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.     

A novel hybrid method based on fuzzy cognitive maps and fuzzy clustering algorithms for grading celiac disease

This paper presents a new method based on fuzzy cognitive map (FCM) and possibilistic fuzzy c-means (PFCM) clustering algorithm for categorizing celiac disease (CD). CD is a complex disorder whose development is affected by genetics (HLA alleles) and gluten ingestion. The celiac patients who are not treated are at a high risk of cancer, malignant lymphoma, and small bowel neoplasia. Therefore, CD diagnosis and grading are of paramount importance. The proposed FCM models human thinking for the purpose of classifying patients suffering from CD. We used the latest grading method where three grades A, B1, and B2 are used. To improve FCM efficiency and classification capability, a nonlinear Hebbian learning algorithm is applied for adjusting the FCM weights. To this end, 89 cases are studied. Three experts extracted seven main determinant characteristics of CD which were considered as FCM concepts. The mutual effects of these concepts on one another and on the final concept were expressed in the form of fuzzy rules and linguistic variables. Using the center of gravity defuzzifier, we obtained the numerical values of these weights and obtained the total weight matrix. Ultimately, combining the FCM model with PFCM algorithm, we obtained the grades A, B1, and B2 accuracies as 88, 90, and 91%, respectively. The main advantage of the proposed FCM is the good transparency and interpretability in the decision-making procedure, which make it a suitable tool for daily usage in the clinical practice.

     

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.     

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.     

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.     

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     

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.     

Complementary case-based reasoning and competitive fuzzy cognitive maps for advanced medical decisions

This paper presents a new hybrid modeling methodology suitable for complex decision making processes. It extends previous work on competitive fuzzy cognitive maps for medical decision support systems by complementing them with case based reasoning methods. The synergy of these methodologies is accomplished by a new proposed algorithm that leads to more dependable advanced medical decision support systems that are suitable to handle situations where the decisions are not clearly distinct. The methodology developed here is applied successfully to model and test two decision support systems, one a differential diagnosis problem from the speech pathology area for the diagnosis of language impairments and the other for decision making choices in external beam radiation therapy.     

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.     

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.     

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。     

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.     

A framework for designing cognitive trajectory controllers using genetically evolved interval type-2 fuzzy cognitive maps

In this paper, we provide a complete framework for the design of genetically evolved cognitive tracking controller based on interval type-2 (IT2) fuzzy cognitive map (FCM). We construct the cognitive controller based on a nonlinear controller by transforming its representation into a FCM. This representation gives the opportunity to prove the stability of the cognitive controller in the framework of nonlinear control theory. Moreover, with the deployment of IT2-fuzzy sets which are known to be capable to handle high level of uncertainty, the proposed cognitive controller has the ability to deal with uncertainty that are encountered in real-time world applications. To accomplish the design of the cognitive controller, we present a systematic approach based on genetic algorithm to optimize its parameters and learn fuzzy rules by extracting them from model space (e.g., a set of rules). Within the paper, all steps in constructing and designing the IT2-FCM-based cognitive controller are presented. We first show the performance improvements of the proposed IT2-FCM-based tracking controller with extensive and comparative simulation results and then with experimental results that were collected on real-world mobile robot. The results clearly show the superiority of proposed cognitive control systems when compared to its conventional and fuzzy controller counterparts. We believe that the proposed genetically evolved design approach of the IT2-FCM-based cognitive controller will provide a bridge between the well-developed cognitive sciences and control theory.     

Data analysis with fuzzy clustering methods

An encompassing, self-contained introduction to the foundations of the broad field of fuzzy clustering is presented. The fuzzy cluster partitions are introduced with special emphasis on the interpretation of the two most encountered types of gradual cluster assignments: the fuzzy and the possibilistic membership degrees. A systematic overview of present fuzzy clustering methods is provided, highlighting the underlying ideas of the different approaches. The class of objective function-based methods, the family of alternating cluster estimation algorithms, and the fuzzy maximum likelihood estimation scheme are discussed. The latter is a fuzzy relative of the well-known expectation maximization algorithm and it is compared to its counterpart in statistical clustering. Related issues are considered, concluding with references to selected developments in the area.     

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.     

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。     

A new methodology for Decisions in Medical Informatics using fuzzy cognitive maps based on fuzzy rule-extraction techniques

In this research work, a novel framework for the construction of augmented Fuzzy Cognitive Maps based on Fuzzy Rule-Extraction methods for decisions in medical informatics is investigated. Specifically, the issue of designing augmented Fuzzy Cognitive Maps combining knowledge from experts and knowledge from data in the form of fuzzy rules generated from rule-based knowledge discovery methods is explored. Fuzzy cognitive maps are knowledge-based techniques which combine elements of fuzzy logic and neural networks and work as artificial cognitive networks. The knowledge extraction methods used in this study extract the available knowledge from data in the form of fuzzy rules and insert them into the FCM, contributing to the development of a dynamic decision support system. The fuzzy rules, which derived by these extraction algorithms (such as fuzzy decision trees, association rule-based methods and neuro-fuzzy methods) are implemented to restructure the FCM model, producing new weights into the FCM model, that initially structured by experts. Concluding, our scope is to present a new methodology through a framework for decision making tasks using the soft computing technique of FCMs based on knowledge extraction methods. A well known medical decision making problem pertaining to the problem of radiotherapy treatment planning selection is presented to illustrate the application of the proposed framework and its functioning.