Evidential cognitive maps
Introduction
The concept of fuzzy cognitive map has received special attention in recent years as a powerful tool to manipulate knowledge by imitating human reasoning and thinking. Many complex problems like fuzzy control [1], [2], [3], approximate reasoning [4], [5], [6], [7], strategic planning [8], [9], [10], [11], data mining [12], virtual worlds and network models [13] have been dealt with using FCMs. Especially, in the field of medical decision making [14], [15], Kannappan [16] models and predicts autistic spectrum disorder using FCM, and an unsupervised non-linear Hebbian learning algorithm is applied to improve it’s efficiency. Papageorgiou [17] presents a novel framework for the construction of augmented FCMs based on fuzzy rule-extraction methods for decisions in medical informatics. The study extracted the available knowledge from data in the form of fuzzy rules and inserted them into the FCM, contributing to the development of a dynamic decision support system. FCM has also been investigated for risk analysis of pulmonary infections during patient admission into the hospital [18], [19], [20], [21], [22], [23].
Although FCM has achieved success in many fields, there are some limitations inherent in FCM, such as lack of adequate capability to handle uncertain information and lack of enough ability to aggregate the information from different sources. Some attention has been paid to the first issue by some researchers. For example, Salmeron [24] proposes an innovative and flexible model based on Grey Systems Theory, called fuzzy grey cognitive maps (FGCM), which can be adapted to a wide range of problems, especially in multiple meaning-based environments. Iakovidis and Papageorgiou [25] propose an approach based on cognitive maps and intuitionistic fuzzy logic, which is called intuitionistic fuzzy cognitive map (IFCM) to extend the existing FCM by considering the expert’s hesitancy in the determination of the causal relations between the concepts of a domain. Similarly, after the introduction of neutrosophic logic (similar to intuitionistic fuzzy sets) by Samarandache [26], indeterminacy has been introduced into causal relationships between some of concepts of FCMs. This is a generalization of FCMs and the structure is called neutrosophic cognitive maps (NCMs) [27]. However, how to extend the ability of FCM to aggregate the information from different sources under uncertain environment is a significant question in the application of FCM and is still an open issue.
Uncertain information fusion has been studied for many years [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], indicating that Dempster Shafer theory (DS theory or evidence theory) is an effective framework to represent and fuse uncertain information. Therefore this paper combines FCM and evidential theory to the concept development of evidential cognitive maps that not only remains the ability to represent uncertainty but also contributes to aggregating knowledge from different sources (experts/commanders). The combination of evidence theory and FCM is shown to be a valuable approach through illustrations.
This paper is organized as follows: Section 2 briefly presents FCM and basic evidence theory and some operations of interval numbers. Section 3 develops the mathematical model of the proposed ECM concept. Section 4 describes the implementation of ECM. Section 5 briefly presents qualitative comparison of ECM with FCM and NCM. An application of ECM to socio-economic model is presented in Section 6.
Section snippets
Preliminaries
In this section, we briefly introduce FCM and evidence theory.
Evidential cognitive map (ECM)
ECM is also a directed graph with feedback, consisting of nodes and weighted arcs. Nodes of the graph stand for the concepts that are used to describe the behavior of the system and they are connected by weighted arcs representing the causal relationships that exist between the concepts. Each concept is characterized by an interval that represents its value, and it results from the transformation of the fuzzy value of the system variable. In this way, the representation of the concept
Application framework of ECMs
The Framework of ECM is shown as Fig. 5, and its application is detailed as follows:
Qualitative comparison with FCM and NCM
Here (in ECM) we use the fact that between any two concepts/nodes the existing relation may be an indeterminate (as) in reality, FCM do not reflect the notion of indeterminacy. Some differences between ECM and FCM are listed as follows:
- (1)
FCM measures the existence of causal relation between two concepts using crisp number between −1 and 1 and if no relation exists it is denoted by 0. ECM measures not only the existence or absence of causal relations between two concepts but also give
An application of ECM to a socio-economic model
This section illustrates the application of the proposed method to a socio-economic model. It is constructed with Population, Crime, Economic condition, Poverty, and Unemployment as nodes or concepts. Our purpose is to evaluate the trend of factors changing with any one factor using ECM.
First, the structure of ECM should be established using several sources of partial knowledge. All the available experts are divided into three groups (group1, group2, and grou3). and the opinions are provided in
Conclusions
Evidential cognitive maps (ECMs) are uncertain-graph structures for representing causal reasoning. They can be considered as the exploration of cognitive maps (CMs) and fuzzy cognitive maps (FCMs). ECMs can not only deal with the uncertain information but can also handle the fuzzy information with the advantage of evidence theory, and can be used in many applications involving decision making and uncertain reasoning. The framework of ECMs is developed in this paper and a simple application is
Acknowledgments
This paper presents results of an on-going research, which is funded by Canada NSERC discovery grant. The work is also partially supported by National Natural Science Foundation of China, Grant Nos. 60874105, 60904099, 61174022, Chongqing Natural Science Foundation for Distinguished Young Scientists, Grant Nos. CSCT, 2010BA2003, Program for New Century Excellent Talents in University, Grant No. NCET-08-0345, Shanghai Rising-Star Program Grant No. 09QA1402900, the Chenxing Scholarship Youth
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