Pair-copula Construction Research Articles

Modeling multivariate dependence by nonparametric pair-copula construction in composite system reliability evaluation

The correlated variation of bus loads, wind powers, etc., has a significant impact on the power system operation risk. The construction of the accurate dependence model becomes vital in the field of power system reliability evaluation. Most currently used methods in estimating the multivariate joint probability density function (PDF) may encounter obstacles such as modeling accuracy and the curse of dimensionality, especially in the high-dimensional case. To address such problems, a nonparametric pair-copula construction (NPCC), which decomposes the joint PDF into a product of marginal PDFs and a set of bivariate copula (also called pair-copula) densities based on graph theoretic algorithm, is used in this paper to achieve an accurate modeling of the multivariate correlation. To construct a unified framework of nonparametric estimation, the marginal PDFs and the bivariate copula densities are both estimated in a data-driven mode. Moreover, a PDF transformation method is also proposed in estimating the bivariate copula densities, aiming to avoid the problem that the distribution range of the transformed variables in the pair-copula densities exceeds its feasible domain. The performance of the proposed NPCC is verified by a modified version of IEEE-RTS79 with complex correlation among bus loads and wind powers.

Probabilistic Load Flow Approach Considering Dependencies of Wind Speed, Solar Irradiance, Electrical Load and Energy Exchange with a Joint Probability Distribution Model

The modelling of stochastic feed-ins and demands becomes essential for transmission grid operation and planning due to the extension of renewable energy sources (RES). Neglecting the correlation between uncertain variables and/or oversimplifying the distribution through the assumption of Normal distributions leads to the inaccurate determination of future network states. Therefore, the uncertainties need to be accurately modelled in order to be used in a probabilistic load flow approach. This paper analyses the characteristics of wind speed and solar irradiance for different locations throughout countries and models the dependencies between them. In addition, the total electrical load and the energy exchange between neighbouring countries are analysed. All of these uncertainties are modelled together in a high-dimensional joint probability distribution using pair-copula constructions. The model is applied to generate samples and determine the probability of extreme events, e.g. high RES production and low demand. The probability for rather high load (>65 GW) and low RES production with wind speed less than 3 m/s and solar irradiance less than 100 W m ² at 90% of all stations is e.g. 0.064%. In addition, the model is integrated in a probabilistic load flow approach in order to analyse the German transmission grid for a future scenario of the year 2025. With the copula, samples are generated as an input for the Monte Carlo simulation approach. The approach enables the assessment of planned HVDC lines. When considering the HVDC lines, the load on the AC lines can be significantly reduced.

Multivariate Quantile Regressions and EVT Approach to Expected Shortfall Estimation

This paper presents a generalisation of quantile regressions to the multivariate case and its possible applications to Expected Shortfall estimation. The suggested methodology leverages on Pair-Copula Construction (PCC) and Extreme Value Theory (EVT). We extend the quantile regression model to the multivariate case making use of PCC. The model we develop allows us to estimate the quantile of a variable given the contemporaneous state of exogeneous variables. That is, we estimate the q% quantile of Y given the values of X1 ... Xn. We then derive the corresponding Expected Shortfall using Extreme value Theory.

A Markov-switching COGARCH approach to cryptocurrency portfolio selection and optimization

Blockchain is a new technology slowly integrating our economy with cryptocurrencies such as Bitcoin and many more applications. Bitcoin and other versions of it (known as Altcoins) are traded everyday at various cryptocurrency exchanges and have drawn the interest of many investors. These new types of assets are characterized by wild swings in prices, and this can lead to large swings in profit and losses. To respond to these dynamics, cryptoinvestors need adequate tools to guide them through their choice of portfolio selection and optimization. Bitcoin returns have shown some form of regime change, suggesting that regime-switching models could more adequately capture the volatility dynamics. This paper presents a two-state Markov-switching COGARCH-R-vine (MSCOGARCH) model for cryptocurrency portfolio selection and compares the performance to the single-regime COGARCH-R-vine (COGARCH). The findings here are in line with the literature where MSCOGARCH outperforms the single-regime COGARCH with regard to the expected shortfall risk. The COGARCH specifications here capture the structural breaks and heavy tailness within each state of the Markov switching in order to achieve a minimal risk and a maximum return. The flexibility of R-vine copula allows adequate bivariate copula selection for each pair of cryptocurrencies to achieve suitable dependence structure through pair-copula construction architecture.

Copula-based multivariate flood probability construction: a review

Basin perspective hydrology and hydraulic water-related queries often demanding an accurate estimation of flood exceedance probabilities or return periods for assessing hydrologic risk. The research on the advancements of flood probability modelling contributed to reduction of flood risk, damage property and human life losses associated with the occurrence of flood events. Higher degree of uncertainty and complex flood dependence structure did not facilitate for their accurate prediction through deterministic approaches, which often demand a probability distribution framework. Unreliability of univariate frequency analysis under parametric or non-parametric framework would be an attribute for underestimation or overestimation of flood risk. Multivariate distribution framework facilitating a comprehensive understanding of flood structure for various possible occurrence combinations among the flood-related random vectors (i.e. flood peak flow, volume and duration). In this literature, copula function is recognized as a highly flexible tool for establishing multivariate joint dependency and their associated return periods in comparison with traditional multivariate functions. The incorporation of vine or pair-copula constructions (or PCC) further exaggerated the efforts of higher dimension copula construction, in terms of precision level in their estimated quantiles, under the minimum information concept. This review explored the efficacy of copula-based methodology for tackling multivariate design problems and can be used as a guideline for water practioner and hydrologist.

A Novel Fault Prediction Method of Wind Turbine Gearbox Based on Pair-Copula Construction and BP Neural Network

Among the various maintenance technologies of wind turbines, online fault prediction technology is a kind of more cost-effective and reliable method. It may also be the most promising method for wind turbines with potential mechanical faults. SCADA data-based online condition monitoring technology has become a hot spot in current researches. Therefore, a novel fault prediction method based on the Pair-Copula model is proposed in this study. First, the conditional mutual information method is introduced to screen out useful variables from a number of variables. Then aiming at the limitation that the conventional Copula model can only deal with two-dimensional variables, the Pair-Copula model is introduced. In addition, the complexity of the prediction model and the dimension of the input variables are greatly reduced by the Pair-Copula model. So, the BP neural network is selected to complete the prediction model. A combined model based on BP neural network and Pair-Copula model is proposed. In order to solve the problem that the conventional Pair-Copula model cannot process real-time data which must be required in fault prediction, a kind of improved Pair-Copula model combined with the kernel density estimation is used to calculate the real-time data. Finally, the proposed method is validated with real data from a 1.5 MW wind turbine, and the effectiveness is confirmed.

Estimation of urban arterial travel time distribution considering link correlations

ABSTRACT This study proposes a pair-copula construction approach to estimate the probability distribution of urban arterial travel times from link travel time distributions (TTDs). First, link correlations are investigated and empirical evidence reveals that adjacent link travel times present multiple relationships. Then, such relationships are captured by a copula mixture model in each pair-copula associated with two connected links. Finally, the arterial TTD is derived as the product of the pair-copulas and link TTDs. The proposed approach is validated with Radio Frequency Identification Data collected from an urban arterial including four links in Nanjing, China. The results indicate that the approach can dynamically capture the positively correlated, negatively correlated, and uncorrelated relationships between link travel times. The considered competing approaches are also used to benchmark the proposed approach, and comparison results confirm the effectiveness and accuracy of the approach, especially when multistate distributions, complex correlations and limited data sample are encountered.

Vine copulas structures modeling on Russian stock market

Pair-copula constructions have proven to be a useful tool in statistical modeling, particularly in the field of finance. The copula-based approach can be used to choose a model that describes the dependence structure and marginal behaviour of the data in efficient way, but is usually applied to pairs of securities. In contrast, vine copulas provide greater flexibility and permit the modeling of complex dependency patterns using the rich variety of bivariate copulas which may be arranged and analysed in a tree structure. However, the number of possible configurations of a vine copula grows exponentially as the number of variables increases, making model selection a major challenge in development. So, to learn the best possible model, one has to identify the best possible structure, which necessitates identifying the connections between the variables and selecting between the multiple bivariate copulas for each pair in the structure. This paper features the use of regular vine copulas in analysis of the co-dependencies of four major Russian Stock Market securities such as Gazprom, Sberbank, Rosneft and FGC UES, represented by the RTS index. For these stocks the D-vine structures of bivariate copulas were constructed, which models are described by Gumbel, Student, BB1and BB7 copulas, and estimates of their parameters were obtained. Computer simulations showed a high accuracy of the approximation of the explored data by D-vine structure of bivariate copulas and the effectiveness of our approach in general.

Estimation of urban route travel time distribution using Markov chains and pair-copula construction

Route travel time information is essential for better travel planning, traffic guidance and congestion avoidance. An available way of representing such information is the estimation of travel time distribution (TTD). However, most methods both in literature and practice prefer to estimate route mean travel times rather than TTDs, especially with less consideration about the spatiotemporal correlations between adjacent links. This study develops a framework of estimating urban route TTD based on Markov chain approach and pair-copula construction with emphasis on capturing the dependence in time and space. The proposed method is validated with Radio Frequency Identification Data collected from an urban arterial in Nanjing, China. The results indicate that the proposed method can dynamically capture the positively correlated, negatively correlated, and uncorrelated relationships between adjacent link travel times. Moreover, the performance of the proposed method produces the least deviation from the route empirical distributions, compared to the considered competing methods.

Physical Understanding of Human-Induced Changes in U.S. Hot Droughts Using Equilibrium Climate Simulations

AbstractAlthough the link between droughts and heat waves is widely recognized, how climate change affects this link remains uncertain. Here we assess how, and by how much, human-induced climate change affects summertime hot drought compound events over the contiguous United States. Results are derived by comparing hot drought statistics in long simulations of a coupled climate model (CESM1) subjected to year-1850 and year-2000 radiative forcings. Within each climate state, a strong and nonlinear dependency of heat-wave intensity on drought severity is found in water-limited regions of the southern Great Plains and southwestern United States whereas heat-wave intensity is found to be insensitive to drought severity in energy-limited regions of the northern and/or northeastern United States. Applying a statistical model that is based on pair-copula constructions, we find that anthropogenic warming leads to enhanced soil moisture–temperature coupling in water-limited areas of the southern Great Plains and/or southwestern United States and consequently amplifies the intensity of extreme heat waves during severe droughts. This strengthened coupling accounts for a substantial fraction of rising temperature extremes related to the long-term climate change in CESM1, highlighting the importance of changes in land–atmosphere feedback in a warmer climate. In contrast, coupling effects remain weak and largely unchanged in energy-limited regions, thereby yielding no appreciable contribution to heat-wave intensification over the northern and/or northeastern United States apart from the long-term warming effects.

A Copula-Based GLMM Model for Multivariate Longitudinal Data with Mixed-Types of Responses

We propose a copula-based generalized linear mixed model (GLMM) to jointly analyze multivariate longitudinal data with mixed types, including continuous, count and binary responses. The association of repeated measurements is modelled through the GLMM model, meanwhile a pair-copula construction (D-vine) is adopted to measure the dependency structure between different responses. By combining mixed models and D-vine copulas, our proposed approach could not only deal with unbalanced data with arbitrary margins but also handle moderate dimensional problems due to the efficiency and flexibility of D-vines. Based on D-vine copulas, algorithms for sampling mixed data and computing likelihood are also developed. Leaving the random effects distribution unspecified, we use nonparametric maximum likelihood for model fitting. Then an E-M algorithm is used to obtain the maximum likelihood estimates of parameters. Both simulations and real data analysis show that the nonparametric models are more efficient and flexible than the parametric models.

Approximate multivariate distribution of key performance indicators through ordered block model and pair-copula construction

Key Performance Indicators (KPIs) play an important role in comprehending and improving a manufacturing system. This article proposes a novel method using Ordered Block Model and Pair-Copula Construction (OBM-PCC) to approximate the multivariate distribution of KPIs. The KPIs are treated as random variables in the OBM and studied under the stochastic queuing framework. The dependence structure of the OBM represents the influence flow from system input parameters to KPIs. Based on the OBM structure, the PCC is employed to simultaneously approximate the joint probability density function represented by KPIs and quantify the KPI values. The OBM-PCC model removes the redundant pair-copulas in traditional modeling, at the same time enjoying the flexibility and desirable analytical properties in KPI modeling, thus efficiently providing the accurate approximation. Extensive numerical studies are presented to demonstrate the effectiveness of the OBM-PCC model.

Simplified vine copula models: Approximations based on the simplifying assumption

Vine copulas, or pair-copula constructions, have become an important tool in high-dimensional dependence modeling. Commonly, it is assumed that the data generating copula can be represented by a simplified vine copula (SVC). In this paper, we study the simplifying assumption and investigate the approximation of multivariate copulas by SVCs. We introduce the partial vine copula (PVC) which is a particular SVC where to any edge a $j$-th order partial copula is assigned. The PVC generalizes the partial correlation matrix and plays a major role in the approximation of copulas by SVCs. We investigate to what extent the PVC describes the dependence structure of the underlying copula. We show that, in general, the PVC does not minimize the Kullback-Leibler divergence from the true copula if the simplifying assumption does not hold. However, under regularity conditions, stepwise estimators of pair-copula constructions converge to the PVC irrespective of whether the simplifying assumption holds or not. Moreover, we elucidate why the PVC is often the best feasible SVC approximation in practice.

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.

Stochastic cosmic ray sources and the TeV break in the all-electron spectrum

Despite significant progress over more than 100 years, no accelerator has been unambiguously identified as the source of the locally measured flux of cosmic rays. High-energy electrons and positrons are of particular importance in the search for nearby sources as radiative energy losses constrain their propagation to distances of about 1 kpc around 1 TeV. At the highest energies, the spectrum is therefore dominated and shaped by only a few sources whose properties can be inferred from the fine structure of the spectrum at energies currently accessed by experiments like AMS-02, CALET, DAMPE, Fermi-LAT, H.E.S.S. and ISS-CREAM . We present a stochastic model of the Galactic all-electron flux and evaluate its compatibility with the measurement recently presented by the H.E.S.S. collaboration. To this end, we have MC generated a large sample of the all-electron flux from an ensemble of random distributions of sources. We confirm the non-Gaussian nature of the probability density of fluxes at individual energies previously reported in analytical computations. For the first time, we also consider the correlations between the fluxes at different energies, treating the binned spectrum as a random vector and parametrising its joint distribution with the help of a pair-copula construction. We show that the spectral break observed in the all-electron spectrum by H.E.S.S. and DAMPE is statistically compatible with a distribution of astrophysical sources like supernova remnants or pulsars, but requires a rate smaller than the canonical supernova rate. This important result provides an astrophysical interpretation of the spectrum at TeV energies and allows differentiating astrophysical source models from exotic explanations, like dark matter annihilation. We also critically assess the reliability of using catalogues of known sources to model the electron-positron flux.

Soil Moisture Drought in Europe: A Compound Event of Precipitation and Potential Evapotranspiration on Multiple Time Scales

AbstractCompound events are extreme impacts that depend on multiple variables that need not be extreme themselves. In this study, we analyze soil moisture drought as a compound event of precipitation and potential evapotranspiration (PET) on multiple time scales related to both meteorological drought and heat waves in wet, transitional, and dry climates in Europe during summer. Drought indices that incorporate PET to account for the effect of temperature on drought conditions are sensitive to global warming. However, as evapotranspiration (ET) is moisture limited in dry climates, the use of such drought indices has often been criticized. We therefore assess the relevance of the contributions of both precipitation and PET to the estimation of soil moisture drought. Applying a statistical model based on pair copula constructions to data from FluxNet sites in Europe, we find at all sites that precipitation exerts the main control over soil moisture drought. At wet sites PET is additionally required to explain the onset, severity, and persistence of drought events over different time scales. At dry sites, where ET is moisture limited in summer, PET does not improve the estimation of soil moisture. In dry climates, increases in drought severity measured by indices incorporating PET may therefore not indicate further drying of soil but the increased availability of energy that can contribute to other environmental hazards such as heat waves and wildfires. We therefore highlight that drought indices including PET should be interpreted within the context of the climate and season in which they are applied in order to maximize their value.

Copula approaches for modeling cross-sectional dependence of data breach losses

Many experts claim that cyber risks are correlated, but there is not much supporting empirical evidence. We consider 3327 data breach events from 2005 to 2016 and identify a significant asymmetric dependence of monthly losses in two cross-sectional settings: cross-industry losses in four categories by breach types (hacking, lost electronic device, unintended disclosure and insider breach) and cross-breach type losses in five categories by industries (banking and insurance, government, medical service, retail/other business and educational institution). To identify the method that best fits the dependence structure of the dataset, we implement copula modeling by separating the dependence into pairwise non-zero losses and zero loss arrivals. We model the former by pair copula construction (PCC) allowing for the flexible choice of copula functions, whereas the latter is modeled by Gaussian copula. We illustrate the usefulness of our results in two applications to risk measurement and pricing. Our findings are important for risk managers and actuaries who are designing cyber-insurance policies.

Generalized Additive Models for Pair-Copula Constructions

ABSTRACTPair-copula constructions are flexible dependence models that use bivariate copulas as building blocks. In this article, we extend them with generalized additive models to allow covariates effects. Borrowing ideas from a traditionally univariate context, we let each pair-copula parameter depend directly on the covariates in a parametric, semiparametric, or nonparametric way. We propose a sequential estimation method that we study by simulation, and apply it to investigate the time-varying dependence structure between the intraday returns on four major foreign exchange rates. An R package, scripts reproducing the results in this article, and additional simulation results are provided as supplementary material.

Generation of Time-Coupled Wind Power Infeed Scenarios Using Pair-Copula Construction

Wind power forecasts for an aggregation of wind farms have become pivotal to many user groups, e.g., system operators and market actors. However, deviations from these forecasts impact system stability and energy prices in power systems with a high share of installed wind power. Hence, modeling such deviations with probabilistic wind power forecasting approaches has become more relevant. Most methods account for different look-ahead timestamps separately and neglect the temporal propagation of wind power forecast errors. This paper presents a new approach that applies the so-called pair-copula constructions or vine copulae to generate time-coupled wind power infeed scenarios for an aggregation of wind farms. By using a copula approach, the modeling of the temporal dependence structure can be separated from the modeling of the wind power uncertainty in each timestamp. A D-vine structure is proposed and compared with a C-vine and a Gaussian copula in a case study on Belgian offshore wind farms. Different pair-copula selections and estimation procedures are tested. The evaluation results appear promising, especially for D-vines.

Probabilistic Modeling of Financial Uncertainties

Since the global financial crash, one of the main trends in the financial engineering discipline has been to enhance the efficiency and flexibility of financial probabilistic risk assessments. Creditors could immensely benefit from such improvements in analysis hoping to minimise potential monetary losses. Analysis of real world financial scenarios require modeling of multiple uncertain quantities with a view to present more accurate, near future probabilistic predictions. Such predictions are essential for an informed decision making. In this article, the authors extend Bayesian Networks Pair-Copula Construction (BN-PCC) further using the minimum information vine model which results in a more flexible and efficient approach in modeling multivariate dependencies of heavy-tailed distribution and tail dependence as observed in the financial data. The authors demonstrate that the extended model based on minimum information Pair-Copula Construction (PCC) can approximate any non-Gaussian BN to any degree of approximation. The proposed method has been applied to the portfolio data derived from a Brazilian case study. The results show that the fitting of the multivariate distribution approximated using the proposed model has been improved compared to other previously published approaches.