Some properties of the cumulative residual entropy of coherent and mixed systems

2017 ◽  
Vol 54 (2) ◽  
pp. 379-393 ◽  
Author(s):  
A. Toomaj ◽  
S. M. Sunoj ◽  
J. Navarro
Keyword(s):  
Shannon Entropy ◽  
Reliability Function ◽  
Residual Entropy ◽  
Dispersion Measure ◽  
Alternative Measure ◽  
Coherent Systems ◽  
Cumulative Residual Entropy ◽  
Cumulative Residual ◽  
Measure Of Uncertainty ◽  
Mixed Systems

Abstract Recently, Rao et al. (2004) introduced an alternative measure of uncertainty known as the cumulative residual entropy (CRE). It is based on the survival (reliability) function F̅ instead of the probability density function f used in classical Shannon entropy. In reliability based system design, the performance characteristics of the coherent systems are of great importance. Accordingly, in this paper, we study the CRE for coherent and mixed systems when the component lifetimes are identically distributed. Bounds for the CRE of the system lifetime are obtained. We use these results to propose a measure to study if a system is close to series and parallel systems of the same size. Our results suggest that the CRE can be viewed as an alternative entropy (dispersion) measure to classical Shannon entropy.

scholarly journals Generalized Entropies, Variance and Applications

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 709 ◽  
Author(s):  
Abdolsaeed Toomaj ◽  
Antonio Di Crescenzo
Keyword(s):  
Residual Entropy ◽  
Dispersion Measure ◽  
Homogeneous Poisson Process ◽  
Cumulative Residual Entropy ◽  
Excess Wealth Order ◽  
Dynamic Version ◽  
Cumulative Residual ◽  
Non Homogeneous Poisson Process ◽  
Generalized Entropies ◽  
Cumulative Entropy

The generalized cumulative residual entropy is a recently defined dispersion measure. In this paper, we obtain some further results for such a measure, in relation to the generalized cumulative residual entropy and the variance of random lifetimes. We show that it has an intimate connection with the non-homogeneous Poisson process. We also get new expressions, bounds and stochastic comparisons involving such measures. Moreover, the dynamic version of the mentioned notions is studied through the residual lifetimes and suitable aging notions. In this framework we achieve some findings of interest in reliability theory, such as a characterization for the exponential distribution, various results on k-out-of-n systems, and a connection to the excess wealth order. We also obtain similar results for the generalized cumulative entropy, which is a dual measure to the generalized cumulative residual entropy.

PROPERTIES FOR GENERALIZED CUMULATIVE PAST MEASURES OF INFORMATION

2018 ◽  
Vol 34 (1) ◽  
pp. 92-111 ◽  
Author(s):  
Camilla Calì ◽  
Maria Longobardi ◽  
Jorge Navarro
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Survival Function ◽  
Residual Entropy ◽  
Coherent Systems ◽  
Underlying Distribution ◽  
Information Measures ◽  
Cumulative Residual Entropy ◽  
Past Entropy ◽  
Cumulative Residual

AbstractThe Shannon entropy based on the probability density function is a key information measure with applications in different areas. Some alternative information measures have been proposed in the literature. Two relevant ones are the cumulative residual entropy (based on the survival function) and the cumulative past entropy (based on the distribution function). Recently, some extensions of these measures have been proposed. Here, we obtain some properties for the generalized cumulative past entropy. In particular, we prove that it determines the underlying distribution. We also study this measure in coherent systems and a closely related generalized past cumulative Kerridge inaccuracy measure.

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

2020 ◽  
pp. 1-26
Author(s):  
Vanderlei da Costa Bueno ◽  
Narayanaswamy Balakrishnan
Keyword(s):  
Reliability Theory ◽  
Poisson Processes ◽  
Residual Entropy ◽  
Coherent Systems ◽  
Component Level ◽  
Martingale Approach ◽  
Information Measures ◽  
Cumulative Residual Entropy ◽  
Nonhomogeneous Poisson Processes ◽  
Cumulative Residual

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.

Generalized Cumulative Residual Entropy of Time Series Based on Permutation Patterns

2021 ◽  
pp. 2150055
Author(s):  
Qin Zhou ◽  
Pengjian Shang
Keyword(s):  
Time Series ◽  
Stock Markets ◽  
Multiple Scales ◽  
Permutation Entropy ◽  
Residual Entropy ◽  
Permutation Patterns ◽  
Cumulative Residual Entropy ◽  
The Us ◽  
Cumulative Residual ◽  
Different Characteristics

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.

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

2020 ◽  
Vol 19 (04) ◽  
pp. 2050038
Author(s):  
Keqiang Dong ◽  
Xiaofang Zhang
Keyword(s):  
Time Series ◽  
Logistic Map ◽  
Higher Order ◽  
Stationary Time Series ◽  
Residual Entropy ◽  
Third Order ◽  
Cumulative Residual Entropy ◽  
Cumulative Residual ◽  
Simulated Time ◽  
Analyze Time Series

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.

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