Cumulative Residual Entropy Research Articles

Worst-case distortion riskmetrics and weighted entropy with partial information

In this paper, we discuss the worst-case distortion riskmetrics for general distributions when only partial information (mean and variance) is known. This result is applicable to a general class of distortion risk measures and variability measures. Furthermore, we also consider the worst-case weighted entropy for general distributions when only partial information is available. Specifically, we provide some applications for entropies, weighted entropies and risk measures. The commonly used entropies include Gini functional, cumulative residual entropy, tail-Gini functional, cumulative Tsallis past entropy, extended Gini coefficient, among others. The risk measures contain some premium principles and shortfalls based on entropy. The shortfalls include the Gini shortfall, extended Gini shortfall, shortfall of cumulative residual entropy and shortfall of cumulative residual Tsallis entropy with order α.

Extended weighted generalized cumulative residual entropy for risk measurement

In this paper, we propose an extended variant of Weighted Generalized Cumulative Residual Entropy (WGCRE), its dynamic version, and a Weighted Generalized Cumulative Kerridge Inaccuracy measure by considering a general continuous non negative weight function. Several properties, including bounds and stochastic ordering, of these measures are obtained under appropriate assumptions. Furthermore, we introduce the empirical extended WGCRE as a non-parametric estimator, demonstrating its almost sure convergence to the proposed measure. A central limit theorem is established, examples are provided for illustrative purposes, and the stability of the empirical measure is examined. Finally, we demonstrate the practicality of this measure using real financial data for risk measurement.

Information Properties of Consecutive Systems Using Fractional Generalized Cumulative Residual Entropy

We investigate some information properties of consecutive k-out-of-n:G systems in light of fractional generalized cumulative residual entropy. We firstly derive a formula to compute fractional generalized cumulative residual entropy related to the system’s lifetime and explore its preservation properties in terms of established stochastic orders. Additionally, we obtain useful bounds. To aid practical applications, we propose two nonparametric estimators for the fractional generalized cumulative residual entropy in these systems. The efficiency and performance of these estimators are illustrated using simulated and real datasets.

High-precision estimation of plant alpha diversity in different ecosystems based on Sentinel-2 data

At present, the accuracy of remote sensing estimation models of plant alpha diversity is generally low, and high-precision estimation models in deciduous broadleaved forest (DBF), deciduous coniferous forest (DCF) and evergreen coniferous forest (ECF) are still lacking. The main purpose of this study is to construct high-precision remote sensing models for plant alpha diversity in multiple ecosystems at global scale. Normalized Difference Vegetation Index (NDVI) were derived from Sentinel-2 data. NDVI and NDVI based spectral diversity/heterogeneity indices were selected as predictive variables, and alpha diversity indices were selected as response variables. Simple linear regression (SLR), partial linear regression (PLSR), and random forest (RF) models were used to evaluate the predictive ability of the predictive variables against the response variables under six ecosystems (evergreen broadleaved forest (EBF), DBF, ECF, DCF, shrub, and grassland), and to compare the estimated robustness of various spectral diversity indices. In terms of prediction accuracy, the SLR models were the worst, and the PLSR model were average. RF performed best, outperforming most current models. Especially in DBF, ECF, shrub and grassland, the determination coefficient R2 of RF models can be as high as 0.9. In terms of the prediction of α-diversity, the prediction effect of species richness was better than that of Shannon index, Simpson index and Pielou index. The higher the vegetation complexity, the more accurate the assessment of vegetation α-diversity tends to be, especially in DBF, shrub and grassland. According to the importance of predictive variables and model stability evaluation results, NDVI, standard deviation of NDVI (SD), and NDVI derived Shannon’s diversity index (Sha), Cumulative Residual Entropy (CRE), Pielou’s evenness index (Pie), Hill’s numbers (Hill), Berger-Parker’s diversity index (Ber), Parametric Rao’s index of quadratic entropy(paRao) are all powerful indicators for predicting plant alpha diversity. Among them, the prediction performance of NDVI and SD is better. This study is not only an exploration of the practicability of R package “rasterdiv”, but also an attempt to construct high-precision remote sensing estimation models of plant alpha diversity at global scale.

Ultrasound normalized cumulative residual entropy imaging: Theory, methodology, and application

Background and Objective:Ultrasound information entropy imaging is an emerging quantitative ultrasound technique for characterizing local tissue scatterer concentrations and arrangements. However, the commonly used ultrasound Shannon entropy imaging based on histogram-derived discrete probability estimation suffers from the drawbacks of histogram settings dependence and unknown estimator performance. In this paper, we introduced the information-theoretic cumulative residual entropy (CRE) defined in a continuous distribution of cumulative distribution functions as a new entropy measure of ultrasound backscatter envelope uncertainty or complexity, and proposed ultrasound CRE imaging for tissue characterization. Methods:We theoretically analyzed the CRE for Rayleigh and Nakagami distributions and proposed a normalized CRE for characterizing scatterer distribution patterns. We proposed a method based on an empirical cumulative distribution function estimator and a trapezoidal numerical integration for estimating the normalized CRE from ultrasound backscatter envelope signals. We presented an ultrasound normalized CRE imaging scheme based on the normalized CRE estimator and the parallel computation technique. We also conducted theoretical analysis of the differential entropy which is an extension of the Shannon entropy to a continuous distribution, and introduced a method for ultrasound differential entropy estimation and imaging. Monte-Carlo simulation experiments were performed to evaluate the estimation accuracy of the normalized CRE and differential entropy estimators. Phantom simulation and clinical experiments were conducted to evaluate the performance of the proposed normalized CRE imaging in characterizing scatterer concentrations and hepatic steatosis (n = 204), respectively. Results:The theoretical normalized CRE for the Rayleigh distribution was π/4, corresponding to the case where there were ≥10 randomly distributed scatterers within the resolution cell of an ultrasound transducer. The theoretical normalized CRE for the Nakagami distribution decreased as the Nakagami parameter m increased, corresponding to that the ultrasound backscattered statistics varied from pre-Rayleigh to Rayleigh and to post-Rayleigh distributions. Monte-Carlo simulation experiments showed that the proposed normalized CRE and differential entropy estimators can produce a satisfying estimation accuracy even when the size of the test samples is small. Phantom simulation experiments showed that the proposed normalized CRE and differential entropy imaging can characterize scatterer concentrations. Clinical experiments showed that the proposed ultrasound normalized CRE imaging is capable to quantitatively characterize hepatic steatosis, outperforming ultrasound differential entropy imaging and being comparable to ultrasound Shannon entropy and Nakagami imaging. Conclusion:This study sheds light on the theory and methodology of ultrasound normalized CRE. The proposed ultrasound normalized CRE can serve as a new, flexible quantitative ultrasound envelope statistics parameter. The proposed ultrasound normalized CRE imaging may find applications in quantified characterization of biological tissues. Our code will be made available publicly at https://github.com/zhouzhuhuang.

Two fractional order cumulative residual time series measures based on Rényi entropy

In the study of time series analysis, Kullback-Leibler divergence can measure the difference between the distribution diverges of time series, and transfer entropy is usually used to quantify the causal influence among time series. In order to analyze the relationship between time series more comprehensively, two novel fractional order times series measures, i.e., Rényi cumulative residual Kullback-Leibler (RCRKL) and Rényi cumulative residual transfer entropy (RCRTE), are proposed in this paper. Some mathematical properties of RCRKL are proved as a distance metric, and the zeros and bounds of RCRTE are studied. Then, some examples on synthetic data are provided to study the effect of parameter changes on metric values. Finally, the proposed two fractional-order time series measures are applied to pavement rutting time series, for which a RCRKL-based clustering is designed and the causal relationships between influence factors and rutting are detected by RCETE. The results demonstrate that the RCRKL-based clustering algorithm performs better than those based on the baseline distance measures. Meanwhile, RCRTE can also detect the true causality between rutting and its influence factors and reject the false causality. In summary, the proposed RCRKL and RCRTE take the advantages of both cumulative residual entropy and Rényi entropy, and provide a more comprehensive insight for time series analysis.

Channel estimation for reconfigurable intelligent surface‐assisted mmWave using multidimensional Runge Kutta orthogonal matching pursuit algorithm in MIMO systems

SummaryIn the context of Reconfigurable Intelligent Surface (RIS) assisted millimeter wave communications within the Multiple Input Multiple Output (MIMO) system, Channel Estimation (CE) poses a significant challenge due to signal sparsity and blockages between transmitters and receivers. Consequently, communication faces difficulties without a direct line‐of‐sight path, leading to increased pilot overhead. To address these obstacles, this research introduces a channel estimation approach for RIS‐aided millimeter wave communication employing the Multidimensional Runge Kutta Orthogonal Matching Pursuit (MRKOMP) algorithm in MIMO systems. The proposed method aims to enhance network coverage and channel capacity for efficient data transmission. By utilizing the multiscale cumulative residual distribution entropy function, the method establishes an indirect path when the line‐of‐sight is obstructed. Furthermore, the multidimensional orthogonal matching pursuit methodology calculates sparsity levels and facilitates sparse recovery. Optimization of the weight parameter through the Runge Kutta optimizer effectively eliminates sparsity, resulting in improved spectral efficiency and RIS gain. Simulation results demonstrate a significant 99.9% enhancement in network capacity and spectral efficiency compared to existing methods.

On the application of compound optimal design strategy in progressively type-II censored life-testing experiments

Progressive censoring schemes allow removal of censoring units at the time of each failure. This flexibility of removing items during the experiment makes progressive censoring very effective but it also increases the number of possible censoring schemes. So the problem of choosing optimal progressive censoring schemes has gained a lot of attention in the literature. In this article, we propose a new criterion based on cumulative residual entropy measure for the determination of optimal progressive type II censoring scheme. The proposed criterion represents an overall information of the progressive type II censored life-testing experiment and we obtain optimal schemes by maximizing this measure. Also, we develop optimal design by maximizing the information subject to a given cost constraint. Finally, we perform a compound optimal design strategy that simultaneously optimizes our proposed criterion and the total cost of the experiment. Our proposed design does not depend on any asymptotic results. Sensitivity analysis for the constraint design and COD are studied for the mis-specification of the input parametrs. A real data set is analyzed for the purpose of illustration.

Dynamic Weighted Cumulative Residual Entropy Estimators for Laplace Distribution: Bayesian Approach

Objectives: To develop Bayesian estimators of dynamic weighted cumulative residual entropy (DWCRE) for Laplace distribution and to investigate posterior risks using various priors and loss functions. Methods: Weighted entropy measure of information is provided by a probabilistic experiment whose basic events are described by their objective probabilities and some qualitative (objective or subjective) weights. In this paper, we have used priors (Jeffrey’s, Hartigan, Uniform and Gumble Type II) and several loss functions. Findings: Bayesian estimators and associated posterior risks for Laplace distribution have been derived for different priors and loss functions. Monte Carlo Simulation study and graphical analyses have also been presented along with the conclusion. Through the comprehensive simulation study in the paper, it has been observed that Hartigan prior is better than other priors in terms of the posterior risk whereas Uniform prior has always higher posterior risk. Novelty: The introduction of new Bayesian estimators and their posterior risks for dynamic weighted cumulative residual entropy (DWCRE) of Laplace distribution. Keywords: Bayesian estimators, Laplace distribution, Fisher information matrix, Loss functions, Priors

Fractional generalized cumulative residual entropy: properties, testing uniformity, and applications to Euro Area daily smoker data

<abstract><p>The fractional generalized cumulative residual entropy, a broader version of the cumulative residual entropy, holds significance in assessing the uncertainty model of random variables and maintains straightforward connections with reliability models and crucial information. This article represents and modifies some novel features of the fractional generalized cumulative residual entropy and discusses the weak convergence. Additionally, the measure is utilized to assess uniformity, involving the derivation of the limit distribution and an approximation of the test statistic's distribution. Furthermore, the concept of stability is addressed. Moreover, the presentation includes the critical points and power analysis against alternative distributions of this test statistic. Furthermore, a simulation study is carried out to compare the power value of the proposed test with that of other tests of uniformity. Moreover, the uniformity test utilizes real data on daily smokers in the countries of the Euro Area. Finally, our model's exponential distribution is applied to our model's empirical form.</p></abstract>

Fractional Cumulative Residual Inaccuracy Information Measure and Its Extensions with Application to Chaotic Maps

The purpose of this work is to introduce fractional cumulative residual inaccuracy (FCRI) information, Jensen-cumulative residual inaccuracy (JCRI), and Jensen-fractional cumulative residual inaccuracy (JFCRI) information measure. Further, we study the FCRI information for some well-known models used in reliability, economics and survival analysis. The associated results reveal some interesting connections between the FCRI information measure and cumulative residual entropy and Gini mean difference measures. Applications to two chaotic discrete-time dynamical systems (Chebyshev and Logistic) are presented to illustrate the behavior of the proposed information measures. FCRI and JFCRI measures allow to determine regions of discrepancy between systems, depending on their respective fractional and chaotic map parameters.

Some Tsallis entropy measures in concomitants of generalized order statistics under iterated FGM bivariate distribution

<p>Shannon differential entropy is extensively applied in the literature as a measure of dispersion or uncertainty. Nonetheless, there are other measurements, such as the cumulative residual Tsallis entropy (CRTE), that reveal interesting effects in several fields. Motivated by this, we study and compute Tsallis measures for the concomitants of the generalized order statistics (CGOS) from the iterated Farlie-Gumbel-Morgenstern (IFGM) bivariate family. Some newly introduced information measures are also being considered for CGOS within the framework of the IFGM family, including Tsallis entropy, CRTE, and an alternative measure of CRTE of order $ \eta $. Applications of these results are given for order statistics and record values with uniform, exponential, and power marginals distributions. In addition, the empirical cumulative Tsallis entropy is suggested as a method to calculate the new information measure. Finally, a real-world data set has been analyzed for illustrative purposes, and the performance is quite satisfactory.</p>

A note on interval cumulative past Renyi entropy

In analogy with cumulative residual entropy, Di Crescenzo and Longobardi (2009) proposed cumulative past entropy which served as the basis for many following works. To this end, we define cumulative past Renyi entropy for double truncated random variable which we call as interval cumulative past Renyi entropy (ICPRE). Some properties, such as effect of monotone transformation and bounds of ICPRE, are studied. Furthermore, it is shown that the proposed measure uniquely determines the distribution function and characterizes certain lifetime distributions. Finally, non-parametric estimators of ICPRE are obtained and the best proposed estimator is investigated with the help of simulated data set and real-life data set.

Bivariate dynamic weighted cumulative residual entropy

Bivariate dynamic weighted cumulative residual entropy

A family of variability measures based on the cumulative residual entropy and distortion functions

Variability measures are important tools in the construction of premium principles and risk aversions. In this paper, we propose a family of such measures based on a distorted weighted cumulative residual entropy, which follows by a sensitivity analysis of distortion risk measures. For this family, we obtain properties, connections with other measures, a covariance representation, and some useful interpretations. Furthermore, we explore an application on premium principles based on beta generated distributions, and we give an empirical estimation. We also provide bounds and numerical illustrations.

Multiscale Cumulative Residual Dispersion Entropy with Applications to Cardiovascular Signals.

Heart rate variability (HRV) is used as an index reflecting the adaptability of the autonomic nervous system to external stimuli and can be used to detect various heart diseases. Since HRVs are the time series signal with nonlinear property, entropy has been an attractive analysis method. Among the various entropy methods, dispersion entropy (DE) has been preferred due to its ability to quantify the time series' underlying complexity with low computational cost. However, the order between patterns is not considered in the probability distribution of dispersion patterns for computing the DE value. Here, a multiscale cumulative residual dispersion entropy (MCRDE), which employs a cumulative residual entropy and DE estimation in multiple temporal scales, is presented. Thus, a generalized and fast estimation of complexity in temporal structures is inherited in the proposed MCRDE. To verify the performance of the proposed MCRDE, the complexity of inter-beat interval obtained from ECG signals of congestive heart failure (CHF), atrial fibrillation (AF), and the healthy group was compared. The experimental results show that MCRDE is more capable of quantifying physiological conditions than preceding multiscale entropy methods in that MCRDE achieves more statistically significant cases in terms of p-value from the Mann-Whitney test.

Covariance Representations and Coherent Measures for Some Entropies.

We obtain covariance and Choquet integral representations for some entropies and give upper bounds of those entropies. The coherent properties of those entropies are discussed. Furthermore, we propose tail-based cumulative residual Tsallis entropy of order α (TCRTE) and tail-based right-tail deviation (TRTD); then, we define a shortfall of cumulative residual Tsallis (CRTES) and shortfall of right-tail deviation entropy (RTDS) and provide some equivalent results. As illustrated examples, the CRTESs of elliptical, inverse Gaussian, gamma and beta distributions are simulated.

Feature recognition of complex systems using cumulative residual Tsallis signal entropy and grey wolf optimized support vector machine

In this paper, the cumulative residual Tsallis singular entropy (CRTSE) is introduced to measure the complex characteristics of nonlinear signals. Firstly, we do singular value decomposition on time series, which can reduce the interference of noise on information extraction, and the singular values represent the information characteristics of the signal. Then the statistical distribution of the signal is described by the cumulative residual function of singular values, and the Tsallis entropy is calculated to quantify the complexity. We verify the effectiveness and robustness of CRTSE through simulation experiments. Finally, we propose a grey wolf optimized support vector machine based on CRTSE called CRTSE-GWOSVM to intelligently diagnose complex systems. The results show that CRTSE can effectively measure the complex characteristics of time series, GWOSVM is superior to SVM and particle swarm optimized support vector machine (PSOSVM) in data identification, and CRTSE-GWOSVM model can identify complex systems more effectively and accurately.

An inaccuracy measure between non-explosive point processes with applications to Markov chains

AbstractInaccuracy and information measures based on cumulative residual entropy are quite useful and have received considerable attention in many fields, such as statistics, probability, and reliability theory. In particular, many authors have studied cumulative residual inaccuracy between coherent systems based on system lifetimes. In a previous paper (Bueno and Balakrishnan, Prob. Eng. Inf. Sci.36, 2022), we discussed a cumulative residual inaccuracy measure for coherent systems at component level, that is, based on the common, stochastically dependent component lifetimes observed under a non-homogeneous Poisson process. In this paper, using a point process martingale approach, we extend this concept to a cumulative residual inaccuracy measure between non-explosive point processes and then specialize the results to Markov occurrence times. If the processes satisfy the proportional risk hazard process property, then the measure determines the Markov chain uniquely. Several examples are presented, including birth-and-death processes and pure birth process, and then the results are applied to coherent systems at component level subject to Markov failure and repair processes.

Generalized cumulative residual Tsallis entropy and its properties

Generalized cumulative residual Tsallis entropy and its properties