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.     

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.     

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.     

基于OWA算子的区间值加权模糊推理

针对如何对区间值模糊产生式规则赋予合理权值的问题,将OWA算子引入到区间值模糊推理中。介绍一种基于OWA算子的区间值赋权方法,根据此方法给出区间值模糊集上的加权模糊产生式规则的推理算法。在采用该算法的过程中,为合理地计算输入事实与规则前件的匹配程度,引入基于OWA算子的区间值模糊匹配函数值和总体贴近度的计算方法。实例分析表明了所给出的区间值模糊推理算法的有效性和可行性。     

A New Family of Induced OWA Operators

A new family of induced ordered weighted averaging (OWA) operators is proposed by invoking the order‐inducing variables at the aggregation step. The objective is to consider the variations in the magnitudes of the order‐inducing variables. The new family of operators include weighted induced OWA, weighted generalized induced OWA, and weighted induced ordered weighted geometric operators. These are further extended to the intuitionistic fuzzy domain. The usefulness of these operators is shown in a supplier selection problem.     

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.     

Generalized point operators for aggregating intuitionistic fuzzy information

We first develop a series of intuitionistic fuzzy point operators, and then based on the idea of generalized aggregation (Yager RR. Generalized OWA aggregation operators. Fuzzy Optim Decis Making 2004;3:93–107 and Zhao H, Xu ZS, Ni MF, Liu SS. Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 2010;25:1–30), we develop various generalized intuitionistic fuzzy point aggregation operators, such as the generalized intuitionistic fuzzy point weighted averaging (GIFPWA) operators, generalized intuitionistic fuzzy point ordered weighted averaging (GIFPOWA) operators, and generalized intuitionistic fuzzy point hybrid averaging (GIFPHA) operators, which can control the certainty degrees of the aggregated arguments with some parameters. Furthermore, we study the properties and special cases of our operators. © 2010 Wiley Periodicals, Inc.     

A C‐OWA operator‐based approach to decision making with interval fuzzy preference relation

Yager [IEEE Trans Syst Man Cybern B 2004;34:1952–1963] introduced a continuous interval argument OWA (C‐OWA) operator, which extends the ordered weighted averaging (OWA) operator, introduced by Yager [IEEE Trans Syst Man Cybern B 1988;18:183–190], to the case in which the given argument is a continuous valued interval rather than a finite set of values. In this article, we utilize the C‐OWA operator to derive the priority vector of an interval fuzzy preference relation and then develop a practical approach to solving the decision‐making problem with interval fuzzy preference relation. Finally, a numerical example is provided to demonstrate the practicability and efficiency of the developed approach. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1289–1298, 2006.     

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.     

The Ordered Weighted Average in the Variance and the Covariance

The ordered weighted average (OWA) is an aggregation operator that provides a parameterized family of aggregation operators between the minimum and the maximum. This paper analyzes the use of the OWA in the variance and the covariance. It presents several extensions by using a unified framework between the weighted average and the OWA. Furthermore, it also develops other generalizations with induced aggregation operators and by using quasi‐arithmetic means. Several measures of correlation by using the OWA are introduced including a new type of Pearson coefficient. The paper ends with some numerical examples focused on the construction of interval and fuzzy numbers with the variance and the covariance.     

Increasing fuzzy rules cooperation based on evolutionary adaptive inference systems

This article presents a study on the use of parametrized operators in the Inference System of linguistic fuzzy systems adapted by evolutionary algorithms, for achieving better cooperation among fuzzy rules. This approach produces a kind of rule cooperation by means of the inference system, increasing the accuracy of the fuzzy system without losing its interpretability. We study the different alternatives for introducing parameters in the Inference System and analyze their interpretation and how they affect the rest of the components of the fuzzy system. We take into account three applications in order to analyze their accuracy in practice. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1035–1064, 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.     

因果关系,贝叶斯网络与认知图

因果关系在预测和推理中具有重要的作用.贝叶斯网络已被用于构建诊断和决策系统.近年来模糊认知图得到了重视.模糊认知图为结构性知识与因果推理提供了又一个理论框架.本文简单介绍贝叶斯网络与认知图及其推理方法在智能系统中的应用.     

The Metric Space of Ordered Weighted Average Operators with Distance Based on Accumulated Entries

Ordered weighted average (OWA) operators with their weighting vectors are very important in many applications. We show that directly taking Minkowski distances (including Manhattan distance and Euclidean distance) as the distances for any two OWA operator is not reasonable. In this study, we propose the standard distance measures for any two OWA operators and then propose a standard metric space for the set of all n‐dimension OWA operators. We analyze and discuss some properties of the introduced OWA metric and further propose a metric space of Choquet integrals represented by the underlying fuzzy measures. Some applications in decision making of OWA distances are also presented in this study.     

A majority model in group decision making using QMA–OWA operators

Group decision‐making problems are situations where a number of experts work in a decision process to obtain a final value that is representative of the global opinion. One of the main problems in this context is to design aggregation operators that take into account the individual opinions of the decision makers. One of the most important operators used for synthesizing the individual opinions in a representative value of majority in the OWA operator, where the majority concept used aggregation processes, is modeled using fuzzy logic and linguistic quantifiers. In this work the semantic of majority used in OWA operators is analyzed, and it is shown how its application in group decision‐making problems does not produce representative results of the concept expressed by the quantifier. To solve this type of problem, two aggregation operators, QMA–OWA, are proposed that use two quantification strategies and a quantified normalization process to model the semantic of the linguistic quantifiers in the group decision‐making process. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 193–208, 2006.     

Fuzzy induced generalized aggregation operators and its application in multi-person decision making

We present a wide range of fuzzy induced generalized aggregation operators such as the fuzzy induced generalized ordered weighted averaging (FIGOWA) and the fuzzy induced quasi-arithmetic OWA (Quasi-FIOWA) operator. They are aggregation operators that use the main characteristics of the fuzzy OWA (FOWA) operator, the induced OWA (IOWA) operator and the generalized (or quasi-arithmetic) OWA operator. Therefore, they use uncertain information represented in the form of fuzzy numbers, generalized (or quasi-arithmetic) means and order inducing variables. The main advantage of these operators is that they include a wide range of mean operators such as the FOWA, the IOWA, the induced Quasi-OWA, the fuzzy IOWA, the fuzzy generalized mean and the fuzzy weighted quasi-arithmetic average (Quasi-FWA). We further generalize this approach by using Choquet integrals, obtaining the fuzzy induced quasi-arithmetic Choquet integral aggregation (Quasi-FICIA) operator. We also develop an application of the new approach in a strategic multi-person decision making problem.     

Futuristic data-driven scenario building: Incorporating text mining and fuzzy association rule mining into fuzzy cognitive map

Fuzzy cognitive maps (FCMs) are one of the representative techniques in developing scenarios that include future concepts and issues, as well as their causal relationships. The technique, initially dependent on deductive modeling of expert knowledge, suffered from inherent limitations of scope and subjectivity; though this lack has been partially addressed by the recent emergence of inductive modeling, the fact that inductive modeling uses a retrospective, historical data that often misses trend-breaking developments. Addressing this issue, the paper suggests the utilization of futuristic data, a collection of future-oriented opinions extracted from online communities of large participation, in scenario building. Because futuristic data is both large in scope and prospective in nature, we believe a methodology based on this particular data set addresses problems of subjectivity and myopia suffered by the previous modeling techniques. To this end, text mining (TM) and latent semantic analysis (LSA) algorithm are applied to extract scenario concepts from futuristic data in textual documents; and fuzzy association rule mining (FARM) technique is utilized to identify their causal weights based on if-then rules. To illustrate the utility of proposed approach, a case of electric vehicle is conducted. The suggested approach can improve the effectiveness and efficiency of scanning knowledge for scenario development.     

Multilayer Cognitive Maps in the Resolution of Problems using the FCM Designer Tool

Cognitive maps are a tool to represent knowledge from a qualitative perspective, allowing to create models of complex systems where an exact mathematical model cannot be used because of the complexity of the system. In the literature, several tools have been proposed to develop cognitive maps and fuzzy cognitive maps (FCMs); one of them is FCM Designer. This paper designs and implements an extension to the FCM Designer tool that allows creating multilayer FCM. With this extension, it is possible to have several FCMs for the same problem, where each one expresses a different level of knowledge of the system under study, but interlinked. Thus, one can have a first level of detailed abstraction of the system with specific information and then more general levels. In addition, we can have different levels where the variables of one level depend on those of another level. That is, the multilayer approach enriches the modeled systems with flow of information between layers, to derive information about the concepts involved in layers from the concepts in other layers. In our multilayer approach, the relationship between the cognitive maps in different layers can be carried out in various ways: with fuzzy rules, connections with weights and with mathematical equations. This work presents the design and the implementation of the extension of the FCM Designer tool, and several test cases in different domains: a FCM to analyze emergent properties of Wikipedia a FCM for medical analysis for diagnosis, and another like recommender system.     

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.