This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Initialisation procedure is influential for ETS forecasting accuracy
User choices to implement Exponential Smoothing (ETS) • initial values, {S0, T0, I0,…,p}
• parameters (fixed or adaptive) {α, β, γ, ϕ}
• loss functions, {MSE, MAD, …}
• decide whether to (re-)normalize the seasonals [Gardner (2006)]
• (individual selection of adequate model form)
ETS Initialisation Relevance of Initial Values
“Parameter selection is
not independent of initial
values and loss functions” 2-stage approach in standard ETS
simultaneous in state-space ETS
DA-M ( aka DM aka ETS(A,D,M)
Damped additive trend &
multiplicative seasonality
Optimisation dominates: full opt./2-stage/heuristic only [Hyndman et al. (2002)], MLE [Broze and Mélard (1990)], nonlinear programming [Segura and Vercher (2001)], etc.
How to determine initial smoothed values? • Question posed since ETS origin [Wade (1967), Cogger (1973), McClain (1981), Taylor (1981)]
• Various „innovative“ approaches in research papers [see, e.g., Bates & Granger (1969)]
• Few guidelines and no empirical evidence [Gardner (1985), Chatfield and Yar (1988)]
Approaches
(1) Least squares estimates [Brown (1959)]
(2) Backcasting [Ledolter and Abraham (1984)].
(3) Global values of Training set [Makridakis et al. (1983)]
(4) Convenient (heuristic) initial values [Makridakis and Wheelwright (1978)]
(5) Zero values (e.g. for all or some) [Makridakis & Hibon, 1991]
• ‘Starting values and loss functions don’t make any difference as optimal smoothing
parameter(s) found compensate for various starting values’ [Gardner (1990b)]
• Makridakis & Hibon (1991): first evidence & propose guidelines on 1001 M-series
(3 nonseasonal ETS methods, 7 types of initial values, multiple loss functions)
”contrary to expectation accuracy is not affected by the type of initial values used”
ETS Initialisation Best practices in Research
“most widely used in practice” [Makridakis & Hibon, 1991]
• Using (longer) averages
• Global Averages of values with
(bounded) parameter Optimisation
• Backcasting of values with
(bounded) parameter Optimisation
• Optimisation of values with
(bounded) parameter Optimisation
From a practical point of view the prevalent use of OLS estimates … seems adequate, … it makes no sense to consider more elaborate alternatives … since such alternatives are more difficult to program and require more computer time.
• Commercial software ignores academic best practices Forecasting practices can be enhanced
• Research use of Aggregate OLS optimised initial values is not the most accurate for all datasets need to consider initialisation in ETS specification
• Forecast Models outside APO can be included in APO use novel algorithms from within APO, incl.
• ETS with damped trend, with additive seasonality etc. • Neural Networks, Support Vector Regression, Decision Trees
• Future Work – Extend to more representative datasets – Assess conditions under which different trend models
work well – Assess heuristics provided (e.g. Hyndman et al) …
Dr. Sven F. Crone Assistant Professor, Director
Lancaster University Management School Research Centre for Forecasting
You may not use the text and images in a paper, tutorial or external
training without explicit prior permission from the centre.
Selected References
• Hyndman, R. J., Koehler, A. B., Snyder, R. D., & Grose, S. (2002). A state space framework for automatic forecasting using exponential smoothing methods. International Journal of Forecasting, 18, 439– 454.
• Makridakis & Hibon (1991) Exponential smoothing: The effect of initial values and loss functions on post-sample forecasting accuracy, IJF, 7, pp. 317-330
• Ledolter, J., & Abraham, B. (1984). Some comments on the initialization of exponential smoothing. Journal of Forecasting, 3, 79– 84.
• Broze, L., & Me´lard, G. (1990). Exponential smoothing: Estimation by maximum likelihood. Journal of Forecasting, 9, 445–455.
• Segura, J. V., & Vercher, E. (2001). A spreadsheet modeling approach to the Holt–Winters optimal forecasting. European Journal of Operational Research, 131, 375– 388.
• SAS Institute Inc., SAS/ETS User’s Guide, Version 8, Cary, NC: SAS Institute Inc., 1999. 1546 pp. (Chapter 12, The FORECAST Procedure)