Deng Entropy Research Articles

Belief Tsallis-Deng Structure Entropy and its uniform framework for analyzing multivariate time-series complexity based on evidence theory

In the framework of evidence theory, this paper proposes a new measure of uncertainty and introduces an entirely new unified framework for analyzing the complexity of multivariate time series. The theoretical foundation of this paper is random sets, viewing multivariate sequences as a new type of set variable, and using the belief theory framework to describe their behavior. Set variables are a natural extension of point variables, thus providing stronger universality in the analysis of multivariate time series. We tested our proposed new analytical framework on the logistic map and used the Tsallis-Deng Structure Entropy introduced in this paper to analyze the generated multivariate time series. Compared to the traditional Deng entropy, Tsallis-Deng Structure Entropy can adjust the sensitivity to time series complexity by tuning its structural parameter p. As the parameters of the logistic map change, the changes in Tsallis-Deng Structure Entropy show a high correlation with the Lyapunov exponent. In addition, we applied the proposed framework to analyze a multivariate epileptic EEG dataset. The results showed that using the novel evidence theory-based analysis framework for the 23-channel multivariate time series, the average classification accuracies of Deng entropy, Tsallis-Deng entropy, and Tsallis-Deng structural entropy for 24 epilepsy patients were 79 %, 74 %, and 82 %, respectively. In contrast, the average recognition accuracy using the traditional probabilistic framework based on multivariate embedding theory was 64 %. The evidence theory-based framework generally outperformed the multivariate embedding theory, with Tsallis-Deng structural entropy achieving the best performance. These results indicate that the proposed evidence theory-based framework not only avoids the dimensionality curse of the multivariate embedding theory but also better quantifies the complexity of multivariate time series from the same system and distinguishes different types of multivariate time series.

Measures of extended fractional Deng entropy and extropy with applications

Recently, Zhang and Shang introduced modifications to the concept of fractional entropy and proved some properties based on the inverse Mittag-Leffler function (MLF). The Deng entropy serves as a valuable measure in the Dempster-Shafer evidence theory (DST) to tackle uncertainty. In this study, we extend the fractional Deng entropy measure, introducing two distinct versions: E fd 1 α ( m ) and E fd 2 α ( m ) . We call this new measure the extended fractional Deng entropy, EFDEn. Additionally, we apply a similar approach to the fractional Deng extropy measure, resulting in E X fd 1 α ( m ) and E X fd 2 α ( m ) . We call this new measure the extended fractional Deng extropy, EFDEx. These two measures are complementary, leading to provide a deeper analysis of known and unknown information. Subsequently, we conduct a comparative analysis of these measures within the DST framework. We also propose the decomposable fractional Deng entropy, an extension of the decomposable entropy for Dempster–Shafer evidence theory, which effectively decomposes fractional Deng entropy. Finally, we delve into a pattern recognition classification problem to highlight the importance of these new measures.

New distance measure-driven flexible linguistic consensus model with application to urban flooding risk assessment

As a novel linguistic representation tool, flexible linguistic expression (FLE) has substantial flexibility in expressing ambiguous and uncertain information from experts. However, FLEs pose the following challenges to developing information measurement and consensus mechanisms for FLEs-based multi-attribute group decision-making (MAGDM). (1) The current studies primarily use linguistic distribution approximation or triangular fuzzy numbers to measure the difference between FLEs indirectly without proposing an exact distance measure. (2) Few studies have been conducted on direct accurate consensus models incorporating adaptive consensus feedback mechanisms. To deal with these issues, the flexible linguistic ordinal Deng entropy (FLODE) is firstly proposed to measure the uncertainty of FLEs, and by combining it with linguistic scale functions, a new distance measure for FLEs is constructed, which can accurately measure the level of group consensus and the difference in pairwise FLEs. Subsequently, by integrating the defined FLODE with the classical entropy weight method, a new method for determining the experts’ personalized attribute weights is constructed, fully considering the impact of experts’ personalized attribute weights on decision-making results. Also, a new consensus model incorporating a minimum adjustment feedback mechanism and dynamic personalized attribute weights is developed, in which the feedback adjustment coefficients and personalized attribute weights are dynamically updated during the consensus reaching process (CRP). Lastly, an urban flooding risk assessment is applied to confirm the effectiveness of the proposed consensus model.

A Multi-Information Fusion Algorithm to Fault Diagnosis of Power Converter in Wind Power Generation Systems

Power electronics-based converters are the major and most vulnerable components in wind power generation systems. Converter faults will affect power quality,damage expensive equipment such asgenerators, or even pose a massive threat to the entire power grid. Fault diagnosis is considered as a powerful means to improve system reliability and reduce maintenance costs. Existing data-driven fault diagnosis methods related to improving feature extraction approaches to obtain high diagnostic accuracy are mostly based on single-scale feature of signals; this neglects the potentially valuable information of other scales. This paper proposes an algorithm of Dempster-Shafer and Deng entropy fusion multi-scale approximate entropy (DSDEMAE) for wind power converter fault diagnosis. Firstly, it calculates the multi-scale approximate entropy of fault signals, mining more potentially valuable information. Secondly, Dempster-Shafer is used to fuse features of various scales and effectively handles the conflicts and uncertainties between different features. Finally,Deng entropyis adopted to measure the uncertainty to adjust weight distribution, reducing the influence of highly conflicting features and mutually benefitting features of different scales. Extensive experimental results on simulated and experimental data demonstrate the effectiveness of the proposed algorithm. Compared with advanced methods, this method can diagnose the faults with higher accuracy and stronger robustness.

Baby cry recognition based on WOA-VMD and an improved Dempster–Shafer evidence theory

Background and objectiveConflict may happen when more than one classifier is used to perform prediction or classification. The recognition model error leads to conflicting evidence. These conflicts can cause decision errors in a baby cry recognition and further decrease its recognition accuracy. Thus, the objective of this study is to propose a method that can effectively minimize the conflict among deep learning models and improve the accuracy of baby cry recognition. MethodsAn improved Dempster–Shafer evidence theory (DST) based on Wasserstein distance and Deng entropy was proposed to reduce the conflicts among the results by combining the credibility degree between evidence and the uncertainty degree of evidence. To validate the effectiveness of the proposed method, examples were analyzed, and applied in a baby cry recognition. The Whale optimization algorithm-Variational mode decomposition (WOA-VMD) was used to optimally decompose the baby cry signals. The deep features of decomposed components were extracted using the VGG16 model. Long Short-Term Memory (LSTM) models were used to classify baby cry signals. An improved DST decision method was used to obtain the decision fusion. ResultsThe proposed fusion method achieves an accuracy of 90.15% in classifying three types of baby cry. Improvement between 2.90% and 4.98% was obtained over the existing DST fusion methods. Recognition accuracy was improved by between 5.79% and 11.53% when compared to the latest methods used in baby cry recognition. ConclusionThe proposed method optimally decomposes baby cry signal, effectively reduces the conflict among the results of deep learning models and improves the accuracy of baby cry recognition.

DENG ENTROPY AND INFORMATION DIMENSION FOR COVID-19 AND COMMON PNEUMONIA CLASSIFICATION

Motivated by previous authors’ work, where Shannon entropy, box covering and information dimension were applied to quantify pulmonary lesions, this paper extends such a contribution in two fashions: (i) Following the approach to quantify pulmonary lesions with Deng entropy and Deng information dimension obtained through box covering method; (ii) exploiting the Shannon and Deng lesion quantification for pulmonary illnesses classification with a bidirectional Long Short Term Memory (bLSTM). The referred pulmonary illnesses are Common Pneumonia (CP) and COVID-19. Shannon entropy and information dimension are performed here and called the Shannon sequence. Then, Deng entropy and Deng information dimension are computed for chest Computed Tomography (CT) images to obtain and combine two data sequences to quantify the pulmonary lesions. The data sequence resulting from the data combination is called the Deng sequence. Both Shannon and Deng sequences are independently used as input for the bLSTM. CT lung scans of 531 healthy subjects, 497 confirmed COVID-19 diagnoses and 516 with CP were analyzed to obtain the Shannon and Deng sequences. The results demonstrate that Deng entropy and Deng information dimension of CT images can differentiate similar lung lesions between COVID-19 and CP. Besides, a statistical analysis shows that: (a) Classification by the bLSTM is better when using the Deng sequence than the Shannon sequence; (b) Deng sequences plus bLSTM significantly outperform DenseNet-201, GoogLeNet and MobileNet-v2 in classifying COVID-19, CP and Normal CT (healthy subjects) in time and accuracy. Hence, the Deng sequence and bLSTM are fast and accurate tools for helping in diagnosing CP and COVID-19.

A d-SUMMABLE APPROACH TO DENG INFORMATION DIMENSION OF COMPLEX NETWORKS

Several new network information dimension definitions have been proposed in recent decades, expanding the scope of applicability of this seminal tool. This paper proposes a new definition based on Deng entropy and d-summability (a concept from geometric measure theory). We will prove to what extent the new formulation will be useful in the theoretical and applied points of view.

A generalized Rényi entropy to measure the uncertainty of a random permutation set

Random permutation set (RPS) introduces a novel set that considers all subsets with ordered elements from a given set. Each subset with ordered elements represents a permutation event within the permutation event space (PES). The permutation mass function (PMF) represents the chance of occurrence of events in the PES. PES and PMF make up RPS, which contains ordered information and also provides a new insight to consider the uncertainty. This characteristic aligns more closely with the occurrence of ordered events in the real world. However, existing entropies cannot measure the uncertainty with ordered information. To address this issue, a generalized Rényi entropy is proposed, it degenerates into different entropies with the changing of scenarios and parameters, in other words, it is compatible with these entropies. When the events in permutation event space are not ordered, Rényi-RPS entropy degenerates into Deng entropy. In addition, Rényi-RPS entropy further degenerates into Rényi entropy under the probability distribution. In a further way, when the parameter α→1, Rényi-RPS entropy evolves into Shannon entropy. Several numerical examples will illustrate the characteristics of the presented Rényi-RPS entropy.

Linearity in Deng entropy

Linearity is considered as a fundamental property of significant importance in numerous domains. A linear mathematical expression is clear, concise, and convenient for solving problems, making it a valuable tool for approximating complex issues. In recent years, Deng entropy has been proposed as a generalization of Shannon entropy, applied to measure the uncertainty degree of the mass function in the power set. There are two main contributions in this paper. One is that the linearity can be observed in Deng entropy. We present a set of specific mass functions named power assigned mass function (PAMF), which could generate linear type Deng entropy (LTDE). The other is that we find the slope is nothing else but the information fractal dimension of mass function. Moreover, we discover that for any given slope within the range of 0 to ln3ln2, at least one mass function that yields Deng entropy corresponding to the given slope can be derived through the mass function generator (MFG), in a strict linear way. Some proofs, numerical examples, discussions and an error analysis are provided to validate the effectiveness of our findings.

Performance evaluation of information fusion systems based on belief entropy

Information fusion systems are widely applied in many fields. However, how to quantitatively evaluate the performance of information fusion systems is still an open issue. To pioneeringly address the issue, in this paper, a performance evaluation model of information fusion systems based on Deng entropy is proposed. The proposed model quantitatively indicates the ability of evidential data fusion algorithms, including Dempster’s combination rule, average combination method, Murphy’s combination method, Yager’s combination rule, and Dubois’s combination rule, to eliminate uncertainty during the fusion process. Deng entropy serves as an indicator to characterize the uncertainty before and after fusion. We define the fusion efficiency parameter η, to numerically evaluate the performance of information fusion systems. Conflict among evidences can also be manifested in the efficiency parameter η. Several examples are presented to illustrate properties of the model. Finally, two real applications in classification are given to verify the practicability of this model.

A new proposed GLCM texture feature: modified Rényi Deng entropy

A new proposed GLCM texture feature: modified Rényi Deng entropy

Multisensor Data Fusion in IoT Environments in Dempster-Shafer Theory Setting: An Improved Evidence Distance-Based Approach.

In IoT environments, voluminous amounts of data are produced every single second. Due to multiple factors, these data are prone to various imperfections, they could be uncertain, conflicting, or even incorrect leading to wrong decisions. Multisensor data fusion has proved to be powerful for managing data coming from heterogeneous sources and moving towards effective decision-making. Dempster-Shafer (D-S) theory is a robust and flexible mathematical tool for modeling and merging uncertain, imprecise, and incomplete data, and is widely used in multisensor data fusion applications such as decision-making, fault diagnosis, pattern recognition, etc. However, the combination of contradictory data has always been challenging in D-S theory, unreasonable results may arise when dealing with highly conflicting sources. In this paper, an improved evidence combination approach is proposed to represent and manage both conflict and uncertainty in IoT environments in order to improve decision-making accuracy. It mainly relies on an improved evidence distance based on Hellinger distance and Deng entropy. To demonstrate the effectiveness of the proposed method, a benchmark example for target recognition and two real application cases in fault diagnosis and IoT decision-making have been provided. Fusion results were compared with several similar methods, and simulation analyses have shown the superiority of the proposed method in terms of conflict management, convergence speed, fusion results reliability, and decision accuracy. In fact, our approach achieved remarkable accuracy rates of 99.32% in target recognition example, 96.14% in fault diagnosis problem, and 99.54% in IoT decision-making application.

Complex-valued Deng entropy

Complex-valued Deng entropy

A new belief entropy and its application in software risk analysis

The measurement of uncertainty has been an important topic of research. In Dempster’s framework, Deng entropy serves as a reliable tool for such measurements. However, it fails to consider more comprehensive information, resulting in the loss of critical data. An improved belief entropy is proposed in this paper, which preserves all the merits of Deng entropy. When there is only a single element, it can be degraded to Shannon entropy. When dealing with multiple elements, the partitioning method employed for mass functions makes it more responsive and efficient than alternative measures of uncertainty. Some numerical examples are given to further illustrate the effectiveness and applicability of the proposed entropy measure. Additionally, a case study is conducted on software risk analysis, demonstrating the practical value and relevance of the proposed method in real-world scenarios.

Information Fusion-Based Fault Diagnosis Method Using Synthetic Indicator

Multisensor information fusion technology plays an essential role in fault diagnosis. Uncertain reasoning is the core of information fusion, and the DS evidence theory (DSE) has founded a general and popular framework for uncertainty reasoning. Under this framework, a novel information fusion method is developed in this article, to solve the problems of counterintuitive results, poor robustness, and “one-vote veto” when fusing highly conflicting evidence using DSE. We proposed a new synthetic indicator, which can effectively eliminate the conflicts between evidences. This synthetic indicator is composed of two indexes: measurement index of indirect conflicts and that of the evidence’s information itself. First, the divergence is used to measure the degree of conflict between bodies of evidence, and the Decision-Making Trial and Evaluation Laboratory (DEMATEL) model is employed to deal with indirect conflicts. Second, the Deng entropy is utilized to quantitatively measure the uncertainty of evidences, and the relative credibility preference of the evidence is expressed by the fuzzy preference relation. Finally, the initial body of evidence frame is weighted and revised by synthetic indicator before adopting the Dempster Shafer (DS) fusion rule. The superiority of the developed method is illustrated by numerical examples and application cases. The results show that fault diagnosis using the developed information fusion method can have a higher diagnostic accuracy.

A novel method for local anomaly detection of time series based on multi entropy fusion

Dynamic Shannon entropy, which is fused by Shannon entropy and permuted distribution entropy (PDE), achieves good results in local anomaly recognition of time series. However, the dynamic Shannon entropy performs poorly in identifying various abnormal behaviors of complex signals with noise and chaotic characteristics and periodic background. It also gets unsatisfying results in anomaly detection of signals containing relatively large uncertainty anomaly information. Deng entropy, an extended entropy of Shannon entropy, expands the information capacity of Shannon entropy and is more excellent at describing uncertainty and complexity. In this paper, dynamic Deng entropy (DyEd) is proposed by fusing Deng entropy and PDE to identify local anomalies of time series. Six numerical experiments and an empirical application are conducted to illustrate the efficiency of the proposed method. Results show that the proposed method is of great success for signal anomaly detection with periodic backgrounds.

Entropy of Random Permutation Set

Recently, a new kind of set, named Random Permutation Set (RPS), has been presented. RPS takes the permutation of a certain set into consideration, which can be regarded as an ordered extension of evidence theory. Uncertainty is an important feature of RPS. A straightforward question is how to measure the uncertainty of RPS. To address this issue, the entropy of RPS (RPS entropy) is presented in this article. The proposed RPS entropy is compatible with Deng entropy and Shannon entropy. In addition, RPS entropy meets probability consistency, additivity, and subadditivity. Numerical examples are designed to illustrate the efficiency of the proposed RPS entropy. Besides, a comparative analysis of the choice of applying RPS entropy, Deng entropy, and Shannon entropy is also carried out.

Normal distribution based on maximum Deng entropy

We define the distribution of uncertainty based on Deng entropy and proved that the limiting form of the distribution is the normal distribution by the characteristic function. We conduct some discussion on the distribution, and we further analyze the error of its normal approximation and give the conditions that should be satisfied for the requirement of given error. Finally, we reveal the relationship between the distribution and the binomial distribution and the advantages of taking it as a prior distribution in evidence theory.

A PSL-based Approach to Human Activity Recognition in Smart Home Environments

Human activity recognition is widely used in smart cities, public safety and other fields, especially in smart home systems where it has a pivotal role. The study addresses the shortcomings of Markov logic networks for human activity recognition and proposes a human activity recognition method in smart home scenarios - an activity recognition framework based on Probabilistic Soft Logic (PSL). The framework is able to deal with logical uncertainty problems and provides expression and inference mechanisms for data uncertainty problems on this basis. The framework utilizes Deng entropy evidence theory to provide an evaluation method for sensor event uncertainty, and combines event calculus for activity modeling. Comparing the PSL method with three other common recognition methods, Ontology, Hidden Markov Model (HMM), and Markov logic network, on a public dataset, it was found that the PSL method has a much better ability to handle data uncertainty than the other three algorithms. The average recognition rates on the ADL and ADL-E sub datasets were 82.87% and 80.33%, respectively. In experiments to verify the ability of PSL to handle temporal complexity, PSL showed the least significant decrease in the average recognition rate and maintained an average recognition rate of 81.02% in the presence of concurrent and alternating activities. The human activity recognition method based on PSL has a better performance in handling both data uncertainty and temporal complexity.

Derive power law distribution with maximum Deng entropy

As one of the typical distributions, power law distribution is widely found in natural world. However, how to derive power law is still an open issue. The main contribution of this paper is to propose a method to derive power law distribution with maximum Deng entropy. In the proposed method, Lagrange multiplier approach, combined with the constraint of two given conditions, is used to obtain power law distribution based on maximum Deng entropy. Some numerical examples are used to illustrate the properties of the distribution.