Multidimensional Copulas Research Articles

Environmental parameter design for jack-up platforms based on FEM-BPNN-JPA coupled approach

This study proposes a novel coupled approach, integrating the non-linear finite element model (FEM), BP neural network (BPNN), and joint probability analysis (JPA), to conduct the ocean environmental parameter design for jack-up platforms. Two bottom constraint methods are involved in the FEM, namely the contact surface model and the hinge constraint model, to conduct the structural response analysis. Subsequently, based on the calculated structural responses from FEM, the BPNN is employed to predict the jack-up response under varying combinations of wind speed, wave height and current velocity. This FEM-BPNN coupled method proves to be an effective and efficient tool for determining the jack-up responses under different environmental load combinations for JPA. Based on the optimal univariate probability distributions of each environmental parameter, multi-dimensional copulas are adopted to construct the conditional and joint probability distributions. These distributions, in turn, yield environmental design parameters under different design criteria and corresponding horizontal displacements of the jack-up platform. Results show that environmental design parameters and corresponding horizontal displacements are smaller compared to those obtained from the univariate probability distribution for the same return period. The FEM-BPNN-JPA coupled approach proves to be effective in estimating environmental design parameters by considering parameter correlations.

A probabilistic approach to provenance studies using whole object elemental composition: Chert (flint) as a case study

Provenancing archeological artefacts and “whole rock” fragments, such as cobbles and pebbles, is important both for studying past societies and reconstructing geomorphological and sedimentary processes. Determining the provenance of archaeological artefacts, for example, is helpful in the study of various aspects of human society such as mobility, trade routes, raw material procurement strategies and societal development, whereas whole rock fragments are useful indicators of sediment transportation routes and landscape evolution. In this paper, we introduce a novel probabilistic approach for exploring the limitations and potential of provenance studies utilizing trace element composition of archaeological artefacts and bulk rock fragments through a set of statistical analyses, demonstrated using a dataset of chert (flint) samples from the eastern Mediterranean as a case study. Because our general goal is to identify sediment provenance at the highest possible resolution, and attribute it to specific outcrops, the robustness of such classifications must consider the inherent variability of the studied samples, which poses a complex problem when an entire suite of elements is considered.In the current study, a scheme for evaluating the possible resolution and potential of provenance studies of flint artefacts is illustrated by applying a set of statistical techniques. The data is investigated across the different labelling hierarchy using Spearman correlation and the Kruskal-Wallis test, and a synthetic dataset that follows the statistical properties of the original data is generated using multidimensional copulas that mimic the correlation of the different elements within each subset. The synthetic dataset is then iteratively sampled, filtered and passed into a classifier that is evaluated using a set of classification metrics (e.g., accuracy, recall, etc.). Using this probabilistic approach, the power of supervised classification is determined for the different hierarchy levels and classes based on their distributions, and the number of specimens required for distinguishing sources at different levels (e.g., era, age, formation and site) is evaluated. We show that 10–20 specimens are needed to determine the geological era and age of the strata from which flint was collected with reasonable certainty (accuracy >0.9), whereas specific outcrops cannot be reliably identified even when a large number of specimens are considered. This approach has the potential to enable the classification of items of unknown origin and the associated uncertainty by comparing them against a dataset of possible sources that can be expanded over time.

Meteorological and Hydrological Drought Risk Assessment Using Multi-Dimensional Copulas in the Wadi Ouahrane Basin in Algeria

A persistent precipitation deficiency (meteorological drought) could spread to surface water bodies and produce a hydrological drought. Meteorological and hydrological droughts are thus closely related, even though they are separated by a time lag. For this reason, it is paramount for water resource planning and for drought risk analysis to study the connection between these two types of drought. With this aim, in this study, both meteorological and hydrological drought were analyzed in the Wadi Ouahrane Basin (Northwest Algeria). In particular, data from six rainfall stations and one hydrometric station for the period 1972–2018 were used to evaluate the Standardized Precipitation Index (SPI) and the Standardized Runoff Index (SRI) at multiple timescales (1, 2, 3, 4, …, 12 months). By means of a copula function, the conditional return period for both types of drought was evaluated. Results evidenced that runoff is characterized by high level of temporal correlation in comparison to rainfall. Moreover, the composite index JDHMI (Joint Deficit Hydro-meteorological Index) was evaluated. This index is able to reflect the simultaneous hydrological and meteorological behavior at different timescales of 1–12 months well and can present the probability of a common hydrological and meteorological deficit situation more accurately and realistically compared to precipitation or runoff-based indicators. It was found that, over the analyzed basin, the average severity of combined hydro-meteorological drought (JDHMI) was 10.19, with a duration of 9 months and a magnitude of 0.93.

Assessing hydrologic drought risk using multi-dimensional copulas: case study in Karkheh River basin

Assessing hydrologic drought risk using multi-dimensional copulas: case study in Karkheh River basin

Construction of Copulas with Hairpin Support

Consider a nondecreasing homeomorphisms f defined on [0, 1] such that \(f(x)<x \) for all \(x\in ]0,1[\). In this paper, we provide necessary and sufficient conditions for such f to be part of a \({\mathcal {C}}\)-hairpin that concentrates the mass of a bivariate copula. In addition, we study when copulas of this kind come from modular functions. Finally, under certain conditions, we give a multidimensional method that generalizes the bivariate case and allows to construct extreme points in the set of multidimensional copulas.

Comprehensive drought characteristics analysis based on a nonlinear multivariate drought index

It is vital to identify drought events and to evaluate multivariate drought characteristics based on a composite drought index for better drought risk assessment and sustainable development of water resources. However, most composite drought indices are constructed by the linear combination, principal component analysis and entropy weight method assuming a linear relationship among different drought indices. In this study, the multidimensional copulas function was applied to construct a nonlinear multivariate drought index (NMDI) to solve the complicated and nonlinear relationship due to its dependence structure and flexibility. The NMDI was constructed by combining meteorological, hydrological, and agricultural variables (precipitation, runoff, and soil moisture) to better reflect the multivariate variables simultaneously. Based on the constructed NMDI and runs theory, drought events for a particular area regarding three drought characteristics: duration, peak, and severity were identified. Finally, multivariate drought risk was analyzed as a tool for providing reliable support in drought decision-making. The results indicate that: (1) multidimensional copulas can effectively solve the complicated and nonlinear relationship among multivariate variables; (2) compared with single and other composite drought indices, the NMDI is slightly more sensitive in capturing recorded drought events; and (3) drought risk shows a spatial variation; out of the five partitions studied, the Jing River Basin as well as the upstream and midstream of the Wei River Basin are characterized by a higher multivariate drought risk. In general, multidimensional copulas provides a reliable way to solve the nonlinear relationship when constructing a comprehensive drought index and evaluating multivariate drought characteristics.

Copula based flexible modeling of associations between clustered event times.

Multivariate survival data are characterized by the presence of correlation between event times within the same cluster. First, we build multi-dimensional copulas with flexible and possibly symmetric dependence structures for such data. In particular, clustered right-censored survival data are modeled using mixtures of max-infinitely divisible bivariate copulas. Second, these copulas are fit by a likelihood approach where the vast amount of copula derivatives present in the likelihood is approximated by finite differences. Third, we formulate conditions for clustered right-censored survival data under which an information criterion for model selection is either weakly consistent or consistent. Several of the familiar selection criteria are included. A set of four-dimensional data on time-to-mastitis is used to demonstrate the developed methodology.

Goodness-of-fit tests for multi-dimensional copulas: Expanding application to historical drought data

The question of how to choose a copula model that best fits a given dataset is a predominant limitation of the copula approach, and the present study aims to investigate the techniques of goodness-of-fit tests for multi-dimensional copulas. A goodness-of-fit test based on Rosenblatt's transformation was mathematically expanded from two dimensions to three dimensions and procedures of a bootstrap version of the test were provided. Through stochastic copula simulation, an empirical application of historical drought data at the Lintong Gauge Station shows that the goodness-of-fit tests perform well, revealing that both trivariate Gaussian and Student t copulas are acceptable for modeling the dependence structures of the observed drought duration, severity, and peak. The goodness-of-fit tests for multi-dimensional copulas can provide further support and help a lot in the potential applications of a wider range of copulas to describe the associations of correlated hydrological variables. However, for the application of copulas with the number of dimensions larger than three, more complicated computational efforts as well as exploration and parameterization of corresponding copulas are required.

Copula analysis of mixture models

Contemporary computers collect databases that can be too large for classical methods to handle. The present work takes data whose observations are distribution functions (rather than the single numerical point value of classical data) and presents a computational statistical approach of a new methodology to group the distributions into classes. The clustering method links the searched partition to the decomposition of mixture densities, through the notions of a function of distributions and of multi-dimensional copulas. The new clustering technique is illustrated by ascertaining distinct temperature and humidity regions for a global climate dataset and shows that the results compare favorably with those obtained from the standard EM algorithm method.

Statistical characterization of temporal structure of storms

The authors present a statistical procedure to estimate the probability distributions of storm characteristics. The approach uses recent advances in stochastic hydrological modeling. The temporal dynamics of rainfall are modeled via a reward alternating renewal process that describes wet and dry phases of storms. In particular, the wet phase is modeled as a rectangular pulse process with dependent random duration and intensity; the global dependence structure is described using multidimensional copulas. The marginal distributions are described by Generalized Pareto laws. The authors derive both the storm volume statistics and the rainfall volume distribution within a fixed temporal window preceding a storm. Based on these results, they calculate the antecedent moisture conditions. The paper includes a thorough discussion of the validity of the assumptions and approximations introduced, and an application to actual rainfall data. The models presented here have important implications for improved design procedures of water resources and hydrologic systems.