Uncertainty Modeling

Abstract

Decisions can never be better than the knowledge upon which they are based. The understanding of knowledge or rather the lack of knowledge, i.e. uncertainty, is thus central in engineering decision making. This chapter, which comprises four lectures, addresses uncertainty and its representation or modeling in engineering. The chapter contains a wide selection of examples of engineering relevance, which illustrate the presented concepts and their application. The first lecture in this chapter, Lecture 4, starts with a philosophical discourse concerning the characteristics of knowledge and the various sources of uncertainty. Thereafter, the mathematical concept of random variables is introduced together with their principal characterizations in terms of probability distribution functions and moments for both discrete and continuous random variables. In Lecture 5, a number of important tools and results of special importance in probabilistic engineering modeling are presented including the expectation operator, random vectors and joint moments, conditional distributions and conditional moments, the distribution of the sum of two random variables and the distribution of functions of random variables in general terms. In Lecture 6, the important central limit theorem, the Normal and Log-normal distributions as well as a selection of the most important distribution functions in engineering modeling are first introduced. Thereafter, the concept of modeling uncertainty associated with discrete events over time is introduced in terms of random sequences and the Bernoulli and Binomial distributions are presented. Lecture 7 addresses the engineering modeling of time variant uncertain phenomena and their extremes through stochastic processes. The important Poisson counting process is introduced and it is explained how and when this provides a useful and strong engineering modeling tool, especially in the context of rare and extreme events which take place at discrete points in time. Thereafter, basic concepts and results are introduced for continuous random processes in terms of the Normal process and important properties such as stationarity and ergodicity are explained in an engineering context. Finally, in this lecture the so-called extreme value distributions for random variables are presented and it is outlined how and when these may be utilized for the representation of extreme events of uncertain time variant phenomena. This lecture closes with the introduction of the concept of return periods and explains how the probability of extreme events may be related to recurrence times.