Single Exponential Smoothing (SES)

SES is a method for smoothing a time series or for forecasting where the mean is either stationary or changes only slowly with time.  The name is misleading, as this is a moving average method where the weight declines as the lag between the current time increases.  As this is smoothing method which relies on previous values, the smoothed value, lags the current value.

The weight given to the current and previous values is given by the formula:

If the smoothing value is small, fluctuations will be heavily damped and the smoothed value will tend towards the mean, if it is large, fluctuations will be significant and the smoothed value will tend towards the current value.

The table below shows the weights applied to current and preceding values using a smoothing factor of 0.5.

Lag	Weight (4 d.p.)	Y	YS
0 (current)	0.5000	10	5.0000
1	0.2500	9	2.2500
2	0.1250	8	1.0000
3	0.0625	7	0.4375
4	0.0313	6	0.1878
5	0.0156	5	0.0780
6	0.0078	4	0.0312
7	0.0039	3	0.0117
8	0.0020	2	0.0040
9	0.0010	1	0.0010
10	0.0005	0	0.0000
Smoothed Value			9.0012

Calculation

The process for calculating the smoothed value simplifies to:

Initial Values

Ideally the initial value of YS in formula 4.1.20 should be calculated using appropriated weights as shown in the above table, however, this means discarding a number of data points at the start of the series, this may not be a problem in long series, but with short ones and small smoothing coefficients, this may not be desirable.  A pragmatic approach may be required:

If the mean value is static or only changing slowly with time, an average of first few values can be used.

If the mean is fluctuating, setting the first smoothed value to the first observed value may be a workable option.

Variation in Smoothing Coefficients

The graphs below show a time series which moves from one level to another, in the first case it is a smooth curve, in the second it is the same function with some added noise.

Time Series without "noise"

Time Series with added "noise"

The two curves illustrate the relationship between smoothing an lag.  A smoothing coefficient of 0.2 produces a smooth curve which lags the source by as much as 6 periods, whilst 0.8 reduces the lag to approx. 1 period, but there are significant fluctuations in the smoothed value.

Forecast

The forecast value is the last smoothed value, this is only meaningful for a stationary time series.

Example

The example is based on the maximum monthly average temperature for London for the period 1860 to 1990.  A smoothing coefficient of 0.05 has been used to damp down the fluctuations to the point where an slow, upward trend can be observed.

It should be pointed out changing the smoothing coefficient gives a significantly different visual impression.

Page modified: 05-Oct-2005

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