An investigation of estimation performance for a multivariate Poisson-gamma model with parameter dependency

Statistical analysis can be overly reliant on naive assumptions of independence between different data generating processes. This results in having greater uncertainty when estimating underlying characteristics of processes as dependency creates an opportunity to boost sample size by incorporating more data into the analysis. However, this assumes that dependency has been appropriately specified, as mis-specified dependency can provide misleading information from the data. The main aim of this research is to investigate the impact of incorporating dependency into the data analysis. Our motivation for this work is concerned with estimating the reliability of items and as such we have restricted our investigation to study homogeneous Poisson processes (HPP), which can be used to model the rate of occurrence of events such as failures. In an HPP, dependency between rates can occur for numerous reasons. Whether it is similarity in mechanical designs, failure occurrence due to a common management culture or comparable failure count across machines for same failure modes. Multiple types of dependencies are considered. Dependencies can take different forms, such as simple linear dependency measured through the Pearson correlation, rank dependencies which capture non-linear dependencies and tail dependencies where the strength of the dependency may be stronger in extreme events as compared to more moderate one. The estimation of the measure of dependency between correlated processes can be challenging. We develop the research grounded in a Bayes or empirical Bayes inferential framework, where uncertainty in the actual rate of occurrence of a process is modelled with a prior probability distribution. We consider prior distributions to belong to the Gamma distribution given its flexibility and mathematical association with the Poisson process. For dependency modelling between processes we consider copulas which are a convenient and flexible way of capturing a variety of different dependency characteristics between distributions. We use a multivariate Poisson – Gamma probability model. The Poisson process captures aleatory uncertainty, the inherent variability in the data. Whereas the Gamma prior describes the epistemic uncertainty. By pooling processes with correlated underlying mean rate we are able to incorporate data from these processes into the inferential process and reduce the estimation error. There are three key research themes investigated in this thesis. First, to investigate the value in reducing estimation error by incorporating dependency within the analysis via theoretical analysis and simulation experiments. We show that correctly accounting for dependency can significantly reduce the estimation error. The findings should inform analysts a priori as to whether it is worth pursuing a more complex analysis for which the dependency parameter needs to be elicited. Second, to examine the consequences of mis-specifying the degree and form of dependency through controlled simulation experiments. We show the relative robustness of different ways of modelling the dependency using copula and Bayesian methods. The findings should inform analysts about the sensitivity of modelling choices. Third, to show how we can operationalise different methods for representing dependency through an industry case study. We show the consequences for a simple decision problem associated with the provision of spare parts to maintain operation of the industry process when depenency between event rates of the machines is appropriately modelled rather than being treated as independent processes.Statistical analysis can be overly reliant on naive assumptions of independence between different data generating processes. This results in having greater uncertainty when estimating underlying characteristics of processes as dependency creates an opportunity to boost sample size by incorporating more data into the analysis. However, this assumes that dependency has been appropriately specified, as mis-specified dependency can provide misleading information from the data. The main aim of this research is to investigate the impact of incorporating dependency into the data analysis. Our motivation for this work is concerned with estimating the reliability of items and as such we have restricted our investigation to study homogeneous Poisson processes (HPP), which can be used to model the rate of occurrence of events such as failures. In an HPP, dependency between rates can occur for numerous reasons. Whether it is similarity in mechanical designs, failure occurrence due to a common management culture or comparable failure count across machines for same failure modes. Multiple types of dependencies are considered. Dependencies can take different forms, such as simple linear dependency measured through the Pearson correlation, rank dependencies which capture non-linear dependencies and tail dependencies where the strength of the dependency may be stronger in extreme events as compared to more moderate one. The estimation of the measure of dependency between correlated processes can be challenging. We develop the research grounded in a Bayes or empirical Bayes inferential framework, where uncertainty in the actual rate of occurrence of a process is modelled with a prior probability distribution. We consider prior distributions to belong to the Gamma distribution given its flexibility and mathematical association with the Poisson process. For dependency modelling between processes we consider copulas which are a convenient and flexible way of capturing a variety of different dependency characteristics between distributions. We use a multivariate Poisson – Gamma probability model. The Poisson process captures aleatory uncertainty, the inherent variability in the data. Whereas the Gamma prior describes the epistemic uncertainty. By pooling processes with correlated underlying mean rate we are able to incorporate data from these processes into the inferential process and reduce the estimation error. There are three key research themes investigated in this thesis. First, to investigate the value in reducing estimation error by incorporating dependency within the analysis via theoretical analysis and simulation experiments. We show that correctly accounting for dependency can significantly reduce the estimation error. The findings should inform analysts a priori as to whether it is worth pursuing a more complex analysis for which the dependency parameter needs to be elicited. Second, to examine the consequences of mis-specifying the degree and form of dependency through controlled simulation experiments. We show the relative robustness of different ways of modelling the dependency using copula and Bayesian methods. The findings should inform analysts about the sensitivity of modelling choices. Third, to show how we can operationalise different methods for representing dependency through an industry case study. We show the consequences for a simple decision problem associated with the provision of spare parts to maintain operation of the industry process when depenency between event rates of the machines is appropriately modelled rather than being treated as independent processes

Is eliciting dependency worth the effort? A study for the multivariate Poisson-Gamma probability model

We examine whether it is worthwhile eliciting subjective judgements to account for dependency in a multivariate Poisson-Gamma probability model. The challenge of estimating reliability during product design motivated the choice of model class. For the multivariate Poisson-Gamma model we adopt an empirical Bayes methodology to present an estimator with improved accuracy. A simulation study investigates the estimation error of this estimator for different degrees of dependency and examines the impact of dependency being mis-specified when assessed by subjective judgement. Our theoretical and simulation findings give analysts insights about the value of eliciting dependency

Mathematical Models of Reliability in Technical Applications

Tato práce popisuje a aplikuje parametrické a neparametrické modely spolehlivosti na cenzorovaná data. Ukazuje implementaci spolehlivosti v metodologii Six Sigma. Metody jsou využity pro přežití/spolehlivost reálných technických dat.This master’s thesis is describing and applying parametric and nonparametric reliability models for censored data. It shows the implementation of reliability in the Six Sigma methodology. The methods are used in survival/reliability of real technical data.

Evaluation of empirical Bayes estimators of correlated event rates and implications for predicting new product reliability

A method for estimating the reliability of a new product based on a comparative analysis of observed data for similar existing products has been motivated by an industry problem; see Quigley and Walls (to appear). Such estimates are required as part of contractual discussions with customers, as well as to inform the reliability program management for the new design. An empirical Bayes inference method has been developed based on a multivariate Poisson-Gamma probability model. The model aims to capture both aleatory and epistemic uncertainties. The latter are those which have the potential to be bought down by gathering more information such as learning about the existence of a potential design weaknesses. The model is multivariate since it involves modelling elements of the new product design reliability in relation to a set of relevant elements from multiple similar products. See, Quigley et al (2013) for more details of probability modelling of correlated events which provides a theoretical framework, and Quigley and Walls (2017) for an approach to construct prior distributions using empirical data