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.     

基于有序加权平均算子的概率模糊认知图

模糊认知图(FCM)与概率模糊认知图(PFCM)使用简单的加权和集结因果推理结果,忽略了原因节点间关联关系的不确定性,阈值函数导致推理结果进一步失真.在继承FCM与PFCM优点的基础上,引入有序加权平均(OWA)算子模拟各种确定的或模糊的与或组合关系,提出了基于有序加权平均算子的概率模糊认知图(OWA-PFCM).通过构建一个动态的攻击效能评估模型,阐述了OWA-PFCM在工程建模中的应用.OWA-PFCM能同时表示因果节点状态的不确定性、因果联系强度的不确定性、与或组合关系的不确定性,具有更强的模拟能力.     

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.     

Weighted fuzzy reasoning using weighted fuzzy Petri nets

This paper presents a Weighted Fuzzy Petri Net model (WFPN) and proposes a weighted fuzzy reasoning algorithm for rule-based systems based on Weighted Fuzzy Petri Nets. The fuzzy production rules in the knowledge base of a rule-based system are modeled by Weighted Fuzzy Petri Nets, where the truth values of the propositions appearing in the fuzzy production rules and the certainty factors of the rules are represented by fuzzy numbers. Furthermore, the weights of the propositions appearing in the rules are also represented by fuzzy numbers. The proposed weighted fuzzy reasoning algorithm can allow the rule-based systems to perform fuzzy reasoning in a more flexible and more intelligent manner     

Autonomous navigation system using Event Driven-Fuzzy Cognitive Maps

This study developed an autonomous navigation system using Fuzzy Cognitive Maps (FCM). Fuzzy Cognitive Map is a tool that can model qualitative knowledge in a structured way through concepts and causal relationships. Its mathematical representation is based on graph theory. A new variant of FCM, named Event Driven-Fuzzy Cognitive Maps (ED-FCM), is proposed to model decision tasks and/or make inferences in autonomous navigation. The FCM??s arcs are updated from the occurrence of special events as dynamic obstacle detection. As a result, the developed model is able to represent the robot??s dynamic behavior in presence of environment changes. This model skill is achieved by adapting the FCM relationships among concepts. A reinforcement learning algorithm is also used to finely adjust the robot behavior. Some simulation results are discussed highlighting the ability of the autonomous robot to navigate among obstacles (navigation at unknown environment). A fuzzy based navigation system is used as a reference to evaluate the proposed autonomous navigation system performance.     

A new hybrid method using evolutionary algorithms to train Fuzzy Cognitive Maps

A novel hybrid method based on evolutionary computation techniques is presented in this paper for training Fuzzy Cognitive Maps. Fuzzy Cognitive Maps is a soft computing technique for modeling complex systems, which combines the synergistic theories of neural networks and fuzzy logic. The methodology of developing Fuzzy Cognitive Maps relies on human expert experience and knowledge, but still exhibits weaknesses in utilization of learning methods and algorithmic background. For this purpose, we investigate a coupling of differential evolution algorithm and unsupervised Hebbian learning algorithm, using both the global search capabilities of Evolutionary strategies and the effectiveness of the nonlinear Hebbian learning rule. The use of differential evolution algorithm is related to the concept of evolution of a number of individuals from generation to generation and that of nonlinear Hebbian rule to the concept of adaptation to the environment by learning. The hybrid algorithm is introduced, presented and applied successfully in real-world problems, from chemical industry and medicine. Experimental results suggest that the hybrid strategy is capable to train FCM effectively leading the system to desired states and determining an appropriate weight matrix for each specific problem.     

Fuzzy clustering with semantic interpretation

In the framework of Axiomatic Fuzzy Set (AFS) theory, we propose a new approach to data clustering. The objective of this clustering is to adhere to some principles of grouping exercised by humans when determining a structure in data. Compared with other clustering approaches, the proposed approach offers more detailed insight into the cluster's structure and the underlying decision making process. This contributes to the enhanced interpretability of the results via the representation capabilities of AFS theory. The effectiveness of the proposed approach is demonstrated by using real-world data, and the obtained results show that the performance of the clustering is comparable with other fuzzy rule-based clustering methods, and benchmark fuzzy clustering methods FCM and K-means. Experimental studies have shown that the proposed fuzzy clustering method can discover the clusters in the data and help specify them in terms of some comprehensive fuzzy rules.     

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     

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.     

A fuzzy inference and categorization approach for supplier selection using compensatory and non-compensatory decision rules

Fuzzy set theory has been used as an approach to deal with uncertainty in the supplier selection decision process. However, most studies limit applications of fuzzy set theory to outranking potential suppliers, not including a qualification stage in the decision process, in which non-compensatory types of decision rules can be used to reduce the set of potential suppliers. This paper presents a supplier selection decision method based on fuzzy inference that integrates both types of approaches: a non-compensatory rule for sorting in qualification stages and a compensatory rule for ranking in the final selection. Fuzzy inference rules model human reasoning and are embedded in the system, which is an advantage when compared to approaches that combine fuzzy set theory with multicriteria decision making methods. Fuzzy inference combined with a fuzzy rule-based classification method is used to categorize suppliers in qualification stages. Classes of supplier performance can be represented by linguistic terms, which allow decision makers to deal with subjectivity and to express qualification requirements in linguistic formats. Implementation of the proposed method and techniques were analyzed and discussed using an illustrative case. Three defuzzification operators were used in the final selection, yielding the same ranking. Factorial design was applied to test consistency and sensitivity of the inference rules. The findings reinforce the argument that including stages of qualification based on fuzzy inference and categorization makes this method especially useful for selecting from a large set of potential suppliers and also for first time purchase.     

一个基于神经网络的专家系统

医疗专家系统主要使用基于知识的技术，其中的决策规则和策略来自于人类的专家。把这些知识和各种推理方法结合，可以建立一个模拟专家决策过程的系统。建立这样一个系统，需要经常与专家磋商，以获取专家的知识，因而需要大量的时间和精力。为此，本文提出直接从数据中提取有效的信息，即用神经网络提取隐含在大量数据中对医疗诊断有效的信息，继之与基于规则的知识，各种推理方法相结合，建立一个神经网络专家系统。     

Optimizing the modified fuzzy ant-miner for efficient medical diagnosis

The advantage of efficient searches belonging to ant-miner over several other approaches leads to prominent achievements on rules mining. Fuzzy ant-miner, an extension of the ant-miner provides a fuzzy mining framework for the automatic extraction of fuzzy rules from labeled numerical data. However, it is easily trapped in local optimal, especially when it applies to medical cases, where real world accuracy is elusive; and the interpretation and integration of medical knowledge is necessary. In order to relieve such a local optimal difficulty, this paper proposes OMFAM which applies simulated annealing to optimize fuzzy set parameters associated with a modified fuzzy ant-miner (MFAM). MFAM employs attributes and training case weighting. The proposed method, OMFAM was experimented with six critical medical cases for developing efficient medical diagnosis systems. The performance measurement relates to accuracy as well as interpretability of the mined rules. The performance of the OMFAM is compared with such references as MFAM, fuzzy ant-miner (FAM), and other classification methods. At last, it indicates the superiority of the OMFAM algorithm over the others.     

The use of fuzzy cognitive maps to simulate the information systems strategic planning process

In the early 1980s articles began to focus on Strategic Planning of Information Systems (SISP) and to argue the critical importance of Information Technology (IT) in today’s organisations. Since then, a large number of models were presented in order to analyse IT from a strategic point of view and suggest new IT projects. However, researchers urge for alternative approaches to SISP, as current ones fall short in taking into consideration both the business and IT perspectives as well as they fail to tackle the complexity of the domain and suggest specific IS opportunities. This article suggests Fuzzy Cognitive Maps (FCM) as an alternative modelling approach and describes how they can be developed and used to simulate the SISP process. FCMs were successfully developed and used in several ill-structured domains, such as decision making, policy making. The proposed FCM contains 165 variables and 210 relationships from both business and IT domains. The strength of this approach lies in its capability not only to comprehensively model qualitative knowledge which dominates strategic decision making, but also to simulate and evaluate several alternative ways of using IT in order to improve organisational performance. This approach introduces computational modelling, as well as it supports scenarios development and simulation in the SISP domain.     

A two-stage evolutionary process for designing TSK fuzzy rule-basedsystems

Nowadays, fuzzy rule-based systems are successfully applied to many different real-world problems. Unfortunately, relatively few well-structured methodologies exist for designing and, in many cases, human experts are not able to express the knowledge needed to solve the problem in the form of fuzzy rules. Takagi-Sugeno-Kang (TSK) fuzzy rule-based systems were enunciated in order to solve this design problem because they are usually identified using numerical data. In this paper we present a two-stage evolutionary process for designing TSK fuzzy rule-based systems from examples combining a generation stage based on a (mu, lambda)-evolution strategy, in which the fuzzy rules with different consequents compete among themselves to form part of a preliminary knowledge base, and a refinement stage in which both the antecedent and consequent parts of the fuzzy rules in this previous knowledge base are adapted by a hybrid evolutionary process composed of a genetic algorithm and an evolution strategy to obtain the final Knowledge base whose rules cooperate in the best possible way. Some aspects make this process different from others proposed until now: the design problem is addressed in two different stages, the use of an angular coding of the consequent parameters that allows us to search across the whole space of possible solutions, and the use of the available knowledge about the system under identification to generate the initial populations of the Evolutionary Algorithms that causes the search process to obtain good solutions more quickly. The performance of the method proposed is shown by solving two different problems: the fuzzy modeling of some three-dimensional surfaces and the computing of the maintenance costs of electrical medium line in Spanish towns. Results obtained are compared with other kind of techniques, evolutionary learning processes to design TSK and Mamdani-type fuzzy rule-based systems in the first case, and classical regression and neural modeling in the second.     

Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets

Fuzzy interpolative reasoning is an important research topic of sparse fuzzy rule-based systems. In recent years, some methods have been presented for dealing with fuzzy interpolative reasoning. However, the involving fuzzy sets appearing in the antecedents of fuzzy rules of the existing fuzzy interpolative reasoning methods must be normal and non-overlapping. Moreover, the reasoning conclusions of the existing fuzzy interpolative reasoning methods sometimes become abnormal fuzzy sets. In this paper, in order to overcome the drawbacks of the existing fuzzy interpolative reasoning methods, we present a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets. The proposed fuzzy interpolative reasoning method can handle the situation of non-normal and overlapping fuzzy sets appearing in the antecedents of fuzzy rules. It can overcome the drawbacks of the existing fuzzy interpolative reasoning methods in sparse fuzzy rule-based systems.     

Multi-objective optimization problem under fuzzy rule constraints using particle swarm optimization

In this paper, a fuzzy multi-objective programming problem is considered where functional relationships between decision variables and objective functions are not completely known to us. Due to uncertainty in real decision situations sometimes it is difficult to find the exact functional relationship between objectives and decision variables. It is assumed that information source from where some knowledge may be obtained about the objective functions consists of a block of fuzzy if-then rules. In such situations, the decision making is difficult and the presence of multiple objectives gives rise to multi-objective optimization problem under fuzzy rule constraints. In order to tackle the problem, appropriate fuzzy reasoning schemes are used to determine crisp functional relationship between the objective functions and the decision variables. Thus a multi-objective optimization problem is formulated from the original fuzzy rule-based multi-objective optimization model. In order to solve the resultant problem, a deterministic single-objective non-linear optimization problem is reformulated with the help of fuzzy optimization technique. Finally, PSO (Particle Swarm Optimization) algorithm is employed to solve the resultant single-objective non-linear optimization model and the computation procedure is illustrated by means of numerical examples.     

Development of a decision making system for selection of dental implant abutments based on the fuzzy cognitive map

Dental implant is a medical operation used to restore the functions of damaged or missing teeth. Correct implantation requires the proper selection of size and shape among the implant structures. In this paper, we propose a method of constructing a Web-based decision making system that enables the selection of a suitable type of abutment by taking into account the patient’s anatomical data and preferences that are based on an expert’s knowledge and experience for those patients. After the classification of the types of abutment that can be connected to fixtures of implants, we built a knowledge base and case base library according to the characteristics of osseous tissue and teeth shape to select optimal abutment. Moreover, we introduce a fuzzy cognitive map that takes into consideration expert’s knowledge for factors that affect implantation. After the determination of the cause-and-effect relationship among the concepts of the fuzzy cognitive map, an osseointegration factor with the highest conceptual concentration weight is inferred from the decision making system. In addition, the selection process for abutment is expressed as a decision making tree and then, it is applied for the rule-based reasoning and case-based reasoning. The optimized selection result is finally extracted based on the fuzzy membership function using fuzzy inference.     

Medical diagnosis with the aid of using fuzzy logic and intuitionistic fuzzy logic

The objective of the present study is to develop/establish a web-based medical diagnostic support system (MDSS) by which health care support can be provided for people living in rural areas of a country. In this respect, this research provides a novel approach for medical diagnosis driven by integrating fuzzy and intuitionistic fuzzy (IF) frameworks. Subsequently, based on the proposed approach a web-based MDSS is developed. The proposed MDSS comprises of a knowledge base (KB) and intuitionistic fuzzy inference system (IFIS). Based on the observation that medical data cannot be described with both precision and certainty, a medical KB is constructed in the form of a set of if-then decision rules by employing both fuzzy and IF logics. After constructing the medical KB, a new set of patients is considered for diagnosing the diseases. For each patient, linguistic values of the patients’ symptoms are considered as inputs of the proposed IFIS and modeled by using the generalized triangular membership functions. Subsequently, integrated fuzzy and IF rule-based inference system is used to find a valid conclusion for the new set of patients. In a nutshell, in this paper fuzzy rule-based and IFS based inference systems are combined for better and more realistic representation of uncertainty of the medical diagnosis problem and for more accurate diagnostic result. The method is composed of following four steps: (1) the modeling of antecedent part of the rules, which consist of linguistic assessments of the patients’ symptoms provided by the doctors/medical experts with their corresponding confidence levels, by using generalized fuzzy numbers; (2) the modeling of consequent part, which reveals the degree of association and the degree of non-association of diseases into the patient, by using IFSs; (3) the use of IF aggregation operator in inference process; (4) the application of relative closeness function to find the final crisp output for a given diagnosis. Finally, the applicability of the proposed approach is illustrated with a suitable case study. This article has also justified the proposed approach by using similarity measurement.