Random Process Analysis With R

Abstract The book targets professionals and graduate students in physics, chemistry, biology, computer science, information theory, economics, environmental science and others, as an introduction to Random Processes Analysis (RPA) using R computer language. It seeks to put RPA within the operational reach of the projected audience, and provides readers with hands-on practical experience in applying RPA with simple numerical examples and applications taken from the targeted disciplines. The book is organized around a practical framework for soundly applying RPA in empirical work. First, consistent with modern trends in university instruction, the book make readers active learners. Second, the book provides readers with an explicit framework—condensed from sound empirical practices recommended in the literature—that details a step-by-step procedure for applying RPA in real-world data application of RPA concepts. Therefore, this book is intended to present concepts, theory and computer code written in R, that helps readers with limited initial knowledge of RPA to become operational with the material. Each subject is described and problems are implemented in the R code, with real data collected in experiments performed by the authors or taken from the literature. Various subjects are described such as a Poisson processes, Markov chains, Random walk, Spectrum Analysis, Montecarlo, Bayesian inference, Genetic Algorithms and Spatial Analysis. The book ends with a Chapter addressing randomness from a mathematical and philosophical standpoint. 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 "The book targets professionals and graduate students in physics, chemistry, biology, computer science, information theory, economics, environmental science and others, as an introduction to Random Processes Analysis (RPA) using R computer language. It seeks to put RPA within the operational reach of the projected audience, and provides readers with hands-on practical experience in applying RPA with simple numerical examples and applications taken from the targeted disciplines. The book is organized around a practical framework for soundly applying RPA in empirical work. First, consistent with modern trends in university instruction, the book make readers active learners. Second, the book provides readers with an explicit framework-condensed from sound empirical practices recommended in the literature-that details a step-by-step procedure for applying RPA in real-world data application of RPA concepts. Therefore, this book is intended to present concepts, theory and computer code written in R, that helps readers with limited initial knowledge of RPA to become operational with the material. Each subject is described and problems are implemented in the R code, with real data collected in experiments performed by the authors or taken from the literature. Various subjects are described such as a Poisson processes, Markov chains, Random walk, Spectrum Analysis, Montecarlo, Bayesian inference, Genetic Algorithms and Spatial Analysis. The book ends with a chapter addressing randomness from a mathematical and philosophical standpoint"-- Provided by publisher 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.