Emulation of Poincaré return maps with Gaussian Kriging models

In this paper we investigate the use of Gaussian emulators to give an accurate and computationally fast method to approximate return maps, a tool used to study the dynamics of differential equations. One advantage of emulators over other approximation techniques is that they encode deterministic data exactly, so where values of the return map are known these are also outputs of the emulator output, another is that emulators allow us to simultaneously emulate a parameterized family of ODEs giving a tool to assess the behavior of perturbed systems. The methods introduced here are illustrated using two well-known dynamical systems: The Rossler equations, and the Billiard system. We show that the method can be used to look at return maps and discuss the further implications for full computation of differential equation outputs

Estimating rate of occurrence of rare events with empirical Bayes : a railway application

Classical approaches to estimating the rate of occurrence of events perform poorly when data are few. Maximum likelihood estimators result in overly optimistic point estimates of zero for situations where there have been no events. Alternative empirical-based approaches have been proposed based on median estimators or non-informative prior distributions. While these alternatives offer an improvement over point estimates of zero, they can be overly conservative. Empirical Bayes procedures offer an unbiased approach through pooling data across different hazards to support stronger statistical inference. This paper considers the application of Empirical Bayes to high consequence low-frequency events, where estimates are required for risk mitigation decision support such as as low as reasonably possible. A summary of empirical Bayes methods is given and the choices of estimation procedures to obtain interval estimates are discussed. The approaches illustrated within the case study are based on the estimation of the rate of occurrence of train derailments within the UK. The usefulness of empirical Bayes within this context is discusse

Expert Elicitation for Reliable System Design

This paper reviews the role of expert judgement to support reliability assessments within the systems engineering design process. Generic design processes are described to give the context and a discussion is given about the nature of the reliability assessments required in the different systems engineering phases. It is argued that, as far as meeting reliability requirements is concerned, the whole design process is more akin to a statistical control process than to a straightforward statistical problem of assessing an unknown distribution. This leads to features of the expert judgement problem in the design context which are substantially different from those seen, for example, in risk assessment. In particular, the role of experts in problem structuring and in developing failure mitigation options is much more prominent, and there is a need to take into account the reliability potential for future mitigation measures downstream in the system life cycle. An overview is given of the stakeholders typically involved in large scale systems engineering design projects, and this is used to argue the need for methods that expose potential judgemental biases in order to generate analyses that can be said to provide rational consensus about uncertainties. Finally, a number of key points are developed with the aim of moving toward a framework that provides a holistic method for tracking reliability assessment through the design process.Comment: This paper commented in: [arXiv:0708.0285], [arXiv:0708.0287], [arXiv:0708.0288]. Rejoinder in [arXiv:0708.0293]. Published at http://dx.doi.org/10.1214/088342306000000510 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Historical Exploration - Learning Lessons from the Past to Inform the Future

This report examines a number of exploration campaigns that have taken place during the last 700 years, and considers them from a risk perspective. The explorations are those led by Christopher Columbus, Sir Walter Raleigh, John Franklin, Sir Ernest Shackleton, the Company of Scotland to Darien and the Apollo project undertaken by NASA. To provide a wider context for investigating the selected exploration campaigns, we seek ways of finding analogies at mission, programmatic and strategic levels and thereby to develop common themes. Ultimately, the purpose of the study is to understand how risk has shaped past explorations, in order to learn lessons for the future. From this, we begin to identify and develop tools for assessing strategic risk in future explorations. Figure 0.1 (see Page 6) summarizes the key inputs used to shape the study, the process and the results, and provides a graphical overview of the methodology used in the project. The first step was to identify the potential cases that could be assessed and to create criteria for selection. These criteria were collaboratively developed through discussion with a Business Historian. From this, six cases were identified as meeting our key criteria. Preliminary analysis of two of the cases allowed us to develop an evaluation framework that was used across all six cases to ensure consistency. This framework was revised and developed further as all six cases were analyzed. A narrative and summary statistics were created for each exploration case studied, in addition to a method for visualizing the important dimensions that capture major events. These Risk Experience Diagrams illustrate how the realizations of events, linked to different types of risks, have influenced the historical development of each exploration campaign. From these diagrams, we can begin to compare risks across each of the cases using a common framework. In addition, exploration risks were classified in terms of mission, program and strategic risks. From this, a Venn diagram and Belief Network were developed to identify how different exploration risks interacted. These diagrams allow us to quickly view the key risk drivers and their interactions in each of the historical cases. By looking at the context in which individual missions take place we have been able to observe the dynamics within an exploration campaign, and gain an understanding of how these interact with influences from stakeholders and competitors. A qualitative model has been created to capture how these factors interact, and are further challenged by unwanted events such as mission failures and competitor successes. This Dynamic Systemic Risk Model is generic and applies broadly to all the exploration ventures studied. This model is an amalgamation of a System Dynamics model, hence incorporating the natural feedback loops within each exploration mission, and a risk model, in order to ensure that the unforeseen events that may occur can be incorporated into the modeling. Finally, an overview is given of the motivational drivers and summaries are presented of the overall costs borne in each exploration venture. An important observation is that all the cases - with the exception of Apollo - were failures in terms of meeting their original objectives. However, despite this, several were strategic successes and indeed changed goals as needed in an entrepreneurial way. The Risk Experience Diagrams developed for each case were used to quantitatively assess which risks were realized most often during our case studies and to draw comparisons at mission, program and strategic levels. In addition, using the Risk Experience Diagrams and the narrative of each case, specific lessons for future exploration were identified. There are three key conclusions to this study: Analyses of historical cases have shown that there exists a set of generic risk classes. This set of risk classes cover mission, program and strategic levels, and includes all the risks encountered in the cases studied. At mission level these are Leadership Decisions, Internal Events and External Events; at program level these are Lack of Learning, Resourcing and Mission Failure; at Strategic Level they are Programmatic Failure, Stakeholder Perception and Goal Change. In addition there are two further risks that impact at all levels: Self-Interest of Actors, and False Model. There is no reason to believe that these risk classes will not be applicable to future exploration and colonization campaigns. We have deliberately selected a range of different exploration and colonization campaigns, taking place between the 15th Century and the 20th Century. The generic risk framework is able to describe the significant types of risk for these missions. Furthermore, many of these risks relate to how human beings interact and learn lessons to guide their future behavior. Although we are better schooled than our forebears and are technically further advanced, there is no reason to think we are fundamentally better at identifying, prioritizing and controlling these classes of risk. Modern risk modeling techniques are capable of addressing mission and program risk but are not as well suited to strategic risk. We have observed that strategic risks are prevalent throughout historic exploration and colonization campaigns. However, systematic approaches do not exist at the moment to analyze such risks. A risk-informed approach to understanding what happened in the past helps us guard against the danger of assuming that those events were inevitable, and highlights those chance events that produced the history that the world experienced. In turn, it allows us to learn more clearly from the past about the way our modern risk modeling techniques might help us to manage the future - and also bring to light those areas where they may not. This study has been retrospective. Based on this analysis, the potential for developing the work in a prospective way by applying the risk models to future campaigns is discussed. Follow on work from this study will focus on creating a portfolio of tools for assessing strategic and programmatic risk

Vine copula approximation : a generic method for coping with conditional dependence

Pair-copula constructions (or vine copulas) are structured, in the layout of vines, with bivariate copulas and conditional bivariate copulas. The main contribution of the current work is an approach to the long-standing problem: how to cope with the dependence structure between the two conditioned variables indicated by an edge, acknowledging that the dependence structure changes with the values of the conditioning variables. This problem is known as the non-simplified vine copula modelling and, though recognized as crucial in the field of multivariate modelling, remains widely unexplored due to its inherent complication, and hence is the motivation of the current work. Rather than resorting to traditional parametric or non-parametric methods, we proceed from an innovative viewpoint: approximating a conditional copula, to any required degree of approximation, by utilizing a family of basis functions. We fully incorporate the impact of the conditioning variables on the functional form of a conditional copula by employing local learning methods. The attractions and dilemmas of the pair-copula approximating technique are revealed via simulated data, and its practical importance is evidenced via a real data set

Applying Bayes linear methods to support reliability procurement decisions

Bayesian methods are common in reliability and risk assessment, however, such methods often demand a large amount of specification and can be computationally intensive. Because of this, many practitioners are unable to take advantage of many of the benefits found in a Bayesian-based approach. The Bayes linear methodology is similar in spirit to a Bayesian approach but offers an alternative method of making inferences. Bayes linear methods are based on the use of expected values rather than probabilities, and updating is carried out by linear adjustment rather than by Bayes Theorem. The foundations of the method are very strong, based as they are in work of De Finetti and developed further by Goldstein. A Bayes linear model requires less specification than a corresponding probability model and for a given amount of model building effort, one can model a more complex situation quicker. The Bayes linear methodology has the potential to allow us to build ''broad-brush' models that enable us, for example, to explore different test setups or analysis methods and assess the benefits that they can give. The output a Bayes linear model is viewed as an approximation to 'traditional' probabilistic models. The methodology has been applied to support reliability decision making within a current United Kingdom Ministry of Defence (MOD) procurement project. The reliability decision maker had to assess different contractor bids and assess the reliability merit of each bid. Currently the MOD assess reliability programmes subjectively using expert knowledge - for a number of reasons, a quantitative method of assessment in some projects is desirable. The Bayes linear methodology was used to support the decision maker in quantifying his assessment of the reliability of each contractor's bid and determining the effectiveness of each contractor's reliability programme. From this, the decision maker was able to communicate to the project leader and contractors, why a specific contractor was chosen. The methodology has been used in other MOD projects and is considered by those within the MOD as a useful tool to support decision making. The paper will contain the following. The paper will introduce the Bayes linear methodology and briefly discuss some of the philosophical implications of adopting a Bayes linear methodology within the context of a reliability programme analysis. The paper will briefly introduce the reliability domain and the reasons why it is believed that the Bayes linear methodology can offer support to decision makers. An in-depth analysis of the problem will then be given documenting the steps taken in the project and how future decision makers can apply the methodology. A brief summary will then be given as to possible future work for those interested in the Bayes linear methodology

On the construction of minimum information bivariate copula families

Copulas have become very popular as modelling tools in probability applications. Given a finite number of expectation constraints for functions defined on the unit square, the minimum information copula is that copula which has minimum information (Kullback-Leibler divergence) from the uniform copula. This can be considered the most ``independent'' copula satisfying the constraints. We demonstrate the existence and uniqueness of such copulas, rigorously establish the relation with discrete approximations, and prove an unexpected relationship between constraint expectation values and the copula density formula

Empirical Bayes Methods for Discrete Event Simulation Performance Measure Estimation

Discrete event simulation (DES) is a widely-used operational research methodology facilitating the analysis of complex real-world systems. Although, generally speaking, simplicity is greatly desirable in DES modelling applications, in many cases the nature of the underlying system results in simulation models which are large in scale, complex, and expensive to run. As such, the careful design and analysis of simulation experiments is essential to ensure valid and efficient inference concerning DES model performance measures. It is envisaged that empirical Bayes (EB) methods, which enable data to be pooled across a set of populations to support inference of the parameters of a single population, may be of use within this context. Despite this potential, EB has so far been neglected within the DES literature. This paper presents a preliminary computational investigation into the efficacy of EB procedures in the estimation of DES performance measures. The results of this investigation, and their significance, are explored. Additionally, likely directions for future research are also addressed

A model for availability growth with application to new generation offshore wind farms

A model for availability growth is developed to capture the effect of systemic risk prior to construction of a complex system. The model has been motivated by new generation offshore wind farms where investment decisions need to be taken before test and operational data are available. We develop a generic model to capture the systemic risks arising from innovation in evolutionary system designs. By modelling the impact of major and minor interventions to mitigate weaknesses and to improve the failure and restoration processes of subassemblies, we are able to measure the growth in availability performance of the system. We describe the choices made in modelling our particular industrial setting using an example for a typical UK Round III offshore wind farm. We obtain point estimates of the expected availability having populated the simulated model using appropriate judgemental and empirical data. We show the relative impact of modelling systemic risk on system availability performance in comparison with estimates obtained (Lesley Walls) from typical system availability modelling assumptions used in offshore wind applications. While modelling growth in availability is necessary for meaningful decision support in developing complex systems such as offshore wind farms, we also discuss the relative value of explicitly articulating epistemic uncertainties