Stationary Stochastic Processes for Scientists and Engineers (Hardback) book cover

Stationary Stochastic Processes for Scientists and Engineers

By Georg Lindgren, Holger Rootzen, Maria Sandsten

© 2013 – Chapman and Hall/CRC

330 pages | 116 B/W Illus.

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Hardback: 9781466586185
pub: 2013-10-11
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pub: 2013-10-09
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About the Book

Stochastic processes are indispensable tools for development and research in signal and image processing, automatic control, oceanography, structural reliability, environmetrics, climatology, econometrics, and many other areas of science and engineering. Suitable for a one-semester course, Stationary Stochastic Processes for Scientists and Engineers teaches students how to use these processes efficiently. Carefully balancing mathematical rigor and ease of exposition, the book provides students with a sufficient understanding of the theory and a practical appreciation of how it is used in real-life situations. Special emphasis is on the interpretation of various statistical models and concepts as well as the types of questions statistical analysis can answer.

The text first introduces numerous examples from signal processing, economics, and general natural sciences and technology. It then covers the estimation of mean value and covariance functions, properties of stationary Poisson processes, Fourier analysis of the covariance function (spectral analysis), and the Gaussian distribution. The book also focuses on input-output relations in linear filters, describes discrete-time auto-regressive and moving average processes, and explains how to solve linear stochastic differential equations. It concludes with frequency analysis and estimation of spectral densities.

With a focus on model building and interpreting the statistical concepts, this classroom-tested book conveys a broad understanding of the mechanisms that generate stationary stochastic processes. By combining theory and applications, the text gives students a well-rounded introduction to these processes. To enable hands-on practice, MATLAB® code is available online.


"This book is designed for a first course in stationary stochastic processes in science and engineering and does a very good job in introducing many concepts and ideas to students in these fields. … the book has probably been tested in the classroom many times, which also manifests itself in its virtual lack of typos. … Another great feature of the book is that it contains a wealth of worked example from many different fields. These help clarify concepts and theorems and I believe students will appreciate them—I certainly did. … The book is well suited for a one-semester course as it contains exercises at the end of each chapter. … I can warmly recommend the book for a first course on stochastic processes as the book introduces many important concepts and illustrates these by interesting examples."

The American Statistician, February 2015

"This book is a lucid and well-paced introduction to stationary stochastic processes, superbly motivated and illustrated through a wealth of convincing applications in science and engineering. It offers a clear guide to the formulation and mathematical properties of these processes and to some non-stationary processes too, without going too deeply into the mathematical foundations; the emphasis throughout is on practical application rather than mathematical development for its own sake. The reader will find tools for analysis and calculation and also—importantly—material to deepen understanding and generate enthusiasm and confidence. An outstanding text."

—Clive Anderson, Department of Probability and Statistics, University of Sheffield, UK

"…a very nicely written book on stationary stochastic processes, with scientists or engineers as audience in mind. There is a careful balance between mathematical rigor and ease of exposition, and the book reads very nicely. Various examples and applications are included along with exercises. It should prove to be a very useful book for engineers and scientists working in the area, but is also interesting for mathematicians as it contains various examples and applications and provides a quick introduction to the theory."

—Alexander M. Lindner, Mathematical Reviews Clippings, June 2015

"… an introduction book to time series analysis … suitable for a one-semester course. It is a well-written and easy to read book which is illustrated by lots of examples. The book is mainly intended for students in science and engineering, but it is also a good source of information for researchers who want to learn about time series analysis."

—Miroslav M. Ristic (Niš), Zentralblatt MATH 1312 — 1

Table of Contents

Stochastic Processes

Some stochastic models

Definition of a stochastic process

Distribution functions

Stationary Processes


Moment functions

Stationary processes

Random phase and amplitude

Estimation of mean value and covariance function

Stationary processes and the non-stationary reality

Monte Carlo simulation from covariance function

The Poisson Process and Its Relatives


The Poisson process

Stationary independent increments

The covariance intensity function

Spatial Poisson process

Inhomogeneous Poisson process

Monte Carlo simulation of Poisson processes

Spectral Representations


Spectrum in continuous time

Spectrum in discrete time

Sampling and the aliasing effect

A few more remarks and difficulties

Monte Carlo simulation from spectrum

Gaussian Processes


Gaussian processes

The Wiener process

Relatives of the Gaussian process

The Lévy process and shot noise process

Simulation of Gaussian process from spectrum

Linear Filters—General Theory


Linear systems and linear filters

Continuity, differentiation, integration

White noise in continuous time

Cross-covariance and cross-spectrum

AR, MA, and ARMA Models


Auto-regression and moving average

Estimation of AR parameters

Prediction in AR and ARMA models

A simple non-linear model—the GARCH process

Monte Carlo simulation of ARMA processes

Linear Filters—Applications


Differential equations with random input

The envelope

Matched filter

Wiener filter

Kalman filter

An example from structural dynamics

Monte Carlo simulation in continuous time

Frequency Analysis and Spectral Estimation


The periodogram

The discrete Fourier transform and the FFT

Bias reduction—data windowing

Reduction of variance

Appendix A: Some Probability and Statistics

Appendix B: Delta Functions and Stieltjes Integrals

Appendix C: Kolmogorov’s Existence Theorem

Appendix D: Covariance/Spectral Density Pairs

Appendix E: A Historical Background



Exercises appear at the end of each chapter.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General
MATHEMATICS / Probability & Statistics / Bayesian Analysis