Use Of Copula Functions Research Articles

Employing the generalized Pareto distribution to analyze extreme rainfall events on consecutive rainy days in Thailand's Chi watershed: implications for flood management

Abstract. Extreme rainfall events in the Chi watershed of northeastern Thailand have significant implications for the safe and economic design of engineered structures and effective reservoir management. This study investigates the characteristics of extreme rainfall events in the watershed and their implications for flood risk management. We apply extreme value theory to historical maximum cumulative rainfall data for consecutive rainy days from 1984 to 2022. The generalized Pareto distribution (GPD) was used to model the extreme rainfall data, with the parameters estimated using maximum likelihood estimation (MLE) and linear moment estimation (L-ME) methods based on specific conditions. The goodness-of-fit tests confirm the suitability of the GPD for the data, with p values exceeding 0.05. Our findings reveal that certain regions, notably Udon Thani, Chaiyaphum, Maha Sarakham, Tha Phra Agromet., Roi Et, and Sisaket provinces, show the highest return levels for consecutive 2 d (CONS-2) and 3 d (CONS-3) rainfall. These results underscore the heightened risk of flash flooding in these regions, even with short periods of continuous rainfall. Based on our findings, we developed 2D return level maps using the Q-geographic information system (Q-GIS) program, providing a visual tool to assist with flood risk management. The study offers valuable insights for designing effective flood management strategies and highlights the need for considering extreme rainfall events in water management and planning. Future research could extend our findings through spatial correlation analysis and the use of copula functions. Overall, this study emphasizes the importance of preparing for extreme rainfall events, particularly in the era of climate change, to mitigate potential flood-related damage.

Classification of hyperspectral images with copulas

In the last decade, supervised learning methods for the classification of remotely sensed images (RSI) have grown significantly, especially for hyper-spectral (HS) images. Recently, deep learning-based approaches have produced encouraging results for the land cover classification of HS images. In particular, the Convolutional Neural Networks (CNN) and Recurrent Neural Networks (RNN) have shown good performance. However, these methods suffer for the problem of the hyperparameter optimization or tuning that requires a high computational cost; moreover, they are sensitive to the number of observations in the learning phase. In this work we propose a novel supervised learning algorithm based on the use of copula functions for the classification of hyperspectral images called CopSCHI (Copula Supervised Classification of Hyperspectral Images). In particular, we start with a dimensionality reduction technique based on Singular Value Decomposition (SVD) in order to extract a small number of relevant features that best preserve the characteristics of the original image. Afterward, we learn the classifier through a dynamic choice of copulas that allows us to identify the distribution of the different classes within the dataset. The use of copulas proves to be a good choice due to their ability to recognize the probability distribution of classes and hence an accurate final classification with low computational cost can be conducted. The proposed approach was tested on two benchmark datasets widely used in literature. The experimental results confirm that CopSCHI outperforms the state-of-the-art methods considered in this paper as competitors.

Structure-to-structure damage correlation for scenario-based regional seismic risk assessment

When performing a regional seismic risk assessment, one is interested in estimating the damage, losses, and, in general, the seismic performance of several structures spatially distributed within a region. In this type of probabilistic risk analysis, it is common to consider the spatial correlation between the ground motion intensities affecting different locations. However, the correlation of damage experienced by different structures conditioned on their levels of intensities, herein referred to as structure-to-structure damage correlation, is usually neglected. This study evaluates the effect of the structure-to-structure damage correlation and proposes a simplified model for incorporating it into regional seismic risk assessment through the use of copula functions. In order to illustrate the influence of the structure-to-structure damage correlation on the regional seismic risk, a case study in San Carlos, California, is analyzed considering and neglecting the structure-to-structure damage correlation. Results show that neglecting the structure-to-structure damage correlation may lead to significant underestimations of large-consequence risks.

Tail Dependence in Financial Markets: A Dynamic Copula Approach

This article is concerned with the study of the tail correlation among equity indices by means of dynamic copula functions. The main idea is to consider the impact of the use of copula functions in the accuracy of the model’s parameters and in the computation of Value-at-Risk (VaR). Results show that copulas provide more sophisticated results in terms of the accuracy of the forecasted VaR, in particular, if they are compared with the results obtained from Dynamic Conditional Correlation (DCC) model.

Применение копул в многомерном анализе обменных курсов на примере развивающихся стран Европы

The work is aimed at modelling and analyzing joint behavior of currency exchange rates in 3 developing European countries – the Check Republic, Hungary, and Poland with the use of Copula functions. The study focuses only on internal relations with no consideration for any additional (external) factors. When building the models, several elliptic and Archimedean Copula families were estimated, and pair-Copula (vine) constructions of different families were considered. The work reveals the most appropriate in terms of consistency to the available data model based on the multivariate elliptical Student Copula. The work estimates the parameters of inherent interconnection narrowness of the currency rates under study based on the selected model built on the daily data in relation to RUB over the period 2007–2017. Besides, the paper considers two approaches to interval forecasting – with the use of correlated multiplier function and wave function which accounts the most likely value range as of the given date. The time-horizon of the study was 30 days. In the conclusion of the work, a comparative analysis of the proposed approaches was carried out, and the comparison was made between the forecasts and actual figures.

Сравнительный анализ использования копула-функций в моделях стохастической границы на примере регионов РФ

Сравнительный анализ использования копула-функций в моделях стохастической границы на примере регионов РФ

Multiterminal Source Coding With Copula Regression for Wireless Sensor Networks Gathering Diverse Data

Efficient data compression at a low processing and communication cost is a key challenge in wireless sensor networks. In this paper, we propose a novel multiterminal source code design, which, contrary to prior work, utilizes both the intra- and the inter-sensor data dependences. The former is exploited by applying simple differential pulse-code modulation followed by arithmetic entropy coding at each distributed encoder. This approach limits the encoding complexity and provides for a flexible design that adapts to the variations in the number of operating sensors. Moreover, we propose a regression method applied at the joint decoder, which aims at leveraging the inter-sensor data dependences. Unlike existing work that focuses on homogeneous data types, the proposed method makes use of copula functions, namely, a statistical model that captures the dependence structure amongst heterogeneous data types. Experimentation using real sensor measurements—taken from the Intel-Berkeley database—shows that the proposed system achieves significant compression improvements compared with the state-of-the-art multiterminal and distributed source coding schemes.

Copula function approaches for the analysis of serial and cross dependence in stock returns

The description of the dynamic behavior of multiple time series represents an important point of departure to obtain accurate forecasts both in economic and financial analysis. We provide a method for the comparison of the out-of-sample performance of portfolios, respectively, ignoring and exploiting serial and cross dependence in stock returns. The serial and cross dependence is modeled using both the classical linear and easy-to-use Vector AutoRegressive and more sophisticated models making use of copula functions. After deriving the classical and copula-based VAR conditional expected returns and covariance, we construct different portfolios and compare them in terms of Sharpe ratio in an out-of-sample period.

Use of copula functions for the reliability of series systems

Purpose– The purpose of this paper is to provide a new method to estimate the reliability of series system by using copula functions. This problem is of great interest in industrial and engineering applications.Design/methodology/approach– The authors introduce copula functions and consider a Bayesian analysis for the proposed models with application to the simulated data.Findings– The use of copula functions for modeling the bivariate distribution could be a good alternative to estimate the reliability of a two components series system. From the results of this study, the authors observe that they get accurate Bayesian inferences for the reliability function considering large samples sizes. The Bayesian parametric models proposed also allow the assessment of system reliability for multicomponent systems simultaneously.Originality/value– Usually, the studies of systems reliability engineering assume independence among the component lifetimes. In the approach the authors consider a dependence structure. Using standard classical inference methods based on asymptotical normality of the maximum likelihood estimators for the parameters the authors could have great computational difficulties and possibly, not accurate inference results, which there is not found in the approach.

Loan default correlation using an Archimedean copula approach: A case for recalibration

Appropriate modelling of loan default correlation capturing the fat tail distributions and non-symmetrical behaviour linked to the sensitivity of the loss correlations is a prerequisite for effective credit risk management, as banks seek to optimally allocate capital. In this study, we provide an insight to the use of copula functions, particularly addressing the key question of why Gaussian copulas caused so much instability during 2007–08. We empirically demonstrate that using an Archimedean copula, particularly the Gumbel, it is more efficient in capturing the top right hand side tail-dependencies, thereby illustrating the impact of fat-tails on non-linear parameters. This finding has significant implications for banks and their capital management requirements, particularly banks employing the Advanced Internal Rate-Based method. This is even more relevant now, with Basel III providing more detailed information as to what constitutes Tier 1, Additional Tier 1 and 2 Capital.

The use of copula functions for modeling the risk of investment in shares traded on world stock exchanges

In this paper the two dimensional model of the investment in shares is presented. The shares prices from five different world stock exchanges (New York, London, Frankfurt, Honk Kong, and Sydney) are examined. The copula functions are used to model the risk of investment. The Hurst threshold exponent derived from the local Detrended Fluctuation Analysis is used to determine the safe investment portfolios with no extreme drops in shares prices. The most important result states that the threshold value is not universal for different markets, however, it is influenced by the subsequent level of market freedom. It was shown, that the level, relatively larger in US, UK, and Australia than in Germany and China, affects the Hurst exponent threshold value.

The use of copula functions for predictive analysis of correlations between extreme storm tides

In this paper we present a method used in quantitative description of weakly predictable hydrological, extreme events at inland sea. Investigations for correlations between variations of individual measuring points, employing combined statistical methods, were carried out. As a main tool for this analysis we used a two-dimensional copula function sensitive for correlated extreme effects. Additionally, a new proposed methodology, based on Detrended Fluctuations Analysis (DFA) and Anomalous Diffusion (AD), was used for the prediction of negative and positive auto-correlations and associated optimum choice of copula functions. As a practical example we analysed maximum storm tides data recorded at five spatially separated places at the Baltic Sea. For the analysis we used Gumbel, Clayton, and Frank copula functions and introduced the reversed Clayton copula. The application of our research model is associated with modelling the risk of high storm tides and possible storm flooding.

The use of copula functions for modeling the risk of investment in shares traded on the Warsaw Stock Exchange

In our work copula functions and the Hurst exponent calculated using the local Detrended Fluctuation Analysis (DFA) were used to investigate the risk of investment made in shares traded on the Warsaw Stock Exchange. The combination of copula functions and the Hurst exponent calculated using local DFA is a new approach. For copula function analysis bivariate variables composed of shares prices of the PEKAO bank (a big bank with high capitalization) and other banks (PKOBP, BZ WBK, MBANK and HANDLOWY in decreasing capitalization order) and companies from other branches (KGHM—mining industry, PKNORLEN—petrol industry as well as ASSECO—software industry) were used. Hurst exponents were calculated for daily shares prices and used to predict high drops of those prices. It appeared to be a valuable indicator in the copula selection procedure, since Hurst exponent’s low values were pointing on heavily tailed copulas e.g. the Clayton one.

Derivatives and Fisher information of bivariate copulas

Data sets with complex relationships between random variables are increasingly studied in statistical applications. A popular approach to model their dependence is the use of copula functions. Our contribution is to derive expressions for the observed and expected information for several bivariate copula families, in particular for the Student’s $$t$$ -copula. Further likelihood derivatives which are required for numerical implementations are computed and a numerically stable implementation is provided in the R-package VineCopula. Using a real world data set of stock returns, we demonstrate the applicability of our approach for the routinely calculation of standard errors. In particular, we illustrate how this prevents overestimating the time-variation of dependence parameters in a rolling window analysis.

Hierarchical structures in the aggregation of premium risk for insurance underwriting

In the valuation of the Solvency II capital requirement, the correct appraisal of risk dependencies acquires particular relevance. These dependencies refer to the recognition of risk diversification in the aggregation process and there are different levels of aggregation and hence different types of diversification. For instance, for a non-life company at the first level the risk components of each single line of business (e.g. premium, reserve, and CAT risks) need to be combined in the overall portfolio, the second level regards the aggregation of different kind of risks as, for example, market and underwriting risk, and finally various solo legal entities could be joined together in a group. Solvency II allows companies to capture these diversification effects in capital requirement assessment, but the identification of a proper methodology can represent a delicate issue. Indeed, while internal models by simulation approaches permit usually to obtain the portfolio multivariate distribution only in the independence case, generally the use of copula functions can consent to have the multivariate distribution under dependence assumptions too. However, the choice of the copula and the parameter estimation could be very problematic when only few data are available. So it could be useful to find a closed formula based on Internal Models independence results with the aim to obtain the capital requirement under dependence assumption. A simple technique, to measure the diversification effect in capital requirement assessment, is the formula, proposed by Solvency II quantitative impact studies, focused on the aggregation of capital charges, the latter equal to percentile minus average of total claims amount distribution of single line of business (LoB), using a linear correlation matrix. On the other hand, this formula produces the correct result only for a restricted class of distributions, while it may underestimate the diversification effect. In this paper we present an alternative method, based on the idea to adjust that formula with proper calibration factors (proposed by Sandström (2007)) and appropriately extended with the aim to consider very skewed distribution too. In the last part considering different non-life multi-line insurers, we compare the capital requirements obtained, for only premium risk, applying the aggregation formula to the results derived by elliptical copulas and hierarchical Archimedean copulas.

Copula-based geostatistical modeling of continuous and discrete data including covariates

It is common in geostatistics to use the variogram to describe the spatial dependence structure and to use kriging as the spatial prediction methodology. Both methods are sensitive to outlying observations and are strongly influenced by the marginal distribution of the underlying random field. Hence, they lead to unreliable results when applied to extreme value or multimodal data. As an alternative to traditional spatial modeling and interpolation we consider the use of copula functions. This paper extends existing copula-based geostatistical models. We show how location dependent covariates e.g. a spatial trend can be accounted for in spatial copula models. Furthermore, we introduce geostatistical copula-based models that are able to deal with random fields having discrete marginal distributions. We propose three different copula-based spatial interpolation methods. By exploiting the relationship between bivariate copulas and indicator covariances, we present indicator kriging and disjunctive kriging. As a second method we present simple kriging of the rank-transformed data. The third method is a plug-in prediction and generalizes the frequently applied trans-Gaussian kriging. Finally, we report on the results obtained for the so-called Helicopter data set which contains extreme radioactivity measurements.

On Default Correlation

This article studies the problem of default correlation. It introduces a random variable called “time-until-default” to denote the survival time of each defaultable entity or financial instrument, and defines the default correlation between two credit risks as the correlation coefficient between their survival times. The author explains why a copula function approach should be used to specify the joint distribution of survival times after marginal distributions of survival times are derived from market information, such as risky bond prices or asset swap spreads. He shows that the current approach to default correlation through asset correlation is equivalent to using a normal copula function. Numerical examples illustrate the use of copula functions in the valuation of some credit derivatives, such as credit default swaps and first-to-default contracts.

On Default Correlation: A Copula Function Approach

This paper studies the problem of default correlation. We first introduce a random variable called time-until-default to denote the survival time of each defaultable entity or financial instrument, and define the default correlation between two credit risks as the correlation coefficient between their survival times. Then we argue why a copula function approach should be used to specify the joint distribution of survival times after marginal distributions of survival times are derived from market information, such as risky bond prices or asset swap spreads. The definition and some basic properties of copula functions are given. We show that the current CreditMetrics approach to default correlation through asset correlation is equivalent to using a normal copula function. Finally, we give some numerical examples to illustrate the use of copula functions in the valuation of some credit derivatives, such as credit default swaps and first-to-default contracts.