Copula Families Research Articles

Study of functional connectivity of central motor system in Parkinson’s disease using copula theory

BackgroundParkinson’s disease (PD) is a neurodegenerative disease characterized by tremor, stiffness, and slowness of movement. Many studies showed an abnormality of functional connectivity in different brain regions at rest. In this study, we aimed to investigate the linear and non-linear connectivity of individual regions to differentiate PD from healthy control. Materials and methodsResting-state fMRI data from 25 Parkinson's individuals and 18 healthy individuals were randomly obtained from the PPMI database. Brain regions involved in the movement and basal ganglia were defined as regions of interest. Functional connectivity between these regions was assessed using correlation, and five types of copula family test (linear and non-linear). Discriminative features were selected from different copula parameters and correlation coefficients statistically and used in the discriminant analysis by clustering and classification methods. ResultsWe find non-linear connections, which may be used as the distinguishing characteristics of PD from controls. The strongest of these were related to the non-linear relationship between the left motor cortex and the right prefrontal cortex, which resulted from three non-linear families of the copula (Gumbel, Frank, Clayton). The best result of k-fold classification belonged to Frank copula with 97.67% accuracy in differentiating PD from healthy controls. ConclusionWe found that non-linear copula tests are superior to simple linear correlation calculation in identifying changes of functional connectivity in PD.

A new family of copulas, with application to estimation of a production frontier system

This paper makes two contributions. The first is to propose a new family of copulas for which the copula arguments are uncorrelated but dependent. Specifically, if w1 and w2 are the uniform random variables in the copula, they are uncorrelated, but w1 is correlated with |w2 − 1/2|. We show how this family of copulas can be applied to the error structure in an econometric production frontier model. The second contribution is to give some general results on how to extend a two-dimensional copula to three or more dimensions. This extension is necessary in our production frontier model when there are multiple inputs, but our results apply more generally to the extension of arbitrary two-dimensional copulas. We also report the results of some simulations and we give an empirical example.

Evaluating the Dependence between Temperature and Precipitation to Better Estimate the Risks of Concurrent Extreme Weather Events

Precipitation and temperature are among major climatic variables that are used to characterize extreme weather events, which can have profound impacts on ecosystems and society. Accurate simulation of these variables at the local scale is essential to adapt urban systems and policies to future climatic changes. However, accurate simulation of these climatic variables is difficult due to possible interdependence and feedbacks among them. In this paper, the concept of copulas was used to model seasonal interdependence between precipitation and temperature. Five copula functions were fitted to grid (approximately 10 km × 10 km) climate data from 1960 to 2013 in southern Ontario, Canada. Theoretical and empirical copulas were then compared with each other to select the most appropriate copula family for this region. Results showed that, of the tested copulas, none of them consistently performed the best over the entire region during all seasons. However, Gumbel copula was the best performer during the winter season, and Clayton performed best in the summer. More variability in terms of best copula was found in spring and fall seasons. By examining the likelihoods of concurrent extreme temperature and precipitation periods including wet/cool in the winter and dry/hot in the summer, we found that ignoring the joint distribution and confounding impacts of precipitation and temperature lead to the underestimation of occurrence of probabilities for these two concurrent extreme modes. This underestimation can also lead to incorrect conclusions and flawed decisions in terms of the severity of these extreme events.

New Families of Bivariate Copulas via Unit Lomax Distortion

This article studies a new family of bivariate copulas constructed using the unit-Lomax distortion derived from a transformation of the non-negative Lomax random variable into a variable whose support is the unit interval. Existing copulas play the role of the base copulas that are distorted into new families of copulas with additional parameters, allowing more flexibility and better fit to data. We present general forms for the new bivariate copula function and its conditional and density distributions. The properties of the new family of the unit-Lomax induced copulas, including the tail behaviors, limiting cases in parameters, Kendall’s tau, and concordance order, are investigated for cases when the base copulas are Archimedean, such as the Clayton, Gumbel, and Frank copulas. An empirical application of the proposed copula model is presented. The unit-Lomax distorted copula models outperform the base copulas.

Multivariate Assessment of Low-Flow Hazards via Copulas: The Case Study of the Çoruh Basin (Turkey)

Bivariate modeling and hazard assessment of low flows are performed exploiting copulas. 7-day low flows observed, respectively, in the upper, middle and lower parts of the Çoruh basin (Turkey) are examined, considering three pairs of certified stations located in different sub-basins. A thorough statistical analysis indicates that the GEV distribution can be used to model the marginal behavior of the low-flow. The joint distributions at each part are modeled via a dozen of copula families. As a result, the Husler–Reiss copula adequately fits the joint low flows in the upper part, while the t-Student copula turns out to best fit the other parts. In order to assess the low-flow hazard, these copulas are then used to compute joint return periods and failure probabilities under a critical bivariate “AND” hazard scenario. The results indicate that the middle and lower parts of the Çoruh basin are likely to experience the largest drought hazards. As a novelty, the statistical tools used allow to objectively quantify drought threatening in a thorough multivariate perspective, which involves distributional analysis, frequency analysis (return periods) and hazard analysis (failure probabilities).

New families of bivariate copulas via unit weibull distortion

This paper introduces a new family of bivariate copulas constructed using a unit Weibull distortion. Existing copulas play the role of the base or initial copulas that are transformed or distorted into a new family of copulas with additional parameters, allowing more flexibility and better fit to data. We present a general form for the new bivariate copula function and its conditional and density distributions. The tail behaviors are investigated and indicate the unit Weibull distortion may result in new copulas with upper tail dependence when the base copula has no upper tail dependence. The concordance ordering and Kendall’s tau are derived for the cases when the base copulas are Archimedean, such as the Clayton and Frank copulas. The Loss-ALEA data are analyzed to evaluate the performance of the proposed new families of copulas.

Bivariate Hydrologic Risk Assessment of Flood Episodes using the Notation of Failure Probability

Floods are becoming the most severe and challenging hydrologic issue at the Kelantan River basin in Malaysia. Flood episodes are usually thoroughly characterized by flood peak discharge flow, volume and duration series. This study incorporated the copula-based methodology in deriving the joint distribution analysis of the annual flood characteristics and the failure probability for assessing the bivariate hydrologic risk. Both the Archimedean and Gaussian copula family were introduced and tested as possible candidate functions. The copula dependence parameters are estimated using the method-of-moment estimation procedure. The Gaussian copula was recognized as the best-fitted distribution for capturing the dependence structure of the flood peak-volume and peak-duration pairs based on goodness-of-fit test statistics and was further employed to derive the joint return periods. The bivariate hydrologic risks of flood peak flow and volume pair, and flood peak flow and duration pair in different return periods (i.e., 5, 10, 20, 50 and 100 years) were estimated and revealed that the risk statistics incrementally increase in the service lifetime and, at the same instant, incrementally decrease in return periods. In addition, we found that ignoring the mutual dependency can underestimate the failure probabilities where the univariate events produced a lower failure probability than the bivariate events. Similarly, the variations in bivariate hydrologic risk with the changes of flood peak in the different synthetic flood volume and duration series (i.e., 5, 10, 20, 50 and 100 years return periods) under different service lifetimes are demonstrated. Investigation revealed that the value of bivariate hydrologic risk statistics incrementally increases over the project lifetime (i.e., 30, 50, and 100 years) service time, and at the same time, it incrementally decreases in the return period of flood volume and duration. Overall, this study could provide a basis for making an appropriate flood defence plan and long-lasting infrastructure designs. Doi: 10.28991/cej-2020-03091599 Full Text: PDF

Monthly drought prediction based on ensemble models

Drought is a natural hazard, which is a result of a prolonged shortage of precipitation, high temperature and change in the weather pattern. Drought harms society, the economy and the natural environment, but it is difficult to identify and characterize. Many areas of Pakistan have suffered severe droughts during the last three decades due to changes in the weather pattern. A drought analysis with the incorporation of climate information has not yet been undertaken in this study region. Here, we propose an ensemble approach for monthly drought prediction and to define and examine wet/dry events. Initially, the drought events were identified by the short term Standardized Precipitation Index (SPI-3). Drought is predicted based on three ensemble models i.e., Equal Ensemble Drought Prediction (EEDP), Weighted Ensemble Drought Prediction (WEDP) and the Conditional Ensemble Drought Prediction (CEDP) model. Besides, two weighting procedures are used for distributing weights in the WEDP model, such as Traditional Weighting (TW) and the Weighted Bootstrap Resampling (WBR) procedure. Four copula families (i.e., Frank, Clayton, Gumbel and Joe) are used to explain the dependency relation between climate indices and precipitation in the CEDP model. Among all four copula families, the Joe copula has been found suitable for most of the times. The CEDP model provides better results in terms of accuracy and uncertainty as compared to other ensemble models for all meteorological stations. The performance of the CEDP model indicates that the climate indices are correlated with a weather pattern of four meteorological stations. Moreover, the percentage occurrence of extreme drought events that have appeared in the Multan, Bahawalpur, Barkhan and Khanpur are 1.44%, 0.57%, 2.59% and 1.71%, respectively, whereas the percentage occurrence of extremely wet events are 2.3%, 1.72%, 0.86% and 2.86%, respectively. The understanding of drought pattern by including climate information can contribute to the knowledge of future agriculture and water resource management.

Bayesian variable selection for non-Gaussian responses: a marginally calibrated copula approach.

We propose a new highly flexible and tractable Bayesian approach to undertake variable selection in non-Gaussian regression models. It uses a copula decomposition for the joint distribution of observations on the dependent variable. This allows the marginal distribution of the dependent variable to be calibrated accurately using a nonparametric or other estimator. The family of copulas employed are "implicit copulas" that are constructed from existing hierarchical Bayesian models widely used for variable selection, and we establish some of their properties. Even though the copulas are high dimensional, they can be estimated efficiently and quickly using Markov chain Monte Carlo. A simulation study shows that when the responses are non-Gaussian, the approach selects variables more accurately than contemporary benchmarks. A real data example in the Web Appendix illustrates that accounting for even mild deviations from normality can lead to a substantial increase in accuracy. To illustrate the full potential of our approach, we extend it to spatial variable selection for fMRI. Using real data, we show our method allows for voxel-specific marginal calibration of the magnetic resonance signal at over 6000 voxels, leading to an increase in the quality of the activationmaps.

Probability pooling for dependent agents in collective learning

The use of copulas is proposed as a way of modelling dependencies between different agents' probability judgements when carrying out probability pooling. This is combined with an established Bayesian model in which pooling is viewed as a form of updating on the basis of probability values provided by different individuals. Adopting the Frank family of copulas we investigate the effect of different assumed levels of comonotonic dependence between individuals, in the context of a collective learning problem in which a population of agents must reach consensus on which of two mutually exclusive and exhaustive hypotheses is true. In this scenario agents receive evidence from two sources; directly from the environment and also from other agents in the form of probability judgements. They then apply Bayesian updating to the former and probability pooling to the latter. We carry out multi-agent simulation experiments and show that optimal population level performance is obtained under the assumption of some degree of comonotonicity between agents, and consequently show that the standard assumption of agent independence is suboptimal. This is found to be particularly true of scenarios where there is a large amount of noise and very low amounts of direct evidence. Finally, we investigate dynamic environments in which the true state of the world changes and show that identifying the optimal level of agent dependency has an even greater effect on performance than for static environments in which the true state remains constant.

Modeling multivariate standardized drought index based on the drought information from precipitation and runoff: a case study of Hare watershed of Southern Ethiopian Rift Valley Basin

Drought is below-normal availability of rainfall, runoff, and or soil moisture for a prolonged period in a given region. Modeling drought index using multiple variables is important for future hydrological drought monitoring and sustainable water resource management. This study aimed to model multivariate standardized drought index (MSDI) based on the drought information from precipitation and runoff. Long year (1980–2014) monthly observed precipitation and runoff data were used to analyzed standardized precipitation index (SPI) and standardized runoff index (SRI) respectively. The best-fit copula family was selected to construct the joint probability distribution (JPD) of the SPI and SRI, and MSDI was developed. SPI, SRI, and MSDI at 6 and 12-month drought time scales were analyzed to characterize hydrological drought properties. The correlation among three drought indices (SPI, SRI, and MSDI) were analyzed using the Pearson correlation method. The goodness-of-fit test result showed that the Clayton copula was found the best-fitted copula function in describing JPD the two drought indices. The MSDI showed that the drought onset most likely similar to the SPI. Moreover, MSDI showed the maximum duration of drought occurred with varying severities about 26–28-months, while the duration of drought is extensive, but the frequency of drought less relative to SPI and SRI. The developed model, MSDI had a high correlation with SPI and SRI $$(R>0.7 \mathrm\,\text{and}\,{R}^{2}>0.5, p \sim 0.0)$$ compared to the correlation between SPI and SRI. Therefore, modeling hydrological drought using multiple variables is better than estimated with a single variable.

On the dependence structure between S&P500, VIX and implicit Interexpectile Differences

We study the dependence structure between the S&P500, the VIX Index, and implicit Interexpectile Differences, that are an alternative measure of implied volatility based on the notion of implicit expectile, recently introduced in Bellini et al. [Implicit expectiles and measures of implied volatility. Quant. Finance, 2018a, 18, 1851–1864]. After filtering the time series of the marginals by ARMA-(E)GARCH models, we fit several parametric families of copulas to the pairwise joint distribution of the residuals, in order to investigate the presence of radial asymmetry and asymptotic tail dependence. We find a negative dependence between S&P500 and both implied volatility indices and a positive dependence between VIX and Interexpectile Differences. The best fitting copulas seem relatively stable over time and display both asymmetry and strong tail dependence, in accordance with the leverage effect.

Drought Risk Analysis in the Eastern Cape Province of South Africa: The Copula Lens

This research study was carried out to investigate the characteristics of drought based on the joint distribution of two dependent variables, the duration and severity, in the Eastern Cape Province, South Africa. The drought variables were computed from the Standardized Precipitation Index for 6- and 12-month accumulation period (hereafter SPI-6 and SPI-12) time series calculated from the monthly rainfall data spanning the last five decades. In this context, the characteristics of climatological drought duration and severity were based on multivariate copula analysis. Five copula functions (from the Archimedean and Elliptical families) were selected and fitted to the drought duration and severity series in order to assess the dependency measure of the two variables. In addition, Joe and Gaussian copula functions were considered and fitted to the drought duration and severity to assess the joint return periods for the dual and cooperative cases. The results indicate that the dependency measure of drought duration and severity are best described by Tawn copula families. The dependence structure results suggest that the study area exhibited low probability of drought duration and high probability of drought severity. Furthermore, the multivariate return period for the dual case is found to be always longer across all the selected univariate return periods. Based on multivariate analysis, the study area (particularly Buffalo City, OR Tambo and Alfred Zoo regions) is determined to have higher/lower risks in terms of the conjunctive/cooperative multivariate drought risk (copula) probability index. The results of the present study could contribute towards policy and decision making through e.g., formulation of the forward-looking contingent plans for sustainable management of water resources and the consequent applications in the preparedness for and adaptation to the drought risks in the water-linked sectors of the economy.

Cost Benefit Analysis of a k-out-of-n: G Type Warm Standby Series System Under Catastrophic Failure Using Copula Linguistics

This paper deals with the study of reliability measures of a complex engineering system consisting three subsystems namely L, M, and N in series configuration. The subsystem-L has three units working under 1-out-of-3: G; policy, the subsystem-M has two units working under 1-out-of-2: G policy and the subsystem-N has one unit working under 1-out-of-1: G; policy. Moreover, the system may face catastrophic failure at any time t. The failure rates of units of all subsystems are constant and assumed to follow the exponential distribution however, their repair supports two types of distribution namely general distribution and Gumbel-Hougaard family copula distribution. The system is analyzed by using the supplementary variable technique, Laplace transformation and Gumbel-Hougaard family of copula to derive the differential equations and to obtain important reliability characteristics such as availability of the system, reliability of the system, MTTF, and profit analysis. The numerical results for reliability, availability, MTTF, and profit function are obtained by taking particular values of various parameters and repair cost using maple. Tables and figures demonstrate the computed results and conclude that copula repair is more effective repair policy for better performance of repairable systems. It gives a new aspect to scientific community to adopt multi-dimension repair in form of copula. Furthermore, the results of the model are beneficial for system engineers and designers, reliability and maintenance managers.

On bivariate Kumaraswamy-distorted copulas

We propose families of bivariate copulas based on the Kumaraswamy distortion of existing copulas. With the additional two parameters in the Kumaraswamy distribution, the induced copulas permit more flexibility in tail behaviors. The framework employed in this paper also provides a method for generating new Archimedean copula models. Two theorems linking the original tail dependence behaviors and those of the distorted copula are derived for distortions that are asymptotically proportional to the power transformation in the lower tail and to the dual-power transformation in the upper tail. We also derive explicit formulas for the Kendall’s τ coefficients, tail order parameters and tail order functions for the induced copulas when Gumbel, Clayton, Frank and Galambos are distorted. An empirical application is also presented.

Spatio-temporal analysis of maximum drought severity using Copulas in Northern Algeria

Abstract Drought mitigation and prevention require a broader knowledge of the spatio-temporal characteristics and return periods of droughts over several years. In this research, drought characteristics (severity, duration, frequency and areal extent) have been analysed in northern Algeria by using the Standardized Precipitation Index to identify drought events from 194 precipitation stations. For frequency analysis, three Archimedean copula families were used to find a relationship between drought duration and severity. The severity–duration–frequency (SDF) and the severity–area–frequency (SAF) curves were obtained. The SDF and SAF curves are then used to build three-dimensional surfaces of drought severity, drought duration and cumulated percentage of the affected area (SDA) for each return period. It has been shown that the return periods of maximum drought events severity vary according to their durations. To address the issue of long-term droughts, a new classification of dry events based on drought severities is proposed. The obtained results show that the western part of Algeria is the most sensitive to severe/extreme droughts of short durations and high probabilities of exceedance. For long-term durations, the study area was sensitive to mild droughts with lower probabilities.

FGM Copula’s Extension Via Polynomial Function

This study is concerned with generalizing the well-known family of FGM copula in terms of a polynomial function. A general form of the FGM copula via a copula function is proposed in order to obtain a new FGM copula family. Various particular cases have been presented according to the degree of the polynomial function. Proofs of necessary and sufficient conditions of the created copulas have been explained so that we guarantee that the desire function is copula. We firstly propose and define a general form that associate the FGM copula with a polynomial function. Graphs and some extra properties are presented as explanatory tool of some degrees of the obtain copula. Finally, we present some important calculations related to the most popular dependences by means of the FGM copula extension.

A comprehensive family of copulas to model bivariate random noise and perturbation

Based on the random vector (X+Z,Y+Z) we study the perturbation CX+Z,Y+Z of the copula CX,Y of the random vector (X,Y) when the random noise Z is independent of both X and Y. We also propose a bivariate copula-based modification of the Irwin-Hall distribution and use it to extend the results of Mesiar et al. (2019) and present several examples for illustration.

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.

Goodness-of-fit testing for copulas: A distribution-free approach

Consider a random sample from a continuous multivariate distribution function $F$ with copula $C$. In order to test the null hypothesis that $C$ belongs to a certain parametric family, we construct an empirical process on the unit hypercube that converges weakly to a standard Wiener process under the null hypothesis. This process can therefore serve as a ‘tests generator’ for asymptotically distribution-free goodness-of-fit testing of copula families. We also prove maximal sensitivity of this process to contiguous alternatives. Finally, we demonstrate through a Monte Carlo simulation study that our approach has excellent finite-sample performance, and we illustrate its applicability with a data analysis.