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Introductory Time Series with R.pdf

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    Fotografía de página completa.pdf
    00.pdf
    Preface
    Contents
    01.pdf
    Time Series Data
    Purpose
    Time series
    R language
    Plots, trends, and seasonal variation
    A flying start: Air passenger bookings
    Unemployment: Maine
    Multiple time series: Electricity, beer and chocolate data
    Quarterly exchange rate: GBP to NZ dollar
    Global temperature series
    Decomposition of series
    Notation
    Models
    Estimating trends and seasonal effects
    Smoothing
    Decomposition in R
    Summary of commands used in examples
    Exercises
    02.pdf
    Correlation
    Purpose
    Expectation and the ensemble
    Expected value
    The ensemble and stationarity
    Ergodic series*
    Variance function
    Autocorrelation
    The correlogram
    General discussion
    Example based on air passenger series
    Example based on the Font Reservoir series
    Covariance of sums of random variables
    Summary of commands used in examples
    Exercises
    03.pdf
    Forecasting Strategies
    Purpose
    Leading variables and associated variables
    Marine coatings
    Building approvals publication
    Gas supply
    Bass model
    Background
    Model definition
    Interpretation of the Bass model*
    Example
    Exponential smoothing and the Holt-Winters method
    Exponential smoothing
    Holt-Winters method
    Four-year-ahead forecasts for the air passenger data
    Summary of commands used in examples
    Exercises
    04.pdf
    Basic Stochastic Models
    Purpose
    White noise
    Introduction
    Definition
    Simulation in R
    Second-order properties and the correlogram
    Fitting a white noise model
    Random walks
    Introduction
    Definition
    The backward shift operator
    Random walk: Second-order properties
    Derivation of second-order properties*
    The difference operator
    Simulation
    Fitted models and diagnostic plots
    Simulated random walk series
    Exchange rate series
    Random walk with drift
    Autoregressive models
    Definition
    Stationary and non-stationary AR processes
    Second-order properties of an AR(1) model
    Derivation of second-order properties for an AR(1) process*
    Correlogram of an AR(1) process
    Partial autocorrelation
    Simulation
    Fitted models
    Model fitted to simulated series
    Exchange rate series: Fitted AR model
    Global temperature series: Fitted AR model
    Summary of R commands
    Exercises
    05.pdf
    Regression
    Purpose
    Linear models
    Definition
    Stationarity
    Simulation
    Fitted models
    Model fitted to simulated data
    Model fitted to the temperature series (1970--2005)
    Autocorrelation and the estimation of sample statistics*
    Generalised least squares
    GLS fit to simulated series
    Confidence interval for the trend in the temperature series
    Linear models with seasonal variables
    Introduction
    Additive seasonal indicator variables
    Example: Seasonal model for the temperature series
    Harmonic seasonal models
    Simulation
    Fit to simulated series
    Harmonic model fitted to temperature series (1970--2005)
    Logarithmic transformations
    Introduction
    Example using the air passenger series
    Non-linear models
    Introduction
    Example of a simulated and fitted non-linear series
    Forecasting from regression
    Introduction
    Prediction in R
    Inverse transform and bias correction
    Log-normal residual errors
    Empirical correction factor for forecasting means
    Example using the air passenger data
    Summary of R commands
    Exercises
    06.pdf
    Stationary Models
    Purpose
    Strictly stationary series
    Moving average models
    MA(q) process: Definition and properties
    R examples: Correlogram and simulation
    Fitted MA models
    Model fitted to simulated series
    Exchange rate series: Fitted MA model
    Mixed models: The ARMA process
    Definition
    Derivation of second-order properties*
    ARMA models: Empirical analysis
    Simulation and fitting
    Exchange rate series
    Electricity production series
    Wave tank data
    Summary of R commands
    Exercises
    07.pdf
    Non-stationary Models
    Purpose
    Non-seasonal ARIMA models
    Differencing and the electricity series
    Integrated model
    Definition and examples
    Simulation and fitting
    IMA(1, 1) model fitted to the beer production series
    Seasonal ARIMA models
    Definition
    Fitting procedure
    ARCH models
    S&P500 series
    Modelling volatility: Definition of the ARCH model
    Extensions and GARCH models
    Simulation and fitted GARCH model
    Fit to S&P500 series
    Volatility in climate series
    GARCH in forecasts and simulations
    Summary of R commands
    Exercises
    08.pdf
    Long-Memory Processes
    Purpose
    Fractional differencing
    Fitting to simulated data
    Assessing evidence of long-term dependence
    Nile minima
    Bellcore Ethernet data
    Bank loan rate
    Simulation
    Summary of additional commands used
    Exercises
    09.pdf
    Spectral Analysis
    Purpose
    Periodic signals
    Sine waves
    Unit of measurement of frequency
    Spectrum
    Fitting sine waves
    Sample spectrum
    Spectra of simulated series
    White noise
    AR(1): Positive coefficient
    AR(1): Negative coefficient
    AR(2)
    Sampling interval and record length
    Nyquist frequency
    Record length
    Applications
    Wave tank data
    Fault detection on electric motors
    Measurement of vibration dose
    Climatic indices
    Bank loan rate
    Discrete Fourier transform (DFT)*
    The spectrum of a random process*
    Discrete white noise
    AR
    Derivation of spectrum
    Autoregressive spectrum estimation
    Finer details
    Leakage
    Confidence intervals
    Daniell windows
    Padding
    Tapering
    Spectral analysis compared with wavelets
    Summary of additional commands used
    Exercises
    10.pdf
    System Identification
    Purpose
    Identifying the gain of a linear system
    Linear system
    Natural frequencies
    Estimator of the gain function
    Spectrum of an AR(p) process
    Simulated single mode of vibration system
    Ocean-going tugboat
    Non-linearity
    Exercises
    11.pdf
    Multivariate Models
    Purpose
    Spurious regression
    Tests for unit roots
    Cointegration
    Definition
    Exchange rate series
    Bivariate and multivariate white noise
    Vector autoregressive models
    VAR model fitted to US economic series
    Summary of R commands
    Exercises
    12.pdf
    State Space Models
    Purpose
    Linear state space models
    Dynamic linear model
    Filtering*
    Prediction*
    Smoothing*
    Fitting to simulated univariate time series
    Random walk plus noise model
    Regression model with time-varying coefficients
    Fitting to univariate time series
    Bivariate time series -- river salinity
    Estimating the variance matrices
    Discussion
    Summary of additional commands used
    Exercises
    13.pdf
    References
    Index
    Use R! Advisors: Robert Gentleman Kurt Hornik Giovanni Parmigiani For other titles published in this series, go to http://www.springer.com/series/6991
    Paul S.P. Cowpertwait · Andrew V. Metcalfe Introductory Time Series with R 123
    Paul S.P. Cowpertwait Inst. Information and Mathematical Sciences Massey University Auckland Albany Campus New Zealand p.s.cowpertwait@massey.ac.nz Andrew V. Metcalfe School of Mathematical Sciences University of Adelaide Adelaide SA 5005 Australia andrew.metcalfe@adelaide.edu.au Series Editors Robert Gentleman Program in Computational Biology Division of Public Health Sciences Fred Hutchinson Cancer Research Center 1100 Fairview Avenue, N. M2-B876 Seattle, Washington 98109 USA Kurt Hornik Department of Statistik and Mathematik Wirtschaftsuniversit¨at Wien Augasse 2-6 A-1090 Wien Austria Giovanni Parmigiani The Sidney Kimmel Comprehensive Cancer Center at Johns Hopkins University 550 North Broadway Baltimore, MD 21205-2011 USA ISBN 978-0-387-88697-8 DOI 10.1007/978-0-387-88698-5 Springer Dordrecht Heidelberg London New York e-ISBN 978-0-387-88698-5 Library of Congress Control Number: 2009928496 c⃝ Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
    In memory of Ian Cowpertwait
    Preface R has a command line interface that o↵ers considerable advantages over menu systems in terms of eciency and speed once the commands are known and the language understood. However, the command line system can be daunting for the first-time user, so there is a need for concise texts to enable the student or analyst to make progress with R in their area of study. This book aims to fulfil that need in the area of time series to enable the non-specialist to progress, at a fairly quick pace, to a level where they can confidently apply a range of time series methods to a variety of data sets. The book assumes the reader has a knowledge typical of a first-year university statistics course and is based around lecture notes from a range of time series courses that we have taught over the last twenty years. Some of this material has been delivered to post- graduate finance students during a concentrated six-week course and was well received, so a selection of the material could be mastered in a concentrated course, although in general it would be more suited to being spread over a complete semester. The book is based around practical applications and generally follows a similar format for each time series model being studied. First, there is an introductory motivational section that describes practical reasons why the model may be needed. Second, the model is described and defined in math- ematical notation. The model is then used to simulate synthetic data using R code that closely reflects the model definition and then fitted to the syn- thetic data to recover the underlying model parameters. Finally, the model is fitted to an example historical data set and appropriate diagnostic plots given. By using R, the whole procedure can be reproduced by the reader, and it is recommended that students work through most of the examples.1 Mathematical derivations are provided in separate frames and starred sec- 1 We used the R package Sweave to ensure that, in general, your code will produce the same output as ours. However, for stylistic reasons we sometimes edited our code; e.g., for the plots there will sometimes be minor di↵erences between those generated by the code in the text and those shown in the actual figures. vii
    viii Preface tions and can be omitted by those wanting to progress quickly to practical applications. At the end of each chapter, a concise summary of the R com- mands that were used is given followed by exercises. All data sets used in the book, and solutions to the odd numbered exercises, are available on the website http://www.massey.ac.nz/⇠pscowper/ts. We thank John Kimmel of Springer and the anonymous referees for their helpful guidance and suggestions, Brian Webby for careful reading of the text and valuable comments, and John Xie for useful comments on an earlier draft. The Institute of Information and Mathematical Sciences at Massey Univer- sity and the School of Mathematical Sciences, University of Adelaide, are acknowledged for support and funding that made our collaboration possible. Paul thanks his wife, Sarah, for her continual encouragement and support during the writing of this book, and our son, Daniel, and daughters, Lydia and Louise, for the joy they bring to our lives. Andrew thanks Natalie for providing inspiration and her enthusiasm for the project. Paul Cowpertwait and Andrew Metcalfe Massey University, Auckland, New Zealand University of Adelaide, Australia December 2008
    Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 1 Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 R language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Plots, trends, and seasonal variation . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.1 A flying start: Air passenger bookings . . . . . . . . . . . . . . . . 1.4.2 Unemployment: Maine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.3 Multiple time series: Electricity, beer and chocolate data 10 1.4.4 Quarterly exchange rate: GBP to NZ dollar . . . . . . . . . . . 14 1.4.5 Global temperature series . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Decomposition of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.3 Estimating trends and seasonal e↵ects . . . . . . . . . . . . . . . 20 1.5.4 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5.5 Decomposition in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Summary of commands used in examples . . . . . . . . . . . . . . . . . . . 24 1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Expectation and the ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Expected value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.2 The ensemble and stationarity . . . . . . . . . . . . . . . . . . . . . . 30 2.2.3 Ergodic series* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.4 Variance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.5 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ix
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