs. Although a few fully automated algorithms to learn these models from data have been introduced, the resulting FCMs are structurally considerably different than those developed by human experts. In particular, maps that were learned from data are much denser (with the density over 90% versus about 40% density of maps developed by humans). The sparseness of the maps is associated with their interpretability: the smaller the number of connections is, the higher is the transparency of the map. To this end, a novel learning approach, sparse real-coded genetic algorithms (SRCGAs), to learn FCMs is proposed. The method utilizes a density parameter to guide the learning toward a formation of maps of a certain predefined density. Comparative tests carried out for both synthetic and real-world data demonstrate that, given a suitable density estimate, the SRCGA method significantly outperforms other state-of-the-art learning methods. When the density estimate is unknown, the new method can be used in an automated fashion using a default value, and it is still able to produce models whose performance exceeds or is equal to the performance of the models generated by other methods.     

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.     

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.     

The challenge of modelling supervisory systems using fuzzy cognitive maps

This paper examines fuzzy cognitive map (FCM) theory and its use in supervisory control systems. An FCM is a graph used to depict cause and effect between concepts that stand for the states and variables of the system. An FCM represents the whole system in a symbolic manner, just as humans have stored the operation of the system in their brains, thus it is possible to help man's intention for more intelligent and autonomous systems. FCM representation, construction and a mathematical model are examined; a generic system is proposed and the implementation of FCM in a process control problem is illustrated and a model for supervisors of manufacturing systems is discussed. Although an FCM seems to be a simple model of system behaviour, it appears to be a powerful and effective tool describing the behaviour of a system and representing the accumulated knowledge of a system.     

Some models of qualitative dynamic cognitive maps with fuzzy states

Some models of dynamic cognitive maps, whose factors are determined in finite linearly ordered qualitative scales, are studied. Notions of fuzzy values and increments of factors and operations over them are determined. Specific features of defuzzification of fuzzy qualitative values are discussed. Basic behavior effects of these models, sources and forms of data fuzziness in the computing process, means for controlling this event, and confidence limits in the simulating process are studied.     

基于模糊认知图的文本分类推理算法

文本分类是信息处理的重要研究方向,现在应用较多的是基于统计计算的分类方法。介绍了利用模糊认知图的文本分类推理理论与算法,该方法是基于数值推理的,实现将统计与规则融合推理,灵活性较大,不需要语料的多次训练,适合于训练不充分和新主题的文本分类和多类分类,并具有一定的鲁棒性。     

Brain tumor characterization using the soft computing technique of fuzzy cognitive maps

The characterization and accurate determination of brain tumor grade is very important because it influences and specifies patient's treatment planning and eventually his life. A new method for characterizing brain tumors is presented in this research work, which models the human thinking approach and the classification results are compared with other computational intelligent techniques proving the efficiency of the proposed methodology. The novelty of the method is based on the use of the soft computing method of fuzzy cognitive maps (FCMs) to represent and model experts’ knowledge (experience, expertise, heuristic). The FCM grading model classification ability was enhanced introducing a computational intelligent training technique, the Activation Hebbian Algorithm. The proposed method was validated for clinical material, comprising of 100 cases. FCM grading model achieved a diagnostic output of accuracy of 90.26% (37/41) and 93.22% (55/59) for brain tumors of low-grade and high-grade, respectively. The results of the proposed grading model present reasonably high accuracy, and are comparable with existing algorithms, such as decision trees and fuzzy decision trees which were tested at the same type of initial data. The main advantage of the proposed FCM grading model is the sufficient interpretability and transparency in decision process, which make it a convenient consulting tool in characterizing tumor aggressiveness for every day clinical practice.     

模糊认知图在股票市场预测中的应用研究

复杂系统中存在着大量的过程依赖、自组织,并且一直是进化的,用传统的方法对其建模十分困难。模糊认知图作为一种模糊逻辑和神经网络相结合的产物,为复杂系统建模提供了一种有效工具。文中根据模糊认知图的特点,提出了用遗传学习算法建立系统的模糊认知图方法,为复杂系统分析及预测提供了一种解决方案。最后,以股票市场的数据为例进行了分析和预测模拟,结果表明,该方法是有效的。     

Improving fuzzy cognitive maps learning through memetic particle swarm optimization

Fuzzy cognitive maps constitute a neuro-fuzzy modeling methodology that can simulate complex systems accurately. Although their configuration is defined by experts, learning schemes based on evolutionary and swarm intelligence algorithms have been employed for improving their efficiency and effectiveness. This paper comprises an extensive study of the recently proposed swarm intelligence memetic algorithm that combines particle swarm optimization with both deterministic and stochastic local search schemes, for fuzzy cognitive maps learning tasks. Also, a new technique for the adaptation of the memetic schemes, with respect to the available number of function evaluations per application of the local search, is proposed. The memetic learning schemes are applied on four real-life problems and compared with established learning methods based on the standard particle swarm optimization, differential evolution, and genetic algorithms, justifying their superiority.     

Augmented fuzzy cognitive maps for modelling LMS critical success factors

This paper proposes to build an Augmented Fuzzy Cognitive Map-based for modelling Critical Success Factors in Learning Management Systems. The study of Critical Success Factors helps decision makers to extract from the multidimensional learning process the core activities that are essential for success. Using Fuzzy Cognitive Maps for modelling Critical Success Factors provides major assistance to the e-learning community, by permitting prediction comparisons to be made between numerous tools measured by multiple factors and its relations.     

Certainty aggregation and the certainty fuzzy measures

Aggregation functions are mostly used in decision‐making situations that require information fusion in a meaningful manner. The main purpose of aggregation is to turn a group of input data into a single and comprehensive one. However, in real decision‐making and system evaluation problems, the decision maker may exhibit only some amount of certainty in her decision inputs. In this study, we show how to aggregate these certainty degrees assigned to a group of inputs in an intuitive and reasonable manner. One of the interesting aspects of the problem is that the value aggregation is independent of their certainty degrees while the certainty aggregation essentially depends on both the input values and the value aggregation function. The construction of the aggregation function gives rise to a fuzzy measure that satisfies some very interesting properties. The technique presented here has wide range of applications.     

Learning and clustering of fuzzy cognitive maps for travel behaviour analysis

In modern society, more and more attention is given to the increase in public transportation or bike use. In this regard, one of the most important issues is to find and analyse the factors influencing car dependency and the attitudes of people in terms of preferred transport mode. Although the individuals’ transport behavioural modelling is a complex task, it has a notable social and economic impact. Thus, in this paper, fuzzy cognitive maps are explored to represent the behaviour and operation of such complex systems. This soft-computing technique allows modelling how the travellers make decisions based on their knowledge of different transport modes properties at different levels of abstraction. These levels correspond to the hierarchy perception including different scenarios of travelling, different benefits of choosing a specific travel mode, and different situations and attributes related to those benefits. We use learning and clustering of fuzzy cognitive maps to describe travellers’ behaviour and change trends in different abstraction levels. Cluster estimations are done before and after the learning of the maps, in order to compare people’s way of thinking if only considering an initial view of a transport mode decision for a daily activity, and when they really have a deeper reasoning process in view of benefits and consequences. The results of this study will help transportation policy decision makers in better understanding of people’s needs and consequently will help them actualizing different policy formulations and implementations.     

Application of evolutionary fuzzy cognitive maps to the long-term prediction of prostate cancer

The prediction of multivariate time series is one of the targeted applications of evolutionary fuzzy cognitive maps (FCM). The objective of the research presented in this paper was to construct the FCM model of prostate cancer using real clinical data and then to apply this model to the prediction of patient's health state. Due to the requirements of the problem state, an improved evolutionary approach for learning of FCM model was proposed. The focus point of the new method was to improve the effectiveness of long-term prediction. The evolutionary approach was verified experimentally using real clinical data acquired during a period of two years. A preliminary pilot-evaluation study with 40 men patient cases suffering with prostate cancer was accomplished. The in-sample and out-of-sample prediction errors were calculated and their decreased values showed the justification of the proposed approach for the cases of long-term prediction. The obtained results were approved by physicians emerging the functionality of the proposed methodology in medical decision making.     

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 context-driven intelligent database processing system using object-oriented fuzzy cognitive maps

Most data sets that describe and evolve from real-world systems are by nature semiquantitative or qualitative rather than quantitative. This can mean large variations in the significance of results that are derived from this data for decision-making processes given that the original database provides training and prototypical examples that reflect systems of events in the real world. In this article we propose a structure for a Knowledge-Based System (KBS) that is derived using significance within given contextual domains. Data that would ordinarily be classified by simple attribute classification techniques are now categorized by understanding patterns and value distributions for attributes and attribute domains that exist within rich and dense databases such as in the case of census databases<‡> and Geographic Information Systems (GIS)<§> rich by the very number of fields and interpretations, depending on the context in which the data are to be reviewed. The structure we have implemented for capturing and structuring semiquantitative information is the Fuzzy Cognitive Map (FCM). We also reduce the number of false patterns labeled “significant” by incorporating the knowledge used by human experts to find significance within the data. We treat this knowledge as initial background knowledge and as a minimal set for distinguishing significance for particular attribute values within a given context. © 1996 John Wiley & Sons, Inc.