Basic Probability Assignment Function Research Articles

An intelligent fault diagnosis approach based on Dempster-Shafer theory for hydraulic valves

Detecting faults in hydraulic valves are of great significance to improve the reliability and security of the whole hydraulic system. However, it is difficult to detect multiple faults in hydraulic valves using existing approaches due to closed structural components and complex hydraulic system itself. Therefore, an intelligent fault diagnosis approach based on Dempster-Shafer (DS) theory is proposed specifically for detecting several faults occurred in hydraulic valves. Actually, it is classified in the ensemble learning in terms of the information fusion theory. In this approach, signal segments containing fault information are selected to structure sample sets firstly. Then sample sets are simultaneously fed into the single classifier including long short-term memory networks (LSTM), convolutional neural network (CNN) and random forests (RF). Through learning spontaneously in these intelligent classification approaches, fault features are concluded and the probability of each type fault is respectively revealed. All probabilities are constructed as basic probability assignment (BPA) functions, which are further calculated in the information fusion process in terms of DS theory. Finally, the fault types are identified by the final fusion results. Experimental investigations are performed to validate performance of the present approach (taken a solenoid controlled pilot operated directional valve as an example). It is shown that the average accuracy ratio of proposed intelligent fault diagnosis approach is 98.5% for six fault types detection. The study does provide an effective access to detect faults in hydraulic valves.

An algorithm based on evidence theory and fuzzy entropy to defend against SSDF

In cognitive radio networks, spectrum sensing is one of the most important functions to identify available spectrum for improving the spectrum utilization. Due to the open characteristic of the wireless electromagnetic environment, the wireless network is vulnerable to be attacked by malicious users (MUs), and spectrum sensing data falsification (SSDF) attack is one of the most harmful attacks on spectrum sensing performance. In this article, an algorithm based on the evidence theory and fuzzy entropy is proposed to resist SSDF attacks. In this algorithm, secondary users (SUs) obtain the corresponding degree of membership function and basic probability assignment function based on the local energy detection result. The new conflicting coefficient is calculated based on the evidence distance and classical conflicting coefficient, and the conflicting weight of the evidence is obtained. The fuzzy weight is calculated by the fuzzy entropy. The credibility weight is obtained by updating the credibility. On this basis, the probability assignment function of the evidence is corrected, and the final result is obtained by using the fusion formula. Simulation results show that the proposed algorithm has a higher detection probability and lower false alarm probability than other algorithms. It can effectively defend against SSDF attacks and improve the performance of spectrum sensing.

Energy Efficient and Reliable Routing Algorithm for Wireless Sensors Networks

In energy-constrained wireless sensor networks, low energy utilization and unbalanced energy distribution are seriously affecting the operation of the network. Therefore, efficient and reasonable routing algorithms are needed to achieve higher Quality of Service (QoS). For the Dempster–Shafer (DS) evidence theory, it can fuse multiple attributes of sensor nodes with reasonable theoretical deduction and has low demand for prior knowledge. Based on the above, we propose an energy efficient and reliable routing algorithm based on DS evidence theory (DS-EERA). First, DS-EERA establishes three attribute indexes as the evidence under considering the neighboring nodes’ residual energy, traffic, the closeness of its path to the shortest path, etc. Then we adopt the entropy weight method to objectively determine the weight of three indexes. After establishing the basic probability assignment (BPA) function, the fusion rule of DS evidence theory is applied to fuse the BPA function of each index value to select the next hop. Finally, each node in the network transmits data through this routing strategy. Theoretical analysis and simulation results show that DS-EERA is promising, which can effectively prolong the network lifetime. Meanwhile, it can also reach a lower packet loss rate and improve the reliability of data transmission.

An Enhanced Cooperative Spectrum Sensing Scheme Against SSDF Attack Based on Dempster-Shafer Evidence Theory for Cognitive Wireless Sensor Networks

Due to the potential applicability of spectrum sensing, cognitive wireless sensor networks have attracted plenty of interest in the research community to improve the bandwidth utilization for practical applications. To alleviate the effect of multi-path fading and resolve the problem of hidden terminal, collaborative spectrum sensing (CSS) is regarded as effective technology to obtain better sensing accuracy. However, CSS is usually vulnerable to the attack behaviors originated from malicious sensor nodes. In this paper, an enhanced cooperative spectrum sensing scheme against SSDF attack based on Dempster-Shafer evidence theory for cognitive wireless sensor networks is introduced. First, the holistic credibility of sensor nodes can be evaluated according to the real-time difference between them and the statistical sensing behaviors. Furthermore, the basic probability assignment function can be defined based on evidence theory, and the credibility of sensor nodes can be estimated. Finally, by using the weighted probability assignment for each cognitive sensor node, the fusion center can reduce the influence of malicious sensor nodes on the final decision and ensure the reliability of reports from cooperative sensor nodes. Simulation results demonstrate that the proposed method can resist SSDF attacks significantly and outperform the traditional secure schemes in aspect of sensing accuracy.

DS/ANP Method: A Simplified Group Analytic Network Process With Consensus Reaching

Analytic network process (ANP) is a significant multiple criteria decision making (MCDM) method. Although lots of efforts have been spent on improving the ANP, there still exist the four aspects of problems to be solved such as too many comparison times, unconsidering expert knowledge structure, ignoring consensus level of experts’ group and ineffectively determining holistic information of the group. This study focuses on providing a simplified group ANP with consensus reaching named as DS/ANP method which incorporates the Dempster-Shafer theory of evidence (DS) with the ANP. An expert information extraction mechanism with the help of knowledge matrix and basic probability assignment function is introduced to simplify comparisons of group ANP and reflect expert knowledge structure. Then Shafer’s discounting and Dempster’s rule are both used to get the holistic information of the group, in which three levels of consensus indices (element level, cluster level, and holism level) are defined in terms of Jousselme distance. A group ANP method with consensus reaching and its corresponding algorithm are both established by following the principles of ANP. Finally, a numerical example is presented to make comparisons and discussions for the proposed method.

An Improved Belief Entropy in Evidence Theory

Uncertainty measurement of the basic probability assignment function has always been a hot issue in Dempster-Shafer evidence. Many existing studies mainly consider the influence of the mass function itself and the size of the frame of discernment, so that the correlation between the subsets is ignored in the power set of the frame of discernment. Without making full use of the information contained in the evidence, the existing methods are less effective in some cases given in the paper. In this paper, inspired by Shannon entropy and Deng entropy, we propose an improved entropy that not only inherits the many advantages of Shannon entropy and Deng entropy, but also fully considers the relationship between subsets, which makes the improved entropy overcome the shortcomings of existing methods and have greater advantages in uncertainty measurement. Many numerical examples are used to demonstrate the validity and superiority of our proposed entropy in this paper.

An Evidence Combination Rule Based on Transferable Belief Model and Application in Reliability Assessment With Multi-Source Data

Increasingly complex system environments have put forward higher requirements for information fusion. Under this background, the application prospects of evidence theory which can function well in uncertain information representation and information fusion is promising. The effect of evidence theory on information fusion is closely related to evidence distance measurement method and evidence combination rules. Considering the difference of evidence information in belief level and that in decision level, this paper redefines the evidence distance and puts forward the measurement method of evidence credibility, thus realizing the correction of the original basic probability assignment function (BPA). Based on the evidence credibility, the evidence weight is generated, which leads to an efficient distribution of evidence conflict in the process of evidence combination. So, a new evidence combination rule is generated. According to a series of examples, the evidence distance proposed in this paper can effectively measure the evidence differences, and the new combination rules can reasonably distribute evidence conflicts thus avoiding evidence paradox. At the end of this paper, the new evidence distance and combination rule, combing with the Transferable Belief Model (TBM), are used to fuse the multi-source reliability test information, then the life cycle evaluation of a product is obtained. In addition, through the comparative analysis of the results, the feasibility and validity of this method being applied to practice are verified.

An Improved Belief Entropy to Measure Uncertainty of Basic Probability Assignments Based on Deng Entropy and Belief Interval

It is still an open issue to measure uncertainty of the basic probability assignment function under Dempster-Shafer theory framework, which is the foundation and preliminary work for conflict degree measurement and combination of evidences. This paper proposes an improved belief entropy to measure uncertainty of the basic probability assignment based on Deng entropy and the belief interval, which takes the belief function and the plausibility function as the lower bound and the upper bound, respectively. Specifically, the center and the span of the belief interval are employed to define the total uncertainty degree. It can be proved that the improved belief entropy will be degenerated to Shannon entropy when the the basic probability assignment is Bayesian. The results of numerical examples and a case study show that its efficiency and flexibility are better compared with previous uncertainty measures.

Three-way decision models and its optimization based on Dempster–Shafer evidence theory and rough sets

Dempster–Shafer evidence, rough set and three-way decision are related to each other. They are all theories and methods of handling uncertain information. First, it is required to combine the Dempster–Shafer evidence with rough set theory, and the rough set theory can therefore be applied to obtain the basic probability assignment function of evidence, which allows it to be obtained more objectively. Second, the conditional probability function in the three-way decision is replaced by a value in the belief interval, and three-way decision models are established to integrate the Dempster–Shafer evidence theory with rough sets. Third, by adjusting the value of the belief factor, the decision inference rules suitable for decision-makers of different risk types are obtained. Finally, an example is provided in this case to prove the validity of this model.

The Classification of EEG Signals with Multi-Domain Fusion Based on D-S Evidence Theory

The classification of electroencephalogram (EEG) signals is a key technique of brain–computer interface (BCI) system. In view of the complexity of EEG signals and the low accuracy in EEG signals recognition, a motor imagery EEG signals classification method with multi-domain fusion based on Dempster–Shafer (D-S) evidence theory is presented in this paper. Firstly, time domain statistics (TDS), autoregressive (AR) model and discrete wavelet transform (DWT) are used to extract features from EEG signals, respectively, and three probabilistic output support vector machine (SVM) classification models are trained based on these three feature sets. Secondly, using the output of each SVM, we construct basic probability assignment (BPA) function and get fusion BPA through D-S evidence theory. Finally, determining the class of test samples based on decision rules. Four databases from BCI competition are employed to evaluate the proposed approach, and the highest classification accuracy reaches 92.83%. Results show that this method acquires higher accuracy and has strong individual adaptability.

Group inference method of attribution theory based on Dempster–Shafer theory of evidence

Kelley’s attribution theory has been widely popular in recent years. Lots of efforts have been spent on improving it with the assumption that there is only one expert to make attributions and the expert is assumed to be omniscient and omnipotent. However, such an assumption hardly exists in reality for the reason that the knowledge of each expert to make judgments is always limited. In order to solve this problem, this paper proposes a group inference method under the framework of Kelley’s attribution theory based on Dempster–Shafer theory of evidence. An information extraction mechanism is introduced to ensure that the real judgments of each expert can be well described. Then Shafer’s discounting is used to generate the basic probability assignment (BPA) functions by integrating the weights of experts on different criteria into the judgments of experts. The Dempster’s rule is employed to make fusion for the BPA functions, and a consensus reaching model which can increase the satisfaction degrees of group decision as much as possible is established to determine the probabilities of external and internal causes for the evaluated behavior. Finally, an algorithm is summarized, and illustrative example and discussion are provided to demonstrate its applicability.

New improved DEMATEL method based on both subjective experience and objective data

The decision-making trial and evaluation laboratory (DEMATEL) method is a hot issue in industrial engineering field for it can help determine critical factors in complex system. Although lots of efforts have been spent on improving the DEMATEL, they are just the extensions from the subjective perspective but lack of the objective perspective. This study focuses on providing a new improved DEMATEL method based on both subjective experience and objective data. In order to reasonably determine the initial direct-relation (IDR) matrix, the basic probability assignment (BPA) function is employed to extract expert experience and the Dempster’s rule with Shafer’s discounting is employed to make combination to derive the subjective IDR matrix. Then the path analysis is suggested to test each possible influence relation included in the subjective IDR matrix, and the objective IDR matrix consisting of path coefficients of any two factors is derived by training the sample data of factors. Following the principle of one-vote negation, the Dempster’s rule is once again used to make combination for two kinds of IDR matrices, based on which an algorithm for the new improved DEMATEL is summarized to find the major factors in a complex system. Finally, numerical comparison and discussion are proposed to demonstrate the applicability and superiority of the prosed method.

Water quality prediction based on recurrent neural network and improved evidence theory: a case study of Qiantang River, China

Water quality prediction is an effective method for managing and protecting water resources by providing an early warning against water quality deterioration. In general, the existing water quality prediction methods are based on a single shallow model which fails to capture the long-term dependence in historical time series and is more likely to cause a high rate of false alarms and false negatives in practical water monitoring application. To resolve these problems, a new model combining recurrent neural network (RNN) with improved Dempster/Shafer (D-S) evidence theory (RNNs-DS) is proposed in this paper. Among them, the RNNs which can handle the long-term dependence in historical time series effectively are used to realize the preliminary prediction of water quality. And the improved D-S evidence theory is used to synthesize the prediction results of RNNs. In addition, an improved strategy based on correlation analysis method is presented for evidence theory to obtain the number of evidence, which reduces uncertainty in evidence selection effectively. Besides, a new basic probability assignment function which based on modified softmax function is proposed. The new function can effectively solve the problems of weight allocation failure in the traditional function. Then, data about permanganate index, pH, total phosphorus, and dissolved oxygen from Jiuxishuichang monitoring station near Qiantang River, Zhejiang Province, China is used to verify the proposed model. Compared with support vector regression (SVR) and backpropagation neural network (BPNN) and three RNN models, the new model shows higher accuracy and better stability as indicated by four indices. Finally, the engineering application of the RNNs-DS algorithm has been realized on the self-developed water environmental monitoring and forecasting system, which can provide effective support for early risk assessment and prevention in water environment.

Group fuzzy comprehensive evaluation method under ignorance

This paper aims at solving such a group fuzzy comprehensive evaluation (FCE) problem that the global or local ignorance may exist in judgments made by experts and the importance degrees of experts are different. The basic probability assignment (BPA) function is used to extract the expert's judgment information and the super fuzzy relationship matrices consisting of the individual type and the general type are constructed by Shafer's discounting and Dempster's rule. Then each type of super fuzzy relationship matrix is combined with factor weight set via a specified fuzzy operator and the comprehensive evaluation result that is a belief distribution on the power set of grade levels is obtained. A multi-objective programming model is established to compute the optimal belief distribution on each grade level and an algorithm is summarized to derive the final grade level that the evaluated alternative belongs to. Moreover, the numerical comparisons between the proposed method and relevant existing methods are given to clarify the advantages of the proposed method. Finally, an illustrative example is provided to demonstrate the applicability of the proposed method and algorithm. It is worth noting that the proposed method can be easily converted into a core algorithm, which is benefit for developing fuzzy expert system from the perspective of ignorance, and thus it has an important impact and significance on expert and intelligent systems.

A New Belief Entropy to Measure Uncertainty of Basic Probability Assignments Based on Belief Function and Plausibility Function.

How to measure the uncertainty of the basic probability assignment (BPA) function is an open issue in Dempster–Shafer (D–S) theory. The main work of this paper is to propose a new belief entropy, which is mainly used to measure the uncertainty of BPA. The proposed belief entropy is based on Deng entropy and probability interval consisting of lower and upper probabilities. In addition, under certain conditions, it can be transformed into Shannon entropy. Numerical examples are used to illustrate the efficiency of the new belief entropy in measurement uncertainty.

A Reliability-Based Consensus Model for Multiattribute Group Decision-Making with Analytically Evidential Reasoning Approach

Expert reliability is the ability to make unmistakable evaluations on attributes for the performance of an alternative in multiattribute group decision making (MAGDM). It has a significant effect on the group consensus calculation and group decision-making; unfortunately the reliability has not been considered in the consensus-reaching model yet. This study focuses on providing a reliability-based consensus model for MAGDM with analytically evidential reasoning (analytical ER for short) approach. The basic probability assignment (BPA) function which can be discounted by expert reliability is introduced to describe the performance judgments of each expert, by combining which of the group judgments could be determined with analytical ER rule. Then the consensus degrees of three levels (attribute level, alternative level, and expert level) are defined by Jousselme distance to identify the experts who should revise their judgments and point out revised suggestions, based on which a decision-making method within interaction is proposed to determine the effective BPA functions of all experts and make final decision-making. Finally, a numerical case study is carried out to illustrate the effectiveness of the method.

Mass Collaboration-Driven Method for Recommending Product Ideas Based on Dempster-Shafer Theory of Evidence

In the mass collaboration mode, there exist a large number of product ideas with low value density and thousands of participants who are differed on their professional backgrounds, knowledge structures, and value orientations. It is impossible for each participant to give a comprehensive evaluation of each idea as that in traditional methods for the reasons as mentioned above. In order to solve this problem, a mass collaboration-driven method for recommending product ideas is proposed based on Dempster-Shafer theory of evidence (DST). Firstly, the method for computing basic probability assignment (BPA) function, which can effectively reflect the facticity of experts’ evaluations, is introduced by discounting belief degrees with weights to extract the evaluation information of product ideas. Then, Dempster’s combination rule is used to combine the derived BPA functions for two times: the first one is to combine the discounted BPA functions on all criteria with respect to a specified expert and the other is to combine the combined BPA functions for all experts with respect to a specified alternative. Finally, the steps of mass collaboration-driven method for recommending product ideas based on the DST are proposed. An illustrative example is provided to demonstrate the applicability of the proposed method.

A hierarchical Dempster-Shafer evidence combination framework for urban area land cover classification

This paper presents a novel evidence combination framework for urban area land cover classification by using Light Detection And Ranging (LIDAR) data fused with co-registered near infrared and color images. The newly developed combination framework is built with a hierarchical structure involving an improved Dempster-Shafer (DS) theory of evidence for decision making. In the framework, a fuzzy basic probability assignment (BPA) function with fuzzy classes is firstly established based on the DS theory of evidence, and a probability is then assigned to each data source, that is derived from the original airborne LIDAR and the co-registered images. Secondly, an interesting approach is to introduce noise removal in an interim stage at the output of the probability distribution, and then the probability assigned to each data source is redistributed with a designated rule. Finally, a decision is made based on a “maximum normal support” rule, leading to the classification results. The proposed framework has been tested on two datasets. The testing results have shown that it can dramatically reduce the computational time in the classification process, and significantly improve the classification accuracy, i.e. 8.22% on Test 1 and 5.76% on Test 2 compared to the basic DS method. Due to its non-iterative and unsupervised nature, the proposed method is fast in computation, does not require training samples, and has achieved high classification accuracy.

IFS/ER-based large-scale multiattribute group decision-making method by considering expert knowledge structure

The large-scale multiattribute group decision-making (LMGDM) consisting of at least 20 experts has been widely popular in recent years. Although lots of efforts have been spent on improving the LMGDM, the subjective factors such as experts’ domain knowledges and bounded rationalities are still not well considered. This study focuses on providing a new LMGDM method by considering expert knowledge structure. An information extraction mechanism providing three kinds of inference ways including singleton attribute inference, local integral inference and global integral inference is introduced to ensure the assessments made by each expert with interval-valued intuitionistic fuzzy values (IVIFV) to be valid. Then a transformation is introduced to derive interval-valued basic probability assignment (BPA) function from the IVIFV, based on which expert reliability and attribute weight can be both reflected by evidential reasoning (ER) discounting. A pair of nonlinear optimization models that are extended by the analytical ER rule are established to make attribute fusion for expert and group fusion for alternative. An algorithm is summarized to solve the LMGDM problems by considering expert knowledge structure. Finally, an illustrative example as well as discussions is provided to demonstrate the applicability of the proposed method and algorithm.

Overall Assessment of Technical Conditions for Vessels

Overall assessment of technical conditions for a vessel is the major guarantees of ships’ combat capability. They are created and completed based on data and information of vessel which are complicated and numerous, so the structure of the vessel system, the subsystems and equipment of the vessel should be investigated accurately and precisely. In this paper, three models, such as the key equipment, the subsystem and the vessel system are given to assess the technical conditions of the vessel by using D-S evidence theory, fuzzy theory and gray relative analysis. The basic probability assignment functions of the key equipment are obtained. The assessment matrix of the vessel for subsystem is given, and the relational degrees of the reference and comparison sequences of the vessel are derived in order to calculate the assessment value. Taking an equipment and a vessel for example, the numerical results show that multi-hierarchy assessment is a useful method for investigating the vessel technical conditions because it can take advantage of the assessment parameters to model and calculate effectively. The influence of the technical parameters of some key equipment for the vessel are considered in the assessment, the reliabilities of the procedures and the results of the assessment are promoted significantly based on D-S evidence theory. The present assessment methods are expected to be useful in the maintenance and management of the vessel and equipment.