Inducing a desired value of correlation between two point-scale variables: a two-step procedure using copulas

Focusing on point-scale random variables, i.e. variables whose support consists of the first m positive integers, we discuss how to build a joint distribution with pre-specified marginal distributions and Pearson’s correlation $$\rho $$ ρ . After recalling how the desired value $$\rho $$ ρ is not free to vary between $$-1$$ - 1 and $$+1$$ + 1 , but generally ranges a narrower interval, whose bounds depend on the two marginal distributions, we devise a procedure that first identifies a class of joint distributions, based on a parametric family of copulas, having the desired margins, and then adjusts the copula parameter in order to match the desired correlation. The proposed methodology addresses a need which often arises when assessing the performance and robustness of some new statistical technique, i.e. trying to build a huge number of replicates of a given dataset, which satisfy—on average—some of its features (for example, the empirical marginal distributions and the pairwise linear correlations). The proposal shows several advantages, such as—among others—allowing for dependence structures other than the Gaussian and being able to accommodate the copula parameter up to an assigned level of precision for $$\rho $$ ρ with a very small computational cost. Based on this procedure, we also suggest a two-step estimation technique for copula-based bivariate discrete distributions, which can be used as an alternative to full and two-step maximum likelihood estimation. Numerical illustration and empirical evidence are provided through some examples and a Monte Carlo simulation study, involving the CUB distribution and three different copulas; an application to real data is also discussed.

Copula-based drought severity-area-frequency curve and its uncertainty, a case study of Heihe River basin, China

Copulas are appropriate tools in drought frequency analysis. However, uncertainties originating from copulas in such frequency analysis have not received significant consideration. This study aims to develop a drought severity-areal extent-frequency (SAF) curve with copula theory and to evaluate the uncertainties in the curve. Three uncertainty sources are considered: different copula functions, copula parameter estimations, and copula input data. A case study in Heihe River basin in China is used as an example to illustrate the proposed approach. Results show that: (1) the dependence structure of drought severity and areal extent can be modeled well by Gumbel; Clayton and Frank depart the most from Gumbel in estimating drought SAF curves; (2) both copula parameter estimation and copula input data contribute to the uncertainties of SAF curves; uncertainty ranges associated with copula input data present wider than those associated with parameter estimations; (3) with the conditional probability decreasing, the differences in the curves derived from different copulas are increasing, and uncertainty ranges of the curves caused by copula parameter estimation and copula input data are also increasing. These results highlight the importance of uncertainty analysis of copula application, given that most studies in hydrology and climatology use copulas for extreme analysis.

Modified Maximum Pseudo Likelihood Method of Copula Parameter Estimation for Skewed Hydrometeorological Data

For multivariate frequency analysis of hydrometeorological data, the copula model is commonly used to construct joint probability distribution due to its flexibility and simplicity. The Maximum Pseudo-Likelihood (MPL) method is one of the most widely used methods for fitting a copula model. The MPL method was derived from the Weibull plotting position formula assuming a uniform distribution. Because extreme hydrometeorological data are often positively skewed, capacity of the MPL method may not be fully utilized. This study proposes the modified MPL (MMPL) method to improve the MPL method by taking into consideration the skewness of the data. In the MMPL method, the Weibull plotting position formula in the original MPL method is replaced with the formulas which can consider the skewness of the data. The Monte-Carlo simulation has been performed under various conditions in order to assess the performance of the proposed method with the Gumbel copula model. The proposed MMPL method provides more precise parameter estimates than does the MPL method for positively skewed hydrometeorological data based on the simulation results. The MMPL method would be a better alternative for fitting the copula model to the skewed data sets. Additionally, applications of the MMPL methods were performed on the two weather stations (Seosan and Yeongwol) in South Korea.

Inducing a Target Association between Ordinal Variables by Using a Parametric Copula Family

The need for building and generating statistically dependent random variables arises in various fields of study where simulation has proven to be a useful tool.In this work, we present an approach for constructing ordinal variables with arbitrarily assigned marginal distributions and value of association or correlation, expressed in terms of either Goodman and Kruskal's gamma or Pearson's linear correlation. The approach first constructs a class of bivariate copula-based distributions matching the assigned margins, and then, within this class, identifies the distribution matching the assigned association or correlation, by calibrating the copula parameter. A numerical example and a possible application are illustrated.

Trivariate copula to design coastal structures

Some coastal structures must be redesigned in the future due to rising sea levels caused by global warming. The design of structures subjected to the actions of waves requires an accurate estimate of the long return period of such parameters as wave height, wave period, storm surge and more specifically their joint exceedance probabilities. The Defra method that is currently used makes it possible to directly connect the joint exceedance probabilities to the product of the univariate probabilities by means of a simple factor. These schematic correlations do not, however, represent all the complexity of the reality and may lead to damaging errors in coastal structure design. The aim of this paper is therefore to remedy the lack of accuracy of these current approaches. To this end, we use copula theory with a copula function that aggregates joint distribution function to its univariate margins. We select a bivariate copula that is adapted to our application by the likelihood method with a copula parameter that is obtained by the error method. In order to integrate extreme events, we also resort to the notion of tail dependence. We can select the copulas with the same tail dependence as data. In the event of an opposite tail dependence structure, we resort to the survival copula. The tail dependence parameter makes it possible to estimate the optimal copula parameter. The most accurate copulas for our practical case with applications in Saint-Malo and Le Havre (France), are the Clayton normal copula and the Gumbel survival copula. The originality of this paper is the creation of a new and accurate trivariate copula. Firstly, we select the fittest bivariate copula with its parameter for the two most correlated univariate margins. Secondly, we build a trivariate function. For this purpose, we aggregate the bivariate function with the remaining univariate margin with its parameter. We show that this trivariate function satisfies the mathematical properties of the copula. We can finally represent joint trivariate exceedance probabilities for a return period of 10, 100 and 1000 years.

Approximate Optimal Transport for Continuous Densities with Copulas

Optimal Transport (OT) formulates a powerful framework by comparing probability distributions, and it has increasingly attracted great attention within the machine learning community. However, it suffers from severe computational burden, due to the intractable objective with respect to the distributions of interest. Especially, there still exist very few attempts for continuous OT, i.e., OT for comparing continuous densities. To this end, we develop a novel continuous OT method, namely Copula OT (Cop-OT). The basic idea is to transform the primal objective of continuous OT into a tractable form with respect to the copula parameter, which can be efficiently solved by stochastic optimization with less time and memory requirements. Empirical results on real applications of image retrieval and synthetic data demonstrate that our Cop-OT can gain more accurate approximations to continuous OT values than the state-of-the-art baselines.

Sequences and antisequences in hypertensive patients under therapyPlatform

Baroreceptor reflex (baroreflex, BRR) is a domineering physiological regulator considering the systolic blood pressure (SBP) and heart rate (HR). It maintains the negative feedback equilibrium: if the blood pressure increase, heart rate decreases and vice versa. The aim of this study is to compare the number of baroreflex sequences in hypertensive patients before and after the drug administration, and to oppose the assumptions about their origin. Other methods that evaluate the mutual connection between the SBP and HR time series are investigated as well, such as cross-entropy, copula parameter, and probability integral transformed entropy. Surrogate data were used as a control.

New Bivariate Pareto Type II Models

Pareto type II distribution has been studied from many statisticians due to its important role in reliability modelling and lifetime testing. In this article, we introduce two bivariate Pareto Type II distributions; one is derived from copula and the other is based on mixture and copula. Parameter Estimates of the proposed distribution are obtained using the maximum likelihood method. The performance of the proposed bivariate distributions is examined using a simulation study. Finally, we analyze one data set under the proposed distributions to illustrate their flexibility for real-life applications.