Reliability-based design optimization of problems with correlated input variables using a Gaussian Copula

The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a transformation from the correlated input random variables into independent standard normal variables. For the transformation with correlated input variables, the two most representative transformations, the Rosenblatt and Nataf transformations, are investigated. The Rosenblatt transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear transformation. Thus, the Nataf transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.     

Stochastic simulation of velocity pulses of near-fault ground motions based on multivariate copula modeling

Due to the featured high-amplitude, long-period pulses in velocity or displacement time–history, the near-fault ground motion may have a stronger destructive effect on engineering structures. Therefore, studying velocity pulses modeling of near-fault ground motions has been an important topic in the earthquake engineering community. However, the variability and probabilistic correlation of pulse model parameters have not received enough attention. To this end, an elegant method for stochastic simulation of velocity pulses of near-fault ground motions is proposed based on the classical Gabor wavelet modeling and probabilistic correlation analysis of associated model parameters by multivariate copula modeling. Especially, an efficient scheme of iteratively determining the optimal copulas pertinent to edges of a complex vine structure is developed, and two parameters relevant to the near-fault ground motion records, i.e., the rupture distance and the strongest pulse orientation, are introduced which provides a physical basis for addressing the dependence structure among variables. For illustrative purposes, the ground accelerograms recorded by near-fault instruments in the 1999 Chi-Chi earthquake are used. It is shown that the multivariate copula modeling enables the efficient representation of probabilistic correlation between model parameters which is consistent with their physical mechanism. The time histories and spectral characteristics of velocity pulses of near-fault ground motions and their stochastic fluctuations with respect to the rupture distance are well simulated. While the empirical linear regression model does not meet the requirements of random vibration analysis and thus cannot guide the safety design of seismic structures.     

Efficient estimation of a semiparametric dynamic copula model

A new semiparametric dynamic copula model is proposed where the marginals are specified as parametric GARCH-type processes, and the dependence parameter of the copula is allowed to change over time in a nonparametric way. A straightforward two-stage estimation method is given by local maximum likelihood for the dependence parameter, conditional on consistent first stage estimates of the marginals. First, the properties of the estimator are characterized in terms of bias and variance and the bandwidth selection problem is discussed. The proposed estimator attains the semiparametric efficiency bound and its superiority is demonstrated through simulations. Finally, the wide applicability of the model in financial time series is illustrated, and it is compared with traditional models based on conditional correlations.     

Partial correlation with copula modeling

We propose a new partial correlation approach using Gaussian copula. Our empirical study found that the Gaussian copula partial correlation has the same value as that which is obtained by performing a Pearson’s partial correlation. With the proposed method, based on canonical vine and d-vine, we captured direct interactions among eight histone genes.     

Robust estimators and tests for bivariate copulas based on likelihood depth

Estimators and tests based on likelihood depth for one-parametric copulas are given. For the Gaussian and Gumbel copulas, it is shown that the maximum depth estimators are biased. They can be corrected and the new estimators are robust against contamination. For testing, simplicial likelihood depth is considered. Because of the bias of the maximum depth estimator, simplicial likelihood depth is not a degenerated U-statistic so that easily asymptotic α-level tests can be derived for arbitrary hypotheses. Tests are in particular investigated for the one-sided alternatives. Simulation studies for the Gaussian and Gumbel copulas show that the power of the first test is rather good, but the latter one has to be improved, which is also done here. The new tests are robust against contamination.     

国内外期货市场相关性研究

利用copula函数和Kendall tau统计量的内在关系，估计出不同copula函数的参数，并选择最优的copula函数来刻画大连大豆期货市场和美国、日本大豆期货市场以及日元／美元汇率之间的相关性结构．基于得到的最优copula函数．研究了上述市场之间的尾部相关性．研究结果对金融风险管理具有重要的现实意义．     

考虑风场高维相依性的电网动态经济调度优化算法

大规模风电并网给电力系统的调度运行带来了巨大的挑战.本文提出改进的二阶段带补偿随机优化算法,用于考虑风场出力高维相依性的电网动态经济调度问题求解.首先,利用Copula函数描述多风场出力的高维相依性,获得多风场出力的联合分布;随后,引入二阶段带补偿随机优化算法解耦求解动态经济调度模型中的常规变量与随机变量;求解过程中,针对补偿费用期望值的计算受限于相依性风场维数,且对迭代方向指导不明确,导致算法收敛耗时长的问题,引入基于整体最小二乘的递推动态多元线性回归法对二阶段带补偿随机优化算法进行改进,通过补偿费用期望值的动态更新,促使两阶段模型的迭代求解快速收敛,克服了传统随机优化方法的"维数灾"弊端,使该算法能够用于考虑风场高维相依性的电网动态经济调度模型求解.最后利用IEEE 118节点系统和某省级实际电网系统验证了所提算法的有效性和实用性.     

A normal copula model for the arrival process in a call center

We propose and examine a probabilistic model for the multivariate distribution of the number of calls in each period of the day (e.g., 15 or 30 min) in a call center, where the marginal distribution of the number of calls in any given period is arbitrary, and the dependence between the periods is modeled via a normal copula. Conditional on the number of calls in a period, their arrival times are independent and uniformly distributed over the period. This type of model has the advantage of being simple and reasonably flexible, and can match the correlations between the arrival counts in different periods much better than previously proposed models. For the situation where the number of periods is large, so the number of correlations to estimate can be excessive, we propose simple parametric forms for the correlations, defined as functions of the time lag between the periods. We test our proposed models on three data sets taken from real‐life call centers and compare their goodness of fit to the best previously proposed methods that we know. In the three cases, the new models provide a much better match of the correlations and coefficients of variation of the arrival counts in individual periods.     

Predicting risk of water quality failures in distribution networks under uncertainties using fault-tree analysis

The water quality in a distribution system is affected by many factors, including operational and environmental conditions as well as the condition in and around the distribution network. Lack of reliable data as well as knowledge gaps with respect to the impact of these factors on water quality make the quantification of water quality failure risk very challenging. Furthermore, the variability inherent in (sometimes) thousands of kilometers of distribution pipes presents added complexities. Major modes of water quality failures can be classified into intrusion of contaminants, regrowth of bacteria (biofilm), water treatment breakthrough, leaching of chemicals or corrosion products from system components, and permeation of organic compounds through plastic pipes. Deliberate contamination and negligence of operators have in recent years become an added concern. In earlier works by Sadiq et al. (2004 Sadiq, R., Kleiner, Y. and Rajani, B. 2004. Aggregative risk analysis for water quality failure in distribution networks. Journal of Water Supply Research and Technology: Aqua, 53(4): 241–261. [Web of Science ®] , [Google Scholar], 2007 Sadiq, R., Kleiner, Y. and Rajani, B. 2007. Water quality failures in distribution networks – risk analysis using fuzzy logic and evidential reasoning. Risk Analysis – An International Journal, 27(5): 1381–1394. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), an aggregative risk analysis approach using hierarchical structure was proposed to describe all possible mechanisms of contamination. In this paper a similar structure is used as a basis for a fault-tree approach. While fault-tree analysis is widely used for many engineering applications, in this paper we specifically explore how interdependencies among factors might impact analysis results. Two types of uncertainties are considered in the proposed analysis. The first is related to the likelihood of risk events, and the second is related to non-linear dependencies among risk events. Each basic risk event (input factor) is defined using a fuzzy probability (likelihood) to deal with its inherent uncertainty. The dependencies among risk events are explored using Frank copula and Frechet's limit. The proposed approach is demonstrated using two well-documented episodes of water quality failures in Canada, namely, Walkerton (ON) and North Battleford (SK).