Holt-Winters Forecasting Algorithm

A Holt-Winters Forecasting Algorithm is an exponential forecasting algorithm that uses a level smoothing factor and a trend smoothing factor.

• AKA: Holt's Linear Trend Procedure.
  • …
• Counter-Example(s):
  • Brown's Forecasting Algorithm.
  • Box-Jenkins Algorithm.
• See: Holt Smoothing.

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References

2015

• https://www.otexts.org/fpp/7/2
  • QUOTE: Holt (1957) extended simple exponential smoothing to allow forecasting of data with a trend. This method involves a forecast equation and two smoothing equations (one for the level and one for the trend):
    • Forecast equation [math]\displaystyle{ \hat{y}_{t+h|t|} = ℓ_t+hb_t }[/math]
    • Level equation [math]\displaystyle{ ℓ_t = αy_t +(1−α)(ℓ_{t−1}+b_{t−1}) }[/math]
    • Trend equation [math]\displaystyle{ b_t = β^*(ℓ_t−ℓ_{t−1})+(1−β^*)b_{t−1} }[/math]
  • where [math]\displaystyle{ ℓ_t }[/math] denotes an estimate of the level of the series at time [math]\displaystyle{ t }[/math], [math]\displaystyle{ b_t }[/math] denotes an estimate of the trend (slope) of the series at time t, [math]\displaystyle{ α }[/math] is the smoothing parameter for the level, [math]\displaystyle{ 0 ≤ α ≤ 1 }[/math] and β∗ is the smoothing parameter for the trend, [math]\displaystyle{ 0 ≤ β^* ≤1 }[/math] (we denote this as [math]\displaystyle{ β^∗ }[/math] instead of β for reasons that will be explained in Section 7/7).
  • As with simple exponential smoothing, the level equation here shows that ℓ_t is a weighted average of observation y_t and the within-sample one-step-ahead forecast for time t, here given by [math]\displaystyle{ ℓ_{t−1} + b_{t−1} }[/math]. The trend equation shows that b_t is a weighted average of the estimated trend at time t based on [math]\displaystyle{ ℓ_t−ℓ_{t−1} }[/math] and b_{t−1}, the previous estimate of the trend.

    The forecast function is no longer flat but trending. The h-step-ahead forecast is equal to the last estimated level plus hh times the last estimated trend value. Hence the forecasts are a linear function of hh.

    The error correction form of the level and the trend equations show the adjustments in terms of the within-sample one-step forecast errors: [math]\displaystyle{ ℓ_t = ℓ_{t−1} + b_{t−1} + αe_t }[/math] :[math]\displaystyle{ b_t = b_{t−1} + αβ^∗e_t }[/math] where : [math]\displaystyle{ e_t = y_t−(ℓ_{t−1}+b_{t−1}) = y_t−\hat{y}_{t|t−1} }[/math].

2014

• http://people.duke.edu/~rnau/411avg.htm#HoltLES
  • QUOTE: Brown’s LES model computes local estimates of level and trend by smoothing the recent data, but the fact that it does so with a single smoothing parameter places a constraint on the data patterns that it is able to fit: the level and trend are not allowed to vary at independent rates. Holt’s LES model addresses this issue by including two smoothing constants, one for the level and one for the trend. At any time t, as in Brown’s model, the there is an estimate Lt of the local level and an estimate Tt of the local trend. Here they are computed recursively from the value of Y observed at time t and the previous estimates of the level and trend by two equations that apply exponential smoothing to them separately.

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
  • QUOTE: Holt-Winters Forecasting: An application of exponential smoothing to a time series that displays a *trend and *seasonality.

1988

• (Chatfield & Yar, 1988) ⇒ Chris Chatfield, and Mohammad Yar. (1988). “Holt-Winters Forecasting: Some Practical Issues." The Statistician
  • ABSTRACT: The Holt-Winters forecasting procedure is a variant of exponential smoothing which is simple, yet generally works well in practice, and is particularly suitable for producing short-term forecasts for sales or demand time-series data. Some practical problems in implementing the method are discussed, including the normalization of seasonal indices, the choice of starting values and the choice of smoothing parameters. There is an important distinction between an automatic and a non-automatic approach to forecasting and detailed suggestions are made for implementing Holt-Winters in both ways. The question as to what underlying model, if any, is assumed by the method is also addressed. Some possible areas for future research are then outlined.

1960

• (Winters, 1960) ⇒ Peter R. Winters. (1960). “Forecasting Sales by Exponentially Weighted Moving Averages." Management Science, 6(3). doi:10.1287/mnsc.6.3.324
  • ABSTRACT: The growing use of computers for mechanized inventory control and production planning has brought with it the need for explicit forecasts of sales and usage for individual products and materials. These forecasts must be made on a routine basis for thousands of products, so that they must be made quickly, and, both in terms of computing time and information storage, cheaply; they should be responsive to changing conditions. The paper presents a method of forecasting sales which has these desirable characteristics, and which in terms of ability to forecast compares favorably with other, more traditional methods. Several models of the exponential forecasting system are presented, along with several examples of application.

1957

• (Holt, 1957) ⇒ Charles C. Holt. (1957). “Forecasting Seasonals and Trends by Exponentially Weighted Moving Averages." Carnegie Inst. of Technology.

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