Deng Entropy Research Articles

Measuring Uncertainty in the Negation Evidence for Multi-Source Information Fusion.

Dempster-Shafer evidence theory is widely used in modeling and reasoning uncertain information in real applications. Recently, a new perspective of modeling uncertain information with the negation of evidence was proposed and has attracted a lot of attention. Both the basic probability assignment (BPA) and the negation of BPA in the evidence theory framework can model and reason uncertain information. However, how to address the uncertainty in the negation information modeled as the negation of BPA is still an open issue. Inspired by the uncertainty measures in Dempster-Shafer evidence theory, a method of measuring the uncertainty in the negation evidence is proposed. The belief entropy named Deng entropy, which has attracted a lot of attention among researchers, is adopted and improved for measuring the uncertainty of negation evidence. The proposed measure is defined based on the negation function of BPA and can quantify the uncertainty of the negation evidence. In addition, an improved method of multi-source information fusion considering uncertainty quantification in the negation evidence with the new measure is proposed. Experimental results on a numerical example and a fault diagnosis problem verify the rationality and effectiveness of the proposed method in measuring and fusing uncertain information.

Maximum entropy of random permutation set

Recently, a new type of set, called random permutation set (RPS), is proposed by considering all the permutations of elements in a certain set. For measuring the uncertainty of RPS, the entropy of RPS is presented. However, the maximum entropy principle of RPS entropy has not been discussed. To address this issue, this paper presents the maximum entropy of RPS. The analytical solution of maximum RPS entropy and its PMF condition are proven and discussed. Besides, numerical examples are used to illustrate the maximum RPS entropy. The results show that the maximum RPS entropy is compatible with the maximum Deng entropy and the maximum Shannon entropy. Moreover, in order to further apply RPS entropy and maximum RPS entropy in practical fields, a comparative analysis of the choice of using Shannon entropy, Deng entropy, and RPS entropy is also carried out.

INFORMATION FRACTAL DIMENSION OF MASS FUNCTION

Fractals play an important role in nonlinear science. The most important parameter when modeling a fractal is the fractal dimension. Existing information dimension can calculate the dimension of probability distribution. However, calculating the fractal dimension given a mass function, which is the generalization of probability, is still an open problem of immense interest. The main contribution of this work is to propose an information fractal dimension of mass function. Numerical examples are given to show the effectiveness of our proposed dimension. We discover an important property in that the dimension of mass function with the maximum Deng entropy is [Formula: see text], which is the well-known fractal dimension of Sierpiski triangle. The application in complexity analysis of time series illustrates the effectiveness of our method.

Information dimension based on [formula omitted] entropy

Information dimension is applied to quantify fractal, and the current methods are improved on the basis of information entropy. Although information entropy is always used to measure the information volume, it fails when dealing with complex and uncertain information. To solve this problem, in this paper, the information dimension is defined by using Deng entropy, due to the reason that Deng entropy as the extension of information entropy has the characteristic of being able to measure uncertain information better. In our method, Deng entropy is considered to measure the information volume in the information dimension. Besides, box coverage is also combined in the process of the box covering to define box coverage Deng entropy information dimension. In addition, the classical information dimension and box dimension of BA scale-free networks, WS and NW small-world networks and four real networks are calculated to compare with the proposed information dimension. The results show that the values of Deng entropy information dimension and box coverage Deng entropy information dimension are bigger than the classical information dimension and box dimension for the same network. Moreover, the information dimension of the networks does not increase linearly with the increase of nodes.

A q-rung orthopair basic probability assignment and its application in medical diagnosis

Dempster-Shafer theory is widely used in decision-making and considered as one of the potential mathematical tools in order to fuse the evidence. However, existing studies in this theory show disadvantage due to conflicting nature of standard evidence set and the combination rule of evidence. In this paper, we have constructed the framework of q-rung evidence set to address the issue of conflicts based on the q-rung fuzzy number due to its more comprehensive range of advantage compared to the other fuzzy or discrete numbers. The proposed q-rung evidence set has the flexibility in assessing a parameter through the q-rung orthopair basic probability assignment consisting of membership and non-membership belief degree. Moreover, as the proposed q-rung orthopair basic probability assignment consists of pair of belief degrees, the possibility of conflicts cannot be ignored entirely. In this regard, a new association coefficient measure is introduced where each component of the belief degrees is modified through the weighted average mass technique. This paper uses various concept such as fuzzy soft sets, Deng entropy, association coefficient measure and score function for decision-making problem. Firstly, to obtain the initial q-rung belief function, we have implemented the Intuitionistic fuzzy soft set to assess the parameter of the alternatives and Deng entropy to find the uncertainty of the parameters. Secondly, the association coefficient measure is used to avoid the conflict through the modified form of evidence. Finally, we combined the evidence and found the score value of the Intuitionistic fuzzy numbers for the ranking of the alternatives based on the score values of alternatives. This study is validated with the case study in the medical diagnosis problem from the existing paper and compared the ranking of alternatives based on the score function of belief measures of the alternatives.

A unified formulation of entropy and its application

In this paper, a general formulation of entropy is proposed. It depends on two parameters and includes Shannon, Tsallis and fractional entropy, all as special cases. This measure of information is referred to as fractional Tsallis entropy and some of its properties are then studied. Furthermore, the corresponding entropy in the context of Dempster–Shafer theory of evidence is proposed and referred to as fractional version of Tsallis–Deng entropy. Finally, an application to two classification problems is presented.

A decomposable Deng entropy

Dempster–Shafer evidence theory is an extension of classical probability theory in the evidential environment. Evidential environment is an environment in which Dempster–Shafer evidence theory is used. The decomposable entropy for the Dempster–Shafer evidence theory can efficiently decompose the Shannon entropy for the Dempster–Shafer evidence theory, and has high theoretical and application value. This article proposes the decomposable Deng entropy, which is an extension of the decomposable entropy for the Dempster–Shafer evidence theory. The decomposable Deng entropy can effectively decompose the Deng entropy. When the cardinalities of all focal elements of a mass function are 1, then the decomposable Deng entropy will collapse to the decomposable entropy for the Dempster–Shafer evidence theory. Many calculation examples are used to verify the performance of the proposed model in decomposing Deng entropy. Experimental results show that the proposed model can efficiently decompose the Deng entropy.

Higher order information volume of mass function

For a certain moment, the information volume of probability space can be accurately expressed by Shannon entropy. But in real life, the distribution of events usually change over time, and the prediction of the information volume for a period of time in the future is still an open question. Deng entropy proposed in recent years is widely applied on measuring the uncertainty, but it is controversial because of its physical explanation and counter-intuitive results. In this paper, we give Deng entropy a new explanation based on the fractal idea, and propose its generalization called time fractal-based belief (TFB) entropy. The TFB entropy is recognized as predicting the uncertainty over a period of time by splitting times, and its maximum value, called higher order information volume of mass function (HOIVMF), which can express the information for a period time reasonably.

INFORMATION VOLUME FRACTAL DIMENSION

There has been immense interest in uncertainty measurement because most real-world problems are accompanied by uncertain events. Therefore, Deng entropy has been proposed to measure the uncertainty in the probability theory and evidence theory. In this paper, we show that the uncertainty of the basic probability assignment (BPA) separated through the maximum Deng entropy separation rule (MDESR) is larger than the maximum Deng entropy of the original BPA. In addition, when the cardinality of the frame of discernment increases, the maximum information volume becomes larger and converges slower. The information volume fractal dimension is then proposed to describe the fractal property of uncertainty about the separated BPA distribution, which indicates the inherent physical meanings of Deng entropy from the perspective of statistics. This work can inspire further research on the fractal property of Deng entropy. Some experiments are applied to show the applicability of our proposed information volume fractal dimension.

An efficient Bayesian inversion method for seepage parameters using a data-driven error model and an ensemble of surrogates considering the interactions between prediction performance indicators

The Bayesian method has been increasingly applied to the inversion of seepage parameters owing to its superiority of considering the uncertainty in the inversion process. However, most of the current Bayesian inversion studies only focus on parameter uncertainty, ignoring the model structure error. In addition, existing research has mostly adopted a single machine learning algorithm or an ensemble of surrogates based on a single prediction performance indicator as a substitute for the seepage forward model and has not considered the interactions between multiple prediction performance indicators, thereby leading to poor accuracy. To address these issues, this study proposed an efficient Bayesian inversion method for seepage parameters using a data-driven error model and an ensemble of surrogates considering the interactions between prediction performance indicators. A data-driven error model based on Gaussian process regression was integrated into the Bayesian inversion model to modify the likelihood function for dealing with the model structure error. For determining the weight coefficients of the ensemble surrogates, the improved Dempster-Shafer (D-S) evidence theory based on the Hellinger distance and Deng entropy was proposed to fuse multiple prediction performance indicators and consider their interactions. Further, an ensemble of surrogates in conjunction with support vector regression, Kriging, and multivariate adaptive regression splines was constructed through weighted summation. The validity and accuracy of the proposed method were verified by applying it to a real hydropower station in China. The results showed that the proposed method can significantly improve the accuracy and efficiency of Bayesian inversion of seepage parameters. The proposed method therefore serves as a new basis for the inversion of seepage parameters and can be applied to parameter inversion in other related fields.

A generalized TODIM-ELECTRE II method based on linguistic Z-numbers and Dempster\u2013Shafer evidence theory with unknown weight information

How to effectively reflect the randomness and reliability of decision information under uncertain circumstances, and thereby improve the accuracy of decision-making in complex decision scenarios, has become a crucial topic in the field of uncertain decision-making. In this article, the loss –aversion behavior of decision-makers and the non-compensation between attributes are considered. Furthermore, a novel generalized TODIM-ELECTRE II method under the linguistic Z-numbers environment is proposed based on Dempster–Shafer evidence theory for multi-criteria group decision-making problems with unknown weight information. Firstly, the evaluation information and its reliability are provided simultaneously by employing linguistic Z-numbers, which have the ability to capture the arbitrariness and vagueness of natural verbal information. Then, the evaluation information is used to derive basic probability assignments in Dempster–Shafer evidence theory, and with the consideration of both inner and outer reliability, this article employed Dempster’s rule to fuse evaluations. Subsequently, a generalized TODIM-ELECTRE II method is conceived under the linguistic Z-numbers environment, which considers both compensatory effects between attributes and the bounded rationality of decision-makers. In addition, criteria weights are obtained by applying Deng entropy which has the ability to deal with uncertainty. Finally, an example of terminal wastewater solidification technology selection is offered to prove this framework’s availability and robustness. The predominance is also verified by a comparative analysis with several existing methods.

A Novel Conflict Management Method Based on Uncertainty of Evidence and Reinforcement Learning for Multi-Sensor Information Fusion.

Dempster–Shafer theory (DST), which is widely used in information fusion, can process uncertain information without prior information; however, when the evidence to combine is highly conflicting, it may lead to counter-intuitive results. Moreover, the existing methods are not strong enough to process real-time and online conflicting evidence. In order to solve the above problems, a novel information fusion method is proposed in this paper. The proposed method combines the uncertainty of evidence and reinforcement learning (RL). Specifically, we consider two uncertainty degrees: the uncertainty of the original basic probability assignment (BPA) and the uncertainty of its negation. Then, Deng entropy is used to measure the uncertainty of BPAs. Two uncertainty degrees are considered as the condition of measuring information quality. Then, the adaptive conflict processing is performed by RL and the combination two uncertainty degrees. The next step is to compute Dempster’s combination rule (DCR) to achieve multi-sensor information fusion. Finally, a decision scheme based on correlation coefficient is used to make the decision. The proposed method not only realizes adaptive conflict evidence management, but also improves the accuracy of multi-sensor information fusion and reduces information loss. Numerical examples verify the effectiveness of the proposed method.

An Improved Approach of Incomplete Information Fusion and Its Application in Sensor Data-Based Fault Diagnosis

The Dempster–Shafer evidence theory has been widely used in the field of data fusion. However, with further research, incomplete information under the open world assumption has been discovered as a new type of uncertain information. The classical Dempster’s combination rules are difficult to solve the related problems of incomplete information under the open world assumption. At the same time, partial information entropy, such as the Deng entropy is also not applicable to deal with problems under the open world assumption. Therefore, this paper proposes a new method framework to process uncertain information and fuse incomplete data. This method is based on an extension to the Deng entropy in the open world assumption, negation of basic probability assignment (BPA), and the generalized combination rule. The proposed method can solve the problem of incomplete information under the open world assumption, and obtain more uncertain information through the negative processing of BPA, which improves the accuracy of the results. The results of applying this method to fault diagnosis of electronic rotor examples show that, compared with the other uncertain information processing and fusion methods, the proposed method has wider adaptability and higher accuracy, and is more conducive to practical engineering applications.

A new correlation coefficient of BPA based on generalized information quality

Dempster rule is commonly used to combine evidence from different sensors because of its excellent mathematical properties (commutativity and associativity). However, the conflict coefficient k of this rule cannot be reasonably and effectively express the conflict between evidence, which makes the Dempster rule greatly questioned. To describe in the relationship between evidence (conflict or correlation) more accurately, an evidence correlation coefficient based on generalized information quality is proposed. First of all, according to Deng entropy to measure the uncertainty of each evidence, and combined with the information quality of Yager, new generalized information quality is proposed, which performs well in measuring the basic probability assignment's certainty. Second, the evidence itself is modified by generalized information quality, and the evidence correlation coefficient is calculated based on the Pearson coefficient formula. A new measurement method of evidence conflict based on evidence correlation coefficient is proposed. Finally, combined with the evidence correlation coefficient and DEMATEL model, the evidence is discounted and combined. Numerous examples are used to analyze and compare the evidence conflict coefficient and the evidence combination results. The experimental results demonstrate that, compared with other evidence conflict measurement methods, the evidence conflict coefficient calculated by this method can reflect the difference between evidence more effectively. The result of evidence combination is more reasonable and accurate.

Fractional Deng Entropy and Extropy and Some Applications.

Deng entropy and extropy are two measures useful in the Dempster–Shafer evidence theory (DST) to study uncertainty, following the idea that extropy is the dual concept of entropy. In this paper, we present their fractional versions named fractional Deng entropy and extropy and compare them to other measures in the framework of DST. Here, we study the maximum for both of them and give several examples. Finally, we analyze a problem of classification in pattern recognition in order to highlight the importance of these new measures.

Measuring the Uncertainty in the Original and Negation of Evidence Using Belief Entropy for Conflict Data Fusion.

Dempster-Shafer (DS) evidence theory is widely used in various fields of uncertain information processing, but it may produce counterintuitive results when dealing with conflicting data. Therefore, this paper proposes a new data fusion method which combines the Deng entropy and the negation of basic probability assignment (BPA). In this method, the uncertain degree in the original BPA and the negation of BPA are considered simultaneously. The degree of uncertainty of BPA and negation of BPA is measured by the Deng entropy, and the two uncertain measurement results are integrated as the final uncertainty degree of the evidence. This new method can not only deal with the data fusion of conflicting evidence, but it can also obtain more uncertain information through the negation of BPA, which is of great help to improve the accuracy of information processing and to reduce the loss of information. We apply it to numerical examples and fault diagnosis experiments to verify the effectiveness and superiority of the method. In addition, some open issues existing in current work, such as the limitations of the Dempster-Shafer theory (DST) under the open world assumption and the necessary properties of uncertainty measurement methods, are also discussed in this paper.

Entropy measure for orderable sets

How to measure the uncertainty of orderable sets is still an open issue. In this paper, a new method based on Deng entropy to measure the uncertainty of orderable sets is proposed. When orderable sets degenerate as unorderable sets, the new entropy would degenerate as Deng entropy. When orderable sets degenerate into singleton, the new entropy would degenerate into Shannon entropy. Some numerical examples are used to illustrate the efficiency and accuracy of the proposed method.

Information Volume of Mass Function

Given a probability distribution, its corresponding information volume is Shannon entropy. However, how to determine the information volume of a given mass function is still an open issue. Based on Deng entropy, the information volume of mass function is presented in this paper. Given a mass function, the corresponding information volume is larger than its uncertainty measured by Deng entropy. In addition, when the cardinal of the frame of discernment is identical, both the total uncertainty case and the BPA distribution of the maximum Deng entropy have the same information volume. Some numerical examples are illustrated to show the efficiency of the proposed information volume of mass function.

Uncertainty measure in evidence theory

As an extension of probability theory, evidence theory is able to better handle unknown and imprecise information. Owing toits advantages, evidence theory has more flexibility and effectiveness for modeling and processing uncertain information.Uncertainty measure plays an essential role both in evidence theory and probability theory.In probability theory, Shannon entropy provides a novel perspective for measuring uncertainty. Various entropies exist for measuring the uncertainty of basic probability assignment (BPA) in evidence theory. However, from the standpoint of the requirements of uncertainty measurement and physics, these entropies are controversial. Therefore, the process for measuring BPA uncertainty currently remains an open issue in the literature.Firstly, this paper reviews the measures of uncertainty in evidence theory followed by an analysis of some related controversies. Secondly, we discuss the development of Deng entropy as an effective way to measure uncertainty, including introducing its definition, analyzing its properties, and comparing it to other measures. We also examine the concept of maximum Deng entropy, the pseudo-Pascal triangle of maximum Deng entropy, generalized belief entropy, and measures of divergence.In addition, we conduct an analysis of the application of Deng entropy and further examine the challenges for future studies on uncertainty measurement in evidence theory. Finally, a conclusion is provided to summarize this study.

Incomplete Information Management Using an Improved Belief Entropy in Dempster-Shafer Evidence Theory.

Quantifying uncertainty is a hot topic for uncertain information processing in the framework of evidence theory, but there is limited research on belief entropy in the open world assumption. In this paper, an uncertainty measurement method that is based on Deng entropy, named Open Deng entropy (ODE), is proposed. In the open world assumption, the frame of discernment (FOD) may be incomplete, and ODE can reasonably and effectively quantify uncertain incomplete information. On the basis of Deng entropy, the ODE adopts the mass value of the empty set, the cardinality of FOD, and the natural constant e to construct a new uncertainty factor for modeling the uncertainty in the FOD. Numerical example shows that, in the closed world assumption, ODE can be degenerated to Deng entropy. An ODE-based information fusion method for sensor data fusion is proposed in uncertain environments. By applying it to the sensor data fusion experiment, the rationality and effectiveness of ODE and its application in uncertain information fusion are verified.