A statistical analysis method for significant wave height and spectral peak frequency considering the random and time-varying effects based on copula function and Bayesian inference

This study investigates the sensitivity and uncertainty of hydrological droughts frequencies and severity in the Weihe Basin, China during 1960–2012, by using six commonly used univariate probability distributions and three Archimedean copulas to fit the marginal and joint distributions of drought characteristics. The Anderson-Darling method is used for testing the goodness-of-fit of the univariate model, and the Akaike information criterion (AIC) is applied to select the best distribution and copula functions. The results demonstrate that there is a very strong correlation between drought duration and drought severity in three stations. The drought return period varies depending on the selected marginal distributions and copula functions and, with an increase of the return period, the differences become larger. In addition, the estimated return periods (both co-occurrence and joint) from the best-fitted copulas are the closet to those from empirical distribution. Therefore, it is critical to select the appropriate marginal distribution and copula function to model the hydrological drought frequency and severity. The results of this study can not only help drought investigation to select a suitable probability distribution and copulas function, but are also useful for regional water resource management. However, a few limitations remain in this study, such as the assumption of stationary of runoff series.

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

This paper investigates the impacts of macroeconomic variables, namely consumer price index, exchange rate, minimum loan rate and oil price movement, on the financials and the property & construction stock index return in the Stock Exchange of Thailand (SET). The monthly data is collected from January 2004 to November 2016, covering 155 observations. We employ a copula based SUR regression as a tool for this study. Ten copula functions are considered in this regression and the best copula function is selected based on Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC). The estimated results show that Gumbel 270 copula is the most appropriate function for being the linkage between the marginal distributions of residuals of financials sector and property & construction sector equations. In addition, the marginal distribution is also tested, and the result shows that normal distribution is the best fit for the marginal distribution for both financials and property & construction equations. Our results suggest that the exchange rate can exert significant impact on both sectors. The dependency parameter also suggests that dependency between financials sector and property & construction sector is negative, and very low dependency, meaning when the impact of macroeconomic variables in one of these two sectors, it just has a little effect to another one sector.

Models of diameter and height distributions play an important role in forest mensuration and inventory, and many models and fitting methods are available for fitting these distributions separately. Very few models and methods appropriate for fitting the joint distribution of diameters and heights are available. The S BB and distributions obtained using Plackett's method are the only appropriate models published in the forestry literature to date. A bivariate joint distribution can be represented in terms of its underlying marginals and a copula function that models the dependency structure. A copula is a function that joins or “couples” a multivariate distribution function to its one-dimensional marginal distributions. The copula approach provides a general way of constructing bivariate distributions, including previously used bivariate models. The basics of the copula representation, the most useful copula functions, and maximum likelihood estimation (MLE) of the copula function are introduced. The emphasis of this article is on the choice of copula function(s) appropriate for modeling the joint distribution of tree diameters and heights. Using an empirical dataset of 102 Chinese fir (Cunninghamia lanceolata Lamb.) plantation plots, we fit each of the copula functions that we considered to each sample-plot joint distribution of tree diameters and heights. A two-stage MLE approach with a nonparametric (empirical) approach to the marginal distributions is adopted. In this case study the normal copula proves to be the best for modeling the joint distribution (dependence) of tree diameters and heights.

Karmakar and Simonovic (2008)describe the methodology of assigning appropriate marginal distributions for three flood characteristics. It is found that the gamma distribution is best fitted for peak flow (P), and a nonparametric distribution from the orthonormal series method best fits to volume (V) and duration (D), based on the root mean square error, Akaike information criterion and Bayesian information criteria. In addition, the chi‐square test is performed to check the significance of fitness. In this paper, a methodology is developed to derive bivariate joint distributions of the flood characteristics using the concept of copulas, considering a set of parametric and nonparametric marginal distributions forP, VandDto mathematically model the correlated structure among them. In the conventional method of flood frequency analysis, the marginal distribution functions of peak flow, volume and duration are assumed to follow some specific parametric distribution function. The concept of copulas relaxes the restriction of traditional flood frequency analysis by selecting marginals from different families of probability distribution functions for flood characteristics. The present study performs a better selection of marginal distribution functions for flood characteristics by parametric and nonparametric estimation procedures, and demonstrates how the concept of copulas may be used for establishing a joint distribution function with mixed marginal distributions. The results obtained are useful for hydrologic design and planning purposes. The methodology is demonstrated with 70 years of stream flow data of Red River at Grand Forks of North Dakota, USA.

Joint probability distribution of winds and waves from wave simulation of 20 years (1989-2008) in Bohai Bay

Groundwater is an essential water resource in the Yarkant River Basin Irrigation District, which is the largest oasis in Xinjiang, China. This study used a novel approach to analyze the relationship between groundwater depth and three driving factors by developing Copula Functions from monthly time series data collected from 16 monitoring wells. More specifically, marginal distribution and joint distribution functions were established, and the conditional probabilities for three data ranges were calculated using both two- and three-dimensional Copula Functions. The developed statistical models showed that groundwater exploitation, runoff, and surface water irrigation significantly affected groundwater depth. The most significant effect on water table declines was associated with groundwater exploitation and lagged 1-month behind the groundwater withdrawals. The amount of runoff and irrigation were both inversely related to water table depth due to groundwater recharge. Frank Copula Functions were found to produce the best fit to the joint distribution of the variables and were used herein, allowing for the establishment of a detailed probability distribution of the change in groundwater depth under the combined effect of multiple controlling factors. The results provided key insights into how irrigation and groundwater withdrawals can be adjusted to effectively manage groundwater resources.

Contemporary financial risk management is significantly based on the analysis of time series of returns. One of the most significant errors frequently committed by analysts is the predominant use of normal distributions when it is clear that the returns are not normal. Copula models and models for non-normal multivariate distributions provide new tools to solve the problem because the obtained results are immediately applicable in portfolio management, option pricing and measuring risk without assuming normality. Therefore, both a theoretician and a practitioner are interested in multivariate models for returns and copula functions. The copula function models provide an effective and interesting technique of constructing multivariate distribution starting from marginal ones. Due to Sklar\'s result established in 1959, we can present any multivariate distribution with a help of corresponding marginal distributions and a selected copula function. In this work we present an application of copula function to construct multivariate conditional distributions of times series. In the last part of this paper dynamic models such as DCC-MVGARCH and conditional copula are analyzed. Moreover, we also present an application of bootstrap in the context of copula function. This work is appended by examples showing practical application of our work.

Partially parametric techniques for multiple imputation

In survival studies when the event times are dependent, performing of the analysis by using of methods based on independent assumption, leads to biased. In this paper, using copula function and considering the dependence structure between the event times, a parametric joint distribution has made fitting to the events, and the effective factors on each of these events would be determined. This retrospective cohort study was conducted from March 2003 to March 2007. The data collected from 256 patients with gastric cancer who underwent surgery and that the event time of the two outcomes of death and recurrence for them was recorded. Akaike Information Criterion (AIC) was used to determine of suitable parametric models. Moreover, applying copula function with regard to the relationships between the events, the effect of the risk factors of each of the two outcomes was determined. The data analysis was done using R2.12.1 software. According to the AIC criterion, the Weibull distribution had the best fitting in both of the event times. The median times for recurrence and survival of the patients were estimated 20.2 and 28.1 months respectively. Furthermore, with a fitting of Weibull distribution to the two event times using Clayton copula function, the variables of gender, tumor size and tumor pathological stage on survival, and tumor size and tumor pathological stage on recurrence were significant (P<0.001). Applying copula function for determining specific risk factors of the semi-competing events produces suitable results opposite the common methods which are based on independent assumption of the events.

Climate and climate related events are still taking the interests of societies as it has a huge impact on live life, energy, economy and sustainability. Meteorological parameters such as precipitation, temperature, humidity provide significant information to humanity to adopt with nature. Within this scope, the aim of this study is to analyze the spatio-temporal variability of mean and maximum temperature, mean relative humidity and total precipitation series and obtain the best fitted probability distributions of them for Karaman, Konya and Ankara Regions of Turkiye considering annual scale by obtaining required data from General Directorate of Meteorology of Turkiye. The spatial distribution of selected meteorological variables has been analyzed by Inverse Distance Weighting (IDW) technique and possible temporal variability of these variables have been examined by the trend tests of Mann Kendall (MKT), Spearman’s Rho (SRT), Sen’ Slope(SSE). The best fitted distributions of parameters have been analyzed by utilizing the most commonly used probability distributions of Normal, Lognormal, Logistic, Gamma, Weibull, Gumbel, Generalized Extreme Value considering their performance in a set of goodness of fit tests which are Kolmogorov-Smirnov, Anderson Darling, Chi Square, Cramer Von Mises, Akaike Information Criterion, Bayesian Information Criterion and Maximum Likelihood Estimations. The trend results of MKT and SRT have demonstrated that although there has been a variety of trend types (increasing or decreasing) in data series, a few times these trends have been significant when considering α=0.01 significant level while SSE results have been generally detected with increasing trend slopes. The best-fit distribution results have shown that generally Generalized Extreme Value, Logistics and Gamma distributions have been detected as the best fitted distributions while Lognormal, Gumbel and Weibull have been detected with less numbers. The results of this study are expected to be beneficial for climate change action and water resources planning and urban management areas as the study presents significant findings.

For smart electricity meter with multiple performance characteristics (PCs) with coupling relationships because of amounts of components experiencing multiple deteriorating processes, we develop a multivariate degradation modeling method via vine copulas to estimate the reliability of products with multiple PCs reflecting degradation states. In the multivariate model, pair‐copula construction and vine graphical representation are used to describe the mutual relationship among PCs to overcome the lack of multivariate copula in high‐dimensional cases. Each PC model of smart electricity meter is built by using drift Brownian motion to describe degradation processes of each PC on the basis of degradation mechanism analysis, and parameters are estimated by using likelihood estimation method. The Pearson correlation coefficient, Kendall's τ and product information are used to analyze correlation among those PCs. Furthermore, on the basis of conditional probability theory, the vine graph is used to construct a multivariate copula which can be decomposed into pair copulas. Akaike information criterion principle is utilized to choose the forms of pair‐copula functions in the correlation model. Finally, the reliability joint distribution of all PCs of smart electricity meter is obtained with combining all PCs' marginal distributions and copula functions. Copyright © 2016 John Wiley &amp; Sons, Ltd.

The frequency‐direction spectra of wind‐generated surface gravity waves are summarized in terms of the spectral peak frequency and the stage of wave development (ratio of wind speed at 10 m height to wave phase speed at the spectral peak frequency). The spectra exhibit an enhanced peak, an equilibrium range at frequencies higher than twice the peak, and increased directional spreading at lower and higher frequencies than the peak. Five parameters completely describe the spectra: four are functions of the stage of wave development and the fifth depends on relative (to the peak) frequency. The spectral formulae provided apply in intermediate and deep water. Marine and offshore engineers will find the formulae useful for design and operational calculations. They require only the wind vector and the spectral peak frequency as input.

We propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions. A joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions. This class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable, and its circumvention of the “curse of dimensionality” associated with purely nonparametric multivariate distributions. We show that the plug-in sieve maximum likelihood estimators (MLEs) of all smooth functionals, including the finite-dimensional copula parameters and the unknown marginal distributions, are semiparametrically efficient, and that their asymptotic variances can be estimated consistently. Moreover, prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve further efficiency gains. Two such cases are studied: (a) the marginal distributions are equal but otherwise unspecified, and (b) some but not all marginal distributions are parametric. Monte Carlo studies indicate that the sieve MLEs perform well in finite samples, especially when prior information on the marginal distributions is incorporated.

This paper explores a nonparametric approach to regression modeling based on copula functions, which overcomes the limitations of classical linear models. The study analyzes the advantages of the copula approach, including the ability to model complex nonlinear, asymmetric, and tail dependencies, as well as the separation of dependence structure and marginal distributions. The main classes of copulas (elliptical and Archimedean), their properties, and their application in bivariate models are discussed. Special attention is paid to practical examples of copula applications in economics and finance. The advantages of the approach, such as flexibility and universality, are highlighted, along with its limitations related to computational complexity and data quality requirements.