Welcome to this forecasting tutorial on Exponential Smoothing.
We will be calculating forecasts using exponential smoothing,
we will also be calculating the mean squared error for the forecasts.
We will be calculating exponentially smoothed forecast for these sales values collected
over a 7-week period. The actual sales values will be represented
by At. Note that some authors use Yt to represent
actual values. Forecast values on the other hand are generally
represented by Ft. The Exponential Smoothing Method uses the
formula Ft+1 = Ft + α(At – Ft) where alpha is a
value between 0 and 1, referred to as the smoothing constant.
This means that the forecast for the current period is obtained by adding the forecast
in the last period to a fraction of the error from the last period.
To make calculations easier, this formula can be rewritten as
Ft+1 = αAt + (1 - α)Ft That is, the new forecast equals the alpha
times the last actual value plus (1 –alpha) times the last forecast value.
We will be using this second formula for our calculations.
Note that both formulas will give the same result. It is just a little easier to use
the second one. So our first objective is to calculate exponential
smoothing forecasts data using α = 0.2.
Since alpha = 0.2, then 1 minus alpha will be 1 - 0.2 which equals 0.8.
The formula then becomes Ft+1 = 0.2 time the actual values + 0.8 times
the forecast values. Since the forecast requires both actual and
forecast values from the last period, we are sometimes given a forecast value for the first
period. If no forecast value for the first period is given, then we assume F1 = A1. That
is, the first forecast value is assumed to be the first actual value. So F1 = 39.
We then calculate F2 as 0.2(39) + 0.8(39) which again will always give the same value,
39. As a result, we just usually assume that F2
= A1 and not bother calculating it at all. That is, if F1 is not given, we simply start
by copying whatever value we have in A1 into F2.
This means that our first real use of the formula begins with F3.
So F3 is going to be .2 times 44 + .8 times 39, which gives 40.
Consequently, F4 will be 0.2(40) + 0.8(40.00) which gives 40.
For F5, we have 0.2(45) + 0.8(40) which gives 41.
For F6, 0.2(38) + 0.8(41.00) which gives 40.4. F7 equals 0.2(43) + 0.8(40.40) and that gives
40.92. We can also forecast week 8 as 0.2(39) + 0.8(40.92)
which gives 40.54. Next we compute the mean squared error.
To obtain the mean squared error, MSE, we first obtain the forecast errors, square them,
and then find the mean or average of the squared errors.
When computing errors for exponential smoothing forecasts, we do not calculate an error for
period 1 unless otherwise stated. So the error for week 2 is 44 - 39 which is
5. For week 3, it is 40 minus 40 which gives
0. For week 4, it is 45 minus 40 which is 5.
For week 5, it is 38 minus 41 which gives -3.
For week 6 it is 2.6, and for week 7 it is -1.92.
Next we square the errors. Starting with week 2,
5 squared is 25 Zero squared is zero
-3 squared is 9 2.6 squared is 6.76
and -1.92 squared is 3.69. Since we only have 6 periods with errors,
we add up these squared errors and divide by 6 to obtain a Mean squared error of 11.58.
And that’s how to calculate forecasts and MSE with exponential smoothing.
Please leave a question or comment below and thanks for watching.