Librería Portfolio

Features

Explains the relationship between a covariance function and spectral density
Illustrates the difference between Fourier analysis of data and Fourier transformation of a covariance function
Covers AR, MA, ARMA, and GARCH models
Details covariance and spectral estimation
Shows how stochastic processes act in linear filters, including the matched, Wiener, and Kalman filters
Describes Monte Carlo simulations of different types of processes
Includes many examples from applied fields as well as exercises that highlight both the theory and practical situations in discrete and continuous time
Provides solutions to exercises and MATLAB code with examples and data on the first author´s website
Summary

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.



Table of Contents

Stochastic Processes
Some stochastic models
Definition of a stochastic process
Distribution functions

Stationary Processes
Introduction
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
Introduction
The Poisson process
Stationary independent increments
The covariance intensity function
Spatial Poisson process
Inhomogeneous Poisson process
Monte Carlo simulation of Poisson processes

Spectral Representations
Introduction
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
Introduction
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
Introduction
Linear systems and linear filters
Continuity, differentiation, integration
White noise in continuous time
Cross-covariance and cross-spectrum

AR, MA, and ARMA Models
Introduction
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
Introduction
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
Introduction
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

References

Index

Exercises appear at the end of each chapter.