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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)

## 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 25 17.4 0.3×25 + 0.7×17.4 = 19.7 15 19.7 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.   