Statistical methods for extreme values and degradation analysis

Liu, Shiyao
Major Professor
William Q. Meeker
Committee Member
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Energy and chemical companies use pipelines to transfer oil, gas and other materials from one place to another, within and between their plants. Pipeline integrity is an important concern because pipeline leakage could result in serious economic or environmental losses. In this dissertation, statistical models and methods motivated by real applications were developed for pipeline reliability using extreme value theory and degradation modeling.

In Chapter 2, interval-censored measurements from a given set of thickness measurement locations (TMLs) along a three-phase pipeline are used to estimate the distribution of the minimum thickness. The block-minima method based on extreme value theory provides a robust approach to estimate the minimum thickness in a pipeline. In the block-minima method using the Gumbel and the generalized extreme value distributions, the choice of the number of blocks involves the trade-off between bias and variance. We conduct a simulation study to compare the properties of different models for estimating minimum pipeline thickness and investigate the effect of block size choice on MSE in the block-minima method.

The pipeline thickness estimation in Chapter 2 is based on data from a single time point at each TML. In the other pipeline applications, longitudinal inspections of the pipeline thickness at particular locations along the pipeline are available. Depending on different mechanisms of corrosion processes, we have observed various types of general degradation paths. In Chapter 3 of this thesis, we propose a degradation model describing corrosion initiation and growth behavior. The parameters in the degradation model are estimated using a Bayesian approach. We derive failure-time and remaining lifetime distributions from the degradation model and compute Bayesian estimates and credible intervals of the failure-time and remaining lifetime distribution. We also develop a hierarchical model to quantify the pipeline corrosion rate for similar circuits within a single facility.

The extreme value theorem suggests that no matter what the underlying parent distribution is, the limiting distribution of minima is the minimum generalized extreme value (GEV) distribution. The likelihood function, as it is usually written as a product of density functions, however, is unbounded in the parameter space. Due to rounding, all data are discrete and the use of densities for ``exact" observations is only an approximation. In Chapter 4 of the thesis, we use the ``correct likelihood'' based on interval censoring to eliminate the problem of an unbounded density-approximation likelihood. We categorize the models that have an unbounded density-approximation likelihood into three groups, which are (1) continuous univariate distributions with both a location and a scale parameter, plus a threshold parameter, (2) discrete mixture models of continuous distributions for which at least one component has both a location and a scale parameter, (3) minimum-type (and maximum-type) models for which at least one of the marginal distributions has both a location and a scale parameter. For each category, we illustrate the density breakdown with specific examples. We also study the effect of the round-off error on estimation using the correct likelihood, and provide a sufficient condition for the joint density to provide the same maximum likelihood estimate as the correct likelihood, as the round-off error goes to zero.