Marc A. Maes

Identifying tails, bounds and end-points of random variables

Abstract The characterization of tails of random variables is of major concern in a safety analysis such as a structural reliability analysis or a quantitative risk analysis of an engineering system. One of the important questions raised is whether the tail is bounded or unbounded. Therefore, in a statistical analysis of a given data set, it makes sense to use only the extreme small or large data in the tail modelling. This raises the important issue of the selection of thresholds above which “tail behaviour” of the data can be justified. In general, thresholds close to the central data will bias the estimation towards the central values which are not informative for the tail. Too extreme thresholds will result in high estimation variances. In this paper we propose to use a finite sample mean square error (MSE) to select such thresholds and to estimate tail characteristics. Estimators for the extreme value index, the end-point and extreme quantiles are based on the so-called generalized quantile plot. This plot is used to discern between bounded and unbounded tail behaviour. A semi-parametric bootstrap technique is used to estimate the MSE at each threshold and to select the optimal threshold at which the MSE is minimized. Confidence limits are obtained using the sampling distribution of estimators at the optimal threshold. In a verification study and an application to wall thickness values of tubes, the MSE-criterion is applied to various extremal properties such as end-points or extreme quantiles and to other parameters that are critically dependent on the tail behaviour of a random variable such as reliability index.

Asymptotic importance sampling

Abstract An importance sampling technique is described which is based on theoretical considerations about the structure of multivariate integrands in domains having small probability content. The method is formulated in the original variable space. Sampling densities are derived for a variety of practical conditions: a single point of maximum loglikelihood; several points; points located at the intersect of several failure surfaces; and, bounded variables. Sampling in the safe domain is avoided and extensive use is made of noncartesian as well as surface coordinates. The parameters of the importance sampling densities are taylored in such a way as to yield asymptotic minimum variance unbiased estimators. The quality and the efficiency of the method improves as the failure probability decreases. Parameter sensitivies are easily computed owing to the use of local surface coordinates. Several examples are provided.

Probabilistic Analysis of Local Ice Pressures

Extensive work in recent years has been carried out on the calculation of global ice loads on a probabilistic basis. An analysis method is presented for local ice pressures, which yields values of pressure for specific values of exceedance probability. In developing this method, particular attention has been paid to problems of exposure (length, position and number of impacts), as well as the area of exposure (area within area versus nominal contact area). The solution has been formulated for a series of discrete impacts, e.g., rams by a vessel, or a series of periods of continuous interactions. Data for the MV CANMAR Kigoriak and USCGC Polar Sea were ranked and curves were fitted through the tail of probability plots for three panel sizes. This allowed determination of exceedance probabilities of the design coefficients for pressure as a junction of area. The method developed was then applied to an example for a ship based on the data and expected number of rams per year. Formulas useful in the design of structures in ice are presented.

A computational framework for risk assessment of RC structures using indicators

The present article starts out by proposing a framework for risk assessment of RC structures utilizing condition indicators. Thereafter, the various building stones of the suggested framework are described. This description includes a summary of the basis for the probabilistic modeling of the initiation phases of chloride-induced corrosion of concrete structures. Furthermore, a probabilistic modeling of condition indicators regarding the condition state of concrete structures is proposed whereby information available at the design stage of concrete structures as well as information obtained through in-service inspections may be utilized for the purpose of reliability updating. Finally, it is described how the probability of localized and spatially distributed degradation of different degrees can be assessed and examples are given on how the various indicators may be used for the purpose of updating the statistical characteristics of the future degradation of RC structures. The presented framework forms a consistent basis for risk assessment of concrete structures subject to chloride-induced corrosion. It can easily be adopted to other degradation phenomena such as carbonation-induced corrosion and it forms a good basis for the development of efficient approaches to Asset Integrity Management of RC structures.

Influence of grade on the reliability of corroding pipelines

This paper focuses on a comparative analysis of the reliability associated with the evolution of corrosion between normal and high-strength pipe material. The use of high strength steel grades such as X100 and X120 for high pressure gas pipeline in the arctic is currently being considered. To achieve this objective, a time-dependent reliability analysis using variable Y/T ratios in a multiaxial finite strain analysis of thin-walled pipeline is performed. This analysis allows for the consideration of longitudinal grooves and the presence of companion axial tension and bending loads. Limit states models are developed based on suitable strain hardening models for the ultimate behavior of corroded medium and high strength pipeline material. In an application, the evolution of corrosion is modeled in pipelines of different grades that have been subjected to an internal corrosion inspection after a specified time which allows for a Bayesian updating of long-term corrosion estimates and, hence, the derivation of annual probabilities of failure as a function of time. The effect of grade and Y/T is clearly demonstrated.

Structural robustness in the light of risk and consequence analysis

Robustness means many things to many people, even amongst civil engineers. In order to describe its essential aspects more clearly within the context of structural systems and critical infrastructure, the concept of robustness in different but related disciplines has been first examined. Both deterministic and probabilistic or risk-based measures of robustness can be developed and used. In the context of systems subject to exceptional hazards, the uncertainty associated with both systems, hazards and consequences plays a prominent role and therefore measures of robustness must express probabilistically how and to what extent certain system performance objectives are affected by external or systemic perturbations. An example is given for a simple structural system.

Hierarchical Modeling of Pipeline Defect Growth Subject to ILI Uncertainty

Pipeline deterioration arises chiefly as the result of various types of internal and external corrosion processes, which are typically subject to several uncertainties. They include material uncertainties, uncertainties in external influences such as loading and environmental variations, uncertainties in operating conditions, various spatial and temporal uncertainties, inspection uncertainties, and modeling uncertainties. Typically, the metal loss time-path at one defect feature may be quite different from the metal loss time-path in a neighboring location even when subject to supposedly similar loading, material and environmental circumstances. On top of that, in-line inspections (ILI) of pipeline systems affected by deterioration are performed infrequently and suffer from considerable uncertainty due to sizing errors and detectability. The present paper provides a Hierarchical Bayes framework for corrosion defect growth. While a full Hierarchical Bayes analysis is practical only for selected critical defect features, we also develop a simplified method based on multi-level generalized least squares. The latter method is useful for scanning large defect inspection data sets. Two detailed examples of the approach are presented.Copyright

Statistics for Modeling Heavy Tailed Distributions in Geology: Part I. Methodology

Many natural phenomena exhibit size distributions that are power laws or power law type distributions. Power laws are specific in the sense that they can exhibit extremely long or heavy tails. The largest event in a sample from such distribution usually dominates the underlying physical or generating process (floods, earthquakes, diamond sizes and values, incomes, insurance). Often, the practitioner is faced with the difficult problem of predicting values far beyond the highest sample value and designing his “system” either to profit from them, or to protect against extreme quantiles. In this paper, we present a novel approach to estimating such heavy tails. The estimation of tail characteristics such as the extreme value index, extreme quantiles, and percentiles (rare events) is shown to depend primarily on the number of extreme data that are used to model the tail. Because only the most extreme data are useful for studying tails, thresholds must be selected above which the data are modeled as power laws. The mean square error (MSE) is used to select such thresholds. A semiparametric bootstrap method is developed to study estimation bias and variance and to derive confidence limits. A simulation study is performed to assess the accuracy of these confidence limits. The overall methodology is applied to the Harvard Central Moment Tensor catalog of global earthquakes.

Bayesian framework for managing preferences in decision-making

A rational decision-making process does not exclude the possibility of decision makers expressing different preferences and disagreeing regarding the effects of consequences and optimal course of actions. This point of view is explored in depth in this paper. A framework is developed that includes several decision makers (instead of just one) and allows for the variability of preferences among these decision makers. The information provided by the varying opinions of decision makers can be used to optimize our own decision-making. To achieve this, likelihood functions are developed for stated preferences among both discrete and continuous alternatives, and stated preference rankings of alternatives. Two applications are pursued: the optimization of the lifecycle utility of a structural system subject to consequences of failure proportional to the intensity of hazards exceeding a variable threshold, and to follow-up consequences. Also, the problem of tight decisions or close calls is investigated in order to explore the efficiency of a Bayesian approach using stated preferences and stated rankings.

Reliability-Based Tail Estimation

In reliability analysis a probabilistic model has two components. One is the mathematical description of its properties; this is part of probability theory. The other one is the description of its relation with reality, i.e. the data and the observations; this is part of statistical inference. Most research up to this date has focused on the problem of the mathematical description of the model, while the second aspect has been somewhat neglected.