Process monitoring method based on correlation variable classification and vine copula

Chemical processes are becoming increasingly complicated, leading to an increase in process variables and more complex relationships among them. The vine copula has a significant advantage in portraying the dependence of high-dimensional variables. However, as the dimensions increase, the vine copula model incurs a high computational load; such pressure greatly reduces model efficiency. Relationships among variables in the industrial process are complex. Different variables may be strongly or weakly associated or even independent. This paper proposes a process monitoring method based on correlation variable classification and vine copula. The weighted correlation measure is first used to divide variables into a correlated subspace and weakly correlated subspace. Then, two vine structures, C-vine and D-vine, are applied to the correlated and weakly correlated subspaces, respectively. This method takes advantage of C-vine for correlated variables and the flexibility of D-vine for weakly correlated variables. Finally, comprehensive statistics are established based on different subspaces. Monitoring results of the numerical system and the Tennessee Eastman process demonstrate the effectiveness and validity of the proposed method.     

基于三维Copula函数的蚀坑预测模型

通过盐雾腐蚀试验研究了钢绞线在不同应力幅作用下的腐蚀规律,并且观测了钢绞线的腐蚀形态.从概率角度出发,得到了钢绞线蚀坑长度、宽度和深度的独立分布形式,然后基于三维Copula函数得到三者之间的联合分布规律,提出了三维蚀坑预测模型,并通过显著性检验对该模型进行了验证.结果 表明:Clayton Copula函数对于模拟蚀坑三维联合分布具有较高的精度;蚀坑使得钢绞线的抗拉强度和延伸率急剧下降,是钢绞线疲劳寿命下降的主要原因;三维蚀坑预测模型的提出,可以为预测钢绞线的腐蚀疲劳寿命提供参考.     

Generalized F-discrepancy-based point selection strategy for dependent random variables in uncertainty quantification of nonlinear structures

In the performance evaluation of structures under disastrous actions, for example, earthquakes, it is important to take into account the randomness of structural parameters. Generally, these random parameters are treated either as independent or perfectly dependent, but practically they are partly dependent. This article aims at developing a point selection strategy for uncertainty quantification of nonlinear structures involving probabilistically dependent random parameters characterized by copula function. For this purpose, the point selection strategy for structures involving independent basic variables is first revisited. As an improvement, a generalized F-discrepancy diminishing oriented iterative screening algorithm is proposed. Then, combining with the conditional sampling method, a conditional point set rearrangement method and a conditional iterative screening-rearrangement method are proposed for probabilistically dependent variables. These new point selection strategies are readily incorporated into the probability density evolution method for uncertainty quantification of nonlinear structures involving dependent random parameters, which is characterized by copula function. The proposed methods are illustrated by two examples including a shear frame with hysteretic restoring forces and a reinforced concrete frame structure with the damage constitutive model of concrete, where the material parameters are probabilistically dependent. The results demonstrate the effectiveness of the proposed method. Problems to be studied are discussed.     

Comorbidity of chronic diseases in the elderly: Patterns identified by a copula design for mixed responses

Joint modeling of multiple health related random variables is essential to develop an understanding for the public health consequences of an aging population. This is particularly true for patients suffering from multiple chronic diseases. The contribution is to introduce a novel model for multivariate data where some response variables are discrete and some are continuous. It is based on pair copula constructions (PCCs) and has two major advantages over existing methodology. First, expressing the joint dependence structure in terms of bivariate copulas leads to a computationally advantageous expression for the likelihood function. This makes maximum likelihood estimation feasible for large multidimensional data sets. Second, different and possibly asymmetric bivariate (conditional) marginal distributions are allowed which is necessary to accurately describe the limiting behavior of conditional distributions for mixed discrete and continuous responses. The advantages and the favorable predictive performance of the model are demonstrated using data from the Second Longitudinal Study of Aging (LSOA II).     

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.     

Do Rosenblatt and Nataf isoprobabilistic transformations really differ?

This article is the third in a series dedicated to the mathematical study of isoprobabilistic transformations and their relationship with stochastic dependence modelling, see [R. Lebrun, A. Dutfoy, An innovating analysis of the Nataf transformation from the viewpoint of copula, Probabilistic Engineering Mechanics (2008). doi: 10.1016/j.probengmech.2008.08.001] for an interpretation of the Nataf transformation in term of normal copula and [R. Lebrun, A. Dutfoy, A generalization of the Nataf transformation to distributions with elliptical copula, Probabilistic Engineering Mechanics (24) (2009), 172–178. doi:10.1016/j.probengmech.2008.05.001] for a generalisation of the Nataf transformation to any elliptical copula.In this article, we explore the relationship between two isoprobabilistic transformations widely used in the community of reliability analysts, namely the Generalised Nataf transformation and Rosenblatt transformation.First, we recall the elementary results of the copula theory that are needed in the remaining of the article, as a preliminary section to the presentation of both the Generalized Nataf transformation and the Rosenblatt transformation in the light of the copula theory.Then, we show that the Rosenblatt transformation using the canonical order of conditioning is identical to the Generalised Nataf transformation in the normal copula case, which is the most usual case in reliability analysis since it corresponds to the classical Nataf transformation. At this step, we also show that it is not possible to extend the Rosenblatt transformation to distributions with general elliptical copula the way the Nataf transformation has been generalised.Furthermore, we explore the effect of the conditioning order of the Rosenblatt transformation on the usual reliability indicators obtained from a FORM or SORM method. We show that in the normal copula case, all these reliability indicators, excepted the importance factors, are unchanged whatever the conditioning order one chooses.In the last section, we conclude the article with two numerical applications that illustrate the previous results: the equivalence between both transformations in the normal copula case, and the effect of the conditioning order in the normal and non-normal copula case.     

A generalization of the Nataf transformation to distributions with elliptical copula

In the first article [Lebrun R, Dutfoy A. An innovating viewpoint of the isoprobabilistic Nataf transformation with the copula theory within exceedance threshold uncertainty analysis. Probabilistic Structure Engineering Structural Reliability. 2008 [in press]], we showed that the Nataf transformation is a way to model the dependence structure of a random vector by a normal copula, parameterized by its correlation matrix. Following this analysis, we propose an extension of this transformation to any elliptical copula, and give the equivalent of the First Order Reliability Method (FORM) and the Second Order Reliability Method (SORM) for this generalization. In particular, we derive the Breitung asymptotic approximation in this new context.     

Efficiently sampling exchangeable Cuadras-Augé copulas in high dimensions

An n-dimensional random vector is constructed whose survival copula is given by a copula that was first presented in Cuadras and Augé [C.M. Cuadras, J. Augé, A continuous general multivariate distribution and its properties, Communications in Statistics - Theory and Methods 10 (4) (1981) 339-353]. This construction adds a Poisson subordinator as mixing variable to initially independent exponentially distributed random variables. It is shown how the choice of Poisson process relates to the parameter of the induced Cuadras-Augé copula. Based on this construction, a sampling algorithm for this multivariate distribution is presented which has average computational efficiency O(nloglogn).     

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).     

条件copula的相关性质

研究条件copula的相关性质,给出生存条件copula的概念研究了它的性质及和条件copula的联系,并研究了几种条件copula之间的关系,为利用条件copula描述条件随机变量的相关性并研究其相关性质提供了方便,最后提出可以进一步讨论的几个问题.