TIME SERIES ANALYSIS AND FORECASTING BY EXAMPLE TIME SERIES ANALYSIS AND FORECASTING BY EXAMPLE Søren Bisgaard Murat Kulahci Technical University of Denmark A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2011 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions 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support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic formats For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Bisgaard, Søren, 1938a Time series analysis and forecasting by example / Søren Bisgaard, Murat Kulahci a p cm (Wiley series in probability and statistics) a Includes bibliographical references and index a ISBN 978-0-470-54064-0 (cloth) Time-series analysis Forecasting I Kulahci, Murat II Title a QA280.B575 2011 a 519.5 dc22 2010048281 Printed in Singapore oBook ISBN: 978-1-118-05694-3 ePub ISBN: 978-1-118-05695-0 10 To the memory of Søren Bisgaard CONTENTS Preface 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 TIME SERIES DATA: EXAMPLES AND BASIC CONCEPTS Introduction Examples of Time Series Data Understanding Autocorrelation The Wold Decomposition The Impulse Response Function Superposition Principle Parsimonious Models Exercises VISUALIZING TIME SERIES DATA STRUCTURES: GRAPHICAL TOOLS Introduction Graphical Analysis of Time Series Graph Terminology Graphical Perception Principles of Graph Construction Aspect Ratio Time Series Plots Bad Graphics Exercises STATIONARY MODELS Basics of Stationary Time Series Models Autoregressive Moving Average (ARMA) Models Stationarity and Invertibility of ARMA Models Checking for Stationarity using Variogram Transformation of Data Exercises NONSTATIONARY MODELS Introduction Detecting Nonstationarity Autoregressive Integrated Moving Average (ARIMA) Models Forecasting using ARIMA Models Example 2: Concentration Measurements from a Chemical Process xi 1 10 12 14 15 18 19 21 21 22 23 24 28 30 34 38 46 47 47 54 62 66 69 73 79 79 79 83 91 93 vii viii CONTENTS 4.6 The EWMA Forecast Exercises 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 SEASONAL MODELS Seasonal Data Seasonal ARIMA Models Forecasting using Seasonal ARIMA Models Example 2: Company X’s Sales Data Exercises TIME SERIES MODEL SELECTION Introduction Finding the “BEST” Model Example: Internet Users Data Model Selection Criteria Impulse Response Function to Study the Differences in Models Comparing Impulse Response Functions for Competing Models ARIMA Models as Rational Approximations AR Versus Arma Controversy Final Thoughts on Model Selection Appendix 6.1: How to Compute Impulse Response Functions with a Spreadsheet Exercises ADDITIONAL ISSUES IN ARIMA MODELS Introduction Linear Difference Equations Eventual Forecast Function Deterministic Trend Models Yet Another Argument for Differencing Constant Term in ARIMA Models Cancellation of Terms in ARIMA Models Stochastic Trend: Unit Root Nonstationary Processes Overdifferencing and Underdifferencing Missing Values in Time Series Data Exercises TRANSFER FUNCTION MODELS Introduction Studying Input–Output Relationships Example 1: The Box–Jenkins’ Gas Furnace Spurious Cross Correlations Prewhitening Identification of the Transfer Function Modeling the Noise The General Methodology for Transfer Function Models 103 104 111 111 116 124 126 152 155 155 155 156 163 166 169 170 171 173 173 174 177 177 177 183 187 189 190 191 194 195 197 201 203 203 203 204 207 207 213 215 222 CONTENTS 8.9 Forecasting Using Transfer Function–Noise Models 8.10 Intervention Analysis Exercises ADDITIONAL TOPICS 9.1 9.2 9.3 9.4 9.5 Spurious Relationships Autocorrelation in Regression Process Regime Changes Analysis of Multiple Time Series Structural Analysis of Multiple Time Series Exercises Appendix A Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.11 A.12 A.13 A.14 A.15 Appendix B Table Table Table Table Table Table Table Table B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 Table B.9 Table Table Table Table Table Table B.10 B.11 B.12 B.13 B.14 B.15 DATASETS USED IN THE EXAMPLES Temperature Readings from a Ceramic Furnace Chemical Process Temperature Readings Chemical Process Concentration Readings International Airline Passengers Company X’s Sales Data Internet Users Data Historical Sea Level (mm) Data in Copenhagen, Denmark Gas Furnace Data Sales with Leading Indicator Crest/Colgate Market Share Simulated Process Data Coen et al (1969) Data Temperature Data from a Ceramic Furnace Temperature Readings from an Industrial Process US Hog Series DATASETS USED IN THE EXERCISES Beverage Amount (ml) Pressure of the Steam Fed to a Distillation Column (bar) Number of Paper Checks Processed in a Local Bank Monthly Sea Levels in Los Angeles, California (mm) Temperature Readings from a Chemical Process (◦ C) Daily Average Exchange Rates between US Dollar and Euro Monthly US Unemployment Rates Monthly Residential Electricity Sales (MWh) and Average Residential Electricity Retail Price (c/kWh) in the United States Monthly Outstanding Consumer Credits Provided by Commercial Banks in the United States (million USD) 100 Observations Simulated from an ARMA (1, 1) Process Quarterly Rental Vacancy Rates in the United States Wăolfer Sunspot Numbers Viscosity Readings from a Chemical Process UK Midyear Population Unemployment and GDP data for the United Kingdom ix 224 238 261 263 263 271 278 285 296 310 311 312 313 314 315 316 317 317 318 319 320 322 323 324 325 326 327 328 329 330 331 334 335 336 337 340 342 343 344 345 346 347 x CONTENTS Table B.16 Table B.17 Monthly Crude Oil Production of OPEC Nations Quarterly Dollar Sales of Marshall Field & Company ($1000) 348 360 Bibliography 361 Index 365 PREFACE Data collected in time often shows serial dependence This, however, violates one of the most fundamental assumptions in our elementary statistics courses where data is usually assumed to be independent Instead, such data should be treated as a time series and analyzed accordingly It has, unfortunately, been our experience that many practitioners found time series analysis techniques and their applications complicated and subsequently were left frustrated Recent advances in computer technology offer some help Nowadays, most statistical software packages can be used to apply many techniques we cover in this book These often user-friendly software packages help the spreading of the use of time series analysis and forecasting tools Although we wholeheartedly welcome this progress, we also believe that statistics welcomes and even requires the input from the analyst who possesses the knowledge of the system being analyzed as well as the shortfalls of the statistical techniques being used in this analysis This input can only enhance the learning experience and improve the final analysis Another important characteristic of time series analysis is that it is best learned by applications (as George Box used to say for statistical methods in general) akin to learning how to swim One can read all the theoretical background on the mechanics of swimming, yet the real learning and joy can only begin when one is in the water struggling to stay afloat and move forward The real joy of statistics comes out with the discovery of the hidden information in the data during the application Time series analysis is no different It is with all these ideas/concerns in mind that Søren and I wrote our first Quality Quandaries in Quality Engineering in 2005 It was about how the stability of processes can be checked using the variogram This led to a series of Quality Quandaries on various topics in time series analysis The main focus has always been to explain a seemingly complicated issue in time series analysis by providing the simple intuition behind it with the help of a numerical example These articles were quite well received and we decided to write a book The challenge was to make a stand-alone book with just enough theory to make the reader grasp the explanations provided with the example from the Quality Quandaries Therefore, we added the necessary amount of theory to the book as the foundation while focusing on explaining the topics through examples In that sense, some readers may find the general presentation approach of this book somewhat unorthodox We believe, however, that this informal and intuition-based approach will help the readers see the time series analysis for what it really is—a fantastic tool of discovery and learning for real-life applications As mentioned earlier, throughout this book, we try to keep the theory to an absolute minimum and whenever more theory is needed, we refer to the seminal xi 316 APPENDIX A DATASETS USED IN THE EXAMPLES TABLE A.5 Company X’s Sales Data Year Month 1965 1965 1965 1965 1965 1965 1965 1965 1965 1965 1965 1965 1966 1966 1966 1966 1966 1966 1966 1966 1966 1966 1966 1966 1967 1967 1967 1967 1967 1967 1967 1967 1967 1967 1967 1967 January February March April May June July August September October November December January February March April May June July August September October November December January February March April May June July August September October November December Sales Year Month 154 96 73 49 36 59 95 169 210 278 298 245 200 118 90 79 78 91 167 169 289 347 375 203 223 104 107 85 75 99 135 211 335 460 488 326 1968 1968 1968 1968 1968 1968 1968 1968 1968 1968 1968 1968 1969 1969 1969 1969 1969 1969 1969 1969 1969 1969 1969 1969 1970 1970 1970 1970 1970 1970 1970 1970 1970 1970 1970 1970 January February March April May June July August September October November December January February March April May June July August September October November December January February March April May June July August September October November December Source: Chatfield and Prothero (1973) Sales Year Month 346 261 224 141 148 145 223 272 445 560 612 467 518 404 300 210 196 186 247 343 464 680 711 610 613 392 273 322 189 257 324 404 677 858 895 664 1971 1971 1971 1971 1971 January February March April May Sales 628 308 324 248 272 APPENDIX A DATASETS USED IN THE EXAMPLES 317 TABLE A.6 Internet Users Data 88 84 85 85 84 85 83 85 88 89 91 99 104 112 126 138 146 151 150 148 147 149 143 132 131 139 147 150 148 145 140 134 131 131 129 126 126 132 137 140 142 150 159 167 170 171 172 172 174 175 172 172 174 174 169 165 156 142 131 121 112 104 102 99 99 95 88 84 84 87 89 88 85 86 89 91 91 94 101 110 121 135 145 149 156 165 171 175 177 182 193 204 208 210 215 222 228 226 222 220 Source: Makridakis et al (2003) TABLE A.7 Historical Sea Level (mm) Data in Copenhagen, Denmark Year Sea Level Year Sea Level Year Sea Level Year Sea Level 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 6939 6977 6933 6949 6986 6950 6966 6952 6952 7009 7051 6947 6927 6950 7021 6943 6976 6982 6960 6921 6964 6962 6993 7000 7030 7011 6945 6994 6956 6951 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 6936 6943 7022 6992 7006 6958 7027 6991 7008 6980 6973 6956 6992 6999 6946 6979 6997 6970 6924 7024 6924 6940 6921 6951 7028 6994 7027 6999 6913 7016 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 7035 6988 6911 6999 7001 6949 7011 6996 7009 6976 6948 6912 7043 7004 6930 6972 6972 6981 7065 6973 6955 6981 7000 6921 7003 6960 6992 6954 6975 6995 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 — — 6967 6968 7061 7012 — 6966 6994 6997 6962 7014 7072 7060 — 7066 6976 6994 7033 6926 7019 7057 — 7038 — 7056 7006 7092 7043 7002 — — Source: www.psmsl.org 318 APPENDIX A DATASETS USED IN THE EXAMPLES TABLE A.8 Gas Furnace Data Xt –0.109 0.000 0.178 0.339 0.373 0.441 0.461 0.348 0.127 –0.180 –0.588 –1.055 –1.421 –1.520 –1.302 –0.814 –0.475 –0.193 0.088 0.435 0.771 0.866 0.875 0.891 0.987 1.263 1.775 1.976 1.934 1.866 1.832 1.767 1.608 1.265 0.790 0.360 0.115 0.088 0.331 0.645 0.960 1.409 2.670 2.834 2.812 2.483 1.929 1.485 1.214 1.239 Yt Xt Yt Xt Yt Xt Yt Xt Yt Xt Yt 53.8 53.6 53.5 53.5 53.4 53.1 52.7 52.4 52.2 52.0 52.0 52.4 53.0 54.0 54.9 56.0 56.8 56.8 56.4 55.7 55.0 54.3 53.2 52.3 51.6 51.2 50.8 50.5 50.0 49.2 48.4 47.9 47.6 47.5 47.5 47.6 48.1 49.0 50.0 51.1 51.8 51.9 51.7 51.2 50.0 48.3 47.0 45.8 45.6 46.0 1.608 1.905 2.023 1.815 0.535 0.122 0.009 0.164 0.671 1.019 1.146 1.155 1.112 1.121 1.223 1.257 1.157 0.913 0.620 0.255 –0.280 –1.080 –1.551 –1.799 –1.825 –1.456 –0.944 –0.570 –0.431 –0.577 –0.960 –1.616 –1.875 –1.891 –1.746 –1.474 –1.201 –0.927 –0.524 0.040 0.788 0.943 0.930 1.006 1.137 1.198 1.054 0.595 –0.080 –0.314 46.9 47.8 48.2 48.3 47.9 47.2 47.2 48.1 49.4 50.6 51.5 51.6 51.2 50.5 50.1 49.8 49.6 49.4 49.3 49.2 49.3 49.7 50.3 51.3 52.8 54.4 56.0 56.9 57.5 57.3 56.6 56.0 55.4 55.4 56.4 57.2 58.0 58.4 58.4 58.1 57.7 57.0 56.0 54.7 53.2 52.1 51.6 51.0 50.5 50.4 –0.288 –0.153 –0.109 –0.187 –0.255 –0.229 –0.007 0.254 0.330 0.102 –0.423 –1.139 –2.275 –2.594 –2.716 –2.510 –1.790 –1.346 –1.081 –0.910 –0.876 –0.885 –0.800 –0.544 –0.416 –0.271 0.000 0.403 0.841 1.285 1.607 1.746 1.683 1.485 0.993 0.648 0.577 0.577 0.632 0.747 0.900 0.993 0.968 0.790 0.399 –0.161 –0.553 –0.603 –0.424 –0.194 51.0 51.8 52.4 53.0 53.4 53.6 53.7 53.8 53.8 53.8 53.3 53.0 52.9 53.4 54.6 56.4 58.0 59.4 60.2 60.0 59.4 58.4 57.6 56.9 56.4 56.0 55.7 55.3 55.0 54.4 53.7 52.8 51.6 50.6 49.4 48.8 48.5 48.7 49.2 49.8 50.4 50.7 50.9 50.7 50.5 50.4 50.2 50.4 51.2 52.3 –0.049 0.060 0.161 0.301 0.517 0.566 0.560 0.573 0.592 0.671 0.933 1.337 1.460 1.353 0.772 0.218 –0.237 –0.714 –1.099 –1.269 –1.175 –0.676 0.033 0.556 0.643 0.484 0.109 –0.310 –0.697 –1.047 –1.218 –1.183 –0.873 –0.336 0.063 0.084 0.000 0.001 0.209 0.556 0.782 0.858 0.918 0.862 0.416 –0.336 –0.959 –1.813 –2.378 –2.499 53.2 53.9 54.1 54.0 53.6 53.2 53.0 52.8 52.3 51.9 51.6 51.6 51.4 51.2 50.7 50.0 49.4 49.3 49.7 50.6 51.8 53.0 54.0 55.3 55.9 55.9 54.6 53.5 52.4 52.1 52.3 53.0 53.8 54.6 55.4 55.9 55.9 55.2 54.4 53.7 53.6 53.6 53.2 52.5 52.0 51.4 51.0 50.9 52.4 53.5 –2.473 –2.330 –2.053 –1.739 –1.261 –0.569 –0.137 –0.024 –0.050 –0.135 –0.276 –0.534 –0.871 –1.243 –1.439 –1.422 –1.175 –0.813 –0.634 –0.582 –0.625 –0.713 –0.848 –1.039 –1.346 –1.628 –1.619 –1.149 –0.488 –0.160 –0.007 –0.092 –0.620 –1.086 –1.525 –1.858 –2.029 –2.024 –1.961 –1.952 –1.794 –1.302 –1.030 –0.918 –0.798 –0.867 –1.047 –1.123 –0.876 –0.395 55.6 58.0 59.5 60.0 60.4 60.5 60.2 59.7 59.0 57.6 56.4 55.2 54.5 54.1 54.1 54.4 55.5 56.2 57.0 57.3 57.4 57.0 56.4 55.9 55.5 55.3 55.2 55.4 56.0 56.5 57.1 57.3 56.8 55.6 55.0 54.1 54.3 55.3 56.4 57.2 57.8 58.3 58.6 58.8 58.8 58.6 58.0 57.4 57.0 56.4 0.185 0.662 0.709 0.605 0.501 0.603 0.943 1.223 1.249 0.824 0.102 0.025 0.382 0.922 1.032 0.866 0.527 0.093 –0.458 –0.748 –0.947 –1.029 –0.928 –0.645 –0.424 –0.276 –0.158 –0.033 0.102 0.251 0.280 0.000 –0.493 –0.759 –0.824 –0.740 –0.528 –0.204 0.034 0.204 0.253 0.195 0.131 0.017 –0.182 –0.262 — — — — 56.3 56.4 56.4 56.0 55.2 54.0 53.0 52.0 51.6 51.6 51.1 50.4 50.0 50.0 52.0 54.0 55.1 54.5 52.8 51.4 50.8 51.2 52.0 52.8 53.8 54.5 54.9 54.9 54.8 54.4 53.7 53.3 52.8 52.6 52.6 53.0 54.3 56.0 57.0 58.0 58.6 58.5 58.3 57.8 57.3 57.0 — — — — Source: BJR series J APPENDIX A DATASETS USED IN THE EXAMPLES 319 TABLE A.9 Sales with Leading Indicator Sales Leading Indicator Sales Leading Indicator Sales Leading Indicator 200.1 199.5 199.4 198.9 199.0 200.2 198.6 200.0 200.3 201.2 201.6 201.5 201.5 203.5 204.9 207.1 210.5 210.5 209.8 208.8 209.5 213.2 213.7 215.1 218.7 219.8 220.5 223.8 222.8 223.8 221.7 222.3 220.8 219.4 220.1 220.6 218.9 217.8 217.7 215.0 215.3 215.9 216.7 216.7 217.7 218.7 222.9 224.9 222.2 220.7 10.01 10.07 10.32 9.75 10.33 10.13 10.36 10.32 10.13 10.16 10.58 10.62 10.86 11.20 10.74 10.56 10.48 10.77 11.33 10.96 11.16 11.70 11.39 11.42 11.94 11.24 11.59 10.96 11.40 11.02 11.01 11.23 11.33 10.83 10.84 11.14 10.38 10.90 11.05 11.11 11.01 11.22 11.21 11.91 11.69 10.93 10.99 11.01 10.84 10.76 220.0 218.7 217.0 215.9 215.8 214.1 212.3 213.9 214.6 213.6 212.1 211.4 213.1 212.9 213.3 211.5 212.3 213.0 211.0 210.7 210.1 211.4 210.0 209.7 208.8 208.8 208.8 210.6 211.9 212.8 212.5 214.8 215.3 217.5 218.8 220.7 222.2 226.7 228.4 233.2 235.7 237.1 240.6 243.8 245.3 246.0 246.3 247.7 247.6 247.8 10.77 10.88 10.49 10.50 11.00 10.98 10.61 10.48 10.53 11.07 10.61 10.86 10.34 10.78 10.80 10.33 10.44 10.50 10.75 10.40 10.40 10.34 10.55 10.46 10.82 10.91 10.87 10.67 11.11 10.88 11.28 11.27 11.44 11.52 12.10 11.83 12.62 12.41 12.43 12.73 13.01 12.74 12.73 12.76 12.92 12.64 12.79 13.05 12.69 13.01 249.4 249.0 249.9 250.5 251.5 249.0 247.6 248.8 250.4 250.7 253.0 253.7 255.0 256.2 256.0 257.4 260.4 260.0 261.3 260.4 261.6 260.8 259.8 259.0 258.9 257.4 257.7 257.9 257.4 257.3 257.6 258.9 257.8 257.7 257.2 257.5 256.8 257.5 257.0 257.6 257.3 257.5 259.6 261.1 262.9 263.3 262.8 261.8 262.2 262.7 12.90 13.12 12.47 12.47 12.94 13.10 12.91 13.39 13.13 13.34 13.34 13.14 13.49 13.87 13.39 13.59 13.27 13.70 13.20 13.32 13.15 13.30 12.94 13.29 13.26 13.08 13.24 13.31 13.52 13.02 13.25 13.12 13.26 13.11 13.30 13.06 13.32 13.10 13.27 13.64 13.58 13.87 13.53 13.41 13.25 13.50 13.58 13.51 13.77 13.40 Source: BJR series M (read down) 320 APPENDIX A DATASETS USED IN THE EXAMPLES TABLE A.10 Crest/Colgate Market Share Year 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 Week Crest 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0.108 0.166 0.126 0.115 0.119 0.176 0.155 0.118 0.136 0.137 0.124 0.131 0.120 0.133 0.067 0.086 0.140 0.122 0.105 0.079 0.130 0.142 0.120 0.115 0.103 0.078 0.093 0.086 0.099 0.078 0.095 0.094 0.056 0.050 0.065 0.091 0.094 0.124 0.153 0.078 0.114 0.088 Colgate Year Week Crest Colgate Year Week Crest Colgate 0.424 0.482 0.428 0.397 0.352 0.342 0.434 0.445 0.428 0.395 0.354 0.497 0.425 0.401 0.363 0.341 0.464 0.431 0.405 0.460 0.410 0.423 0.310 0.413 0.411 0.452 0.405 0.290 0.342 0.311 0.327 0.413 0.400 0.380 0.371 0.344 0.345 0.363 0.392 0.379 0.349 0.337 43 44 45 46 47 48 49 50 51 52 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0.235 0.250 0.360 0.282 0.257 0.345 0.344 0.358 0.322 0.332 0.315 0.316 0.341 0.387 0.402 0.347 0.414 0.426 0.322 0.372 0.381 0.339 0.405 0.304 0.439 0.336 0.405 0.359 0.379 0.303 0.340 0.312 0.291 0.259 0.342 0.458 0.275 0.340 0.385 0.338 0.370 0.290 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 10 11 12 13 14 15 16 17 18 19 20 21 22 0.325 0.337 0.338 0.323 0.357 0.381 0.371 0.294 0.384 0.286 0.335 0.310 0.304 0.305 0.403 0.365 0.305 0.172 0.321 0.343 0.354 0.316 0.292 0.305 0.294 0.289 0.301 0.304 0.306 0.405 0.344 0.353 0.383 0.349 0.374 0.411 0.287 0.420 0.470 0.354 0.392 0.421 1958 1958 1958 1958 1958 1958 1958 1958 1958 1958 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 0.165 0.160 0.075 0.118 0.100 0.102 0.131 0.148 0.137 0.090 0.088 0.172 0.111 0.097 0.098 0.090 0.127 0.116 0.137 0.111 0.107 0.097 0.134 0.160 0.147 0.104 0.128 0.128 0.165 0.184 0.172 0.207 0.221 0.159 0.198 0.197 0.251 0.146 0.133 0.243 0.192 0.150 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1959 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 0.221 0.183 0.136 0.206 0.127 0.139 0.189 0.194 0.114 0.229 0.148 0.155 0.106 0.156 0.053 0.112 0.084 0.191 0.149 0.143 0.094 0.184 0.205 0.206 0.191 0.195 0.179 0.272 0.203 0.165 0.138 0.216 0.132 0.120 0.083 0.118 0.125 0.109 0.119 0.154 0.122 0.126 APPENDIX A DATASETS USED IN THE EXAMPLES 321 TABLE A.10 (Continued) Year 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1960 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 Week Crest 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 10 11 12 0.126 0.130 0.158 0.141 0.145 0.127 0.171 0.152 0.211 0.309 0.242 0.380 0.362 0.328 0.359 0.352 0.322 0.333 0.365 0.367 0.305 0.298 0.307 0.318 0.280 0.298 0.336 0.339 0.344 0.310 0.317 0.369 0.320 0.290 0.361 0.235 0.320 0.337 0.289 0.339 0.187 0.414 Colgate Year 0.435 0.424 0.344 0.369 0.364 0.386 0.406 0.439 0.345 0.291 0.292 0.249 0.283 0.301 0.280 0.251 0.303 0.274 0.328 0.244 0.323 0.288 0.293 0.321 0.330 0.273 0.304 0.292 0.251 0.350 0.302 0.306 0.272 0.296 0.265 0.364 0.284 0.330 0.351 0.336 0.383 0.214 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1961 1962 1962 Week Crest 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 0.373 0.265 0.316 0.245 0.328 0.368 0.287 0.369 0.406 0.316 0.362 0.308 0.286 0.420 0.299 0.383 0.354 0.418 0.425 0.445 0.408 0.282 0.410 0.425 0.358 0.393 0.375 0.273 0.237 0.331 0.335 0.395 0.357 0.296 0.307 0.390 0.298 0.381 0.354 0.436 0.357 0.427 Colgate Year Week Crest Colgate 0.260 0.298 0.248 0.308 0.356 0.278 0.314 0.214 0.253 0.287 0.238 0.253 0.336 0.255 0.249 0.195 0.269 0.201 0.184 0.203 0.193 0.322 0.261 0.183 0.289 0.243 0.302 0.350 0.401 0.332 0.351 0.280 0.308 0.299 0.199 0.283 0.333 0.233 0.296 0.267 0.253 0.239 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 0.155 0.247 0.201 0.266 0.290 0.231 0.255 0.242 0.271 0.266 0.244 0.204 0.213 0.295 0.254 0.242 0.228 0.181 0.264 0.277 0.284 0.248 0.280 0.249 0.279 0.282 0.267 0.252 0.190 0.284 0.207 0.327 0.259 0.286 0.275 0.244 0.341 0.331 0.250 0.220 0.293 0.205 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 1962 0.432 0.450 0.530 0.431 0.420 0.411 0.423 0.433 0.393 0.389 0.387 0.439 0.421 0.363 0.401 0.394 0.459 0.441 0.388 0.373 0.385 0.314 0.347 0.408 0.341 0.361 0.414 0.380 0.274 0.352 0.439 0.355 0.435 0.408 0.383 0.357 0.374 0.366 0.346 0.381 0.329 0.474 (continued overleaf) 322 APPENDIX A DATASETS USED IN THE EXAMPLES TABLE A.10 (Continued) Year 1962 1962 1962 1962 1962 1962 1962 1962 Week Crest 45 46 47 48 49 50 51 52 0.397 0.436 0.417 0.430 0.388 0.453 0.316 0.414 Colgate Year 0.254 0.268 0.211 0.203 0.271 0.290 0.323 0.253 1963 1963 1963 1963 1963 1963 1963 1963 Week Crest 0.396 0.420 0.432 0.453 0.430 0.327 0.388 0.377 Colgate Year Week Crest Colgate 0.230 0.220 0.235 0.228 0.216 0.324 0.268 0.257 10 11 12 13 14 15 16 0.194 0.212 0.218 0.216 0.276 0.190 0.249 0.172 1963 1963 1963 1963 1963 1963 1963 1963 0.466 0.478 0.365 0.472 0.399 0.391 0.473 0.384 Source: Wichern and Jones (1977) TABLE A.11 Simulated Process Data −0.38 −0.86 −4.39 −5.43 −3.43 −1.14 0.15 6.91 2.42 3.71 4.08 9.19 6.78 −1.82 0.53 −5.11 −4.77 −4.61 −2.53 0.32 7.23 2.55 3.17 6.97 9.38 6.73 −2.52 1.77 −5.30 −4.85 −3.27 −3.43 0.45 6.54 4.13 1.15 7.67 9.62 7.12 −3.94 2.65 −4.24 −6.76 −3.01 −2.45 0.33 5.12 4.98 0.15 8.26 8.36 7.32 −5.10 1.50 −3.07 −6.57 −3.58 −0.44 0.21 5.00 4.86 −0.82 9.77 6.63 — −4.80 −1.54 −3.97 −5.07 −2.65 −0.67 0.48 2.88 4.47 −1.05 10.55 6.34 — −3.64 −2.71 −3.45 −3.79 −2.09 −0.98 1.83 1.29 2.73 0.61 10.80 5.74 — −2.14 −3.70 −3.97 −2.55 −1.56 0.17 4.38 2.53 2.25 2.21 10.03 6.39 — −0.97 2.72 4.30 6.47 9.50 13.59 7.87 11.71 −1.84 −2.75 0.57 −4.14 −1.61 −1.68 6.31 3.37 5.37 12.16 13.76 8.09 9.64 −0.63 −2.93 1.83 −4.93 — −2.40 6.83 4.37 6.19 14.48 13.29 7.38 6.93 −0.25 −3.05 0.19 −4.60 — −1.93 4.90 5.61 6.83 14.44 11.96 8.11 5.14 −1.01 −2.58 −0.59 −4.49 — −1.45 2.38 6.22 7.40 13.93 11.72 7.42 2.66 −2.15 −2.05 −3.33 −5.58 — The simulated xt data recorded row-wise −0.94 0.13 1.75 6.74 7.64 14.39 10.99 8.03 1.11 −3.41 −1.21 −4.66 −6.17 −0.89 0.87 1.71 6.38 7.60 14.36 9.72 10.02 0.62 −4.31 −1.54 −4.55 −3.93 −0.91 0.00 3.04 7.09 8.15 13.96 7.74 11.71 −0.17 −3.69 −1.41 −4.35 −2.68 The simulated yt data recorded row-wise APPENDIX A DATASETS USED IN THE EXAMPLES 323 TABLE A.12 Coen et al (1969) Data F T F T F T Commodity UK Car F T Commodity UK Car Observation Index Index Production Observation Index Index Production 1∗ 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 148.5 165.4 178.5 187.3 195.9 205.2 191.5 183 184.5 181.1 173.7 185.4 201.8 198 168 161.6 170.2 184.5 211 218.3 231.7 247.4 301.9 323.8 314.1 321 96.21 93.74 91.37 86.31 84.98 86.46 90.04 94.74 92.43 92.41 91.65 89.38 91.05 89.89 90.16 86.78 88.45 90.69 86.03 84.85 84.07 81.96 80.03 79.8 80.19 80.13 121,874 126,260 145,248 160,370 163,648 178,195 187,197 195,916 199,253 227,616 215,363 231,728 231,767 211,211 185,200 152,404 156,163 151,567 213,683 244,543 253,111 266,580 253,543 261,675 249,407 246,248 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 312.9 323.7 349.3 310.4 295.8 301.2 285.8 271.7 283.6 295.7 309.3 295.7 342 335.1 344.4 360.9 346.5 340.6 340.3 323.3 345.6 349.3 359.7 320 299.9 80.42 82.67 82.78 82.61 82.47 81.86 79.7 77.89 77.61 78.9 79.72 78.08 77.54 76.99 76.25 78.13 80.38 81.78 82.81 84.99 86.31 85.95 90.73 92.42 87.18 293,062 285,809 366,265 374,241 375,764 354,411 249,527 206,165 258,410 279,342 264,824 312,983 300,932 323,424 312,780 363,336 378,275 414,457 459,158 460,397 462,279 434,255 475,890 439,365 413,666 ∗ The first observation for Financial Times index corresponds to the second quarter of 1954, whereas the first obser- vations for Financial Times commodity index and for UK car production correspond to the third and fourth quarters of 1952, respectively 324 APPENDIX A DATASETS USED IN THE EXAMPLES TABLE A.13 Temperature Data from a Ceramic Furnace Observation x x Observation x x Observation x x Observation x x6 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 828 828 829 833 836 841 840 842 843 842 842 842 840 842 842 844 843 839 838 838 839 837 841 840 839 836 835 835 833 832 832 832 831 830 832 832 832 831 830 830 830 830 829 829 829 830 829 829 829 831 831 830 830 831 831 833 833 834 833 831 831 832 833 833 831 832 832 832 831 831 829 829 830 830 829 830 829 827 826 824 824 823 825 826 826 827 828 829 829 830 829 828 829 829 829 829 829 829 830 830 829 828 828 828 828 829 835 837 836 834 835 835 836 837 836 837 838 839 837 837 838 838 838 838 843 847 848 850 851 849 847 842 842 841 841 841 840 841 841 827 828 828 832 834 839 838 840 841 840 840 840 839 840 836 839 838 832 831 831 832 830 833 834 833 832 829 829 827 826 826 826 824 824 826 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 826 827 825 824 824 823 824 822 822 823 824 823 823 823 824 824 823 823 824 824 827 827 828 827 825 825 826 826 826 824 825 826 825 825 824 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 823 823 823 823 823 823 822 821 820 818 817 817 819 820 819 821 821 823 822 823 822 822 822 823 822 822 822 822 823 823 822 821 820 821 820 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 822 825 826 823 822 822 822 822 823 821 821 822 822 820 819 820 820 819 819 824 826 827 829 829 828 826 821 821 820 819 819 819 819 819 APPENDIX A DATASETS USED IN THE EXAMPLES TABLE A.14 Temperature Readings from an Industrial Process x1 1016.1 1017.5 1017.5 1019.2 1018.1 1017.3 1016.6 1018.2 1018.5 1018.5 1017.6 1018.5 1016.6 1017.7 1018.3 1017.5 1017.6 1018.6 1018.8 1017.0 1019.4 1020.6 1021.7 1021.8 1021.8 1021.8 1021.6 1023.3 1023.8 1023.0 1017.7 1017.6 1018.4 1017.2 1016.6 1016.7 1016.8 1016.2 1016.6 1019.0 1020.7 1019.9 1019.1 1017.8 1017.8 1017.8 1016.2 1016.5 1017.3 1014.7 x2 x3 x4 x5 1022.5 1023.8 1024.3 1024.0 1022.6 1022.3 1022.2 1021.3 1021.3 1022.1 1021.1 1021.5 1021.6 1024.5 1023.3 1022.6 1022.1 1022.3 1022.8 1022.7 1023.9 1024.1 1022.7 1023.2 1022.8 1022.8 1024.6 1025.6 1025.7 1026.9 1025.4 1023.8 1023.3 1023.8 1024.4 1024.3 1024.7 1024.9 1024.6 1023.9 1023.5 1023.4 1023.4 1024.2 1023.7 1023.7 1023.0 1022.2 1022.4 1021.7 1027.9 1029.9 1030.1 1030.9 1030.4 1031.3 1031.6 1031.5 1032.2 1032.5 1031.4 1031.1 1029.6 1029.4 1029.3 1028.8 1028.4 1028.1 1028.8 1031.7 1032.1 1031.2 1032.2 1030.5 1030.4 1030.7 1029.5 1029.1 1029.3 1029.1 1028.3 1027.3 1029.3 1027.0 1026.0 1025.3 1024.8 1024.5 1024.8 1024.1 1023.6 1024.4 1025.6 1025.8 1025.8 1023.8 1024.7 1024.9 1025.0 1026.6 1030.6 1033.4 1032.4 1032.3 1032.0 1031.5 1031.3 1032.2 1032.1 1031.8 1032.6 1032.7 1032.4 1031.5 1031.7 1031.5 1031.9 1032.5 1032.3 1032.4 1032.7 1033.4 1033.6 1032.8 1033.2 1033.2 1031.8 1032.0 1031.8 1031.8 1030.8 1030.5 1029.9 1028.4 1029.5 1028.7 1028.3 1029.2 1029.9 1030.5 1032.1 1032.2 1032.0 1031.2 1031.7 1031.0 1031.5 1032.9 1033.0 1034.3 1020.0 1022.2 1020.5 1021.1 1020.5 1020.1 1019.6 1020.4 1020.4 1019.3 1020.2 1020.7 1020.1 1019.0 1018.4 1018.2 1018.7 1019.1 1018.7 1017.6 1016.7 1015.4 1015.9 1016.4 1016.7 1016.5 1017.1 1017.4 1016.4 1016.9 1017.3 1018.6 1018.2 1015.9 1013.7 1014.1 1014.0 1012.6 1012.7 1014.6 1014.6 1012.9 1014.0 1013.3 1012.7 1012.3 1012.5 1012.0 1011.0 1011.3 325 326 APPENDIX A DATASETS USED IN THE EXAMPLES TABLE A.15 US Hog Series z1 z2 z3 z4 z5 z1 z2 z3 538 522 513 529 565 594 600 584 554 553 595 637 641 647 634 629 638 662 675 658 629 625 648 682 676 655 640 668 678 692 710 727 712 708 705 680 682 713 726 729 752 509 663 751 739 598 556 594 667 776 754 689 498 643 681 778 829 751 704 633 663 709 763 681 627 667 804 782 707 653 639 672 669 729 784 842 886 784 770 783 877 777 944 841 911 768 718 634 735 858 673 609 604 457 612 642 849 733 672 594 559 604 678 571 490 747 651 645 609 705 453 382 466 506 525 595 829 654 673 691 660 643 754 900 964 893 1051 1057 1107 1003 1025 1161 1170 1181 1194 1244 1232 1095 1244 1218 1289 1313 1251 1205 1352 1361 1218 1368 1278 1279 1208 1404 1427 1359 1371 1423 1425 1234 1443 1401 1429 1470 1482 1417 719 716 724 732 740 748 756 748 740 732 744 756 778 799 799 799 799 801 803 806 806 806 810 813 810 806 771 771 780 789 799 820 834 848 863 884 906 928 949 960 971 766 720 682 743 743 730 723 753 782 760 799 808 779 770 777 841 823 746 717 744 791 771 746 739 773 793 768 592 633 634 649 699 786 735 782 869 923 774 787 754 810 957 970 903 995 1022 998 928 1073 1294 1346 1301 1134 1024 1090 1013 1119 1195 1235 1120 1112 1129 1055 787 624 612 800 1104 1075 1052 1048 891 921 1193 1352 1243 1314 1380 1556 1632 813 790 712 831 742 847 850 830 1056 1163 1182 1180 805 714 865 911 1027 846 869 928 924 903 777 507 500 716 911 816 1019 714 687 754 791 876 962 1050 1037 1104 1193 1334 Source: Quenouille (1957) (read down) z4 1409 1417 1455 1394 1469 1357 1402 1452 1385 1464 1388 1428 1487 1467 1432 1459 1347 1447 1406 1418 1426 1401 1318 1411 1467 1380 1161 1362 1178 1422 1406 1412 1390 1424 1487 1472 1490 1458 1507 1372 z5 982 987 991 1004 1013 1004 1013 1053 1149 1248 1316 1384 1190 1179 1228 1238 1246 1253 1253 1253 1255 1223 1114 982 929 978 1013 1045 1100 1097 1090 1100 1188 1303 1422 1498 1544 1582 1607 1629 BIBLIOGRAPHY Abraham, B and J Ledolter (1983) Statistical Methods for Forecasting, New York: John Wiley and Sons Akaike, H (1974) “A new look at the statistical model identification,” IEEE Transactions Automatic Control , AC-19, pp 716–723 Anscombe, F J (1973) “Graphs in statistical analysis,” The 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integrated moving average process,” Biometrika, 60, pp 235–239 364 BIBLIOGRAPHY Wichern, D W and R H Jones (1977) “Assessing the impact of market disturbances using intervention analysis,” Management Science, 24(3), pp 329–337 Wilson, G T (1973) “Contribution to discussion of “Box–Jenkins seasonal forecasting: problems in a case study,” by C Chatfield and D L Prothero,” Journal of the Royal Statistical Society, Series A, 136, pp 315–319 Winters, P R (1960) “Forecasting sales by exponentially weighted moving averages,” Management Science, 6, pp 235–239 Wold, H O (1938) A Study in the Analysis of Stationary Time Series, 2nd Edition, 1954, Uppsala, Sweden: Almquist & Wiksell Yule, G U (1927) “On a method of investigating periodicities in disturbed series, with reference to Wolfer’s Sunspot Numbers,” Philosophical Transactions of the Royal Society of London, Series A, 226, pp 267–298 ... 40 40 Time (hours) 120 Time (hours) 120 Time (hours) 120 Time (hours) 120 Time (hours) 120 80 Time (hours) 120 Time series plot of FrontCool_11 80 Time series plot of FrontCool_9 80 Time series. .. theory and attempt to present major concepts in time series analysis via numerous examples, some of which are quite well known in the literature 1.2 EXAMPLES OF TIME SERIES DATA Examples of time series. .. 40 40 40 40 40 Time (hours) 120 Time (hours) 120 Time (hours) 120 Time (hours) 120 80 Time (hours) 120 Time series plot of FrontCool_10 80 Time series plot of FrontCool_8 80 Time series plot of