Holt-Winters Method

In the Holt Winters Method (aka Triple Exponential Smoothing), we add a seasonal component to the Holt’s Linear Trend model.

Let c be the length of a seasonal cycle. Thus c = 12 for months in a year, c = 7 for days in a week and c = 4 for quarters in a year. The model takes the following form for all i > c

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where 0 < α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ γ ≤ 1–α. The initial values, i.e. where 1 ≤c, are given by

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An alternative form of these equations is given by

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whereimage016z

Note that if γ = 0, then the Holt-Winters model is equivalent to the Holt model and if β = 0 and γ = 0, then the Holt-Winters model is equivalent to the Single Exponential Smoothing model.

Example 1: Calculate the forecasted values of the time series shown in range C4:C19 of Figure 1 using the Holt-Winter method with α = .5, β = .5 and γ = .5.

The result is shown in Figure 1. First we calculate s1, s2, s3, s4, where c = 4, as shown in range F4:F7. We do this by inserting the formula =C4/AVERAGE(C$4:C$7) in cell F4, highlighting the range F4:F7 and pressing Ctrl-D.

Next, we calculate uc and vc by placing the formula =C7/F7 in cell D7 and the value 0 in cell E7.

We now insert the formula =C$22*C8/F4+(1-C$22)*(D7+E7) in cell D8, the formula =D$22*(D8-D7)+(1-D$22)*E7 in cell E8, =E$22*(C8/D8)+(1-E$22)*F4 in cell F8 and the formula =(D7+E7)*F4 in cell G8, and then highlight the range D8:F19 and press Ctrl-D.

Holt-Winters method Excel

Figure 1 – Holt-Winters Method

Observation: For any value of i, the forecast at time i+h is given by the formula

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Example 2: Forecast the y values for 2014 from Example 1 (i.e. the next 4 quarters).

The result is shown in Figure 2. The values through 2013 are copied from Figure 1. The forecasted value for Q1 of 2014 is 36.87209 (cell N20), as calculated by the following formula with reference to cells in Figure 1.

=(D$19+(L20-L$19)*E$19)*F16

The other three forecasted values are calculated by highlighting the range N20:N23 and pressing Ctrl-D.

Holt-Winters forecast

Figure 2 – Holt-Winters’ Forecast

Observation: As we have done in Example 2 of Holt’s Linear Trend, we can use Solver to determine which values of alpha, beta and gamma yield the best Holt-Winters’ fit for the data in Example 1. In this case you need to assign the value α+γ to some cell and add a constraint to Solver making this value less than 1 (as well as constraining α and β to be less than 1). This turns out to be α = .184124, β = .439756 and γ = .417974, which yields a value for MSE of 19.9562.

7 Responses to Holt-Winters Method

  1. debadrita panda says:

    very nice, just need a clarification, for prediction the formula is =(D$18+(L20-L$19)*E$18)*F16, why D$18 and E$18 NOT D$19 E$19 as, it predicts the value of 20.

    • Charles says:

      Debadrita Panda,
      You are correct. Thank you very much for identifying this error. I have now corrected the error on the referenced webpage.
      I really appreciate your help in improving the Real Statistics website.
      Charles

  2. PVL says:

    Hi Charles,
    Thanks for the help.

    If the time series data has y=0 as some entry the calculation for S and u, v start giving DIV#0 error. If i replace 0 with a very small number like 0.001 the value for initial S goes very low and value of u,v gets multiplied by same factor and the forecast has huge errors.
    Sample Data : YEAR Month mm
    2012 Jan 16.7
    2012 Feb 0
    2012 Mar 0
    2012 Apr 8.2
    2012 May 0
    2012 Jun 2.1
    2012 Jul 438.4
    2012 Aug 367.6
    2012 Sep 71.5
    2012 Oct 0
    2012 Nov 0
    2012 Dec 0

    What do you suggest can be done?

  3. rodrigo oyarzun says:

    Awesome work! Crystal clear.
    where can I download the excel file?

    Thanks!

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