A novel copula-based probabilistic model is proposed to establish the temperature difference analysis model for a long-span suspension bridge’s steel box girder. The key idea is to express a ...two-dimensional function of the temperature difference in flat steel box girder by using copulas. The maximum and minimum values of daily temperature difference model was developed using long-terms structural health monitoring data. Then, the correlation between adjacent temperature differences is investigated using five types of copulas. Akaike information criterion (AIC) is used to select an optimal model from five types of copulas, and the optimal joint function (two-dimensional function) for steel box girder’s temperature difference is established. Finally, the structure’s temperature gradient model is extrapolated for the service life of the structure by using Monte Carlo method. Moreover, this paper discusses the temperature gradient models using five types of common copulas and four types of time-varying copulas. The result shows that the t-copula is the optimal function to build the two-dimensional functions for steel box girder’s temperature difference, and the temperature model along the transverse direction can offer useful information that is not available in the design codes.
We derive upper and lower bounds on the expectation of f(S) under dependence uncertainty, that is, when the marginal distributions of the random vector S = (S1,..., Sd) are known but their dependence ...structure is partially unknown. We solve the problem by providing improved Fréchet–Hoeffding bounds on the copula of S that account for additional information. In particular, we derive bounds when the values of the copula are given on a compact subset of 0, 1d, the value of a functional of the copula is prescribed or different types of information are available on the lower dimensional marginals of the copula. We then show that, in contrast to the two-dimensional case, the bounds are quasi-copulas but fail to be copulas if d > 2. Thus, in order to translate the improved Fréchet–Hoeffding bounds into bounds on the expectation of f(S), we develop an alternative representation of multivariate integrals with respect to copulas that admits also quasi-copulas as integrators. By means of this representation, we provide an integral characterization of orthant orders on the set of quasi-copulas which relates the improved Fréchet–Hoeffding bounds to bounds on the expectation of f(S). Finally, we apply these results to compute model-free bounds on the prices of multi-asset options that take partial information on the dependence structure into account, such as correlations or market prices of other traded derivatives. The numerical results show that the additional information leads to a significant improvement of the option price bounds compared to the situation where only the marginal distributions are known.
The accurate understanding of the dependence structure implied by the parametric models studied in statistical and financial literature has drawn growing attention in recent times. In particular, ...tail dependence is crucial in this analysis. We study the general tail dependence function in some of the most common copula models found in literature, in which this function has not been obtained. These models are used by statisticians and practitioners alike. With the general tail dependence function, we cover positive and non-positive tail dependence often overlooked. The use of the general tail dependence function generalises the well known approach of using the survival copula to tackle upper tail dependence. We present relevant results regarding tail dependence related functions. We include a broad guide for bivariate families and Hierarchical Archimedean copulas in dimensions 3 and 4. In the multivariate case we study the Marshall-Olkin copula, examples of models based on Max-Id distributions and Extreme Value copulas among others. In an empirical section we exemplify with real data the usefulness of our results by modelling arbitrary types of tail dependence. Furthermore we show how our approach can yield better estimates of Value at Risk than other standard approaches.
We present a regression technique for data-driven problems based on polynomial chaos expansion (PCE). PCE is a popular technique in the field of uncertainty quantification (UQ), where it is typically ...used to replace a runnable but expensive computational model subject to random inputs with an inexpensive-to-evaluate polynomial function. The metamodel obtained enables a reliable estimation of the statistics of the output, provided that a suitable probabilistic model of the input is available.
Machine learning (ML) regression is a research field that focuses on providing purely data-driven input-output maps, with the focus on pointwise prediction accuracy. We show that a PCE metamodel purely trained on data can yield pointwise predictions whose accuracy is comparable to that of other ML regression models, such as neural networks and support vector machines. The comparisons are performed on benchmark datasets available from the literature. The methodology also enables the quantification of the output uncertainties, and is robust to noise. Furthermore, it enjoys additional desirable properties, such as good performance for small training sets and simplicity of construction, with only little parameter tuning required.
•Data-driven polynomial chaos expansion (PCE) is an effective regression algorithm.•PCE yields both accurate point predictions and estimates of the output statistics.•Point prediction accuracy similar to neural networks and SVMs in considered real data.•Sparse PCE is applicable to high-dimensional feature spaces and small sample sets.•Simple to build, requires little parameter tuning, easy to interpret and portable.
In recent years, copula models are being used in all areas of human endeavors including energy, environment, social, natural and physical sciences. Copulas are the most powerful tools that can model ...dependent structures between various complex correlated variables. In this paper, we specifically examine the development of copula models and their applications in the areas of energy, fuel cells, forestry and environmental sciences. It reviews the latest literature on the types of copula models, including Gumbel, Clayton, Frank, Gaussian, vine, and the theoretical development of a mixture of bivariate and multivariate copula distributions, in terms of both static and dynamic applications. A comparative review of literature is done using ARMA, DCC, and GARCH models relative to copulas.
•Paper review complex dependent structures of correlated variables using copula.•It presents copula application in energy, fuel cells, forestry and environmental sciences.•Reviews literature related to copula and ARMA, DCC, BEKK and GARCH models.•Paper opens new application of copula in forestry, fuel cells and environment.
This contribution presents a concrete example of uncertainty propagation in a stereo matching pipeline. It considers the problem of matching pixels between pairs of images whose radiometry is ...uncertain and modeled by possibility distributions. Copulas serve as dependency models between variables and are used to propagate the imprecise models. The propagation steps are detailed in the simple case of the Sum of Absolute Difference cost function for didactic purposes. The method results in an imprecise matching cost curve. To reduce computation time, a sufficient condition for conserving possibility distributions after the propagation is also presented. Finally, results are compared with Monte Carlo simulations, indicating that the method produces envelopes capable of correctly estimating the matching cost.
•Multivariate Extreme value models based on Gumbel Copulas are used to estimate crash frequency-by-severity from traffic conflict indicators.•Traffic conflict indicators are extracted from videos by ...applying an automated computer vision technique.•The accuracy and precision of crash predictions are not proportional to the number of conflict indicators used in the extreme value models.•Modified Time to Collision (MTTC) and Deceleration Rate to Avoid a Crash (DRAC) is the best combination of indicators for rear-end crash frequency estimation.•A trivariate model with MTTC, DRAC and Delta-V efficiently estimates crash frequency-by-severity.
Traffic conflict techniques are a viable alternative to crash-based safety assessments and are particularly well suited to evaluating emerging technologies such as connected and automated vehicles for which crash data are sparsely available. Recently, the use of multiple traffic conflict indicators has become common in methodological studies, yet it is often difficult to determine which conflict indicators are appropriate given the application context, and the net benefit, in terms of improved crash prediction accuracy, of considering additional conflict indicators. Addressing these concerns, this study investigates the potential benefits of multiple conflict indicators for conflict-based crash estimation models by using a multivariate extreme value modeling framework (with Gumbel-Hougaard copulas) to estimate crash frequency by severity. The selected conflict indicators include Modified Time-To-Collision (MTTC), Deceleration Rate to Avoid a Collision (DRAC), Proportion of Stopping Distance (PSD) and expected post-collision change in velocity (Delta-V). The proposed framework was applied to estimate the total, severe (Maximum Abbreviated Injury Scale ≥ 3; MAIS3+), and non-severe (MAIS < 3) rear-end crash frequencies at three four-legged signalized intersections in Brisbane, Australia. Rear-end traffic conflicts were extracted from video data using state-of-the-art Computer Vision analytics. Results show that the prediction performance improvements are not necessarily proportional to the number of conflict indicators used in extreme value models. MTTC and DRAC, combined with the severity indicator Delta-V, were the most suitable predictors of rear-end crashes at signalized intersections. Results suggest that instead of adding more and more conflict indicators, careful selection of compatible conflict indicators (considering their functional differences and empirical correlations) is the best way to enhance the predictive performance of conflict-based models.
This study examines the characteristics of the risk spillover under extreme market scenarios between the US stock market and precious metals (gold, silver, platinum) and oil using a copula approach ...for tail dependence and conditional value-at-risk (CoVaR) spillover measures. The results indicate asymmetric tail dependence of the US stock market with silver and platinum, profound during market downturns. Gold and oil symmetrically co-move with the US stock market under normal and extreme market scenarios. Silver and platinum most strongly influence US stock market in the downside, while oil does it on the upside. The US stock market most strongly influences oil and silver under both market downturns and upturns. Gold weakly spillover to the US stock market, suggesting that investors can use gold as an equity portfolio diversifier.
•This study examines whether the US stock market spillovers on oil, gold, silver, and platinum under extreme market scenarios.•The US stock market most strongly impacts prices and returns of oil and silver during market downturns and upturns.•Investors can use gold as an equity portfolio diversifier.
In actuarial science, collective risk models, in which the aggregate claim amount of a portfolio is defined in terms of random sums, play a crucial role. In these models, it is common to assume that ...the number of claims and their amounts are independent, even if this might not always be the case. We consider collective risk models with different dependence structures. Due to the importance of such risk models in an actuarial setting, we first investigate a collective risk model with dependence involving the family of multivariate mixed Erlang distributions. Other models based on mixtures involving bivariate and multivariate copulas in a more general setting are then presented. These different structures allow to link the number of claims to each claim amount, and to quantify the aggregate claim loss. Then, we use Archimedean and hierarchical Archimedean copulas in collective risk models, to model the dependence between the claim number random variable and the claim amount random variables involved in the random sum. Such dependence structures allow us to derive a computational methodology for the assessment of the aggregate claim amount. While being very flexible, this methodology is easy to implement, and can easily fit more complicated hierarchical structures.
A multivariate data set, which exhibit complex patterns of dependence, particularly in the tails, can be modelled using a cascade of lower-dimensional copulae. In this paper, we compare two such ...models that differ in their representation of the dependency structure, namely the nested Archimedean construction (NAC) and the pair-copula construction (PCC). The NAC is much more restrictive than the PCC in two respects. There are strong limitations on the degree of dependence in each level of the NAC, and all the bivariate copulas in this construction has to be Archimedean. Based on an empirical study with two different four-dimensional data sets; precipitation values and equity returns, we show that the PCC provides a better fit than the NAC and that it is computationally more efficient. Hence, we claim that the PCC is more suitable than the NAC for hich-dimensional modelling.
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