Probabilistic Performance Analysis of Fault Diagnosis Schemes

Chapter 2
Background
2.1 Introduction
The purpose of this chapter is to establish the context and background for our discussion
of probabilistic fault diagnosis problems. First, we provide a brief summary of the key
definitions of probability theory. Then, we review some standard terminology and definitions
from reliability theory. Finally, we provide a brief survey of fault diagnosis. This survey
includes a list of commonly-used terminology, an outline of the key techniques used to
design fault diagnosis schemes, and some comments on existing performance analyses for
fault diagnosis problems.
2.2 Probability Theory
In this section, we review the basic definitions of probability theory and establish some
notation. A complete survey of probability theory is beyond the scope of this dissertation,
and the informal definitions stated here are only meant to clarify the subsequent usage of
probability notation. See Rosenthal [81] or Williams [99] for a rigorous measure-theoretic
introduction to probability theory, and see Papoulis and Pillai [72] or Jazwinski [50] for an
introduction to stochastic processes.
2.2.1 Foundations
Suppose that Ω is a nonempty set called the sample space. Each point ω ∈ Ω is an outcome.
Assume that F is a σ-algebra of subsets of Ω. Each set E ∈ F is called a event. Let P be a
measure on the measurable space (Ω,F ), such that P(Ω) = 1. Then, P is called a probability
measure and the triple (Ω,F , P) is called a probability space.
Given a space S, let T be a topology defined on S. Then, a Borel set is any subset of S that
can be formed by taking a countable union, a countable intersection, or the complement of
open sets in T . The collection of Borel sets in S, denoted B(S), forms a σ-algebra known as
the Borel σ-algebra. We use the simpler notation B when the space S is clear from context.
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