Cumulative Residual Entropy Research Articles

English

In this paper, we obtain lower bounds for the variance of a function of random variables in terms of measures of reliability and entropy. Also based on the obtained characterization via the lower bounds for the variance of a function of random variable X, we find a characterization of the weighted function corresponding to density function f(x), in terms of Chernoff-type inequalities. Subsequently, we obtain monotonic relationships between variance residual life and dynamic cumulative residual entropy and between variance past lifetime and dynamic cumulative past entropy. Moreover, we find lower bounds for the variance of functions of weighted random variables with specific weight functions applicable in reliability under suitable conditions.

Generalized Cumulative Residual Entropy of Time Series Based on Permutation Patterns

Cumulative residual entropy (CRE) has been suggested as a new measure to quantify uncertainty of nonlinear time series signals. Combined with permutation entropy and Rényi entropy, we introduce a generalized measure of CRE at multiple scales, namely generalized cumulative residual entropy (GCRE), and further propose a modification of GCRE procedure by the weighting scheme — weighted generalized cumulative residual entropy (WGCRE). The GCRE and WGCRE methods are performed on the synthetic series to study properties of parameters and verify the validity of measuring complexity of the series. After that, the GCRE and WGCRE methods are applied to the US, European and Chinese stock markets. Through data analysis and statistics comparison, the proposed methods can effectively distinguish stock markets with different characteristics.

Some new findings on the cumulative residual Tsallis entropy

Several generalizations of Shannon entropy have been introduced in the literature. One of such a measure is the cumulative residual Tsallis entropy (CRTE) which can be viewed as an alternative dispersion measure. In this paper, we obtain some further results for such a measure, in relation with the cumulative residual Tsallis entropy and with the variance of random variables. Specifically, we present new equivalent expressions, bounds, normalized CRTE, connection to the differential entropy and excess wealth transform and stochastic comparisons involving such measures. Besides, the dynamic version of such under consideration measure is elaborated and some monotonicity results are also given. Finally, we consider the problem of estimating the CRTE by means of the empirical CRTE. In this regard, we use two different empirical estimators of cumulative distribution function to estimate CRTE. Then, we study practical results of the second estimator in blind image quality assessment.

Cumulative residual entropy of equilibrium distribution of order n

In the present work, we discuss the cumulative residual entropy of equilibrium distribution of order n. The work provides a generalization of cumulative residual entropy and its dynamic version and also acts as the entropy of a hierarchy of weighted distributions. We provide a comparison of the new entropy with the corresponding Shannon’s entropy and discuss their properties.

Bounds on the Lifetime Expectations of Series Systems with IFR Component Lifetimes

We consider series systems built of components which have independent identically distributed (iid) lifetimes with an increasing failure rate (IFR). We determine sharp upper bounds for the expectations of the system lifetimes expressed in terms of the mean, and various scale units based on absolute central moments of component lifetimes. We further establish analogous bounds under a more stringent assumption that the component lifetimes have an increasing density (ID) function. We also indicate the relationship between the IFR property of the components and the generalized cumulative residual entropy of the series system lifetime.

Relation Between Cumulative Residual Entropy and Excess Wealth Transform with Applications to Reliability and Risk

Abstract Dynamic cumulative residual entropy is a new addition to the class of information measures. In the present paper, we study its relationship with excess wealth transform and derive some identities connecting the two using the quantile-based approach. Some theoretical results that have applications to infer ageing properties and risk measures are presented. These are used as tools to analyse real life data.

Dynamic cumulative residual Rényi entropy for Lomax distribution: Bayesian and non-Bayesian methods

<abstract> An alternative measure of uncertainty related to residual lifetime function is the dynamic cumulative residual entropy which plays a significant role in reliability and survival analysis. This article deals with estimating dynamic cumulative residual Rényi entropy (DCRRE) for Lomax distribution using maximum likelihood and Bayesian methods of estimation. The maximum likelihood estimates and approximate confidence intervals of DCRRE are derived. Bayesian estimates and Bayesian credible intervals are derived based on gamma priors for the DCRRE under squared error, linear exponential (LINEX) and precautionary loss functions. The Metropolis-Hastings algorithm is employed to generate Markov chain Monte Carlo samples from the posterior distributions. The Bayes estimates are compared through Monte Carlo simulations. Regarding simulation results, we observe that the maximum likelihood and Bayesian estimates of the DCRRE are decreasing function on time. Further, maximum likelihood and Bayesian estimates of the DCRRE perform well as the sample size increases. Bayesian estimate of the DCRRE under LINEX loss function is more convenient than the other estimates in the most of the situations. Real data set is analyzed for clarifying purposes. </abstract>

Multiscale Rényi cumulative residual distribution entropy: Reliability analysis of financial time series

In the study of the complexity of time series, information measurement is an effective method to quantify the reliability of dynamic systems, such as financial markets, and its practical use is to identify the state of systems. In this paper, we propose a modification of cumulative residual entropy (CRE) based on cumulative distribution of a random variable, called multiscale Rényi cumulative residual distribution entropy (MRCE), to investigate information content found in more general cases. The CRE is a relevant dynamic measure of uncertainty in reliability studies. Rényi entropy and distribution entropy (DistEn) present diverse means to characterize different complexity behaviors of time series. Compared with the previous complex dynamics methods, the MRCE has larger range showing time series patterns in the field of parameterized transformation. Therefore, MRCE combines the multiscale theory and Rényi cumulative residual distribution entropy (RCE). It is applied to classical discrete distributions, synthetic series and real-world data. Results reveal that MRCE allows a high sensitivity to the predetermined parameters. The improved method enables us to further analyse the complexity of different time series at different scales. Simultaneously, financial time series of stock markets in the same region exhibit obvious similarities.

A CUMULATIVE RESIDUAL INACCURACY MEASURE FOR COHERENT SYSTEMS AT COMPONENT LEVEL AND UNDER NONHOMOGENEOUS POISSON PROCESSES

Inaccuracy and information measures based on cumulative residual entropy are quite useful and have attracted considerable attention in many fields including reliability theory. Using a point process martingale approach and a compensator version of Kumar and Taneja's generalized inaccuracy measure of two nonnegative continuous random variables, we define here an inaccuracy measure between two coherent systems when the lifetimes of their common components are observed. We then extend the results to the situation when the components in the systems are subject to failure according to a double stochastic Poisson process.

Higher-Order Information Measures from Cumulative Densities in Continuous Variable Quantum Systems

A definition of three-variable cumulative residual entropy is introduced, and then used to obtain expressions for higher order or triple-wise correlation measures, that are based on cumulative residual densities. These information measures are calculated in continuous variable quantum systems comprised of three oscillators, and their behaviour compared to the analogous measures from Shannon information theory. There is an overall consistency in the behaviour of the newly introduced measures as compared to the Shannon ones. There are, however, differences in interpretation, in the case of three uncoupled oscillators, where the correlation is due to wave function symmetry. In interacting systems, the cumulative based measures are shown in order to detect salient features, which are also present in the Shannon based ones.

Investigating the Crisis Forecasting Ability of the Cumulative Residual Entropy Measure by using Logistic Map Simulation Data and Tehran Stock Exchange Overall Index

اهمیت پیش‌بینی در مباحث سرمایه‌گذاری موجب شده است که طیف وسیعی از روش‌ها در علوم مختلف جهت پیش‌بینی روندها و قیمت‌های آتی به کار گرفته شوند. پیش‌بینی بحران نیز در سال‌های گذشته نمود بیشتری پیدا کرده است. این موضوع نیازمند بررسی‌های بیشتر دینامیکی سیستم‌های پیچیده می‌باشد تا مشخص شود به هنگام بحران چه تغییراتی در دینامیک سیستم رخ می‌دهد. پژوهش پیش‌رو به بررسی توانایی پیش‌بینی بحران با معیار آنتروپی باقی‌مانده تجمعی در بورس اوراق بهادار تهران (به عنوان یک سیستم پیچیده) پرداخته است. در این پژوهش از داده‌های شبیه‌ساز بحران که بوسیله نقشه لوجستیک ساخته می‌شوند استفاده شده است تا توانایی معیار در حالت نظری بررسی گردد. در حالت کاربردی نیز از داده های شاخص کل بورس اوراق بهادار تهران از مهر 1389 تا شهریور 1398 استفاده شده است. نتایج پژوهش نشان داده است که معیار پیشنهادی توانایی پیش‌بینی بحران را دارد.

On a family of coherent measures of variability

Risk measures are important and widely used tools in quantitative risk management of insurance companies and financial institutions. In this paper, we will introduce a family of coherent variability measures with comonotonic additivity, which is based on Lr-metric between a probability distribution and its distortion. One of its special cases is the cumulative residual entropy of a distribution. Further properties and potential applications of these coherent variability measures are presented. More attention is paid on composing a new coherent risk measure from expected shortfall and tail cumulative residual entropy to capture tail risk.

Characterization of continuous symmetric distributions using information measures of records

In this paper, several characterizations of continuous symmetric distributions are provided. The results are based on the properties of some information measures of k-records. These include cumulative residual (past) entropy, Shannon entropy, Renyi entropy, Tsallis entropy, also some common Kerridge inaccuracy measures. It is proved that the equality of information in upper and lower k-records is a characteristic property of continuous symmetric distributions. Completeness properties of certain function sequences are also used to obtain some characterization results.

Some Statistical Properties of a Weighted Distribution and Its Application to Waiting Time Data

The weighted distribution is mainly used in various real life fields such as ecology, reliability engineering, medical science etc. Some statistical properties are derived such as moment generating function, characteristic function, cumulative generating function, the hazard function, Renyi entropy, Cumulative residual entropy. For parameter estimation maximum likelihood estimation method is used. The considered weighted probability distribution is applied to two real data sets of waiting time to examine the suitability and applicability.

Some Reliability Properties of Bivariate Cumulative Residual Tsallis Entropy

Rajesh and Sunoj (Stat Pap 60(3):583–593, 2019) have introduced an alternative form of cumulative Tsalli’s Entropy for continuous random variables and studied its importance in reliability studies. The present study aims at extending this measure to the bivariate case based on two types of conditioning, viz. conditionally specified and survival models. In a two component system, it measures the uncertainty associated with one component when the other component is either failed or survived a specified period of time. Characterization results are established for important bivariate lifetime models. We also propose an empirical and kernel-based plug-in estimators for conditional cumulative Tsalli’s entropy and study its performance using simulated data. A real data analysis is also carried out to evaluate the usefullness of the empirical estimator of a bivariate dynamic cumulative Tsalllis entropy and compared its performance with that of similar estimator due to Sati and Singh (J Appl Math Inf 35(1–2):45–58, 2017).

Multiscale cumulative residual distribution entropy and its applications on heart rate time series

Distribution entropy has been proved to reveal stability for short time series and to distinguish different classes of series by complexity. However, there still exists some drawbacks. For example, it does not consider the possible causality underlying the data, which may not precisely identify deterministic from stochastic processes. In addition, cumulative residual entropy can successfully solve such problems and identify randomness and complexity of time series quite clearly. We therefore combine distribution entropy with cumulative residual entropy named cumulative residual distribution entropy (CRDE), aiming at considering both distribution and values of distances in the state space. CRDE can detect the temporal and spatial structures of the series after adding multiscale analysis. Results show that the combined method can characterize series from stochastic system (white noise and 1/f noise) and deterministic system (chaotic and periodic series). Then, we apply it to physiological signals, and the result is consistent with the one that loss of complexity at larger scales is related to aging and disease.

Cumulative residual entropy applied to testing uniformity

We present properties of the cumulative residual entropy and then propose a test for uniformity. An approximation of distribution of the test statistic is derived. The mean, variance, and the asymptotic null distribution of the test statistic are presented. Percentage points and power against seven alternatives are reported. The results of a simulation study show that the test is competitive in terms of power. Moreover, the proposed statistic is easy to compute and can be approximated by the beta distribution.

Multiscale Fractional Cumulative Residual Entropy of Higher-Order Moments for Estimating Uncertainty

The fractional cumulative residual entropy is not only a powerful tool for the analysis of complex system, but also a promising way to analyze time series. In this paper, we present an approach to measure the uncertainty of non-stationary time series named higher-order multiscale fractional cumulative residual entropy. We describe how fractional cumulative residual entropy may be calculated based on second-order, third-order, fourth-order statistical moments and multiscale method. The implementation of higher-order multiscale fractional cumulative residual entropy is illustrated with simulated time series generated by uniform distribution on [0, 1]. Finally, we present the application of higher-order multiscale fractional cumulative residual entropy in logistic map time series and stock markets time series, respectively.

Effective subspace detection based on the measurement of both the spectral and spatial information for hyperspectral image classification

ABSTRACT Subspace detection from high dimensional hyperspectral image (HSI) data cube has become an important area of research for efficient identification of ground objects. Standard feature extraction method such as Principal Component Analysis (PCA) has some drawbacks as it depends solely on global variance of the dataset generated. Folded-PCA (FPCA), an improvement of PCA, offers more benefits over PCA as it envisages both local and global structures of image contents and requires less computation and memory. These superior properties make FPCA more effective for feature extraction in high dimensional remote sensing images e.g. HSIs. Therefore, the proposed feature reduction method combines FPCA feature extraction with Normalized Cross Cumulative Residual Entropy (NCCRE) feature selection, termed as FPCA-NCCRE, for efficient features’ subspace detection. NCCRE is utilised as a means of feature selection over the new features generated from FPCA to obtain a more informative subspace. It is experimented on a real mixed agricultural and an urban hyperspectral dataset. Finally, Kernel Support Vector Machine (KSVM) is implemented to calculate the classification accuracy using the detected subspace. From the experiments, it is observed that the proposed method outperforms the baseline approaches and obtains the highest accuracy of 97.67 and 98.57% on the two real hyperspectral images.

Connections between Weighted Generalized Cumulative Residual Entropy and Variance

A shift-dependent information measure is favorable to handle in some specific applied contexts such as mathematical neurobiology and survival analysis. For this reason, the weighted differential entropy has been introduced in the literature. In accordance with this measure, we propose the weighted generalized cumulative residual entropy as well. Despite existing apparent similarities between these measures, however, there are quite substantial and subtle differences between them because of their different metrics. In this paper, particularly, we show that the proposed measure is equivalent to the generalized cumulative residual entropy of the cumulative weighted random variable. Thus, we first provide expressions for the variance and the new measure in terms of the weighted mean residual life function and then elaborate on some characteristics of such measures, including equivalent expressions, stochastic comparisons, bounds, and connection with the excess wealth transform. Finally, we also illustrate some applications of interest in system reliability with reference to shock models and random minima.