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Exponential smoothing

Formula for exponential smoothing -

St  =  a.Xt +(1-a).S(t-1)
where

  • St is the value of the smoothed series at time t,
  • St-1 is the value of the smoothed series at time t-1,
  • Xt is the value of the smoothed series at time t,
  • a is a constant between 0 and 1

Worked example, where a=0.3 & 1-a=0.7

Raw data at time t Smoothed data at time t-1 Smoothed data at time t
12 (12)
30(12) 0.3×30 + 0.7×(12) = 17.4
2517.4 0.3×25 + 0.7×17.4 = 19.7
1519.7 0.3×15 + 0.7×19.7 = 18.3
etc.18.3 etc.

  • Notice that, because there are no earlier data, the first point (in this case 12), is unsmoothed - and has a far greater influence than all the other observations. This effect is obvious on the first graph of this set.
  • Notice also that, because only observations prior to time t are used, this form of smoothing tends to introduce a lag - the smaller a is, the greater this lag tends to be.
  • The following code produces much the same sort of result using R.