Marginal Distribution Functions Research Articles

A copula-based approach to instream ecological flow determination considering inter- and intra-annual runoff variability

ABSTRACT Instream ecological flow (IEF) provides a flow reference value for maintaining the stability of river ecosystems and enriching biodiversity. Existing methods for determining the IEF often consider only one of the intra-annual runoff distribution characteristics or the inter-annual runoff distribution characteristics. In this study, a copula-based ecological runoff determination method is proposed to consider both annual and monthly runoff magnitudes. The marginal distribution functions of annual and monthly runoff are first determined using the great likelihood method. Then the copula function is used to construct the joint distribution function of annual and monthly runoff series. Using the flow duration curve method, the IEF under different annual and monthly runoff exceedance probabilities is calculated. The probability values are combined with the initial IEF values to determine the final IEF process. A case study is conducted using real-world data from the Tongtian River in China. The results show that the total annual basic ecological water demands of the Tongtian River under abundant, flat, and dry water years are 138, 102 and 72. 2 billion m3. The proposed method effectively avoids the disturbance of extreme runoff.

PyCoSMoS: An advanced toolbox for simulating real-world hydroclimatic data

Simulation models are a fundamental tool for investigating hydrological processes and for water resource management. In this study, we introduce PyCoSMoS, a Python toolbox that enables researchers to simulate observed univariate time series mimicking hydroclimatic processes. This toolbox preserves arbitrary marginal distribution and autocorrelation functions, while significantly reducing computational burden. PyCoSMoS is built upon the mixed-Uniform CoSMoS method recently proposed by Papalexiou et al. (2023). The toolbox is designed to minimize the user’s input, requiring only observed time series, marginal distribution, correlation function, and the number of lags. The output provides both visual and quantitative comparisons between the observed and simulated time series. We evaluate the performance of the package using various synthetic case studies and the results demonstrate satisfactory accuracy. Furthermore, we apply the toolbox to three real case studies: precipitation, temperature, and relative humidity, for which the toolbox can successfully simulate the observed time series in each case.

Phase II control chart for monitoring Gumbel’s bivariate exponential distribution

Schemes for monitoring two or more time between events (TBE) data are becoming increasingly significant in statistical process monitoring. In practice, parameters must be estimated prior to the start of process monitoring. In this paper, we first propose a multivariate time between events (MTBE) control chart based on the one-sample log-likelihood ratio test for monitoring the Gumbel’s bivariate exponential (GBE) distribution, and then we analyze the effect of parameter estimation on this chart. To avoid the effect of parameter estimation, we then propose a method based on the two-sample log-likelihood ratio test for monitoring the shifts occurring in either the scale parameter or the dependence parameter or both for the GBE distribution process. The proposed method is designed with two-sample log-likelihood ratio test for univariate marginal distribution function and Gumbel copula function. A Max-type function-based approach is implemented for joint monitoring of margin distribution and copula-related structure. In contrast to the traditional schemes used to monitor bivariate processes, the strength of our proposed technique lies in the ability to identify exactly which component of the scale parameter or dependence parameter is responsible for the source of the signal. The performance of the proposed chart is analyzed via Monte Carlo simulations. Finally, two examples are mentioned to illustrate the implementation process of the proposed scheme.

Data-driven early warning indicator for the overall stability of rock slopes: An example in hydropower engineering

In hydropower engineering, monitoring the instability of rock slopes is a crucial undertaking. Current early warning models of rock slopes lack consideration of the overall stability and cannot reflect the stability differences of different structure forms. Addressing these issues, we have successfully developed an integrative early warning indicator for the overall stability of rock slopes, utilizing the copula function and the safety stability rate. Initially, we introduce the safety stability rate to quantify the impact of rock structure on stability. Subsequently, the ISSR-MDF model, which integrates the safety stability rate with the marginal distribution function, is proposed. On this basis, we established the early warning indicator using the copula function. The results show that the indicator can reflect the structural characteristics of rock slopes and the trend of changes in multi-point residuals. The early warning model can provide valuable references for the stability assessment of slopes in hydropower engineering.

Quantifying endogenous and exogenous shocks to financial sector systemic risk: A comparison of GFC and COVID-19

In this study, we use segregated endogenous and exogenous shocks to large banks’ returns to compare the effect of each on financial sector systemic risk. We use the copula-CoVaR methodology and GARCH (1,1) with time-varying moments to model the marginal distribution function and bivariate probability distribution of the tail returns. We find that endogenous risk dominates exogenous risk in the financial system. A comparison of the 2008 global financial crisis and COVID-19 reveals that the crisis aggravates only as exogenous shocks to the system persist. Additionally, we find that large banks reduce the total risk of the system in normal times but increase the risk of the financial system in crisis times. Our findings have important implications for policymakers, investors, and portfolio managers.

Empirical likelihood based confidence regions for functional of copulas

In the present paper, we are mainly concerned with the statistical inference for the functional of nonparametric copula models satisfying linear constraints. The asymptotic properties of the obtained estimates and test statistics are given. Finally, a general notion of bootstrap for the proposed estimates and test statistics, constructed by exchangeably weighting sample, is presented, which is of its own interest. These results are proved under some standard structural conditions on some classes of functions and some mild conditions on the model, without assuming anything about the marginal distribution functions, except continuity. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.

Bivariate regional drought frequency analysis using multivariate approaches: a case study in southwestern Iran

Abstract Bivariate approaches in Regional Frequency Analysis (RFA) address two issues: first, to evaluate the homogeneity of regions, and second, to estimate the joint return periods. This study was conducted to investigate the joint return period of a severe historical drought in southwestern Iran. Fifty-nine rain gauges were first clustered into three, four, and five regions using the fuzzy c-means clustering (FCM) algorithm. Then bivariate discordancy and homogeneity tests were applied to adjust the initial clusters. The results showed that only in the case of three clusters were all the regions homogeneous. Therefore, it can be inferred that combining clustering analysis and discordancy test is insufficient to form homogeneous regions. Finally, the joint return period, by choosing Generalized Logistic and Wakeby as marginal distributions and Clayton as a copula, was estimated for all the sites in the three regions. Since no three-parameter distribution function fitted well to the variable severity, the bivariate homogeneity index does not necessarily attest to region homogeneity regarding the marginal distribution functions. It is also deduced that sites with higher mean annual precipiataion (MAP) and, correspondingly, higher elevation are more likely to experience shorter return periods of same drought events, in contrast to sites with lower MAP or lower elevation.

Stochastic dominance and statistical preference for random variables coupled by arbitrary copulas

Recently, results have been published showing that first order stochastic dominance implies statistical preference and diff-stochastic dominance, when the copula relating the compared variables is either Archimedean, the product copula, or one of the Fréchet-Hoeffding bounds.In the present paper, we rely on known results on multivariate stochastic orders to extend these results and simplify the proofs. The results are expanded in two directions: First, we show that it suffices for the copula to be symmetric. Second, we reveal that first stochastic dominance entails a wider range of stochastic preferences beyond statistical preference and diff-stochastic dominance.We further analyze whether first stochastic dominance implies statistical preference for the case of asymmetric copulas. We observe that, when at least one of the marginal cumulative distribution functions has no discontinuity jumps, the family of asymmetric copulas for which the implication holds is at least as large as the one for which it does not.

Vine Copula Model: Application to Chemical Elements in Water Samples

Copula can link the bivariate distribution function with marginal distribution functions without requiring specific information about the interdependence among random variables. There are several types of copulas, such as elliptical copulas, Archimedean copulas, and extreme value copulas. However, in multivariate modeling, each type of copula has limitations in modeling complex dependence structures in terms of symmetry and tail dependence properties. The class of vine copulas overcomes these limitations by constructing multivariate models using bivariate copulas in a tree-like structure. The bivariate copulas used in this study include the Clayton, Gumbel, Frank, Gaussian, and Student's t copula families. This study discusses the construction of vine copula models, parameter estimation, and their applications. The construction of vine copulas is done through the decomposition of conditional probability density functions and substituting bivariate copula density functions into the decomposition results. The data used in the study is the logarithm of the concentration of chemical elements in water samples in Colorado. The parameter estimation method used is pseudo-maximum likelihood with sequential estimation. Model selection is then performed using the Akaike information criterion (AIC) to determine the most suitable model. The results indicate that Caesium and Titanium have a dependency relationship with Scandium. Moreover, Scandium and Titanium exhibit the strongest dependence compared to other variable pairs.

A comprehensive review on the development of copulas in financial field

The Copula concept has long been used in many applications, especially in the financial field. This concept was first used in 1959 by Sklar in his mathematical work and greatly assisted in the applications of financial and insurance areas. The copula functions have been widely used in dependence modeling. In this study, we look at how the copula began to develop from a basic form to a more advanced form through studies that previous researchers have made. Throughout this study, we find various types of the copula, and each exhibits its own characteristics lying under two main families, Elliptical and Archimedean copulas. Our findings suggest that copula is vital in solving problems in statistical dependence measures and joint marginal distribution functions. This comprehensive study served as a review paper on the development of copulas from their initial existence to their latest evolution.

Simulating non-homogeneous non-Gaussian corrosion fields on pipelines based on inline inspection data

This study presents a methodology to simulate non-homogeneous non-Gaussian corrosion fields on the external surface of buried steel pipelines by using inline inspection (ILI) data. It is assumed that the non-homogeneous non-Gaussian corrosion field consists of multiple homogeneous non-Gaussian anomalies that can be characterized by the marginal distribution and spatial autocorrelation function of the corresponding corrosion depths. Based on corrosion data collected from in-service pipelines, empirical relationships are developed to estimate parameters of the marginal distribution and autocorrelation function from the ILI information. To generate a synthetic corrosion field, one first generates realizations of corrosion anomalies and then merge the generated anomalies into a single non-homogeneous field by applying a spatial modulating function to each anomaly. The proposed methodology will improve the accuracy of the fitness-for-service assessment of corroded pipelines in practice as the burst capacity of the corroded pipeline can be more accurately evaluated by using the synthetic corrosion field than using the idealized corrosion field obtained from the ILI data.

An iterative multi-fidelity scheme for simulating multi-dimensional non-Gaussian random fields

This paper presents an efficient multi-fidelity scheme to simulate multi-dimensional non-Gaussian random fields that are specified by covariance functions and marginal probability distribution functions. To develop the multi-fidelity scheme, two numerical algorithms are proposed in turn. The first algorithm is used to simulate random samples of the random field. In this algorithm, initial random samples are first generated to meet the marginal distribution and an iterative procedure is adopted to change the ranking of the random samples to match the target covariance function. By using the above algorithm, random samples satisfying the target covariance function and the target marginal distribution are obtained, but it is computationally intensive and is not suitable to simulate large-scale random fields. By taking advantage of Karhunen–Loève expansion, a multi-fidelity algorithm is then proposed to reduce the computational effort. The random variables in Karhunen–Loève expansion are calculated via performing the first algorithm on the low-fidelity model and the deterministic functions in Karhunen–Loève expansion are solved on the high-fidelity model. In this way, the proposed method has low computational effort and a high fidelity simultaneously. Three numerical examples, including two- and three-dimensional non-Gaussian random fields, are used to verify the good performance of the proposed algorithms.

On some non parametric estimators of the quantile density function for a stationary associated process

In this article, we consider smooth estimators for the quantile density function (qdf) for a sequence of stationary non negative associated random variables with a common marginal distribution function. The qdf is given by , representing the corresponding quantile function. The smooth estimators of considered here are adapted from those of considered in Chaubey, Dewan, and Li (2021). A few asymptotic properties of these estimators are established parallel to those in the i.i.d. case. A numerical study comparing the mean squared errors of various estimators indicates the advantages and a few limitations of various estimators. The smoothing parameter is selected based on the BCV and RLCV (a variation of likelihood cross-validation) criteria. It is concluded, based on the numerical studies, that the RLCV criterion may produce over-smoothing, hence BCV criterion may be preferable. The numerical studies also suggest that, overall, the estimator proposed by Soni, Dewan, and Jain (2012) seems to have some advantage over the other estimators considered in this article.

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.

Effect of nonlinearity and persistence on multiscale irreversibility, non-stationarity, and complexity of time series—Case of data generated by the modified Langevin model

Stochastic models of a time series can take the form of a nonlinear equation and have a built-in memory mechanism. Generated time series can be characterized by measures of certain features, e.g., non-stationarity, irreversibility, irregularity, multifractality, and short/long-tail distribution. Knowledge of the relationship between the form of the model and features of data seems to be the key to model time series. The paper presents a systematic analysis of the multiscale behavior of selected measures of irreversibility, irregularity, and non-stationarity vs degree of nonlinearity and persistence. As a time series generator, the modified nonlinear Langevin equation with built-in persistence is adopted. The modes of nonlinearity are determined by one parameter and do not change the half-Gaussian form of the marginal distribution function. The expected direct dependencies (sometimes non-trivial) were found and explained using the simplicity of the model. It has been shown that the change in nonlinearity, although subjected to a strong constraint (the same marginal distribution), causes significant changes in the tested markers of irregularity and non-stationarity. However, a synergy of non-linearity and persistence is needed to induce greater changes in irreversibility.

INTERCEPT ESTIMATION IN NONLINEAR SELECTION MODELS

We propose various semiparametric estimators for nonlinear selection models, where slope and intercept can be separately identified. When the selection equation satisfies a monotonic index restriction, we suggest a local polynomial estimator, using only observations for which the marginal cumulative distribution function of the instrument index is close to one. Data-driven procedures such as cross-validation may be used to select the bandwidth for this estimator. We then consider the case in which the monotonic index restriction does not hold and/or the set of observations with a propensity score close to one is thin so that convergence occurs at a rate that is arbitrarily close to the cubic rate. We explore the finite sample behavior in a Monte Carlo study and illustrate the use of our estimator using a model for count data with multiplicative unobserved heterogeneity.

Stochastic simulation of wind wave parameters for energy production

A combination of stochastic and deterministic models is applied for the study of ocean wind waves. Timeseries of significant wave height and mean zero up-crossing period, obtained from globally scattered floating buoys, are analyzed in order to construct a double periodic model, and select an optimal marginal distribution and dependence function for the description of the stochastic structure of wind waves. It is concluded that wind waves, in contrast to the atmospheric wind speed process, are mostly governed by the seasonal periodicity rather than the diurnal periodicity, which is often weak and can be neglected. Also, the Pareto-Burr-Feller distribution is found to be a fair selection among other common three-parameter marginal distributions. The dependence function is simulated through the Hurst-Kolmogorov (HK) dynamics using the climacogram (i.e., variance of the averaged process in the scale domain), a stochastic tool that can robustly estimate both the short-term fractal and long-range dependence behaviors both apparent at the wind wave process. To test the validity of the model, a stochastic synthesis of the wind wave process is performed through the Symmetric Moving Average scheme, focused on the explicit preservation of the probabilistic and the dependence structures. Finally, the stochastic model is applied for simulation to an offshore station southeast of Australia having one of the largest record lengths. The energy potential is also estimated through the significant wave height and the mean zero up-crossing period of both the synthetic and observed timeseries, and the effectiveness of the model is further discussed.

Joint probability analysis of streamflow and sediment load based on hybrid copula.

Statistical analysis of streamflow and sediment is very important for integrated watershed management and the design of water infrastructure, especially in silt-rich rivers. Here, we propose a bivariate joint distribution framework based on nonparametric kernel density estimation (KDE) and a hybrid copula function to describe the complex streamflow-sediment dependent structure. In this framework, the non-parametric KDE is used to fit the marginal distribution function of streamflow and sediment variables, and then the hybrid copula function is constructed by using the linear combination of Clayton, Frank, and Gumbel copulas, and compared with five commonly used single copulas (Clayton, Frank, Gumbel, Gaussian, and t). We use the Jinsha River Basin (JRB) in the Yangtze River's (JR) upper reaches to verify the proposed method. The results show the following: (1) Compared with the gamma distribution (Gamma) and generalized extreme value (GEV) distribution of parameters, the marginal distribution function of streamflow and sediment variables can be effectively obtained based on nonparametric KDE. (2) Compared with the single copula, the hybrid copula function more fully reflects the complex dependent structure of streamflow and sediment variables. (3) Compared with the best single copula, the precision of return period based on hybrid copula can be increased by 7.41%. In addition, the synchronous probability of streamflow and sediment in JRB is 0.553, and the asynchronous probability of streamflow and sediment is 0.447. This study can not only improve the accuracy of streamflow and sediment statistical analysis in JRB, but also provide a useful framework for other bivariate joint probability analysis.

Modeling climate extremes using the four-parameter kappa distribution for [formula omitted]-largest order statistics

Accurate estimation of the T-year return levels of climate extremes using statistical distribution is a critical step in the projection of future climate and in engineering design for disaster response. We show how the estimation of such quantities can be improved by fitting the four-parameter kappa distribution for r-largest order statistics (rK4D), which was developed in this study. The rK4D is an extension of the generalized extreme value distribution for r-largest order statistics (rGEVD), similar to the four-parameter kappa distribution (K4D), which is an extension of the generalized extreme value distribution (GEVD). This new distribution (rK4D) can be useful not only for fitting data when three parameters in the GEVD are not sufficient to capture the variability of the extreme observations, but also in reducing the estimation uncertainty by making use of the r-largest extreme observations instead of only the block maxima. We derive a joint probability density function (PDF) of rK4D and the marginal and conditional cumulative distribution functions and PDFs. To estimate the parameters, the maximum likelihood estimation and the maximum penalized likelihood estimation methods were considered. The usefulness and practical effectiveness of the rK4D are illustrated by the Monte Carlo simulation and by an application to the Bangkok extreme rainfall data. A few new distributions for r-largest order statistics are also derived as special cases of the rK4D, such as the r-largest logistic, the r-largest generalized logistic, and the r-largest generalized Gumbel distributions. These distributions for r-largest order statistics would be useful in modeling extreme values for many research areas, including hydrology and climatology.

Probabilistic Life Prediction of Tunnel Boring Machine under Wearing Conditions with Incomplete Information

This paper developed a data analysis approach to estimate the probabilistic life of an earth pressure balance (EPB) tunnel boring machine (TBM) under wearing conditions with incomplete information. The marginal reliability function of each system component of TBM is derived based on data collected from the site. The structure of the failure framework was determined based on the evaluation of influencing factors, including the wearing of the cutter head panel and screw conveyor. The joint distribution model was built by utilizing the best-fit copula function and the remaining reliable mining distance can be predicted from this model. Real data of the remaining thickness of the wearing resistance structure of the cutter head panel and screw conveyor from an earth pressure balance (EPB) TBM were captured. A realistic metro tunneling project in China was utilized to examine the applicability and effectiveness of the developed approach. The results indicate that: (1) With the selection of normal distribution and Gumbel copula as the best-fit marginal distribution function and copula function, the reliable mining distance was predicted as 4.0834 km when the reliability equaled 0.2. (2) The copula function was necessary to be considered to assess the joint distribution of the reliability function, as the predicted mining distance reduces significantly to 3.9970 km if assumed independent. (3) It enables the user to identify the weak component in the machinery and significantly improve the reliable mining distance to 4.5075 km by increasing the initial thickness of the screw conveyor by 0.5 mm. This approach can be implemented to minimize the risk of unintended TBM breakdown and improve the tunneling efficiency by reducing unnecessary cutter head intervention during the mining process.