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- [Forecasting: Principles and Practice](https://otexts.com/fpp2/) - [Preface](https://otexts.com/fpp2/index.html) - [**1** Getting started](https://otexts.com/fpp2/intro.html) - [**1\.1** What can be forecast?](https://otexts.com/fpp2/what-can-be-forecast.html) - [**1\.2** Forecasting, planning and goals](https://otexts.com/fpp2/planning.html) - [**1\.3** Determining what to forecast](https://otexts.com/fpp2/determining-what-to-forecast.html) - [**1\.4** Forecasting data and methods](https://otexts.com/fpp2/data-methods.html) - [**1\.5** Some case studies](https://otexts.com/fpp2/case-studies.html) - [**1\.6** The basic steps in a forecasting task](https://otexts.com/fpp2/basic-steps.html) - [**1\.7** The statistical forecasting perspective](https://otexts.com/fpp2/perspective.html) - [**1\.8** Exercises](https://otexts.com/fpp2/intro-exercises.html) - [**1\.9** Further reading](https://otexts.com/fpp2/intro-reading.html) - [**2** Time series graphics](https://otexts.com/fpp2/graphics.html) - [**2\.1** `ts` objects](https://otexts.com/fpp2/ts-objects.html) - [**2\.2** Time plots](https://otexts.com/fpp2/time-plots.html) - [**2\.3** Time series patterns](https://otexts.com/fpp2/tspatterns.html) - [**2\.4** Seasonal plots](https://otexts.com/fpp2/seasonal-plots.html) - [**2\.5** Seasonal subseries plots](https://otexts.com/fpp2/seasonal-subseries-plots.html) - [**2\.6** Scatterplots](https://otexts.com/fpp2/scatterplots.html) - [**2\.7** Lag plots](https://otexts.com/fpp2/lag-plots.html) - [**2\.8** Autocorrelation](https://otexts.com/fpp2/autocorrelation.html) - [**2\.9** White noise](https://otexts.com/fpp2/wn.html) - [**2\.10** Exercises](https://otexts.com/fpp2/graphics-exercises.html) - [**2\.11** Further reading](https://otexts.com/fpp2/graphics-reading.html) - [**3** The forecaster’s toolbox](https://otexts.com/fpp2/toolbox.html) - [**3\.1** Some simple forecasting methods](https://otexts.com/fpp2/simple-methods.html) - [**3\.2** Transformations and adjustments](https://otexts.com/fpp2/transformations.html) - [**3\.3** Residual diagnostics](https://otexts.com/fpp2/residuals.html) - [**3\.4** Evaluating forecast accuracy](https://otexts.com/fpp2/accuracy.html) - [**3\.5** Prediction intervals](https://otexts.com/fpp2/prediction-intervals.html) - [**3\.6** The forecast package in R](https://otexts.com/fpp2/the-forecast-package-in-r.html) - [**3\.7** Exercises](https://otexts.com/fpp2/toolbox-exercises.html) - [**3\.8** Further reading](https://otexts.com/fpp2/toolbox-reading.html) - [**4** Judgmental forecasts](https://otexts.com/fpp2/judgmental.html) - [**4\.1** Beware of limitations](https://otexts.com/fpp2/judgmental-limitations.html) - [**4\.2** Key principles](https://otexts.com/fpp2/judgmental-principles.html) - [**4\.3** The Delphi method](https://otexts.com/fpp2/delphimethod.html) - [**4\.4** Forecasting by analogy](https://otexts.com/fpp2/analogies.html) - [**4\.5** Scenario forecasting](https://otexts.com/fpp2/scenarios.html) - [**4\.6** New product forecasting](https://otexts.com/fpp2/new-products.html) - [**4\.7** Judgmental adjustments](https://otexts.com/fpp2/judgmental-adjustments.html) - [**4\.8** Further reading](https://otexts.com/fpp2/judgmental-reading.html) - [**5** Time series regression models](https://otexts.com/fpp2/regression.html) - [**5\.1** The linear model](https://otexts.com/fpp2/regression-intro.html) - [**5\.2** Least squares estimation](https://otexts.com/fpp2/least-squares.html) - [**5\.3** Evaluating the regression model](https://otexts.com/fpp2/regression-evaluation.html) - [**5\.4** Some useful predictors](https://otexts.com/fpp2/useful-predictors.html) - [**5\.5** Selecting predictors](https://otexts.com/fpp2/selecting-predictors.html) - [**5\.6** Forecasting with regression](https://otexts.com/fpp2/forecasting-regression.html) - [**5\.7** Matrix formulation](https://otexts.com/fpp2/regression-matrices.html) - [**5\.8** Nonlinear regression](https://otexts.com/fpp2/nonlinear-regression.html) - [**5\.9** Correlation, causation and forecasting](https://otexts.com/fpp2/causality.html) - [**5\.10** Exercises](https://otexts.com/fpp2/regression-exercises.html) - [**5\.11** Further reading](https://otexts.com/fpp2/regression-reading.html) - [**6** Time series decomposition](https://otexts.com/fpp2/decomposition.html) - [**6\.1** Time series components](https://otexts.com/fpp2/components.html) - [**6\.2** Moving averages](https://otexts.com/fpp2/moving-averages.html) - [**6\.3** Classical decomposition](https://otexts.com/fpp2/classical-decomposition.html) - [**6\.4** X11 decomposition](https://otexts.com/fpp2/x11.html) - [**6\.5** SEATS decomposition](https://otexts.com/fpp2/seats.html) - [**6\.6** STL decomposition](https://otexts.com/fpp2/stl.html) - [**6\.7** Measuring strength of trend and seasonality](https://otexts.com/fpp2/seasonal-strength.html) - [**6\.8** Forecasting with decomposition](https://otexts.com/fpp2/forecasting-decomposition.html) - [**6\.9** Exercises](https://otexts.com/fpp2/decomposition-exercises.html) - [**6\.10** Further reading](https://otexts.com/fpp2/decomposition-reading.html) - [**7** Exponential smoothing](https://otexts.com/fpp2/expsmooth.html) - [**7\.1** Simple exponential smoothing](https://otexts.com/fpp2/ses.html) - [**7\.2** Trend methods](https://otexts.com/fpp2/holt.html) - [**7\.3** Holt-Winters’ seasonal method](https://otexts.com/fpp2/holt-winters.html) - [**7\.4** A taxonomy of exponential smoothing methods](https://otexts.com/fpp2/taxonomy.html) - [**7\.5** Innovations state space models for exponential smoothing](https://otexts.com/fpp2/ets.html) - [**7\.6** Estimation and model selection](https://otexts.com/fpp2/estimation-and-model-selection.html) - [**7\.7** Forecasting with ETS models](https://otexts.com/fpp2/ets-forecasting.html) - [**7\.8** Exercises](https://otexts.com/fpp2/expsmooth-exercises.html) - [**7\.9** Further reading](https://otexts.com/fpp2/expsmooth-reading.html) - [**8** ARIMA models](https://otexts.com/fpp2/arima.html) - [**8\.1** Stationarity and differencing](https://otexts.com/fpp2/stationarity.html) - [**8\.2** Backshift notation](https://otexts.com/fpp2/backshift.html) - [**8\.3** Autoregressive models](https://otexts.com/fpp2/AR.html) - [**8\.4** Moving average models](https://otexts.com/fpp2/MA.html) - [**8\.5** Non-seasonal ARIMA models](https://otexts.com/fpp2/non-seasonal-arima.html) - [**8\.6** Estimation and order selection](https://otexts.com/fpp2/arima-estimation.html) - [**8\.7** ARIMA modelling in R](https://otexts.com/fpp2/arima-r.html) - [**8\.8** Forecasting](https://otexts.com/fpp2/arima-forecasting.html) - [**8\.9** Seasonal ARIMA models](https://otexts.com/fpp2/seasonal-arima.html) - [**8\.10** ARIMA vs ETS](https://otexts.com/fpp2/arima-ets.html) - [**8\.11** Exercises](https://otexts.com/fpp2/arima-exercises.html) - [**8\.12** Further reading](https://otexts.com/fpp2/arima-reading.html) - [**9** Dynamic regression models](https://otexts.com/fpp2/dynamic.html) - [**9\.1** Estimation](https://otexts.com/fpp2/estimation.html) - [**9\.2** Regression with ARIMA errors in R](https://otexts.com/fpp2/regarima.html) - [**9\.3** Forecasting](https://otexts.com/fpp2/forecasting.html) - [**9\.4** Stochastic and deterministic trends](https://otexts.com/fpp2/stochastic-and-deterministic-trends.html) - [**9\.5** Dynamic harmonic regression](https://otexts.com/fpp2/dhr.html) - [**9\.6** Lagged predictors](https://otexts.com/fpp2/lagged-predictors.html) - [**9\.7** Exercises](https://otexts.com/fpp2/dynamic-exercises.html) - [**9\.8** Further reading](https://otexts.com/fpp2/dynamic-reading.html) - [**10** Forecasting hierarchical or grouped time series](https://otexts.com/fpp2/hierarchical.html) - [**10\.1** Hierarchical time series](https://otexts.com/fpp2/hts.html) - [**10\.2** Grouped time series](https://otexts.com/fpp2/gts.html) - [**10\.3** The bottom-up approach](https://otexts.com/fpp2/bottom-up.html) - [**10\.4** Top-down approaches](https://otexts.com/fpp2/top-down.html) - [**10\.5** Middle-out approach](https://otexts.com/fpp2/middle-out.html) - [**10\.6** Mapping matrices](https://otexts.com/fpp2/mapping-matrices.html) - [**10\.7** The optimal reconciliation approach](https://otexts.com/fpp2/reconciliation.html) - [**10\.8** Exercises](https://otexts.com/fpp2/hierarchical-exercises.html) - [**10\.9** Further reading](https://otexts.com/fpp2/hierarchical-reading.html) - [**11** Advanced forecasting methods](https://otexts.com/fpp2/advanced.html) - [**11\.1** Complex seasonality](https://otexts.com/fpp2/complexseasonality.html) - [**11\.2** Vector autoregressions](https://otexts.com/fpp2/VAR.html) - [**11\.3** Neural network models](https://otexts.com/fpp2/nnetar.html) - [**11\.4** Bootstrapping and bagging](https://otexts.com/fpp2/bootstrap.html) - [**11\.5** Exercises](https://otexts.com/fpp2/advanced-exercises.html) - [**11\.6** Further reading](https://otexts.com/fpp2/advanced-reading.html) - [**12** Some practical forecasting issues](https://otexts.com/fpp2/practical.html) - [**12\.1** Weekly, daily and sub-daily data](https://otexts.com/fpp2/weekly.html) - [**12\.2** Time series of counts](https://otexts.com/fpp2/counts.html) - [**12\.3** Ensuring forecasts stay within limits](https://otexts.com/fpp2/limits.html) - [**12\.4** Forecast combinations](https://otexts.com/fpp2/combinations.html) - [**12\.5** Prediction intervals for aggregates](https://otexts.com/fpp2/aggregates.html) - [**12\.6** Backcasting](https://otexts.com/fpp2/backcasting.html) - [**12\.7** Very long and very short time series](https://otexts.com/fpp2/long-short-ts.html) - [**12\.8** Forecasting on training and test sets](https://otexts.com/fpp2/forecasting-on-training-and-test-sets.html) - [**12\.9** Dealing with missing values and outliers](https://otexts.com/fpp2/missing-outliers.html) - [**12\.10** Further reading](https://otexts.com/fpp2/further-reading.html) - [Appendix: Using R](https://otexts.com/fpp2/appendix-using-r.html) - [Appendix: For instructors](https://otexts.com/fpp2/appendix-for-instructors.html) - [Appendix: Reviews](https://otexts.com/fpp2/appendix-reviews.html) - [Translations](https://otexts.com/fpp2/translations.html) - [About the authors](https://otexts.com/fpp2/about-the-authors.html) - [Buy a print or downloadable version](https://otexts.com/fpp2/buy-a-print-or-downloadable-version.html) - [Help and feedback](https://otexts.com/fpp2/help-and-feedback.html) - [Bibliography](https://otexts.com/fpp2/bibliography.html) - [Published by OTexts™ with bookdown](https://otexts.com/) # [Forecasting: Principles and Practice (2nd ed)](https://otexts.com/fpp2/) # Chapter 7 Exponential smoothing Exponential smoothing was proposed in the late 1950s ([Brown, 1959](https://otexts.com/fpp2/expsmooth.html#ref-Brown59); [Holt, 1957](https://otexts.com/fpp2/expsmooth.html#ref-Holt57); [Winters, 1960](https://otexts.com/fpp2/expsmooth.html#ref-Winters60)), and has motivated some of the most successful forecasting methods. Forecasts produced using exponential smoothing methods are weighted averages of past observations, with the weights decaying exponentially as the observations get older. In other words, the more recent the observation the higher the associated weight. This framework generates reliable forecasts quickly and for a wide range of time series, which is a great advantage and of major importance to applications in industry. This chapter is divided into two parts. In the first part (Sections [7\.1](https://otexts.com/fpp2/ses.html#ses)–[7\.4](https://otexts.com/fpp2/taxonomy.html#taxonomy)) we present the mechanics of the most important exponential smoothing methods, and their application in forecasting time series with various characteristics. This helps us develop an intuition to how these methods work. In this setting, selecting and using a forecasting method may appear to be somewhat ad hoc. The selection of the method is generally based on recognising key components of the time series (trend and seasonal) and the way in which these enter the smoothing method (e.g., in an additive, damped or multiplicative manner). In the second part of the chapter (Sections [7\.5](https://otexts.com/fpp2/ets.html#ets)–[7\.7](https://otexts.com/fpp2/ets-forecasting.html#ets-forecasting)) we present the statistical models that underlie exponential smoothing methods. These models generate identical point forecasts to the methods discussed in the first part of the chapter, but also generate prediction intervals. Furthermore, this statistical framework allows for genuine model selection between competing models. ### Bibliography Brown, R. G. (1959). *Statistical forecasting for inventory control*. McGraw/Hill. Holt, C. C. (1957). *Forecasting seasonals and trends by exponentially weighted averages* (O.N.R. Memorandum No. 52). Carnegie Institute of Technology, Pittsburgh USA. [\[DOI\]](https://doi.org/10.1016/j.ijforecast.2003.09.015) Winters, P. R. (1960). Forecasting sales by exponentially weighted moving averages. *Management Science*, *6*(3), 324–342. [\[DOI\]](https://doi.org/10.1287/mnsc.6.3.324)