Sampling from Complex Probability Distributions: A Monte Carlo Primer for Engineers

Abstract

Models which are constructed to represent the uncertainty arising in engineered systems can often be quite complex to ensure they provide a reasonably faithful reflection of the real-world system. As a result, even computation of simple expectations, event probabilities, variances, or integration over utilities for a decision problem can be analytically intractable. Indeed, such models are often sufficiently high dimensional that even traditional numerical methods perform poorly. However, access to random samples drawn from the probability model under study typically simplifies such problems substantially. The methodologies to generate and use such samples fall under the stable of techniques usually referred to as ‘Monte Carlo methods’. This chapter provides a motivation, simple primer introduction to the basics, and sign-posts to further reading and literature on Monte Carlo methods, in a manner that should be accessible to those with an engineering mathematics background. There is deliberately informal mathematical presentation which avoids measure-theoretic formalism. The accompanying lecture can be viewed at https://www.louisaslett.com/Courses/UTOPIAE/.

A Wind Power Scenario Generation Method Based on Copula Functions and Forecast Errors

“…The Monte Carlo sampling method uses probability distributions derived from historical data and random numbers to generate samples [8]. The Monte Carlo sampling method is a commonly used scenario generation method thanks to its simplicity and computational efficiency [9]. Hence, numerous studies have employed the method for scenario generation across diverse research objectives.…”