Random Processes

Overview

Class Central Tips

Probability and statistics provide an excellent tool for understanding, modeling and communicating uncertainty in engineering systems. In many applications there is the added challenge of considering random quantities that vary over time and/or space. Examples can be found in seismic applications, financial markets, heterogeneous materials, and image processing, among many others. This course provides an introduction into some of the ways in which random processes and random fields are measured, quantified and communicated. Through video lectures, activities, and interactive content, students will learn about correlation functions, spectral density functions, local average processes and Monte Carlo simulation. There will be an emphasis on understanding each concept, estimating these quantities from data, and using this data as the basis for generating realistic sample random processes. By the end of this course, you will be able to: - Explain the meaning of the correlation function, the spectral density function, homogeneity, ergodicity. - Identify parameters of a random process based on available data. - Relate random process descriptors to reliability via maximum value distributions. - Simulate a random process with desired correlation and/or spectral density function.

Syllabus

Basic Definitions of Random Fields

In this module, you will be introduced to some basic definitions of random processes and examples of engineering applications in which they are important. There will also be a review of probability density functions to introduce the marginal distribution that describes a random process.

Correlation Function of Random Processes

In this module, you will be introduced to the correlation function and correlation length as a means to describe random processes. You will learn to recognize how changes in the correlation function affect the random process, and vice versa. Finally, there will be a case study in which the correlation function & length are calculated based on a given set of data.

Spectral Analysis of Random Processes

In this module, you will be introduced to the spectral density function as an alternative means to describe random processes. You will learn to recognize how changes in the spectral density function affect the random process, and vice versa. Finally, there will be a case study in which the spectral density function & moments are calculated based on a given set of data.

Monte Carlo Simulation & Reliability

In this module, you will work with simulation-based approaches to generate random processes, based on the correlation function or the spectral density function. The approach will be applied in the context of reliability.