Introduction to Time Series Analysis and Forecasting E-Book

Douglas C. Montgomery

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: WILEY SERIES IN PROB & STATISTICS/see 1345/6,6214/5
• Sprache: Englisch

Seitenzahl: 856

Veröffentlichungsjahr: 2024

Beschreibung

Bring the latest statistical tools to bear on predicting future variables and outcomes

A huge range of fields rely on forecasts of how certain variables and causal factors will affect future outcomes, from product sales to inflation rates to demographic changes. Time series analysis is the branch of applied statistics which generates forecasts, and its sophisticated use of time oriented data can vastly impact the quality of crucial predictions. The latest computing and statistical methodologies are constantly being sought to refine these predictions and increase the confidence with which important actors can rely on future outcomes.

Time Series Analysis and Forecasting presents a comprehensive overview of the methodologies required to produce these forecasts with the aid of time-oriented data sets. The potential applications for these techniques are nearly limitless, and this foundational volume has now been updated to reflect the most advanced tools. The result, more than ever, is an essential introduction to a core area of statistical analysis.

Readers of the third edition of Time Series Analysis and Forecasting will also find:

• Updates incorporating JMP, SAS, and R software, with new examples throughout
• Over 300 exercises and 50 programming algorithms that balance theory and practice
• Supplementary materials in the e-book including solutions to many problems, data sets, and brand-new explanatory videos covering the key concepts and examples from each chapter.

Time Series Analysis and Forecasting is ideal for graduate and advanced undergraduate courses in the areas of data science and analytics and forecasting and time series analysis. It is also an outstanding reference for practicing data scientists.

Leseprobe

Table of Contents

COVER

TABLE OF CONTENTS

TITLE PAGE

COPYRIGHT

PREFACE

ABOUT THE COMPANION WEBSITE

CHAPTER 1: INTRODUCTION TO TIME SERIES ANALYSIS AND FORECASTING

1.1 THE NATURE AND USES OF FORECASTS

1.2 SOME EXAMPLES OF TIME SERIES

1.3 THE FORECASTING PROCESS

1.4 DATA FOR FORECASTING

1.5 RESOURCES FOR FORECASTING

EXERCISES

CHAPTER 2: STATISTICS BACKGROUND FOR TIME SERIES ANALYSIS AND FORECASTING

2.1 INTRODUCTION

2.2 GRAPHICAL DISPLAYS

2.3 NUMERICAL DESCRIPTION OF TIME SERIES DATA

2.4 USE OF DATA TRANSFORMATIONS AND ADJUSTMENTS

2.5 GENERAL APPROACH TO TIME SERIES MODELING AND FORECASTING

2.6 EVALUATING AND MONITORING FORECASTING MODEL PERFORMANCE

2.7 R COMMANDS FOR CHAPTER 2

EXERCISES

CHAPTER 3: REGRESSION ANALYSIS AND FORECASTING

3.1 INTRODUCTION

3.2 LEAST SQUARES ESTIMATION IN LINEAR REGRESSION MODELS

3.3 STATISTICAL INFERENCE IN LINEAR REGRESSION

3.4 PREDICTION OF NEW OBSERVATIONS

3.5 MODEL ADEQUACY CHECKING

3.6 VARIABLE SELECTION METHODS IN REGRESSION

3.7 GENERALIZED AND WEIGHTED LEAST SQUARES

3.8 REGRESSION MODELS FOR GENERAL TIME SERIES DATA

3.9 ECONOMETRIC MODELS

3.10 R COMMANDS FOR CHAPTER 3

EXERCISES

CHAPTER 4: EXPONENTIAL SMOOTHING METHODS

4.1 INTRODUCTION

4.2 FIRST-ORDER EXPONENTIAL SMOOTHING

4.3 MODELING TIME SERIES DATA

4.4 SECOND-ORDER EXPONENTIAL SMOOTHING

4.5 HIGHER-ORDER EXPONENTIAL SMOOTHING

4.6 FORECASTING

4.7 EXPONENTIAL SMOOTHING FOR SEASONAL DATA

4.8 EXPONENTIAL SMOOTHING OF BIOSURVEILLANCE DATA

4.9 EXPONENTIAL SMOOTHERS AND ARIMA MODELS

4.10 R COMMANDS FOR CHAPTER 4

EXERCISES

NOTE

CHAPTER 5: AUTOREGRESSIVE INTEGRATED MOVING AVERAGE (ARIMA) MODELS

5.1 INTRODUCTION

5.2 LINEAR MODELS FOR STATIONARY TIME SERIES

5.3 FINITE ORDER MOVING AVERAGE PROCESSES

5.4 FINITE ORDER AUTOREGRESSIVE PROCESSES

5.5 MIXED AUTOREGRESSIVE–MOVING AVERAGE PROCESSES

5.6 NONSTATIONARY PROCESSES

5.7 TIME SERIES MODEL BUILDING

5.8 FORECASTING ARIMA PROCESSES

5.9 SEASONAL PROCESSES

5.10 ARIMA MODELING OF BIOSURVEILLANCE DATA

5.11 FINAL COMMENTS

5.12 R COMMANDS FOR CHAPTER 5

EXERCISES

CHAPTER 6: TRANSFER FUNCTIONS AND INTERVENTION MODELS

6.1 INTRODUCTION

6.2 TRANSFER FUNCTION MODELS

6.3 TRANSFER FUNCTION–NOISE MODELS

6.4 CROSS-CORRELATION FUNCTION

6.5 MODEL SPECIFICATION

6.6 FORECASTING WITH TRANSFER FUNCTION–NOISE MODELS

6.7 INTERVENTION ANALYSIS

6.8 R COMMANDS FOR CHAPTER 6

EXERCISES

CHAPTER 7: OTHER TIME SERIES ANALYSIS AND FORECASTING METHODS

7.1 MULTIVARIATE TIME SERIES MODELS AND FORECASTING

7.2 STATE SPACE MODELS

7.3 ARCH AND GARCH MODELS

7.4 DIRECT FORECASTING OF PERCENTILES

7.5 COMBINING FORECASTS TO IMPROVE PREDICTION PERFORMANCE

7.6 AGGREGATION AND DISAGGREGATION OF FORECASTS

7.7 NEURAL NETWORKS AND FORECASTING

7.8 SPECTRAL ANALYSIS

7.9 BAYESIAN METHODS IN FORECASTING

7.10 SOME COMMENTS ON PRACTICAL IMPLEMENTATION AND USE OF STATISTICAL FORECASTING PROCEDURES

7.11 R COMMANDS FOR CHAPTER 7

EXERCISES

APPENDIX A: STATISTICAL TABLES

APPENDIX B: DATA SETS FOR EXERCISES

APPENDIX C: INTRODUCTION TO R

BASIC CONCEPTS IN R

BIBLIOGRAPHY

INDEX

END USER LICENSE AGREEMENT

List of Tables

Chapter 2

TABLE 2.1 Chemical Process Viscosity Readings

TABLE 2.2 Calculation of Forecast Accuracy Measures

TABLE 2.3 One-Step-Ahead Forecast Errors

TABLE 2.4 Sample ACF of the One-Step-Ahead Forecast Errors in Table 2.3

TABLE E2.1 One-Step-Ahead Forecast Errors for Exercise 2.44

TABLE E2.2 One-Step-Ahead Forecast Errors for Exercise 2.45

Chapter 3

TABLE 3.1 Cross-Section Data for Multiple Linear Regression

TABLE 3.2 Patient Satisfaction Survey Data

TABLE 3.3 JMP Output for the Patient Satisfaction Data in Table 3.2

TABLE 3.4 Analysis of Variance for Testing Significance of Regression

TABLE 3.5 JMP Output for the Second-Order Model for the Patient Satisfaction...

TABLE 3.6 JMP Calculations of the Standard Errors of the Fitted Values and P...

TABLE 3.7 Residuals and Other Diagnostics for the Regression Model for the P...

TABLE 3.8 Expanded Patient Satisfaction Data

TABLE 3.9 JMP Forward Selection for the Patient Satisfaction Data in Table 3...

TABLE 3.10 JMP Backward Elimination for the Patient Satisfaction Data in Tab...

TABLE 3.11 JMP Stepwise (Mixed) Variable Selection for the Patient Satisfact...

TABLE 3.12 JMP All Possible Models Regression for the Patient Satisfaction D...

TABLE 3.13 Connector Strength Data

TABLE 3.14 Soft Drink Concentrate Sales Data

TABLE 3.15 Minitab Output for the Soft Drink Concentrate Sales Data

TABLE 3.16 Expanded Soft Drink Concentrate Sales Data for Example 3.13

TABLE 3.17 Minitab Output for the Soft Drink Concentrate Data in Example 3.1...

TABLE 3.18 Toothpaste Market Share Data

TABLE 3.19 Minitab Regression Results for the Toothpaste Market Share Data

TABLE 3.20 Minitab Regression Results for Fitting the Transformed Model to t...

TABLE 3.21 SAS PROC AUTOREG Output for the Toothpaste Market Share Data, Ass...

TABLE 3.22 SAS PROC AUTOREG Output for the Toothpaste Market Share Data, Ass...

TABLE 3.23 Minitab Results for Fitting Model (3.119) to the Toothpaste Marke...

TABLE 3.24 Minitab Results for Fitting Model (3.120) to the Toothpaste Marke...

TABLE E3.1 Days that Ozone Levels Exceed 20 ppm and Seasonal Meteorological ...

TABLE E3.2 Monthly Steam Usage and Average Ambient Temperature

TABLE E3.3 Number of Retained Impressions and Advertising Expenditures

TABLE E3.4 Wine Quality Data (Found in Minitab)

TABLE E3.5 Market Share and Price of Canned Peaches

TABLE E3.6 Cosmetic Sales Data for Exercise 3.12

TABLE E3.7 Catapult Experiment Data for Exercise 3.18

TABLE E3.8 Major League Team Performance Data for 2016

Chapter 4

TABLE 4.1 Dow Jones Index at the End of the Month from June 1999 to June 200...

TABLE 4.2 Consumer Price Index from January 1995 to December 2004

TABLE 4.3 Second-Order Exponential Smoothing of the US Consumer Price Index ...

TABLE 4.4 Second-Order Exponential Smoothing of the Dow Jones Index (with λ ...

TABLE 4.5 The Weekly Average Speed During Nonrush Hours

TABLE 4.6

SS

E

for Different

λ

Values for the Average Speed Data

TABLE 4.7 The Predictions and Prediction Errors for Various λ Values for CPI...

TABLE 4.8 JMP Output for the CPI Data

TABLE 4.9 The Trigg–Leach Smoother for the Dow Jones Index

TABLE 4.10 US Clothing Sales from January 1992 to December 2003

TABLE 4.11 Liquor Store Sales from January 1992 to December 2004

TABLE 4.12 Counts of Respiratory Complaints at a Metropolitan Hospital

TABLE 4.13 First-Order Simple Exponential Smoothing Applied to the Respirato...

TABLE 4.14 Second-Order Simple Exponential Smoothing Applied to the Respirat...

TABLE 4.15 Winters' Additive Seasonal Exponential Smoothing Applied to the R...

TABLE E4.1 Data for Exercise 4.1

TABLE E4.2 Data for Exercise 4.4

TABLE E4.3 Data for Exercise 4.7

TABLE E4.4 Data for Exercise 4.8

TABLE E4.5 Soft Drink Demand Data

Chapter 5

TABLE 5.1 Behavior of Theoretical ACF and PACF for Stationary Processes

TABLE 5.2 Sample ACFs and PACFs for Some Realizations of MA(1) and MA(2) Mod...

TABLE 5.3 Sample ACFs and PACFs for Some Realizations of AR(1) and AR(2) Mod...

TABLE 5.4 Sample ACFs and PACFs for Some Realizations of ARMA(1,1) Models

TABLE 5.5 Weekly Total Number of Loan Applications for the Last 2 Years

TABLE 5.6 Minitab Output for the AR(2) Model for the Loan Application Data

TABLE 5.7 Minitab Output for the AR(1) Model for the Dow Jones Index

TABLE 5.8 JMP AR(2) Output for the Loan Application Data

TABLE 5.9 Minitab Output for the ARIMA(0, 1, 1) (0, 1, 1)12 Model for the U...

TABLE 5.10 Summary of Models fit to the Respiratory Syndrome Count Data

TABLE E5.1 Data for Exercise 5.2

TABLE E5.2 Data for Exercise 5.3

Chapter 6

TABLE 6.1 Impulse Response Function with , , and

TABLE 6.2 The viscosity,

y

(

t

) and temperature,

x

(

t

)

TABLE 6.3 AR(1) Model for Temperature,

x

(

t

)

TABLE 6.4 AR(1) Model for

N

t

TABLE 6.5 JMP Output for the Viscosity-Temperature Transfer Function-Noise M...

TABLE 6.6 Impulse Response Function for Example 6.3

TABLE 6.7 Model Summary Statistics for the Two-Input Transfer Function Model...

TABLE 6.8 Estimated standard deviations of the prediction error for the tran...

TABLE 6.9 Output responses to step and pulse inputs.

TABLE 6.10 Weekly Cereal Sales Data

TABLE 6.11 JMP Output for the Intervention Analysis in Example 6.5

TABLE 6.12 Model Summary Statistics for Example 6.6

TABLE 6.13 Model Summary Statistics for the Alternate Intervention Model for...

TABLE E6.1 Time Series Data for Exercise 6.21 (100 observations, read down t...

TABLE E6.2 Time Series Data for Exercise 6.22 (100 observations, read down t...

Chapter 7

TABLE 7.1 Pressure Readings at Both Ends of the Furnace

TABLE 7.2 SAS Commands to Fit a VAR(1) Model to the Pressure Data

TABLE 7.3 SAS Output for the VAR(1) Model for the Pressure Data

TABLE 7.4 Weekly Closing Values for the S&P500 Index from 1995 to 1998

TABLE 7.5 SAS Commands to Fit the ARCH(3) Model

a

TABLE 7.6 SAS output for the ARCH(3) model

TABLE 7.7 Distribution of New Automobile Loan Applications

TABLE E7.1 Spare Part Demand Information for Exercise 7.3

TABLE E7.2 Luxury Car Rental Demand Information for Exercise 7.4

TABLE E7.3 Forecast Errors for Exercise 7.15

TABLE E7.4 Data for Exercise 7.24

TABLE E7.5 Property Crime Data for Exercise 7.27

Appendix A

TABLE A.1 Cumulative Standard Normal Distribution

TABLE A.2 Percentage Points of the Distribution

TABLE A.3 Percentage Points of the Chi-Square Distribution

TABLE A.4 Percentage Points of the Distribution

TABLE A.5 Critical Values of the Durbin–Watson Statistic

Appendix B

TABLE B.1 Market Yield on US Treasury Securities at 10-Year Constant Maturit...

TABLE B.2 Pharmaceutical Product Sales

TABLE B.3 Chemical Process Viscosity

TABLE B.4 US Production of Blue and Gorgonzola Cheeses

TABLE B.5 US Beverage Manufacturer Product Shipments, Unadjusted

TABLE B.6 Global Mean Surface Air Temperature Anomaly and Global CO2 Concent...

TABLE B.7 Whole Foods Market Stock Price, Daily Closing Adjusted for Splits...

TABLE B.8 Unemployment Rate—Full-Time Labor Force, Not Seasonally Adjusted...

TABLE B.9 International Sunspot Numbers

TABLE B.10 United Kingdom Airline Miles Flown

TABLE B.11 Champagne Sales

TABLE B.12 Chemical Process Yield, with Operating Temperature (Uncontrolled)

TABLE B.13 US Production of Ice Cream and Frozen Yogurt

TABLE B.14 Atmospheric CO

2

Concentrations at Mauna Loa Observatory

TABLE B.15 US National Violent Crime Rate

TABLE B.16 US Gross Domestic Product

TABLE B.17 Total Annual US Energy Consumption

TABLE B.18 Annual US Coal Production

TABLE B.19 Arizona Drowning Rate, Children 1–4 Years Old

TABLE B.20 US Internal Revenue Tax Refunds

TABLE B.21 Arizona Average Retail Price of Residential Electricity (Cents pe...

TABLE B.22 Denmark Crude Oil Production (In Thousands of Tons)

TABLE B.23 US Influenza Positive Tests (Percentage)

TABLE B.24 Mean Daily Solar Radiation in Zion Canyon, Utah (Langleys)

TABLE B.25 US Motor Vehicle Traffic Fatalities

TABLE B.26 Single-Family Residential New Home Sales and Building Permits (In...

TABLE B.27 Best Airline On-Time Arrival Performance (Percentage)

TABLE B.28 US Automobile Manufacturing Shipments (Dollar in Millions)

TABLE B.29 Nile River Flow at Aswan (CMS)

TABLE B.30 Lead Production

TABLE B.31 Particle Size

TABLE B.32 Unemployment

TABLE B.33 Southern Oscillation (Read down then across)

TABLE B.34 Steel Shipments (Thousand of tons)

TABLE B.35 Australia Long-Term Visitors Departures

TABLE B.36 Atmospheric CO2 Concentrations at Mauna Loa Observatory (Monthly)...

TABLE B.37 Number of Electrical Workers (Monthly)

TABLE B.38 US GNP Data

TABLE B.39 Monthly Sales in $1000

List of Illustrations

Chapter 1

FIGURE 1.1 Time series plot of the market yield on US Treasury Securities at...

FIGURE 1.2 Five years of Bitcoin prices (from Yahoo Finance).

FIGURE 1.3 Daily new cases of COVID-19.

FIGURE 1.4 Daily deaths from COVID-19.

FIGURE 1.5 Pharmaceutical product sales.

FIGURE 1.6 Chemical process viscosity readings.

FIGURE 1.7 The US annual production of blue and gorgonzola cheeses.

FIGURE 1.8 The US beverage manufacturer monthly product shipments, unadjuste...

FIGURE 1.9 Global mean surface air temperature annual anomaly.

FIGURE 1.10 Whole foods market stock price, daily closing adjusted for split...

FIGURE 1.11 Monthly unemployment rate—full-time labor force, unadjusted.

FIGURE 1.12 The international sunspot number.

FIGURE 1.13 Pharmaceutical product sales.

FIGURE 1.14 Chemical process viscosity readings, with sensor malfunction.

FIGURE 1.15 The forecasting process.

Chapter 2

FIGURE 2.1 Time series plot and histogram of chemical process viscosity read...

FIGURE 2.2 Time series plot and histogram of beverage production shipments....

FIGURE 2.3 Scatter plot of temperature anomaly versus CO concentrations....

FIGURE 2.4 Open-high/close-low chart of Whole Foods Market stock price.

FIGURE 2.5 Time series plot of global mean surface air temperature anomaly, ...

FIGURE 2.6 Time series plot of Whole Foods Market stock price, with 50-day m...

FIGURE 2.7 Viscosity readings with (a) moving average and (b) moving median....

FIGURE 2.8 Pharmaceutical product sales.

FIGURE 2.9 Chemical process viscosity readings.

FIGURE 2.10 Scatter diagram of pharmaceutical product sales at lag .

FIGURE 2.11 Scatter diagram of chemical viscosity readings at lag .

FIGURE 2.12 Sample autocorrelation function for chemical viscosity readings,...

FIGURE 2.13 Listing of sample autocorrelation functions for first 25 lags of...

FIGURE 2.14 Autocorrelation function for pharmaceutical product sales, with ...

FIGURE 2.15 Autocorrelation function for Whole Foods Market stock price, wit...

FIGURE 2.16 JMP output for the sample variogram of the chemical process visc...

FIGURE 2.17 JMP sample variogram of the chemical process viscosity data from...

FIGURE 2.18 JMP output for the sample variogram of the Whole Foods Market st...

FIGURE 2.19 Sample variogram of the Whole Foods Market stock price data from...

FIGURE 2.20 Yearly International Sunspot Number, (a) untransformed and (b) n...

FIGURE 2.21 Blue and gorgonzola cheese production, with fitted regression li...

FIGURE 2.22 Residual plots for simple linear regression model of blue and go...

FIGURE 2.23 Blue and gorgonzola cheese production, with one difference.

FIGURE 2.24 Residual plots for one difference of blue and gorgonzola cheese ...

FIGURE 2.25 Time series plots of seasonal- and trend-differenced beverage da...

FIGURE 2.26 Residual plots for linear trend model of differenced beverage sh...

FIGURE 2.27 Time series plot of decomposition model for beverage shipments....

FIGURE 2.28 Seasonal analysis for beverage shipments.

FIGURE 2.29 Component analysis of beverage shipments.

FIGURE 2.30 Residual plots for additive model of beverage shipments.

FIGURE 2.31 Time series plot of decomposition model for transformed beverage...

FIGURE 2.32 Component analysis of transformed beverage data.

FIGURE 2.33 Residual plots from decomposition model for transformed beverage...

FIGURE 2.34 JMP output for the

X

-11 procedure.

FIGURE 2.35 Normal probability plot of forecast errors from Table 2.2.

FIGURE 2.36 Sample ACF of forecast errors from Table 2.4.

FIGURE 2.37 Normal probability plot of forecast errors from Table 2.3.

FIGURE 2.38 Individuals and moving range control charts of the one-step-ahea...

FIGURE 2.39 CUSUM control chart of the one-step-ahead forecast errors in Tab...

FIGURE 2.40 EWMA control chart of the one-step-ahead forecast errors in Tabl...

Chapter 3

FIGURE 3.1 Plots of residuals for the patient satisfaction model.

FIGURE 3.2 Scatter diagram of connector strength versus age from Table 3.12....

FIGURE 3.3 Plot of residuals versus weeks.

FIGURE 3.4 Scatter plot of absolute residuals versus weeks.

FIGURE 3.5 Plot of residuals versus time for the soft drink concentrate sale...

FIGURE 3.6 Plot of residuals versus time for the soft drink concentrate sale...

Chapter 4

FIGURE 4.1 The process of smoothing a data set.

FIGURE 4.2 The Dow Jones Index from June 1999 to June 2001.

FIGURE 4.3 The Dow Jones Index from June 1999 to June 2006.

FIGURE 4.4 The Dow Jones Index from June 1999 to June 2006 with moving avera...

FIGURE 4.5 The Dow Jones Index from June 1999 to June 2006 with first-order ...

FIGURE 4.6 The Dow Jones Index from June 1999 to June 2006 with first-order ...

FIGURE 4.7 The Dow Jones Index from June 1999 to June 2006 with first-order ...

FIGURE 4.8 The Dow Jones Index from June 1999 to June 2006 with first-order ...

FIGURE 4.9 The Dow Jones Index from February 2003 to February 2004.

FIGURE 4.10 The Dow Jones Index from February 2003 to February 2004 with sim...

FIGURE 4.11 The Dow Jones Index from June 1999 to June 2006 using exponentia...

FIGURE 4.12 The Dow Jones Index from February 2003 to February 2004 with sec...

FIGURE 4.13 US Consumer Price Index from January 1995 to December 2004.

FIGURE 4.14 Single exponential smoothing of the US Consumer Price Index (wit...

FIGURE 4.15 Second-order exponential smoothing of the US Consumer Price Inde...

FIGURE 4.16 Double exponential smoothing of the US Consumer Price Index (wit...

FIGURE 4.17 Second-order exponential smoothing of the Dow Jones Index (with

FIGURE 4.18 The weekly average speed during nonrush hours.

FIGURE 4.19 Plot of

SS

E

for various

λ

values for average speed data.

FIGURE 4.20 Forecasts for the weekly average speed data for weeks 79–90.

FIGURE 4.21 Scatter plot of the sum of the squared one-step-ahead prediction...

FIGURE 4.22 ACF plot for the CPI data (with 5% significance limits for the a...

FIGURE 4.23 The 1- to 12-step-ahead forecasts of the CPI data for 2004.

FIGURE 4.24 The one-step-ahead forecasts of the CPI data for 2004.

FIGURE 4.25 Time series plot of the Dow Jones Index from June 1999 to June 2...

FIGURE 4.26 Time series plot of US clothing sales from January 1992 to Decem...

FIGURE 4.27 Smoothed data for the US clothing sales from January 1992 to Dec...

FIGURE 4.28 Forecasts for 2003 for the US clothing sales.

FIGURE 4.29 Time series plot of liquor store sales data from January 1992 to...

FIGURE 4.30 Smoothed data for the liquor store sales from January 1992 to De...

FIGURE 4.31 Smoothed data for the liquor store sales from January 1992 to De...

FIGURE 4.32 Forecasts for the liquor store sales for 2004 using the multipli...

FIGURE 4.33 Time series plot of daily respiratory syndrome count, with kerne...

FIGURE 4.34 Box plots of residuals from the kernel-smoothed line fit to dail...

FIGURE 4.35 Respiratory syndrome counts using first-order exponential smooth...

FIGURE 4.36 Respiratory syndrome counts using second-order exponential smoot...

FIGURE 4.37 Respiratory syndrome counts using winters' additive seasonal exp...

Chapter 5

FIGURE 5.1 Realizations of (a) stationary, (b) near nonstationary, and (c) n...

FIGURE 5.2 A realization of the MA(1) process, .

FIGURE 5.3 A realization of the MA(1) process, .

FIGURE 5.4 A realization of the MA(2) process, .

FIGURE 5.5 A realization of the AR(1) process, .

FIGURE 5.6 A realization of the AR(1) process, .

FIGURE 5.7 A realization of the AR(2) process, .

FIGURE 5.8 A realization of the AR(2) process, .

FIGURE 5.9 Partial autocorrelation functions for the realizations of (a) MA(...

FIGURE 5.10 Two realizations of the ARMA(1,1) model: (a) and (b) . (c) Th...

FIGURE 5.11 A realization of the ARIMA(0, 1, 0) model, , its first differen...

FIGURE 5.12 A realization of the ARIMA(0, 1, 1) model, , its first differen...

FIGURE 5.13 Time series plot of the weekly total number of loan applications...

FIGURE 5.14 ACF and PACF for the weekly total number of loan applications.

FIGURE 5.15 The sample ACF and PACF of the residuals for the AR(2) model in ...

FIGURE 5.16 Residual plots for the AR(2) model in Table 5.4.

FIGURE 5.17 Time series plot of the actual data and fitted values for the AR...

FIGURE 5.18 Time series plot of the Dow Jones Index from June 1999 to June 2...

FIGURE 5.19 Sample ACF and PACF of the Dow Jones Index.

FIGURE 5.20 Sample ACF and PACF of the residuals from the AR(1) model for th...

FIGURE 5.21 Residual plots from the AR(1) model for the Dow Jones Index data...

FIGURE 5.22 Time series plot of the first difference () of the Dow Jones I...

FIGURE 5.23 Sample ACF and PACF plots of the first difference of the Dow Jon...

FIGURE 5.24 Time series plot and forecasts for the weekly loan application d...

FIGURE 5.25 Sample ACF and PACF plots of the US clothing sales data.

FIGURE 5.26 Time series plot of for the US clothing sales data.

FIGURE 5.27 Sample ACF and PACF plots of .

FIGURE 5.28 Sample ACF and PACF plots of residuals from the ARIMA(0, 1, 1)

FIGURE 5.29 Residual plots from the ARIMA(0, 1, 1) (0, 1, 1) model for th...

FIGURE 5.30 Time series plot of the actual data and fitted values from the A...

FIGURE 5.31 ACF, PACF, and variogram for daily respiratory syndrome counts....

FIGURE 5.32 ACF, PACF, and variogram for the first difference of the daily r...

FIGURE 5.33 ACF, PACF, and variogram for the residuals of ARIMA(1, 1, 1) fit...

FIGURE 5.34 Plots of residuals from ARIMA(1, 1, 1) fit to daily respiratory ...

FIGURE 5.35 ACF, PACF, and variogram for residuals of ARIMA(2, 1, 1) fit to ...

FIGURE 5.36 Plots of residuals from ARIMA(2, 1, 1) fit to daily respiratory ...

FIGURE 5.37 ACF, PACF, and variogram of residuals from ARIMA(2, 1, 1)  (0, ...

FIGURE 5.38 Plots of residuals from ARIMA(2, 1, 1) (0, 0, 1) fit to daily...

Chapter 6

FIGURE 6.1 The stable region for the impulse response function for .

FIGURE 6.2 Example of an impulse response function.

FIGURE 6.3 Time series plots of the viscosity,

y

(

t

) and temperature,

x

(

t

).

FIGURE 6.4 Sample ACF and PACF of the temperature.

FIGURE 6.5 Sample ACF and PACF of the residuals from the AR(1) model for the...

FIGURE 6.6 Residual plots from the AR(1) model for the temperature.

FIGURE 6.7 Sample cross-correlation function between and .

FIGURE 6.8 Plot of the impulse response function for the viscosity data.

FIGURE 6.9 Time series plot of .

FIGURE 6.10 Sample ACF and PACF of .

FIGURE 6.11 Sample ACF and PACF of the residuals of the AR(1) model for .

FIGURE 6.12 Residual plots of the AR(l) model for .

FIGURE 6.13 Sample cross-correlation function between α

1

and the residuals o...

FIGURE 6.14 Hourly readings of final product viscosity , incoming raw mater...

FIGURE 6.15 The time series plots of the actual and 1- to 6-step ahead forec...

FIGURE 6.16 Time series plot of the weekly sales data.

FIGURE 6.17 Sample ACF and PACF pulse of the sales data for weeks 1–87.

FIGURE 6.18 Sample ACF and PACF plots of the first difference of the sales d...

FIGURE 6.19 Natural logarithm of monthly electric energy consumption in mega...

Chapter 7

FIGURE 7.1 Time series plots of the pressure readings at both ends of the fu...

FIGURE 7.2 The sample ACF plot for: (a) the pressure readings at the front e...

FIGURE 7.3 Time series plots of the residuals from the VAR(1) model.

FIGURE 7.4 Actual and fitted values for the pressure readings at the front e...

FIGURE 7.5 Actual and fitted values for the pressure readings at the back en...

FIGURE 7.6 Time series plot of S&P500 Index weekly close from 1995 to 1998....

FIGURE 7.7 Time series plot of the first difference of the log transformatio...

FIGURE 7.8 ACF and PACF plots of the first difference of the log transformat...

FIGURE 7.9 ACF and PACF plots of the square of the first difference of the l...

FIGURE 7.10 Cumulative distribution of the number of loan applications, week...

FIGURE 7.11 Cumulative distribution of the number of loan applications, week...

FIGURE 7.12 Artificial neural network with one hidden layer.

FIGURE 7.13 A single node of an autoregressive neural network.

FIGURE 7.14 Recurrent neural network with a feedback loop.

FIGURE 7.15 Expanding the feedback loop of an RNN.

FIGURE 7.16 Long short-term memory (LSTM) model.

FIGURE 7.17 A single cell of an LSTM where

σ

and

tanh

represent the sig...

FIGURE 7.18 The spectrum of a white noise process.

FIGURE 7.19 The spectrum of AR(1) processes.

FIGURE 7.20 The spectrum of MA(1) processes.

FIGURE 7.21 The spectrum of the seasonal process.

FIGURE 7.22 JMP output showing the spectrum, ACF, PACF, and variogram for th...

Guide

Cover

Table of Contents

Title Page

Copyright

Preface

About the Companion Website

Begin Reading

Appendix A: Statistical Tables

Appendix B: Data Sets for Exercises

Appendix C: Introduction to R

Bibliography

Index

End User License Agreement

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WILEY SERIES IN PROBABILITY AND STATISTICS

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Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Geof H. Givens, Harvey Goldstein, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford WeisbergEditors Emeriti: J. Stuart Hunter, Iain M. Johnstone, Joseph B. Kadane, Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume.

INTRODUCTION TO TIME SERIES ANALYSIS AND FORECASTING

Third Edition

DOUGLAS C. MONTGOMERY

Arizona State University

Tempe, Arizona, USA

CHERYL L. JENNINGS

Arizona State University

Tempe, Arizona, USA

MURAT KULAHCI

Technical University of Denmark

Lyngby, Denmark

and

Luleå University of Technology

Luleå, Sweden

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PREFACE

Analyzing time-oriented data and forecasting future values of a time series are among the most important problems that analysts face in many fields, ranging from finance and economics to managing production operations, to the analysis of political and social policy decisions, to investigating the impact of humans and the policy decisions that they make on the environment. Consequently, there is a large group of people in a variety of fields, including finance, economics, science, engineering, statistics, and public policy who need to understand some basic concepts of time series analysis and forecasting. Unfortunately, most basic statistics and operations management books give little if any attention to time-oriented data and little guidance on forecasting. There are some very good high level books on time series analysis. These books are mostly written for technical specialists who are taking a doctoral-level course or doing research in the field. They tend to be very theoretical and often focus on a few specific topics or techniques. We have written this book to fill the gap between these two extremes.

We have made a number of changes in this revision of the book. New material has been added on data preparation for forecasting, including dealing with outliers and missing values, use of the variogram and sections on the spectrum, and an introduction to Bayesian methods in forecasting. We have added many new exercises and examples, including new data sets inAppendix B, and edited many sections of the text to improve the clarity of the presentation.

Like the previous editions, this book is intended for practitioners who make real-world forecasts. We have attempted to keep the mathematical level modest to encourage a variety of users for the book. Our focus is on short- to medium-term forecasting where statistical methods are useful. Since many organizations can improve their effectiveness and business results by making better short- to medium-term forecasts, this book should be useful to a wide variety of professionals. The book can also be used as a textbook for an applied forecasting and time series analysis course at the advanced undergraduate or first-year graduate level. Students in this course could come from engineering, business, statistics, operations research, mathematics, computer science, and any area of application where making forecasts is important. Readers need a background in basic statistics (previous exposure to linear regression would be helpful, but not essential), and some knowledge of matrix algebra, although matrices appear mostly in the chapter on regression, and if one is interested mainly in the results, the details involving matrix manipulation can be skipped. Integrals and derivatives appear in a few places in the book, but no detailed working knowledge of calculus is required.

Successful time series analysis and forecasting requires that the analyst interact with computer software. The techniques and algorithms are just not suitable to manual calculations. We have chosen to demonstrate the techniques presented using three packages: Minitab®, JMP®, and R, and occasionally SAS®. We have selected these packages because they are widely used in practice and because they have generally good capability for analyzing time series data and generating forecasts. Because R is increasingly popular in statistics courses, we have included a section in each chapter showing the R code necessary for working some of the examples in the chapter. We have also added a brief appendix on the use of R. The basic principles that underlie most of our presentation are not specific to any particular software package. Readers can use any software that they like or have available that has basic statistical forecasting capability. While the text examples do utilize these particular software packages and illustrate some of their features and capability, these features or similar ones are found in many other software packages.

There are three basic approaches to generating forecasts: regression-based methods, heuristic smoothing methods, and general time series models. Because all three of these basic approaches are useful, we give an introduction to all of them. Chapter 1 introduces the basic forecasting problem, defines terminology, and illustrates many of the common features of time series data. Chapter 2 contains many of the basic statistical tools used in analyzing time series data. Topics include plots, numerical summaries of time series data including the autocovariance and autocorrelation functions, transformations, differencing, and decomposing a time series into trend and seasonal components. We also introduce metrics for evaluating forecast errors and methods for evaluating and tracking forecasting performance over time. Chapter 3 discusses regression analysis and its use in forecasting. We discuss both crosssection and time series regression data, least squares and maximum likelihood model fitting, model adequacy checking, prediction intervals, and weighted and generalized least squares. The first part of this chapter covers many of the topics typically seen in an introductory treatment of regression, either in a stand-alone course or as part of another applied statistics course. It should be a reasonable review for many readers. Chapter 4 presents exponential smoothing techniques, both for time series with polynomial components and for seasonal data. We discuss and illustrate methods for selecting the smoothing constant(s), forecasting, and constructing prediction intervals. The explicit time series modeling approach to forecasting that we have chosen to emphasize is the autoregressive integrated moving average (ARIMA) model approach. Chapter 5 introduces ARIMA models and illustrates how to identify and fit these models for both nonseasonal and seasonal time series. Forecasting and prediction interval construction are also discussed and illustrated. Chapter 6 extends this discussion into transfer function models and intervention modeling and analysis. Chapter 7 surveys several other useful topics from time series analysis and forecasting, including multivariate time series problems, ARCH and GARCH models, and combinations of forecasts. We also give some practical advice for using statistical approaches to forecasting and provide some information about realistic expectations. The last two chapters of the book are somewhat higher in level than the first five.

Each chapter has a set of exercises. Some of these exercises involve analyzing the data sets given in Appendix B. These data sets represent an interesting cross section of real time series data, typical of those encountered in practical forecasting problems. Most of these data sets are used in exercises in two or more chapters, an indication that there are usually several approaches to analyzing, modeling, and forecasting a time series. There are other good sources of data for practicing the techniques given in this book. Some of the ones that we have found very interesting and useful include the U.S. Department of Labor—Bureau of Labor Statistics (http://www.bls.gov/data/home.htm), the U.S. Department of Agriculture—National Agricultural Statistics Service, Quick Stats Agricultural Statistics Data (http://www.nass.usda.gov/Data_and_Statistics/Quick_Stats/index.asp), the U.S. Census Bureau (http://www.census.gov), and the U.S. Department of the Treasury (http://www.treas.gov/offices/domestic-finance/debt-management/interest-rate/). The time series data library created by Rob Hyndman at Monash University (http://www-personal.buseco.monash.edu.au/∼hyndman/TSDL/index.htm) and the time series data library at the Mathematics Department of the University of York (http://www.york.ac.uk/depts/maths/data/ts/) also contain many excellent data sets. Some of these sources provide links to other data. Data sets and other materials related to this book can be found at ftp://ftp.wiley.com/public/scitechmed/ timeseries.

We would like to thank the many individuals who provided feedback and suggestions for improvement to the previous editions. We found these suggestions most helpful. We are indebted to Clifford Long who generously provided the R codes he used with his students when he taught from the book. We found his codes very helpful in putting the end-of-chapter R code sections together. We also have placed a premium in the book on bridging the gap between theory and practice. We have not emphasized proofs or technical details and have tried to give intuitive explanations of the material whenever possible. The result is a book that can be used with a wide variety of audiences, with different interests and technical backgrounds, whose common interests are understanding how to analyze time-oriented data and constructing good short-term statistically based forecasts.

We express our appreciation to the individuals and organizations who have given their permission to use copyrighted material. These materials are noted in the text. Portions of the output contained in this book are printed with permission of Minitab Inc. All material remains the exclusive property and copyright of Minitab Inc. All rights reserved.

DOUGLAS C. MONTGOMERY

CHERYL L. JENNINGS

MURATKULAHCI

ABOUT THE COMPANION WEBSITE

This book is accompanied by the instructor and student companion websites, and the sites can be accessed from the page at

www.wiley.com/go/montgomery/timeseriesforecasting3e

The student companion website includes:

Data sets from the book.

Chapter-wise lecture videos.

Solutions to the text problems.

The instructor's website for the book includes the following:

Solutions to the text problems.

Chapter-wise lecture videos.

Chapter-wise PowerPoint lecture slides.

CHAPTER 1INTRODUCTION TO TIME SERIES ANALYSIS AND FORECASTING

It is difficult to make predictions, especially about the future

NEILS BOHR, Danish physicist

1.1 THE NATURE AND USES OF FORECASTS

A forecast is a prediction of some future event or events. As suggested by Neils Bohr, making good predictions is not always easy. Famously “bad” forecasts include the following from the book Bad Predictions:

“The population is constant in size and will remain so right up to the end of mankind.”

L'Encyclopedie

, 1756.

“1930 will be a splendid employment year.” U.S. Department of Labor,

New Year's Forecast

in 1929, just before the market crash on October 29.

“Computers are multiplying at a rapid rate. By the turn of the century there will be 220,000 in the U.S.”

Wall Street Journal

, 1966.

Forecasting is an important problem that spans many fields including business and industry, government, economics, environmental sciences, public health, medicine, social science, politics, and finance. Forecasting problems are often classified as short-term, medium-term, and long-term. Short-term forecasting problems involve predicting events only a few time periods (days, weeks, and months) into the future. Medium-term forecasts extend from 1 to 2 years into the future, and long-term forecasting problems can extend beyond that by many years. Short- and medium-term forecasts are required for activities that range from operations management to budgeting and selecting new research and development projects. Long-term forecasts impact issues such as strategic planning. Short- and medium-term forecasting is typically based on identifying, modeling, and extrapolating the patterns found in historical data. Because these historical data usually exhibit inertia and do not change dramatically very quickly, statistical methods are very useful for short- and medium-term forecasting. This book is about the use of these statistical methods.

Most forecasting problems involve the use of time series data. A time series is a time-oriented or chronological sequence of observations on a variable of interest. For example, Figure 1.1 shows the market yield on US Treasury Securities at 10-year constant maturity from April 1953 through December 2006 (data in Appendix B, Table B.1). This graph is called a time series plot. The rate variable is collected at equally spaced time periods, as is typical in most time series and forecasting applications. Many business applications of forecasting utilize daily, weekly, monthly, quarterly, or annual data, but any reporting interval may be used. Furthermore, the data may be instantaneous, such as the viscosity of a chemical product at the point in time where it is measured; it may be cumulative, such as the total sales of a product during the month; or it may be a statistic that in some way reflects the activity of the variable during the time period, such as the daily closing price of a specific stock on the New York Stock Exchange.

FIGURE 1.1 Time series plot of the market yield on US Treasury Securities at 10-year constant maturity.

Source: US Treasury.

Because time series data exhibits the inertial effects mentioned previously, they usually do not satisfy the usual assumptions made in most statistical methods. That is, time series data are usually not independent. This means that special statistical methods that takes this into account are required for most forecasting and time series analysis problems. Those methods are the focus of this work.

The reason that forecasting is so important is that prediction of future events is a critical input into many types of planning and decision-making processes, with application to areas such as the following:

Operations Management

. Business organizations routinely use forecasts of product sales or demand for services in order to schedule production, control inventories, manage the supply chain, determine staffing requirements, and plan capacity. Forecasts may also be used to determine the mix of products or services to be offered and the locations at which products are to be produced.

Marketing

. Forecasting is important in many marketing decisions. Forecasts of sales response to advertising expenditures, new promotions, or changes in pricing polices enable businesses to evaluate their effectiveness, determine whether goals are being met, and make adjustments.

Finance and Risk Management

. Investors in financial assets are interested in forecasting the returns from their investments. These assets include but are not limited to stocks, bonds, and commodities; other investment decisions can be made relative to forecasts of interest rates, options, and currency exchange rates. Financial risk management requires forecasts of the volatility of asset returns so that the risks associated with investment portfolios can be evaluated and insured, and so that financial derivatives can be properly priced.

FIGURE 1.2 Five years of Bitcoin prices (from Yahoo Finance).

As an example of financial data, consider the Bitcoin price history for the most recent five years shown in Figure 1.2. Bitcoin is a cryptocurrency introduced in early 2009 although standard pricing did not begin until about a year later. The graph shows that there has been considerable growth in Bitcoin prices, but also considerable volatility. Investors and currency traders would be interested in modeling and forecasting the performance of this asset. However, the inherent volatility in Bitcoin price would make this a very challenging task.

Economics

. Governments, financial institutions, and policy organizations require forecasts of major economic variables, such as gross domestic product, population growth, unemployment, interest rates, inflation, job growth, production, and consumption. These forecasts are an integral part of the guidance behind monetary and fiscal policy, and budgeting plans and decisions made by governments. They are also instrumental in the strategic planning decisions made by business organizations and financial institutions.

Industrial Process Control

. Forecasts of the future values of critical quality characteristics of a production process can help determine when important controllable variables in the process should be changed, or if the process should be shut down and overhauled. Feedback and feedforward control schemes are widely used in monitoring and adjustment of industrial processes, and predictions of the process output are an integral part of these schemes.

Demography

. Forecasts of population by country and regions are made routinely, often stratified by variables such as gender, age, and race. Demographers also forecast births, deaths, and migration patterns of populations. Governments use these forecasts for planning policy and social service actions, such as spending on health care, retirement programs, and antipoverty programs. Many businesses use forecasts of populations by age groups to make strategic plans regarding developing new product lines or the types of services that will be offered.

Public Health Applications

. As an example of the use of time series analysis in the public health arena, let us consider the recent COVID-19 pandemic. This is also known as the

coronavirus pandemic

, caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The virus was first identified in an outbreak in the China in December 2019. Attempts to contain it there failed, allowing the virus to spread worldwide in 2020. The pandemic triggered social and economic disruption around the world, including a global recession. There were widespread supply shortages, including food shortages, resulting from supply chain disruptions. Mitigation strategies including travel restriction, business and school closures, social distancing measures, masking mandates, testing, contact tracing of infected individuals, and remote working were widespread. COVID-19 vaccines became available in late 2020 and have been widely deployed. As of mid-2023, the pandemic had caused over 700,000,000 cases and approximately 6.9 million deaths.

Figure 1.3 shows a time series plot of daily new cases from early 2020 to mid-2023. Figure 1.4 shows a plot of daily deaths from mid-2020 through mid-2023. The number of deaths declined rapidly over the last year shown in the graph due to the various mitigation strategies and the widespread availability and use of effective vaccines. Public health agencies at the national, state, and local level frequently made data such as this available. There was also interest in hospitalizations arising from the disease, as there was some concerns that hospital resources would be overwhelmed by the number of cases requiring that level of care.

FIGURE 1.3 Daily new cases of COVID-19.

FIGURE 1.4 Daily deaths from COVID-19.

A time series analysist could use this data to predict cases, deaths, and hospitalizations. It could also be possible to include the introduction of the mitigation measures including vaccines in the analysis to determine the potential effectiveness of these measures in reducing the spread of severity of the disease. A technique called intervention analysis can be used to do this. Intervention analysis is discussed in this book in Chapter 6.

These are only a few of the many different situations where forecasts are required in order to make good decisions. Despite the wide range of problem situations that require forecasts, there are only two broad types of forecasting techniques—qualitative methods and quantitative methods.

Qualitative forecasting techniques are often subjective in nature and require judgment on the part of experts. Qualitative forecasts are often used in situations where there is little or no historical data on which to base the forecast. An example would be the introduction of a new product, for which there is no relevant history. In this situation, the company might use the expert opinion of sales and marketing personnel to subjectively estimate product sales during the new product introduction phase of its life cycle. Sometimes qualitative forecasting methods make use of marketing tests, surveys of potential customers, and experience with the sales performance of other products (both their own and those of competitors). However, although some data analysis may be performed, the basis of the forecast is subjective judgment.

Perhaps the most formal and widely known qualitative forecasting technique is the Delphi Method. This technique was developed by the RAND Corporation (see Dalkey [1967]). It employs a panel of experts who are assumed to be knowledgeable about the problem. The panel members are physically separated to avoid their deliberations being impacted either by social pressures or by a single dominant individual. Each panel member responds to a questionnaire containing a series of questions and returns the information to a coordinator. Following the first questionnaire, subsequent questions are submitted to the panelists along with information about the opinions of the panel as a group. This allows panelists to review their predictions relative to the opinions of the entire group. After several rounds, it is hoped that the opinions of the panelists converge to a consensus, although achieving a consensus is not required and justified differences of opinion can be included in the outcome. Qualitative forecasting methods are not emphasized in this book.

Quantitative forecasting techniques make formal use of historical data and a forecasting model. The model formally summarizes patterns in the data and expresses a statistical relationship between previous and current values of the variable. Then the model is used to project the patterns in the data into the future. In other words, the forecasting model is used to extrapolate past and current behavior into the future. There are several types of forecasting models in general use. The three most widely used are regression models, smoothing models, and general time series models. Regression models make use of relationships between the variable of interest and one or more related predictor variables. Sometimes regression models are called causal forecasting models, because the predictor variables are assumed to describe the forces that cause or drive the observed values of the variable of interest. An example would be using data on house purchases as a predictor variable to forecast furniture sales. The method of least squares is the formal basis of most regression models. Smoothing models typically employ a simple function of previous observations to provide a forecast of the variable of interest. These methods may have a formal statistical basis, but they are often used and justified heuristically on the basis that they are easy to use and produce satisfactory results. General time series models employ the statistical properties of the historical data to specify a formal model and then estimate the unknown parameters of this model (usually) by least squares. In subsequent chapters, we will discuss all three types of quantitative forecasting models.

The form of the forecast can be important. We typically think of a forecast as a single number that represents our best estimate of the future value of the variable of interest. Statisticians would call this a point estimate or point forecast. Now these forecasts are almost always wrong; that is, we experience forecast error. Consequently, it is usually a good practice to accompany a forecast with an estimate of how large a forecast error might be experienced. One way to do this is to provide a prediction interval (PI) to accompany the point forecast. The PI is a range of values for the future observation, and it is likely to prove far more useful in decision-making than a single number. We will show how to obtain PIs for most of the forecasting methods discussed in the book.

Other important features of the forecasting problem are the forecast horizon and the forecast interval. The forecast horizon is the number of future periods for which forecasts must be produced. The horizon is often dictated by the nature of the problem. For example, in production planning, forecasts of product demand may be made on a monthly basis. Because of the time required to change or modify a production schedule, ensure that sufficient raw material and component parts are available from the supply chain, and plan the delivery of completed goods to customers or inventory facilities, it would be necessary to forecast up to 3 months ahead. The forecast horizon is also often called the forecast lead time. The