Estimation of Rare Event Probabilities in Complex Aerospace and Other Systems

A Practical Approach

Jerome Morio, Mathieu Balesdent

• Publisher: Woodhead Publishing
• Year: 2015
• Print ISBN:
  9780081000915
• eBook ISBN:
  9780081001110
• Topic: Technologie et ingénierie
• Subtopic: Aérodynamisme

Rare event probability (10-4 and less) estimation has become a large area of research in the reliability engineering and system safety domains. A significant number of methods have been proposed to reduce the computation burden for the estimation of rare events from advanced sampling approaches to extreme value theory. However, it is often difficult in practice to determine which algorithm is the most adapted to a given problem.Estimation of Rare Event Probabilities in Complex Aerospace and Other Systems: A Practical Approach provides a broad up-to-date view of the current available techniques to estimate rare event probabilities described with a unified notation, a mathematical pseudocode to ease their potential implementation and finally a large spectrum of simulation results on academic and realistic use cases. - Provides a broad overview of the practical approach of rare event methods. - Includes algorithms that are applied to aerospace benchmark test cases - Offers insight into practical tuning issues

1

Introduction to rare event probability estimation

J. Morio; M. Balesdent

Abstract

Crude Monte Carlo methods are well suited to characterize events of which associated probabilities are not too low with respect to a given simulation budget. For very seldom observed events, such as the collision probability between two aircraft in airspace, Monte Carlo based approaches do not lead to accurate results. Indeed, the number of available samples is often insufficient to estimate such low probabilities (at least 10⁶ samples are needed to estimate a probability of order 10⁻⁴ with 10% relative error with Monte Carlo simulations). The objective of this book is to propose some guidelines to determine the most efficient estimation techniques for a given problem.

Keywords

Rare event

Simulation

Aerospace system

Probability

Statistics

1.1 The book purposes

Rare event probability estimation has become a large area of research in the reliability engineering and system safety domains. A significant number of methods have been proposed to reduce the computational burden for the estimation of rare events from advanced sampling approaches to extreme value theory. However, it is often difficult in practice to determine which algorithm is the most adapted to a given problem. The purposes of this book are thus to provide a broad up-to-date view of the currently available techniques to estimate rare event probabilities described with unified notations, mathematical pseudocodes to ease their potential implementation, and finally, a large spectrum of simulation results on academic and realistic use cases. We detail the pros and cons of the different algorithms depending on the problem characteristics to answer this question: Which rare event probability estimation techniques are the most adapted to the reader’s situation?

1.2 What are the events of interest considered in this book?

Let us first specify the main characteristics of the events that are considered in this book:

• If the event of interest happens, it has a highly severe affect on the performances or the safety of the system. If we want to analyze the risk of the event, a measure of damage must be defined (Kaplan & Garrick, 1981).

• The event probability is low; more precisely, it is difficult to observe realizations of this event of interest considering the available simulation budget. Thus, such rare event probability cannot be estimated easily with direct Monte Carlo simulations.

• The rare event probability must be assessed very accurately because an error on the rare event probability could have serious consequences. The underestimation of the rare event probability must be completely avoided.

The issue of estimating rare event probabilities corresponds to a very large number of applicative situations. In this book, we focus on rare event probability estimation in aerospace systems. For instance, Chapter 11 presents the estimation of the probability that a satellite and a spatial debris collide. This probability must be estimated with accuracy because a collision will end the mission of the satellite. Thus, it must be avoided at all costs because of the required financial budget to launch a satellite. Typical probabilities of collision between satellite and debris are lower than 10⁻⁴ and thus cannot be estimated with direct Monte Carlo simulations with a sufficient accuracy.

1.3 The book organization

The first part of the book provides the essential background in probability and statistics and describes the different complex systems considered for the estimation of rare event probabilities. In the second part, we review the main approaches to estimate rare event probabilities. For each technique, its principle and its main theoretical and experimental features are presented. We also propose a mathematical pseudocode and some generally efficient tuning parameters. Some application results on academic test cases are finally given to evaluate the potential of the described method on different situations (high dimension, multiple failure domain, etc.). The performances of the different algorithms to estimate rare event probabilities are then compared in a third part on realistic test cases from the aerospace field. This comparison enables us in the last part of this book to propose some guidelines to determine which methods are the most adapted to a given system, depending on its main characteristics (e.g., dimension, the available simulation budget).

References

Kaplan S, Garrick BJ. On the quantitative definition of risk. Risk Analysis. 1981;1(1):11–27.

Part One

Essential Background in Mathematics and System Analysis

2

Basics of probability and statistics

J. Morio; M. Balesdent

Abstract

It is necessary to have a minimal knowledge of some fundamental definitions and theorems of probability and statistics to understand the principles of the different rare event probability estimation methods. The main goal of this chapter...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Dedication
• Preface
• Foreword
• Biography of the external contributors to this book
• Abbreviations
• 1: Introduction to rare event probability estimation
• Part One: Essential Background in Mathematics and System Analysis
• Part Two: Practical Overview of the Main Rare Event Estimation Techniques
• Part Three: Benchmark of the Methods to Aerospace Problems
• Part Four: Practical Guidelines of Rare Event Probability Estimation
• Index