Entropy Copula Research Articles

Maximum entropy copula for bivariate drought analysis

Copula functions are widely used to derive multivariate probability distributions in hydrometeorology. One of the key steps in the copula method is the derivation of marginal distributions of individual variables which can be accomplished using the principle of maximum entropy where the distribution parameters are estimated from the specified constraints. This study, investigated two drought variables (severity and duration) by coupling the principle of maximum entropy with parametric and empirical copulas. So, homogeneous climatic zones were first identified by applying the fuzzy clustering method to data from 39 synoptic stations in Iran and then drought severity and duration were determined with the standardized precipitation index. These two variables were scaled and their marginal probability distribution functions were derived using the principle of maximum entropy as well as empirically. Then, the joint probability distribution of drought severity and duration was determined using maximum entropy-copula, and parametric and empirical copulas. Thereafter, bivariate conditional return periods were determined for each homogeneous region. Results showed that 1) univariate and bivariate distributions can be obtained by maximizing entropy; 2) the dependence structure via Spearman's rho, which directly affects the Lagrange parameters of entropy copula, was a controlling factor to optimize the objective function; 3) for a given set of constraints, the maximum entropy copula is independent of the types of marginals; 4)The entropy-entropy copula (with entropy marginals) is considered a better method than alternatives, because it has a similar result to the parametric methods while it only needs to fit a single model.

The Type II Quasi Lambert Family

Probability distributions and their families play an effective role in statistical modeling and statistical analysis. Recently, researchers have been increasingly interested in generating new families with high flexibility and low number of milestones. We propose and study a new family of continuous distributions. Relevant properties are presented. Many bivariate versions of the new family are derived under the Farlie-Gumbel-Morgenstern copula, modified Farlie-Gumbel-Morgenstern copula, Clayton copula, entropy copula and Ali-Mikhail-Haq copula. We present two characterizations of the new family. Different estimation methods such as the maximum likelihood estimation, maximum product spacing estimation, least squares estimation, weighted least squares estimation, Anderson-Darling estimation and the Cramer-von Mises estimation methods are considered. Simulation studies for comparing estimation methods are performed based on the baseline Lindley model. Two real data sets are analyzed for comparing the competitive models.

A maximum entropy copula model for mixed data: representation, estimation and applications

A new nonparametric model of maximum-entropy (MaxEnt) copula density function is proposed, which offers the following advantages: (i) it is valid for mixed random vector. By ‘mixed’, we mean the method works for any combination of discrete or continuous variables in a fully automated manner; (ii) it yields a bonafide density estimate with intepretable parameters. By ‘bonafide’, we mean the estimate guarantees to be a non-negative function, integrates to 1; and (iii) it plays a unifying role in our understanding of a large class of statistical methods for mixed . Our approach utilises modern machinery of nonparametric statistics to represent and approximate log-copula density function via LP-Fourier transform. Several real-data examples are also provided to explore the key theoretical and practical implications of the theory.

Ranking and optimizing the rain-gauge networks using the entropy–copula approach (Case study of the Siminehrood Basin, Iran)

The existence of reliable rain-gauge networks is a necessity in managing water sources and the relevant environmental issues in any basin. The present study aimed to investigate the rain-gauge network and rank the rain-gauge stations in the Tazehkand sub-basin in the southwest of Lake Urmia, Iran, using the entropy–copula approach using the data obtained from six rain-gauge stations in the Siminehrood basin in Northwestern Iran between 1981 and 2019. The interaction effects of the stations were investigated using the copula simulation approach instead of the multivariate regression analysis. The R-vine was selected as the most convenient copula after investigating various vines using the statistics related to error investigation. Then, the interaction effects of the stations were investigated using the R-vine. The rainfall in each station was estimated by considering the rainfall of other stations. Moreover, the more prevalent multivariate regression analysis was implemented, and its results were compared with the results obtained from the vine simulation approach. The results of the comparison according to the NSE statistic showed that the vine simulation approach was above 90% across all stations. The estimates for the ITI and N(i) indices of each station using the entropy theory showed the shortage of stations in the investigated area. The stations were in average, above-average, and surplus modes according to the ITI index. The rating concerning the N(i) index indicated that the Dashband station was the most appropriate one in terms of communicating with other stations and conveniently covering the plain.

The Type I Quasi Lambert Family

A new G family of probability distributions called the type I quasi Lambert family is defined and applied for modeling real lifetime data. Some new bivariate type G families using "Farlie-Gumbel-Morgenstern copula", "modified Farlie-Gumbel-Morgenstern copula", "Clayton copula" and "Renyi's entropy copula" are derived. Three characterizations of the new family are presented. Some of its statistical properties are derived and studied. The maximum likelihood estimation, maximum product spacing estimation, least squares estimation, Anderson-Darling estimation and Cramer-von Mises estimation methods are used for estimating the unknown parameters. Graphical assessments under the five different estimation methods are introduced. Based on these assessments, all estimation methods perform well. Finally, an application to illustrate the importance and flexibility of the new family is proposed.

A new reciprocal Rayleigh extension: properties, copulas, different methods of estimation and a modified right-censored test for validation

Abstract In this article, a new reciprocal Rayleigh extension called the Xgamma reciprocal Rayleigh model is defined and studied. The relevant statistical properties are derived, and the useful results related to the convexity and concavity are addressed. We discussed the estimation of the parameters using different estimation methods such as the maximum likelihood estimation method, the ordinary least squares estimation method, the weighted least squares estimation method, the Cramer-Von-Mises estimation method, and the bootstrapping method. A simulation study was conducted to assess the performances of the proposed estimation methods are investigated through a simulation study. Many bivariate and multivariate type model have also been derived based on Farlie-Gumbel-Morgenstern copula, the Clayton copula, Renyi’s entropy copula and the Ali-Mikhail-Haq copula. A modified Nikulin-Rao-Robson test for right-censored validation is applied to a censored real data set.

A New Family of Continuous Distributions: Properties, Copulas and Real Life Data Modeling

A new family of distributions called the Kumaraswamy Rayleigh family is defied and studied. Some of its relevant statistical properties are derived. Many new bivariate type G families using the of Farlie-Gumbel-Morgenstern, modified Farlie-Gumbel-Morgenstern copula, Clayton copula and Renyi’s entropy copula are derived. The method of the maximum likelihood estimation is used. Some special models based on log-logistic, exponential, Weibull, Rayleigh, Pareto type II and Burr type X, Lindley distributions are presented and studied. Three dimensional skewness and kurtosis plots are presented. A graphical assessment is performed. Two real life applications to illustrate the flexibility, potentiality and importance of the new family is proposed.

A New Family of Continuous Probability Distributions.

In this paper, a new parametric compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. Relevant mathematical properties are derived. Some new bivariate G families using the theorems of “Farlie-Gumbel-Morgenstern copula”, “the modified Farlie-Gumbel-Morgenstern copula”, “the Clayton copula”, and “the Renyi’s entropy copula” are presented. Many special members are derived, and a special attention is devoted to the exponential and the one parameter Pareto type II model. The maximum likelihood method is used to estimate the model parameters. A graphical simulation is performed to assess the finite sample behavior of the estimators of the maximum likelihood method. Two real-life data applications are proposed to illustrate the importance of the new family.

Expanded Fréchet Model: Mathematical Properties, Copula, Different Estimation Methods, Applications and Validation Testing

The extreme value theory is expanded by proposing and studying a new version of the Fréchet model. Some new bivariate type extensions using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula, and Renyi’s entropy copula are derived. After a quick study for its properties, different non-Bayesian estimation methods under uncensored schemes are considered, such as the maximum likelihood estimation method, Anderson–Darling estimation method, ordinary least square estimation method, Cramér–von-Mises estimation method, weighted least square estimation method, left-tail Anderson–Darling estimation method, and right-tail Anderson–Darling estimation method. Numerical simulations were performed for comparing the estimation methods using different sample sizes for three different combinations of parameters. The Barzilai–Borwein algorithm was employed via a simulation study. Three applications were presented for measuring the flexibility and the importance of the new model for comparing the competitive distributions under the uncensored scheme. Using the approach of the Bagdonavicius–Nikulin goodness-of-fit test for validation under the right censored data, we propose a modified chi-square goodness-of-fit test for the new model. The modified goodness-of-fit statistic test was applied for the right censored real data set, called leukemia free-survival times for autologous transplants. Based on the maximum likelihood estimators on initial data, the modified goodness-of-fit test recovered the loss in information while the grouping data and followed chi-square distributions. All elements of the modified goodness-of-fit criteria tests are explicitly derived and given.

Co-localization analysis in fluorescence microscopy via maximum entropy copula.

Co-localization analysis is a popular method for quantitative analysis in fluorescence microscopy imaging. The localization of marked proteins in the cell nucleus allows a deep insight into biological processes in the nucleus. Several metrics have been developed for measuring the co-localization of two markers, however, they depend on subjective thresholding of background and the assumption of linearity. We propose a robust method to estimate the bivariate distribution function of two color channels. From this, we can quantify their co- or anti-colocalization. The proposed method is a combination of the Maximum Entropy Method (MEM) and a Gaussian Copula, which we call the Maximum Entropy Copula (MEC). This new method can measure the spatial and nonlinear correlation of signals to determine the marker colocalization in fluorescence microscopy images. The proposed method is compared with MEM for bivariate probability distributions. The new colocalization metric is validated on simulated and real data. The results show that MEC can determine co- and anti-colocalization even in high background settings. MEC can, therefore, be used as a robust tool for colocalization analysis.

A MCMC-based maximum entropy copula method for bivariate drought risk analysis of the Amu Darya River Basin

In this study, a MCMC-based maximum entropy copula (abbreviated as MCMC-MEC) method is developed for assessing drought risk under uncertainty. MCMC-MEC has advantages in analyzing the bivariate drought risk (i.e. duration and severity) through runs-theory, improving the fitting processes of drought variables based on maximum entropy, and reflecting the uncertainties in bivariate drought risk assessment through MCMC. Then, MCMC-MEC is applied to the Amu Darya River Basin, a typical arid and semi-arid basin in Central Asia. Results disclose that, during 1935–2015, the study basin experienced about 42 drought events, average duration of each drought was 5 months (severity 4.38), and the most severe drought event lasted for 25.3 months (severity 27.9). Results also reveal that uncertainty in parameter of copula has effects on drought risk, and duration and severity would increase with confidence level of univariate drought probability. At confidence level of 0.9, the bivariate drought risk would be 4%-6% (duration ≥ 7.0 months and severity ≥ 5.8); at confidence level of 0.95, the drought risk would be 1%-2% (duration ≥ 10.0 months and severity ≥ 10.6). During 2005–2015, drought risk for the study basin was increased due to the decreasing precipitation and runoff, and their contribution rates to the drought risk were 55.8% and 44.2%, respectively. The findings are helpful for providing decision support of drought management and disaster prevention.

A New Parametric Life Family of Distributions: Properties, Copula and Modeling Failure and Service Times

A new family of probability distributions is defined and applied for modeling symmetric real-life datasets. Some new bivariate type G families using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula and Renyi’s entropy copula are derived. Moreover, some of its statistical properties are presented and studied. Next, the maximum likelihood estimation method is used. A graphical assessment based on biases and mean squared errors is introduced. Based on this assessment, the maximum likelihood method performs well and can be used for estimating the model parameters. Finally, two symmetric real-life applications to illustrate the importance and flexibility of the new family are proposed. The symmetricity of the real data is proved nonparametrically using the kernel density estimation method.

A New Log-Logistic Lifetime Model with Mathematical Properties, Copula, Modified Goodness-of-Fit Test for Validation and Real Data Modeling

After defining a new log-logistic model and studying its properties, some new bivariate type versions using “Farlie-Gumbel-Morgenstern Copula”, “modified Farlie-Gumbel-Morgenstern Copula”, “Clayton Copula”, and “Renyi’s entropy Copula” are derived. Then, using the Bagdonavicius-Nikulin goodness-of-fit (BN-GOF) test for validation, we proposed a goodness-of-fit test for a new log-logistic model. The modified test is applied for the “right censored” real dataset of survival times. All elements of the modified test are explicitly derived and given. Three real data applications are presented for measuring the flexibility and the importance of the new model under the uncensored scheme. Two other real datasets are analyzed for censored validation.

Transmuted Topp-Leone Weibull Lifetime Distribution: Statistical Properties and Different Method of Estimation

In this work we focus on proposing a new lifetime Weibull type model called the transmuted Topp-Leone Weibull and studying its properties. We derive some new bivariate and multivariate transmuted Topp-Leone Weibull versions using “Farlie Gumbel Morgenstern (FGM) Copula”, “modified FGM Copula”, “Clayton Copula” and “Renyi's entropy Copula”. The estimation of its unknown parameters is carried out by considering different method of estimation. The statistical performances of all methods are studied by two real data sets and a numerical Monte Carlo simulation. The Cramer-Von Mises method is the best method for modeling the carbon fibers data. The maximum likelihood method is the best method for modeling the Leukemia data, however all other methods performed well.

Statistical modelling of extreme storms using copulas: A comparison study

Coastal risk assessment and structure design heavily rely on the statistical analysis of the extreme wave climate, such as the wave height, storm duration, wave period and surge height. Due to the dependence among these variables, the multivariate dependence structure should be taken into account for statistical simulation of the storm events. Here, three modelling framework are described and compared for probabilistic modelling of extreme storms, i.e. the Gaussian copula, conditional mixture method and entropy copula. This paper demonstrates the functionality of the approaches in simulating the sea storm while maintaining their statistical characteristics. None of the three methods can fit all the bivariate cases best, and the Gaussian copula gives the best overall fitting quality for the four dimensional data in the case study, while the conditional mixture method gives the lowest fitting quality. The entropy copula gives a comparable simulation results and shows its great potential to be universally applicable for modelling the multivariable joint distribution by avoiding the procedure of assigning any copula family before fitting data.

Entropy and the variability of dependence: System of profound knowledge (SPK) and induced entropy copula as a measure of variability for copula density

Deming’s System of Profound Knowledge (SPK) is a framework for creating, disseminating, and applying knowledge in business. One of its basic principles is the knowledge of variation. SPK is based on a systemic approach in which each part of the system is interconnected. Process control seeks to determine whether a process is in control, meaning whether the distribution of some critical measured variable is stable. Complex systems may vary through time and their variability can be measured by its entropy and its interrelatedness. The entropy copula or interrelatedness is interpreted as a measure of reduction of independence measured by the entropy of independent variables. This paper provides the expression of the entropy copula for multivariate t distribution and introduces the concept of induced entropy copula as a measure of variability for copula densities. A test for uncorrelatedness based on the entropy copula is also developed. A manufacturing example, using multivariate control chart, is analyzed using the entropy copula approach. Tables are provided in the appendix for testing at different significance levels.

Probabilistic modelling of flood events using the entropy copula

The estimation of flood frequency is vital for the flood control strategies and hydraulic structure design. Generating synthetic flood events according to statistical properties of observations is one of plausible methods to analyze the flood frequency. Due to the statistical dependence among the flood event variables (i.e. the flood peak, volume and duration), a multidimensional joint probability estimation is required. Recently, the copula method is widely used for multivariable dependent structure construction, however, the copula family should be chosen before application and the choice process is sometimes rather subjective. The entropy copula, a new copula family, employed in this research proposed a way to avoid the relatively subjective process by combining the theories of copula and entropy. The analysis shows the effectiveness of the entropy copula for probabilistic modelling the flood events of two hydrological gauges, and a comparison of accuracy with the popular copulas was made. The Gibbs sampling technique was applied for trivariate flood events simulation in order to mitigate the calculation difficulties of extending to three dimension directly. The simulation results indicate that the entropy copula is a simple and effective copula family for trivariate flood simulation.

The Structure of the Class of Maximum Tsallis–Havrda–Chavát Entropy Copulas

A maximum entropy copula is the copula associated with the joint distribution, with prescribed marginal distributions on [ 0 , 1 ] , which maximizes the Tsallis–Havrda–Chavát entropy with q = 2 . We find necessary and sufficient conditions for each maximum entropy copula to be a copula in the class introduced in Rodríguez-Lallena and Úbeda-Flores (2004), and we also show that each copula in that class is a maximum entropy copula.

Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis

Entropy is a measure of uncertainty and has been commonly used for various applications, including probability inferences in hydrology. Copula has been widely used for constructing joint distributions to model the dependence structure of multivariate hydrological random variables. Integration of entropy and copula theories provides new insights in hydrologic modeling and analysis, for which the development and application are still in infancy. Two broad branches of integration of the two concepts, entropy copula and copula entropy, are introduced in this study. On the one hand, the entropy theory can be used to derive new families of copulas based on information content matching. On the other hand, the copula entropy provides attractive alternatives in the nonlinear dependence measurement even in higher dimensions. We introduce in this study the integration of entropy and copula theories in the dependence modeling and analysis to illustrate the potential applications in hydrology and water resources.

Maximum entropy copula with given diagonal section

We consider copulas with a given diagonal section and compute the explicit density of the unique optimal copula which maximizes the entropy. In this sense, this copula is the least informative among the copulas with a given diagonal section. We give an explicit criterion on the diagonal section for the existence of the optimal copula and give a closed formula for its entropy. We also provide examples for some diagonal sections of usual bivariate copulas and illustrate the differences between these copulas and the associated maximum entropy copula with the same diagonal section.