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The quality and quantity of groundwater resources are often assessed by sampling a limited number of wells drilled sparsely across a plain. So far, various methods have been developed for assessing and modeling groundwater quality, each of which has its capabilities and limitations. In the present study, the copula functions were applied to multivariate analysis of groundwater quality variables (including SAR, K, Mg, Na, Ca, Cl, EC, pH, TDS, SO4, TH and HCO3) in Shahrekord plain, Iran. For this purpose, the quality data from 24 wells distributed across the Shahrekord plain during the period of 1990–2020 were used. For bivariate analyses of groundwater quality, first the fitness of some common distributions in hydrology were examined to the quality variables and the appropriate marginal distribution was determined by Kolmogorov–Smirnov test (K–S). The results revealed that the Generalized Extreme Value distribution has an acceptable fitness on groundwater quality variables of Shahrekord plain. The results of correlation analysis indicated that the highest correlation based on Spearman Rho, Kendall Tau and Pearson correlation coefficient is related to the paired variables of SAR-Na and EC-TDS with correlation coefficient greater than 0.9. Therefore, these variables were selected for further analysis. In the next step, bivariate distribution of two different quality variables in a well (point analysis) and also bivariate distribution of a quality variable in two different wells (inter-well analysis) were created using copula functions. To do this, the fitness of different copulas (including Plackett, Joe, Clayton, Frank, Farlie–Gumbel–Morgenstern, Ali–Mikhail–Haq, Gumbel, Gumbel–Hougaard, Gumbel–Barnett, Philip–Gumbel and Galambos) was tested to construct a bivariate distribution of quality variables. For choosing the best fitted copula on studied paired variables, the theoretical copula values was compared with the corresponding values of empirical copula based on the goodness of fit criteria. Based on the results of point analysis, the Joe copula function was chosen as the best fitted copula function for multivariate analysis of two quality variables in a well. For inter-well analysis (IWA), the Clayton copula function was selected for creating bivariate distribution of a quality variable in two different wells. Comparing the performance of the proposed IWA method with geostatistical methods showed that in addition to the IWA method having acceptable accuracy, it has a higher efficiency than geostatistical methods, especially in areas where the number of sampling wells is less. [ABSTRACT FROM AUTHOR]

After a transmission line is covered by ice in winter, ice-shedding and vibration occurs under special meteorological and external dynamic conditions, which leads to intense transmission line shaking. Transmission line ice-shedding and vibration often cause line flashover trips and outages. In January 2018, three 500 kV transmission lines, namely, the 500 kV Guanli line, the 500 kV Dushan line, and the 500 kV Guanqiao line, tripped and cut off due to ice-shedding and vibration in Anhui province, seriously threatening the safe operation of a large power grid. Current studies mainly focus on analyzing the influence factors and characteristics of line ice-shedding and investigating suppression measures, but they only analyze the correlation between each influencing factor and icing or shedding, and do not consider the coupling effects between multiple factors. In this paper, the key influencing factors and the probability distribution of transmission line ice-shedding were analyzed, and a multiple-factor coupling fault probability calculation model of line ice-shedding based on Copula function was proposed. The fault probability was calculated directly by considering multiple influence factors at the same time, which effectively overcame the error caused by multi-factor transformation in fuzzy membership degree and other methods. It provided an important decision-making basis for preventing and controlling transmission line ice-shedding faults. [ABSTRACT FROM AUTHOR]

In this paper, we study the estimation of a scale parameter from a sample of lifetimes of coherent systems with a fixed structure. We assume that the components are independent and identically distributed having a common distribution which belongs to a scale parameter family. Some results are obtained as well for dependent (exchangeable) components. To this end, we will use the representations for the distribution function of a coherent system based on signatures. We prove that the efficiency of the estimators depends on the structure of the system and on the scale parameter family. In the dependence case, it also depends on the baseline copula function. [ABSTRACT FROM AUTHOR]

The lack of effective failure correlation analysis is one main reason for the gap between the reliability models and the actual complex systems with mixed static and dynamic characteristics. Takagi and Sugeno (T-S) dynamic fault tree is one powerful tool to analyze the static and dynamic failure logic relationship but it assumes the failure probability of the event is independent. Therefore, this paper proposes a multi-dimensional T-S dynamic fault tree analysis method involving failure correlation. The method integrates the failure probability distribution function of basic events with multi-factors and the multi-dimensional copula function, and the important measure of this method is also deduced. The reliability model expression for systems with failure correlations, both in series and in parallel, is discussed and verified. Compare the proposed method with the assumption that the probability of a failure event is independent. This method solves the problem of a large error when ignoring the failure correlation between parts and the degree of the correlation between variables can be characterized. The reliability analysis can be conducted on complex systems affected both by multi-factors and failure correlations. The proposed method is applied to the reliability analysis of a hydraulic height adjustment system and the correctness and superiority of the method are verified. [ABSTRACT FROM AUTHOR]

This study introduces methodologies for constructing joint probability distribution functions utilizing the Copula function and neural networks, and evaluates their efficacy in marine and civil engineering projects. Through an analytical comparison of both models using a numerical example, it is revealed that the neural network model exhibits superior adaptability to large sample sizes. This adaptability is attributed to the neural network's ability to learn complex relationships within the data, which is especially beneficial when dealing with large datasets. The neural network model also demonstrates higher accuracy in constructing joint probability distribution functions compared to the Copula function model. In marine and civil engineering, the adaptability and accuracy of neural networks are of paramount importance due to the variable and complex nature of weather patterns. A practical engineering application is presented, wherein a joint probabilistic distribution neural network model of wind velocity and rain intensity is established for the Lanzhou–Xinjiang high-speed railroad in China. This model illustrates the promising application of neural networks in engineering projects where weather factors play a critical role. Subsequent to the construction of the joint probability distribution functions, a feature importance analysis is incorporated to quantify the contribution of different weather parameters such as wind velocity and rain intensity to the joint distribution function. This analysis provides an objective assessment of the relative importance of various weather factors and offers data-driven insights that are essential for engineering applications where weather conditions are a significant consideration. The study concludes by highlighting the potential benefits of neural network models in marine and civil engineering, suggesting areas for future exploration. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Rainfall and temperature extremes have become more frequent and severe in recent times due to changing climate. Since these catastrophic occurrences directly affect a region's hydrology, it is imperative to develop models that can project and explain the joint behavior of climate variables. Copula functions have been used relatively successfully to capture multivariate processes. With climate being a multifaceted process, there is interdependence between variables, making copula use desirable since traditional bivariate distributions do not account for the dependent structure. In this study, we introduced a bivariate exponentiated Teissier distribution based on a Clayton copula. For parameter estimation, the maximum likelihood and inference functions for margin approaches are used. A simulation study that considered various sets of parameters is also conducted in order to select the most efficient parameter estimation method. Last, the applicability of the proposed model is demonstrated using real-world data from flood and temperature processes. After fitting, the log-likelihood, Akaike information criteria (AIC), and Bayesian information criteria (BIC) values of the proposed model are −145.00, 300.00, and 311.71 for flood data, respectively, and −128.71, 267.42, and 275.98 for temperature data, respectively. Estimated parameters are α ^ 1 = 1.730 038 4 , α ^ 2 = 2.164 941 2 , θ ^ 1 = 0.429 843 6 , θ ^ 2 = 0.598 815 7 , δ ^ = 1.662 378 9 for flood data and α ^ 1 = 4745.48 , θ ^ 1 = 0.0988 , α ^ 2 = 26 056.65 , θ ^ 2 = 0.0583 , δ ^ = 0.454 for temperature data. It is concluded that this model may be effectively used for modeling the hydrological processes for calculating the probabilities of flood and extreme temperature events. [ABSTRACT FROM AUTHOR]

This paper examines the preservation of several aging classes of lifetime distributions in the formation of coherent and mixed systems with independent and identically distributed (i.i.d.) or identically distributed (i.d.) component lifetimes. The increasing mean inactivity time class and the decreasing mean time to failure class are developed for the lifetime of systems with possibly dependent and i.d. component lifetimes. The increasing likelihood ratio property is also discussed for the lifetime of a coherent system with i.i.d. component lifetimes. We present sufficient conditions satisfied by the signature of a coherent system with i.i.d. components with exponential distribution, under which the decreasing mean remaining lifetime, the increasing mean inactivity time, and the decreasing mean time to failure are all satisfied by the lifetime of the system. Illustrative examples are presented to support the established results. [ABSTRACT FROM AUTHOR]

Stochastic characterizations of functions subject to constraints result in treating them as functions with non-independent variables. By using the distribution function or copula of the input variables that comply with such constraints, we derive two types of partial derivatives of functions with non-independent variables (i.e., actual and dependent derivatives) and argue in favor of the latter. Dependent partial derivatives of functions with non-independent variables rely on the dependent Jacobian matrix of non-independent variables, which is also used to define a tensor metric. The differential geometric framework allows us to derive the gradient, Hessian, and Taylor-type expansions of functions with non-independent variables. [ABSTRACT FROM AUTHOR]

In this article, we introduce a novel bivariate probability distribution that is absolutely continuous. Considering the Farlie–Gumbel–Morgenstern (FGM) copula and the unit-Weibull distribution, we can obtain a bivariate unit-Weibull distribution. We evaluate the main properties of the new proposal and use two estimation methods to estimate the parameter for the bivariate probability distribution. A brief Monte Carlo simulation study is conducted to assess the behavior of the employed estimation method and the characteristics of the estimators. Ultimately, as an illustration, a real-life application is presented, demonstrating the utility of the proposal. [ABSTRACT FROM AUTHOR]

Frequency analysis is a widely used method in hydrology; however, the conventional approaches to frequency analysis do not consider the spatial dependence between stations. This study proposes a novel approach for multisite frequency analysis using the copula method. This approach uses the framework of pair copula construction to extended bivariate copula functions for multivariate analysis, allowing for non-homogeneous marginal distribution and spatial dependence. Our approach was applied to analyze hourly rainfall in Taiwan's Lanyang watershed from 1996 to 2011, constructing the multisite rainfall distribution to estimate the frequency of extreme rainfall events. The results showed the nonlinear relationships and distinct frequency distributions of rainfall across the study area due to changes in elevations. Based on the copula-based multisite rainfall distribution, the study simulated extreme rainfall events at multiple sites with predetermined frequencies, preserving the spatial characteristics of rainfall more accurately than previous methods. This approach avoids potential underestimation and overestimation of hourly rainfall intensity at specified return periods resulting from assumptions of homogeneity and spatial independence. This study also evaluated possible multisite rainfall combinations at a given frequency using Monte-Carlo simulation. These simulations characterized spatial uncertainties in design multisite rainfall, providing critical information for water resources management and hydrological design. [ABSTRACT FROM AUTHOR]

Droughts have far-reaching detrimental impacts on the environment, society, and economy, ranging from regional to national levels. As the drought characteristics are interrelated, multivariate analysis of those is necessary to understand the actual drought situation in a region. However, such studies are limited. Hence, this study aimed to develop a framework to investigate the meteorological and hydrological droughts using a multivariate analysis of drought characteristics in the Pennar River basin (a semi-arid region) of India, dominated by agricultural activity. Long-term observations (1980–2015) of precipitation, temperature, and streamflow were used to calculate the Standardized Precipitation Evapotranspiration Index (SPEI) and Streamflow Drought Index (SDI) at a 3-month timescale. Based on these indices, drought duration, severity, and peak were abstracted using the Run Theory. The best-fit marginal distribution was determined for every drought characteristic to establish the bivariate joint probability distribution (the copula function). Based on the best-fit copula function, the joint probabilities and the joint return periods were estimated. Results revealed that, for meteorological drought, Frank Copula and Survival Clayton copula were the best-fit copula function for the joint risk of drought duration and severity and the combination of drought duration and peak as well as drought severity and peak, respectively. The joint return period of a drought event with characteristics above the 25th percentile of the same varies between 1 and 2 years, whereas it spans between 1 and 45 years for the 50th percentile. This analysis will provide vital information for water management and planning in a region to mitigate droughts. [ABSTRACT FROM AUTHOR]

Efficient slope reliability analysis that accounts for the interdependencies of random variables is challenging. In this study, an efficient reliability method is proposed based on importance sampling (IS) and copulas. First, the IS method based on the design point is introduced to construct the marginal IS density. Copulas are subsequently adopted to capture the interdependencies and establish both the initial joint distributions and the joint IS densities of random variable pairs. The IS algorithm is modified to consider the dependencies through not only the estimation of the probability distribution but also random sampling. Finally, a framework of the copula‐based IS method is constructed. The applicability, accuracy, efficiency, and robustness of the proposed method were validated by a classical slope case. The results showed that with the proposed method, slope reliability could be efficiently assessed with a relatively high accuracy. The dependencies of the variables significantly affected the IS and final slope reliability, whereas the IS density had a significantly lower effect. The proposed method combines the advantages of IS and copulas well. Interdependencies of the variables can be properly characterized during sampling. Moreover, the proposed method was shown to be robust regardless of the system failure probability level. It can be further applied to improve the random finite element method in the future. [ABSTRACT FROM AUTHOR]

In production design processes, multiple correlated responses with different distributions are often encountered. The existing literature usually assumes that they follow normal distributions for computational convenience, and then analyzes these responses using traditional parametric methods. A few research papers assume that they follow the same type of distribution, such as the t-distribution, and then use a multivariate joint distribution to deal with the correlation. However, these methods give a poor approximation to the actual problem and may lead to the recommended settings that yield substandard products. In this article, we propose a new method for the robust parameter design that can solve the above problems. Specifically, a semiparametric model is used to estimate the margins, and then a joint distribution function is constructed using a multivariate copula function. Finally, the probability that the responses meet the specifications simultaneously is used to obtain the optimal settings. The advantages of the proposed method lie in the consideration of multiple correlation patterns among responses, the absence of restrictions on the response distributions, and the use of nonparametric smoothing to reduce the risk of model misspecification. The results of the case study and the simulation study validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Multiple flood-generating mechanisms can affect the communities and infrastructure systems across coastal areas. Reliable design flood estimation and risk assessment in these regions demand characterization of the interrelationships between different drivers as well as the corresponding compounding effects. In this study, we assess the compound flood risks across Canada's coasts considering eight bivariate flooding scenarios acquired from four major drivers including precipitation, streamflow, skew surge, and total water level. For each scenario, an initial dependency test based on Kendall's Tau is conducted. The individual variability of each driver, at different locations, is characterized through parametric extreme value distributions followed by developing the bivariate joint distributions using copula functions. Parameters of the copula functions and the marginal distributions are inferred based on the Bayesian approach. Further, compound flood risks and failure probabilities are analyzed considering the OR, AND, Kendall, and conditional hazard scenarios. Results suggest that most locations can be affected by compound flooding associated with at least two bivariate events. Among the studied scenarios, the concurrent occurrence of extreme precipitation and sea level is the most common across ~ 80% of the coastal locations. In addition, critical sites that can be affected by the majority of the bivariate events (six out of eight combinations) are identified at the Pacific, south of Lake Huron, Lake Erie, and Lake Ontario. Besides, the estimated failure probabilities show higher hydrologic risks associated with the bivariate analysis of flood drivers compared to the univariate estimates. [ABSTRACT FROM AUTHOR]

Sklar's theorem represents a single multivariate distribution function through its univariate marginal distributions and a copula. However, it fails in general when it comes to the representation of the envelopes of a family F of multivariate distribution functions (i.e. its point-wise infimum and supremum), even if we allow more general representing functions than copulas. In this paper we develop an alternative representation which describes the envelopes of a family F in terms of the copulas that correspond to the members of F. Our representation is reminiscent of Sklar's representation but is based on the idea of corner patches of copulas. We prove a series of four representation theorems; (1) a general one, which holds even if the envelopes of the family do not possess a Sklar type representation, (2) a version tailored for the case when the envelopes do possess a Sklar type representation, (3) a theorem for families of continuous distribution functions, and (4) a variant appropriate for modeling dependencies of absolutely continuous random variables. In addition, we demonstrate how our results can be applied to give a Sklar type theorem in the imprecise probability setting, which offers a new approach to multivariate probability boxes. [ABSTRACT FROM AUTHOR]

The paper introduces a function defined by a power series in the probabilities of multivariate records in a sequence of independent identically distributed random vectors with continuous components. In the univariate case, this function is always the same, but in the multivariate case there exists a broad variety of such functions determined by distribution copulas. The probabilistic meaning of this function and its derivatives is presented. A calculation method using the Kendall distribution function is given. The concepts of favorableness of copulas for records, record time distribution, and mean record time (without taking into account the order) are introduced. Examples are given. [ABSTRACT FROM AUTHOR]

Let Λ be the collection of all probability distributions for (X , X ˜) , where X is a fixed random vector and X ˜ ranges over all possible knockoff copies of X (in the sense of Candes et al. (2018)). Three topics are developed in this paper: (i) A new characterization of Λ is proved; (ii) A certain subclass of Λ , defined in terms of copulas, is introduced; (iii) The (meaningful) special case where the components of X are conditionally independent is treated in depth. In real problems, after observing X = x , each of points (i)–(ii)–(iii) may be useful to generate a value x ˜ for X ˜ conditionally on X = x. • The paper provides some new theoretical insights on the knockoffs procedure introduced by Barber and Candes (2015). Let Λ be the set of all possible knockofffs. In Section 2, a new characterization of Λ is proved. • In Section 3, a certain (proper) subclass Λ 0 ⊂ Λ is introduced. The elements of Λ 0 admit a simple and explicit representation in terms of copulas. • In Section 4, we focus on the case where the variables X 1 , ... , X p are conditionally independent. Under this assumption, we show how to build a knockoff X ˜. [ABSTRACT FROM AUTHOR]

In recent times, copula families have been mostly employed in bivariate drought duration and severity. Archimedean copulas are mainly used conveniently among various potential families for hydrologic design and water resource management. In this study, drought events are defined by standard precipitation index (SPI3 time series) series for 24-gauge stations in Ceyhan Basin Turkey considering different elevation levels (high, average, and low). Both duration and severity are tested for their dependence with Mann–Kendall and Spearman Rho trend test at a 0.05 significant level. While Gamma and Weibull perform better than the other distributions at average elevation, Log-normal and Weibull distributions are found to fit the DD and DS series better for most stations located in low and high elevations, respectively. The best fit copula is found to construct a joint distribution function. The influence of upper tail dependencies is tested to select the most appropriate copula function. Gumbel and BB1 copulas are often regarded as the best copulas for the DD and DS series, respectively. After acquiring the best fit copula, the joint return periods are modeled for each station. Finally, considering drought risk categories (mild, moderate, severe, and extreme drought), spatial distributions of drought risk-return period are constructed. High risks are observed in the middle part of the basin based on TDS and T′DS under mild drought categories. Apart from using univariate frequency analysis, bivariate frequency analysis is performed to model the joint return period based on copula theory which is great of importance for hydrology design and water resource management. [ABSTRACT FROM AUTHOR]

In this paper, we make use of meta-elliptical copula functions to build a new multivariate distribution with fixed marginal distributions and dependence structure to analyze bounded data. Specifically, we present a flexible p-elliptical multivariate probability distribution in the hypercube (0 , 1) p p with fixed marginal GF-quantile distributions. We then present some illustrative examples and a meta-elliptical multivariate regression model as a flexible alternative to the multivariate normal regression model on unit intervals. A simulation study and real-life data analysis using a Bayesian framework with the extreme-value quantile functions show the flexibility of the proposed meta-multivariate normal regression model for modeling the observed proportion response variables. [ABSTRACT FROM AUTHOR]

With the gradual depletion of surface resources, rock instability caused by deep high stress and mining disturbance seriously affects safe mining. To create effective risk management, a rock instability risk field model using microseismic monitoring data is proposed in this study. Rock instability risk was presented visually in 3D visualization. The in-situ microseismic monitoring data was collected and analyzed to make calculation of peak ground velocity (PGV), peak ground acceleration (PGA), energy flux, energy and seismic moment. Indicator weights of PGV, PGA, energy flux are confirmed by using the analytic hierarchy process (AHP) to calculate risk severity. The Copula function is then used to solve the joint probability distribution function of energy and seismic moment. Then the spatial distribution characteristics of risk can be obtained by data fitting. Subsequently, the three-dimensional (3D) risk field model was established. Meanwhile, the established risk field is verified by comparing monitoring data without disturbance and the blasting data with disturbance. It is suggested that the proposed risk field method could evaluate the regional risk of rock instability reasonably and accurately, which lays a theoretical foundation for the risk prediction and management of rock instability in deep mining. [ABSTRACT FROM AUTHOR]

Recent studies have shown that the univariate exponentiated Teissier distribution is an effective model for analysing rainfall data. A bivariate probability distribution offers greater insight into a process like a flood or drought than a univariate method for analysing the characteristics of environmental occurrences. As a result, in this study we developed a bivariate exponentiated Teissier distribution. The marginals of the proposed distribution model follow the exponentiated Teissier distribution. The cumulative distribution function of this bivariate model consists of continuous and singular components. The proposed model is shown to have several fundamental statistical properties, including joint cumulative distribution, joint density, marginals, and conditional distributions. Additionally, some copula functions are examined. The proposed model's parameter estimations are derived using the maximum likelihood method. Furthermore, the asymptotic confidence intervals are given for these estimates. A Monte Carlo simulation study with various sample sizes was performed to evaluate the behaviour of maximum likelihood estimates. Finally, we used football and flood datasets to demonstrate how the proposed model may be employed. [ABSTRACT FROM AUTHOR]

Flash drought (FD) is characterized by the rapid onset and development of drought conditions. It usually occurs during the growing seasons, causing more severe impacts on agriculture and society than the slowly evolving droughts. Based on the Standardized Evaporative Stress Ratio (SESR), this study presents an assessment of the spatiotemporal variability of the joint return periods of FD characteristics in the Pearl River basin (PRB), southern China. Three FD characteristics (i.e., duration D, intensity I, peak P) are extracted at each 0.25° × 0.25° grid point over the PRB by the Runs theory. Four marginal distribution functions (gamma, exponential, generalized extreme value, and lognormal) are used to fit FD characteristics, while three Archimedean copula functions (Clayton, Frank, and Gumbel) are used for generating the joint distributions of various paired FD characteristics. The results indicate that lognormal is the best-fitted marginal distribution function of FD characteristics in most parts of PRB, while Frank and Clayton are the best-fitted copula of the joint distributions of three pairs of FD characteristics in most parts of PRB. During 1953–2013, the FD events are more frequent in eastern PRB (> 40 events) than western PRB (< 10 events), and larger FD characteristics (D and I) are also found in eastern PRB than western PRB. The return period of each FD characteristic is smaller in eastern PRB than western PRB, leading to smaller joint return periods of three paired FD characteristics (D-I, D-P, P-I) in eastern PRB than western PRB. Overall, our results suggest that the risk of FD is gradually increased from the west to the east of the PRB. [ABSTRACT FROM AUTHOR]

For a large class of distribution functions we study properties of the product of random variables X and Y. We take into account the dependency structure between X and Y by making assumptions about the asymptotic equality P(X > x|Y = y) ~ h(y)P(X > x) as x → ∞, uniformly for y in the range of Y. As particular consequences, some well-known results concerning the product of random variables are reviewed, among them the Breiman's theorem. An application is made in the case where the dependence between X and Y is characterized by asymptotic conditions on their copula. [ABSTRACT FROM AUTHOR]

The problem of estimation of multivariate survival function under dependent random right-censoring of observations is considered. To construct estimators, Archimedean copula functions are used. Consistency properties of estimators are proved by martingale techniques. The possibility of application of estimators to integral-type functionals is discussed. [ABSTRACT FROM AUTHOR]

We study an n-player game with random payoffs and continuous strategy sets. The payoff function of each player is defined by its expected value and the strategy set of each player is defined by a joint chance constraint. The random constraint vectors defining the joint chance constraint are dependent and follow elliptically symmetric distributions. The Archimedean copula is used to capture the dependence among random constraint vectors. We propose a reformulation of the joint chance constraint of each player. Under mild assumptions on the players' payoff functions and 1-dimensional spherical distribution functions, we show that there exists a Nash equilibrium of the game. [ABSTRACT FROM AUTHOR]

Investigation of homogeneity regions using univariate characteristics is an important step in the regional frequency analysis method. However, some hydrological phenomena have multivariate characteristics that cannot be studied by univariate methods. Droughts are one of these phenomena their definition as univariate will not be effective for risk assessment, decision-making, and management. Therefore, in this study, the regional frequency analysis of drought was studied in multivariate methods using SEI (Standardized Evapotranspiration Index), SSI (Standardized Soil Moisture Index), and SRI (Standardized Runoff Index) indices in the Karkheh River basin from 1996 to 2019. The indices calculated probabilistic distribution between the variables of evapotranspiration, runoff, and soil moisture using multivariate L-moments method and Copula functions and considered meteorological, agricultural, and hydrological droughts simultaneously. The results of multivariate regional frequency analysis considering the Copula Gumbel as the regional Copula showed that the basin is homogeneous in terms of severity of SEI-SSI combined drought indices and is heterogeneous in terms of severity of SEI-SSI combined drought indices. However, after clustering the basin into four homogeneous areas in terms of characteristics of SPI (Standardized Precipitation Index), the basin is homogeneous in all areas in terms of univariate SEI, SSI, and SRI indices and is heterogeneous in the third and fourth clusters of SRI and SSI drought indices. Pearson Type (III), Pareto, normal, and general logistics distribution functions were found suitable to investigate the characteristics of SEI, SSI, and SRI drought indices in this case. Finally, large estimates of the types of combined droughts and their probability of occurrence showed that the northern and southern parts of the Karkheh River basin will experience short and consecutive droughts in the next years. Droughts in areas without meteorological data can be predicted in terms of joint probability using the multivariate regional frequency analysis method proposed in this study. [ABSTRACT FROM AUTHOR]

We further develop the recently proposed event-by-event cumulants of azimuthal angles. The role of reflection symmetry, permutation symmetry, frame independence, and relabeling of particle indices in the cumulant expansion is discussed in detail. We argue that mathematical and statistical properties of cumulants are preserved if cumulants of azimuthal angles are defined event-by-event in terms of single-event averages of azimuthal angles, while they are violated in the traditional approach in which cumulants are defined in terms of all-event averages. We derive for the first time the example analytic solutions for the contribution of combinatorial background in the measured two- and three-particle correlations. We demonstrate that these solutions for the combinatorial background are universal, as they can be written generically in terms of multiplicity-dependent combinatorial weights and marginal probability density functions of starting multivariate distribution. The new general results between multiparticle azimuthal correlators and flow amplitudes and symmetry planes are presented. [ABSTRACT FROM AUTHOR]

Under climate change, the sea surface temperature and salinity change greatly, which poses a considerable threat to sustainable food security. Sea surface temperature and salinity (SST/SSS) are selected to examine the annual output of swimming crab in 24 cities along the eastern China. The Copula-based function was used to construct the probability distribution model of the swimming crab yield with SST and SSS. The pure premium rate of the swimming crab production in these 24 cities are also examined. The results show that 1) There is significant positive correlations between the yield of swimming crab with temperature and salinity over the study area. The only exception is that the correlation between yield of swimming crab and salinity is not significant in the south of study area. 2) The span of the pure insurance premium rate of swimming crab in 24 cities increases rapidly with the increase of the protection level, the maximum span up to 2.04%, and the minimum span is only 1.6%. 3) The distribution of the swimming crab insurance premium rate is various in space. The insurance premium rate of 8 cities in the south of Taizhou is low with the highest premium rate at 5.6%. The insurance premium rate of 16 cities in north of Taizhou is relatively high with the rate between 6%-22%. The research can provide a theoretical basis for the pricing of insurance products for swimming crab in 24 cities in the typical aquaculture areas in eastern China. [ABSTRACT FROM AUTHOR]

Seismic fragility analysis methods commonly used for assessing structural components and systems have certain limitations. This paper introduces a new semi-parametric approach for structural component fragility analysis, the TKC method. This method is based on log-transformed kernel density estimation (log-TKDE) and copula functions, which establish the joint probability distribution (JPD) of a structural response value (d) and a disaster-inducing factor strength parameter (α) using log-TKDE and match the marginal cumulative distributions (MCDs) of d and a via bivariate copula functions. A novel method for evaluating structural system fragility is also proposed based on the R-vine copula function. This approach accounts for the intricate correlations among different components, providing a precise estimation of the multidimensional JPD. The reliability and accuracy of the proposed methods are demonstrated through a seismic fragility analysis of a sea-crossing cable-stayed bridge under earthquake load. Furthermore, a multi-hazard fragility analysis of bridge components and systems is conducted considering hazards such as earthquake, wind, and wave loads. A comparison is made with the seismic fragility of bridges subjected to earthquake loads alone. The results show that the median and 95% confidence of the damage probability of structural components calculated by the TKC method and Monte Carlo simulation (MCS) closely match, which validates the proposed method. The TKC method is also more accurate and computationally efficient than the MCS method with the same Bootstrap sample size and confidence level (95%). The proposed structural system fragility assessment method based on the R-vine copula accounts for complex correlations between components and yields more accurate results than the first-order boundary method. Load spectrum characteristics, structural component characteristics, and other factors cause the multi-hazard fragility of bridge components and systems to differ substantially from their seismic fragility. The difference rate of system damage probability can reach 28.82%. In evaluating the anti-disaster performance of sea-crossing bridges, it is necessary to incorporate the marine environmental load and consider the dynamic response coupling effect of the bridge excited by seafloor ground motions with extreme wind and wave loads. This approach can guide the optimization of bridge designs and facilitate multi-hazard fragility analysis in accordance with real damage probabilities. • A novel method for fragility analysis of bridge components based on log-TKDE and copula. • A novel fragility assessment method for structural systems using R-vine copula function. • Multi-hazard fragility analysis of sea-crossing cable-stayed bridge under combined earthquake-wind-wave action. • Difference in structural anti-disaster performance between multi-hazards action and seafloor earthquakes action. [ABSTRACT FROM AUTHOR]

Two-dimensional distributions whose marginal distributions are uniform are called bivariate copulas. Among them, the one that satisfies given constraints on expectation and is closest to being an independent distribution in the sense of Kullback–Leibler divergence is called the minimum information bivariate copula. The density function of the minimum information copula contains a set of functions called the normalizing functions, which are often difficult to compute. Although a number of proper scoring rules for probability distributions having normalizing constants such as exponential families have been proposed, these scores are not applicable to the minimum information copulas due to the normalizing functions. In this paper, we propose the conditional Kullback–Leibler score, which avoids computation of the normalizing functions. The main idea of its construction is to use pairs of observations. We show that the proposed score is strictly proper in the space of copula density functions and therefore the estimator derived from it has asymptotic consistency. Furthermore, the score is convex with respect to the parameters and can be easily optimized by the gradient methods. [ABSTRACT FROM AUTHOR]

In this work, we consider the joint distribution function as well as the copula of (X + Z; Y ) where the random vector (X; Y;Z) is characterized by a copula CX;Y;Z . We use this copula to analyze a Berkson measurement error model. By presenting a general form of a Berkson measurement error model with copula-dependent random variables, we investigate some of its special cases. Some theoretical results, several examples as well as a simulation study, are proposed for illustration. [ABSTRACT FROM AUTHOR]

Accurate prediction of wind speed plays an important role in increasing the power generation of wind turbines and realizing efficient use of wind energy. However, due to the large number of wind turbines in the wind farm and the complex wake effects between the units, the coupling degree and spatial correlation of the wind speed of the wind turbines are increased. Accordingly, this paper proposes a wind speed prediction model based on spatio-temporal dependency analysis. The proposed model first uses long short-term memory(LSTM) neural network to predict the wind speed of each wind turbine to obtain its residuals, which can extract the time correlation of the wind-speed series; Then by applying the Clayton Copula function to analyze the correlation between the residual series and wind-speed series to get the joint-distribution function. The joint-distribution function can be used to calculate the prediction error of the wind speed and complete the wind speed prediction. The validity of the method in this work is verified using the measured wind-speed data of a wind farm. Experimental results show that the method effectively solves the problem of finding spatio-temporal dependency of wind speed and significantly improves the prediction accuracy of wind speed. [ABSTRACT FROM AUTHOR]

The fitted Graph HT ht conditional bivariate probability distributions clearly depict that the acceleration characteristics vary with the type of subject vehicle involved. Further, the modal values of the bivariate Graph HT ht copula models were estimated corresponding to speeds exceeding 20 kmph for all vehicle types. [Extracted from the article]

Merging different flood characteristics in a distribution function is provided by copula structures. In this study, the nested copula structure was used to construct a trivariate distribution of flood duration (D), peak (P), and volume (V). The required data were obtained by screening the flood events recorded at Armand Gauging Station, Iran. The characteristics of selected 63 flood events (1993–2018) were extracted and the best marginal distribution function of each was determined by Kolmogorov–Smirnov test. Then the fitness of six different copula functions (Frank, Clayton, Joe, Gumbel–Hougaard, Gaussian and Student's t were examined for creating the joint distribution function. The best fitted marginal distribution is Johnson SB, for flood duration, and Lognormal (3p), for flood peak and flood volume. The best-fitted function for creating bivariate and trivariate distributions of flood characteristics in Armand Basin is Frank copula. In the next phase, the bivariate and trivariate joint return periods (at two states of AND, OR), Kendall return period and conditional return periods were calculated. The results revealed that the conditional return period of one flood variable given two other flood variables is greater than the corresponding values for the conditional return period of two flood variables given the third flood variable. [ABSTRACT FROM AUTHOR]

Over the decades that have passed since they were introduced, copulae still remain a very powerful tool for modeling and estimating multivariate distributions. This work gives an overview of copula theory and it also summarizes the latest results. This article recalls the basic definition, the most important cases of bivariate copulae, and it then proceeds to a sketch of how multivariate copulae are developed both from bivariate copulae and from scratch. Regarding higher dimensions, the focus is on hierarchical Archimedean, vine, and factor copulae, which are the most often used and most flexible ways to introduce copulae to multivariate distributions. We also provide an overview of how copulae can be used in various fields of data science, including recent results. These fields include but are not limited to time series and machine learning. Finally, we describe estimation and testing methods for copulae in general, their application to the presented copula structures, and we give some specific testing and estimation procedures for those specific copulae. This article is categorized under:Statistical Models > Multivariate ModelsStatistical Models > Semiparametric ModelsStatistical and Graphical Methods of Data Analysis > Multivariate Analysis [ABSTRACT FROM AUTHOR]

Risk management of stock portfolios is a fundamental problem for the financial analysis since it indicates the potential losses of an investment at any given time. The objective of this study is to use bivariate static conditional copulas to quantify the dependence structure and to estimate the risk measure Value-at-Risk (VaR). There were selected stocks that have been performing outstandingly on the Brazilian Stock Exchange to compose pairs trading portfolios (B3, Gerdau, Magazine Luiza, and Petrobras). Due to the flexibility that this methodology offers in the construction of multivariate distributions and risk aggregation in finance, we used the copula-APARCH approach with the Normal, T-student, and Joe-Clayton copula functions. In most scenarios, the results showed a pattern of dependence at the extremes. Moreover, the copula form seems not to be relevant for VaR estimation, since in most portfolios the appropriate copulas lead to significant VaR estimates. It has found that the best models fitted provided conservative risk measures, estimates at 5% and 1%, in a scenario more aggressive. [ABSTRACT FROM AUTHOR]

The instability of seabed slope sediments is the main factor influencing the safety of marine resource development. Therefore, to ensure the safe operation of submarine pipelines under complex and uncertain seabed rock and soil conditions, a reliability model was developed to elucidate the trend of impact-related pipeline damage due to submarine slides. Then, a risk assessment of the damage process of submarine slides impacting pipelines was conducted, which is of great significance for the in-depth safety assessment of pipelines impacted by submarine slides. Based on the copula function, a joint probability distribution model considering the correlation among risk variables was established for rational correlation characterization. A probability analysis method of impact-related pipeline damage attributed to submarine slides based on the copula function was proposed. The Monte Carlo simulation (MCS) method was employed to simulate the random uncertainty in limited observation values and accurately determine the reliability of safe pipeline operation under the action of submarine slides. The conclusions were as follows: (1) Based on the copula function, a joint probability distribution model of risk variables with any marginal distribution function and related structure could be developed. (2) The copula function could reasonably characterize relevant nonnormal distribution characteristics of risk variables and could simulate samples conforming to the distribution pattern of the risk variables. (3) The failure probability calculated with the traditional independent normal distribution model was very low, which could result in a notable overestimation of the reliability of submarine pipelines. [ABSTRACT FROM AUTHOR]

The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well. [ABSTRACT FROM AUTHOR]

The software Randentropy is designed to estimate inequality in a random system where several individuals interact moving among many communities and producing dependent random quantities of an attribute. The overall inequality is assessed by computing the Random Theil's Entropy. Firstly, the software estimates a piecewise homogeneous Markov chain by identifying the change-points and the relative transition probability matrices. Secondly, it estimates the multivariate distribution function of the attribute using a copula function approach and finally, through a Monte Carlo algorithm, evaluates the expected value of the Random Theil's Entropy. Possible applications are discussed as related to the fields of finance and human mobility. [ABSTRACT FROM AUTHOR]

The present study compares the daily and monthly precipitation estimates of the CHIRPS satellite data with the in situ measurements at four stations scattered over the Kosar Dam basin in southwestern Iran. The uncertainty of the satellite precipitation estimates was calculated through simulation with the Copula functions. For this purpose, 55% of the stations, daily and monthly rainfall data relative to the 1987–2012 period were used for training (simulation), and the other 45% were used for testing (validation) the performance of the Copula model. First, the daily, monthly, and annual satellite precipitation estimates were statistically compared with precipitation observed at the stations and the whole basin using the Pearson correlation coefficient (CC), root mean square error (RMSE), and Bias statistics. The computed CC between the areal average of observed and satellite precipitation estimation at the basin is 0.49, 0.82, and 0.33 for daily, monthly, and annual time scales, respectively. The difference (biases) between the satellite estimates and in situ measurements was then calculated for daily, monthly, and annual time scales over the training period. The obtained biases were subsequently fitted with the General Extreme Value distribution function coupled with the Gaussian Copula model to generate a series of similar random biases for all precipitation events. Then, the generated random biases were summed with the original satellite estimates to correct the associated biases. The bias-corrected precipitation for the training period was then compared to the original estimates of the satellite at the stations and the whole basin using the P-factor, R-factor, Bias, RMSE, and CC statistics. The statistics show that the random biases generated by the Copula method for the monthly CHIRPS satellite data relative to the 14-year training period have reduced the error rate of the satellite data by 74 to 95 percent when compared to observations. The satellite precipitation estimates of the 11-year test period were also corrected using the generated random biases in the training period. The results show that the bias correction considerably improved the monthly estimates and reduced the error rate of the satellite estimation by about 76 percent. In general, the simulation of the satellite precipitation with the Gaussian Copula model was performed satisfactorily at the monthly time scale, but it was less efficient at the daily time scale. [ABSTRACT FROM AUTHOR]

Modern power sector requires grid observability under all scenarios for its ideal functioning. This enforces the operator to incorporate state estimation solutions based on a priori measurements to deduce the corresponding operating states of the grid. The key principle for such aforementioned algorithms lies on an occurrence of an over determined class of system having an ample redundancy in the measurements. Operators employ state forecasting solutions to counter the loss of real-time measurements. This work encompasses a critical comparison between several machine learning models along with ARIMA and time delayed neural network architecture for proper forecasting of operating states under normal as well as contingency scenarios. To showcase the efficacy of the proposed approach, this work incorporates a comprehensive comparison between them based on RMSE, MSE and MAE index. Copula-based synthetic data generation based on Gaussian multivariate distribution of the a priori measurements and operating states along with optimal hyper-parameter tuning of the models have shown the effectiveness of such algorithms in predicting future state estimates. The proposed machine learning models can be also seen to showcase an effective forecasting strategy under varying noise scenarios. This work also showcases the implementation of the models for real-time state forecasting strategy having computational times in the order of micro seconds. All the simulations have been carried out on the standard IEEE 14 bus test bench to support the former proposals. [ABSTRACT FROM AUTHOR]

• STMF is developed for system reliability analysis considering failure correlation. • The joint optimum of empirical risk and expected loss is realized by MSM of STMF. • The correlation measurement hierarchy is established by MTM of STMF. • STMF is verified by analysis of turbine bladed disk. • STMF holds superior accuracy and efficiency in reliability analysis. To improve the efficiency and accuracy of complex system reliability analysis considering failure correlation, a selective transmit modeling framework (STMF) is proposed by integrating multi-stage selective modeling (MSM) strategy with multi-failure transmit modeling (MTM) architecture. In MSM, the modeling quality of multi-learner is discussed to establish the optimal functional relationship between multi-variable and multi-response by considering the suitability of modeling samples, and the fitting performance of copula functions is quantified to construct the optimal correlation model between multi-failure through distance measure. For MTM, the correlation sequence measurement structure of multiple modes is established to construct the joint reliability analysis model of complex structure by updating iteratively the margin distribution functions with the strongest correlation. Taking the turbine bladed disk as study object, the validation of the proposed STMF is verified by comparing with other methods. The analytical results of study case can be expressed as that (i) STMF holds the superior performance in modeling accuracy and time-consuming, at 2.6308 × 10-2 and 0.4638 s respectively; (ii) STMF also has the best property in simulation, separately at 2.2123 s in simulation efficiency and at 99.9899 % in simulation precision; (iii) the reliability 0.9876 of study case obtained by STMF is higher than the reliability 0.9855 that depends on the mutual independence assumption. The effort of this work is prominent to promote effectively the quantification of multi-failure correlation and the improvement of reliability analysis performance for complex systems. [ABSTRACT FROM AUTHOR]

A novel method for analyzing the reliability of structures under non-stationary stochastic seismic excitations, considering the combined effect of multiple structural demand extreme values, is proposed. The spectral representation method is employed to establish a non-stationary stochastic seismic excitation model, and based on the theory of first-passage probability, multiple integral formulas for seismic reliability under multidimensional limit states are derived. The extreme value distribution is established using the Gamma mixture model (GMM). The equations for estimating the model parameters are derived based on both fractional moments and moment-generating functions, while the determination of the number of gamma distribution components is guided by the probability distribution and statistical characteristics of the samples. The joint probability density function (JPDF) for multiple demand extreme values is established by incorporating copula functions to account for correlation, and the fitting accuracy of different copula functions is assessed. The proposed method is illustrated using reinforced concrete (RC) frame structures. The results demonstrate that the fitting accuracy of extreme value distribution can be enhanced by adjusting the number of gamma distribution components in the GMM, which exhibits high accuracy in fitting both the main and tail regions. The presence of correlation can induce variations in the JPDF, thereby exerting an influence on the failure probability. Consequently, disregarding correlation is not conducive to reliability analysis. [ABSTRACT FROM AUTHOR]

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. • Develop an early warning indicator for rock slopes in engineering geology. • Introduce the safety stability rate to quantify the impact of rock structure on stability. • Construct the ISSR-MDF model to alter the influence of each measurement point. • Establish a multidimensional residual joint distribution function using copula theory. • Case study demonstrates indicator's efficacy in overall stability assessment. [ABSTRACT FROM AUTHOR]

• The proposed approach can characterize multivariate, asymmetric, and multimodal distributions of Multivariate, Uncertain, and Irregular (MUI) geotechnical data. • The optimal, robust, and hyper-robust predictions on the posterior distribution and posterior predictive distribution are developed based on fusing the Copula theory and Bayesian inference and performing operations of marginalization and conditioning. • A unified approach for credible region, using the idea of the HDR (Highest Density Region), can be constructed by performing the MCS based on the developed optimal, robust, and hyper-robust predictions. • The proposed approach can be utilized for the visualization-aided analysis of the statistical joint behaviors of the multivariate random variables of MUI geotechnical data. Geotechnical data are typically Multivariate, Uncertain, and Irregular (MUI), so the probability distribution of geotechnical data is Multivariate, Asymmetric, and Multimodal (MAM). Probability Density Function (PDF) modelling and Credible Region (CR) construction are two key issues for a MAM distribution. There are two fundamental difficulties in characterizing a MAM distribution. The first is on joint PDF modelling as many traditional approaches collapse for a MAM distribution. Copula theory has attracted special attention for this purpose but very few works attempted to tackle the critical problem of probabilistic prediction on target variables using available information of remaining variables based on the copula-based joint PDF. The second is on CR construction of a MAM distribution as it cannot find a unique CR of a MAM distribution given an exceedance probability only. There is still a lack of a unified approach for CR construction for a MAM distribution of geotechnical data. Aiming to resolve these two fundamental difficulties, we propose the BAyeSIan Copula-based Highest density region/contour (BASIC-H) for providing a systematic framework of PDF modelling and CR construction of a MAM distribution. This framework contains Stage-PDF and Stage-CR. Stage-PDF fuses the copula theory and Bayesian inference to develop optimal, robust, and hyper-robust predictions on the posterior distribution and posterior predictive distribution. Stage-CR adopts the constraint for the CR that the probability density of every point inside the CR is at least as large as the probability density of any point outside, which is the same as the idea of the HDR (Highest Density Region). The Monte Carlo Simulation (MCS), based on the developed optimal, robust, and hyper-robust posterior distributions and posterior predictive distributions, is performed for estimation of the probability density boundary, which is a key parameter for constructing the HDR. Examples using simulated data and Quaternary clay data are presented to illustrate the capabilities of the BASIC-H in PDF modelling and CR construction of MAM distributions of geotechnical data. [ABSTRACT FROM AUTHOR]

• The integrated framework can reveal joint encounter and state evolution of short-duration rainstorms. • It's more sensitive to the changes in their probability of 0.2–0.7 and the JRPRE of 15-30freq. • The probability of synchronous F-L between P and R is greater. • The transition probability and limit probability of the F/H and F-H are large. • The P and R usually appear in the evolutionary state with the highest risk. In this study, in order to reveal the joint encounter and state evolution characteristics of urban short-duration rainstorms, a case study is conducted in Zhengzhou city, China. The joint probability distribution model of total rainfall (P) and peak rainfall (R) is constructed by using the Copula function, and their joint encounter and joint return period are analyzed. Additionally, the state evolution characteristics of P and R are explored by incorporating with the Markov chain. The results show that: (1) there is a good correlation between P and R under different rainfall durations. The Clayton Copula function is found suitable for 0.5 h and 1 h rainfall durations, while the Frank Copula function is selected for the 1.5 h rainfall duration. The P and R are more densely distributed in the joint probability interval of 0.2–0.7 and the joint return period of rainfall events (JRPRE) of 15-30freq, which are particularly sensitive to the changes in the corresponding intervals. (2) The probability of synchronous rich (F)-low (L) between P and R is greater than that of asynchronous F-L. The probability of synchronous F-L increases with longer rainfall durations, reaching the highest value of 74.92 % for the 1.5 h rainfall duration. (3) The transition probability and limit probability of the F/high (H) state and F-H combination state are relatively large, indicating that P and R exhibit strong self-conservation. These states are associated with the highest risk and tend to occur more frequently with shorter average return periods of rainfall events (ARPRE). The findings are valuable in uncovering the occurrence patterns of short-duration rainstorms and facilitating measures to mitigate waterlogging, thereby supporting urban sustainable development. [ABSTRACT FROM AUTHOR]

• NTDI (nonlinear trivariate drought index) performs better than single type drought indices for drought monitoring. • NTDI can effectively monitor the drought onset and duration. • NTDI shows good potential in the agricultural drought assessment. Due to the complexity of meteorological and hydrological conditions in a changing environment, previous drought indices for monitoring a specific drought type do not reflect the overall regional situation of water scarcity. Therefore, in order to obtain accurate and reliable drought monitoring, a more integrated drought index should be developed to identify drought events comprehensively. In this paper, a non-linear trivariate drought index (NTDI) was constructed based on the joint probability distribution of parametric copulas, combining precipitation (P), potential evapotranspiration (PET), and root zone soil moisture (SM) variables. Subsequently, it was respectively compared with four drought indices, SPEI, SSMI, SC-PDSI and TVDI, and cross-validated with actual recorded drought events and annual crop yield to evaluate its applicability in arid Central Asia (ACA). The results indicated that: (1) Frank copula (1-,3-month scale) and Gumbel copula (6-,12-month scale) were considered to be the best-fitted copula functions for constructing joint probability distributions in the ACA. (2) The NTDI integrated the P-PET and SM drought signals to sensitively capture drought onset and duration, reflecting the combined characteristics of meteorological and agricultural drought. (3) The drought information expressed by NTDI was generally consistent with recorded drought events, and the monitoring results are accurate. (4)The NTDI performed better in agricultural drought monitoring than other drought indices. This study provides a reliable multivariate composite indicator which is significant for drought monitoring, prevention and risk assessment in ACA. [ABSTRACT FROM AUTHOR]

Flash floods cause great harm to people's lives and property safety. Rainfall is one of the main causes of flash floods in small watersheds. The uncertainty of rainfall events results in inconsistency between the traditional single rainfall pattern and the actual rainfall process, which poses a great challenge for the early warning and forecasting of flash floods. To carry out the effective flash flood early warning and forecasting, this paper proposes a novel rainfall pattern by coupling total rainfall and peak rainfall intensity based on copula functions, i.e., the rainfall pattern of risk probability combination (RPRPC). On this basis, the Hydrologic Engineering Center-Hydrologic Modeling System (HEC-HMS) hydrological model is used to simulate the rainfall-runoff process, the trial algorithm is used to calculate the critical rainfall (CR), and the optimistic-general-pessimistic (O–G-P) early warning mode considering the decision maker's risk preference is proposed. The small watershed of Xinxian in Henan province, China, is taken as a case study for calculation. The results show that the RPRPC is feasible and closer to the actual rainfall process than the traditional rainfall pattern, Frank copula function is the best for determining the joint distribution function of total rainfall and peak rainfall intensity, and the HEC-HMS model can be applied to small watersheds in hilly areas. Additionally, both RPRPC and antecedent soil moisture condition (ASMC) have influence on CR, and the variation of RPRPC will change the influence of ASMC on CR. Finally, the effectiveness of O–G-P early warning mode is verified. [ABSTRACT FROM AUTHOR]

A novel validation frame for deterministic solar irradiance forecasts is presented. It bases on the perception that a perfect forecast of a random variable must be indistinguishable from its observation. In mathematical terms: The random variables observation and forecast must be exchangeable. This implies that they have the same probability distribution and that the pertaining copula is symmetric. The validation frame tallies the equality of the probability distributions of observed and forecast solar irradiance, the symmetry of the pertaining copula, and the forecasting skill with well-defined measures, namely the Kolmogorov-Smirnov test statistic, copula asymmetry, and Spearman's ρ. The three measures are scale-invariant under strictly increasing transformations of observation and forecast. Thus, the frame is particularly suited for benchmarking solar irradiance forecasts from different locations, different calendar days, and over different periods. The frame is demonstrated on 24-h-ahead solar irradiance forecasts from two sites with different climates. Yet, it can be applied to any forecast of a continuous random variable. Additional theoretical insights connect the method to present-day practice, such as the Murphy-Winkler framework and the skill score. • A perfect forecast of a random variable is indistinguishable from its observation. • Forecast and observation must be exchangeable. • Forecast and observation have the same distribution, and the copula is symmetric. • KS test statistic, copula asymmetry, and Spearman's ρ tally key properties. • The three measures are scale-invariant under strictly increasing transformations. [ABSTRACT FROM AUTHOR]