Random process analysis (RPA) is used as a mathematical model in physics, chemistry, biology, computer science, information theory, economics, environmental science, and many other disciplines. Over time, it has become more and more important for the provision of computer code and data sets. This book presents the key concepts, theory, and computer code written in R, helping readers with limited initial knowledge of random processes to become confident in their understanding and application of these principles in their own research. Consistent with modern trends in university education, the authors make readers active learners with hands-on computer experiments in R code directing them through RPA methods and helping them understand the underlying logic.
Each subject is illustrated with real data collected in experiments performed by the authors or taken from key literature. As a result, the reader can promptly apply the analysis to their own data, making this book an invaluable resource for undergraduate and graduate students, as well as professionals, in physics, engineering, biophysical and environmental sciences, economics, and social sciences.
Author(s): Marco Bittelli, Roberto Olmi, Rodolfo Rosa
Publisher: Oxford University Press
Year: 2023
Language: English
Pages: 512
City: Oxford
Cover
Titlepage
Copyright
Dedication
Preface
Acknowledgements
Contents
Introduction
Historical Background
The Philosopher and the Gambler
Comments
Exercises
Introduction to Stochastic Processes
Basic notion
Stationary processes
Ergodic processes
Markov processes
Predicting the future
Stationarity
Classification of states
Periodic and aperiodic states
Stopping time and other relevant random times
Strong Markov property
Recurrent and transient states
Mean recurrence time and stationary distribution
Sojourn time
Summing up
Continuous-time Markov chain
Matrix transition probability function
Transition intensity matrix
Embedded matrix
Poisson retrouvé
Birth-death process
Probability and determinism: the Buffon's needle
Ehrenfest urn model
Exercises
Poisson Processes
Counting process
Poisson process from counting process
Poisson process from Bernoulli process
Poisson process through the inter-arrival time
Poisson processes simulations
Merging of independent Poisson processes
Nonhomogeneous Poisson process
Exercises
Random Walk
Definitions and examples
Barriers
Gambler's ruin
Reflecting barriers
Two-dimensional random walk
Some topics on Brownian motion
Brownian motion as limit of random walks
Exercises
ARMA Processes
White noise and other useful definitions
The lag operator
Moving-average processes
Moving-average processes of higher order
Autoregressive processes
Low-order autoregressive processes
Autocorrelation structure and model analysis
Autoregressive moving-average processes (ARMA)
An introduction to non stationary and seasonal time series
Integrated ARMA models
Seasonal ARIMA models
An example
A physical application
Runoff-rainfall relationship in a real case: the Loire river
Exercises
Spectrum Analysis
Spectrum of stochastic signals
Periodogram and power spectral density (PSD) estimation
Consistent estimation of power spectral density
Noise spectrum
Red and blue spectrum
Applications of spectrum analysis
Searching for hidden periodicity
Singular Spectrum Analysis
Application to real data: the average temperatures in Switzerland in one century
An SSA application for beer lovers
Exercises
Markov Chain Monte Carlo
Mother Nature's minimization algorithm
From physical birth to statistical development
The travelling salesman problem
Exercises
Bayesian Inference and Stochastic Processes
Application of MCMC in a regression problem with auto-correlated errors
MCMC implementation of Bayesian regression
Bayesian spectral analysis applied to RADAR target detection
Bayesian analysis of a Poisson process: the waiting-time paradox
Bayesian analysis applied to a lighthouse
Description
Solution
Numerical procedure
Results
Exercises
Genetic algorithms: an evolutionary-based global random search
Introduction
Terminology and basics of GA
Biological terms
Representation of the tentative solutions
Genetic operators
Simple genetic algorithm
GA at work: selection and reproduction
An optimization problem: the Prof. Koza fast-food chain
Schemata theory. In other words, why genetic algorithms work
The schemata theorem
A simple application: non linear fitting
Solution using a standard method
Genetic solution
Advanced genetic algorithms
Elitism
Inseminated and variable-size populations
Other genetic operators
Real coded GA
Parameter estimation of ARMA models
Solving the travelling salesman problem
Concluding remarks
Exercises
The Problem of Accuracy
Estimating accuracy
Averaging time series
The batch means method
The moving block bootstrap method
Introduction to the MBB
The MBB in R
Convergence diagnostic with the MBB method
The Gelman and Rubin method
Exercises
Spatial Analysis
Geostatistical perspective
Stationarity in spatial processes
Correlation coefficient and correlogram
Semivariogram
Variogram model
Spatial prediction
Kriging
Spacetime analysis
On the optimization of the spatio-temporal variogram
Exercises
How Random is a Random Process?
Random hints about randomness
Characterizing mathematical randomness
Randomness and complexity
Entropy
Shannon's entropy
Sƿ Entropy
Approximated entropy
A final note
Appendix A Bootstrap
Bootstrap standard error
Parametric bootstrap
Appendix B JAGS
The JAGS language
Extracting samples from a distribution
Regression example
List of Symbols
List of R Codes
References
Index
