Exponential Smoothing

Exponential smoothing is one of the most popular techniques used in time series smoothing and forecasting. The smoothing procedure is based on calculating exponential moving averages of the smoothed series.

The main advantage of a forecast model based on exponential averages is in the following: it can adapt to a new process level without being excessively responsive to random deviations.

The method was developed by Messrs. Brown and Halt independently of one another. Holt also suggested exponential smoothing models for processes with constant level, linearly growing processes, and processes with seasonal effects.

The simplest form of exponential smoothing is given by the formulas:

where:

Xt-1. Actual observation at the moment t-1.
St. Value of exponential average at the moment t.
α. Example of smoothing, α = const, α ϵ (0; 1].

Exponential average at the moment t is shown as the weighted sum of the current observation and exponential average of the previous observation with the weights α and (1 - α) respectively. If this recurrence relation is used consistently, the value St can be expressed via values of the time series X:

Thus, the value St is the weighted sum of all series members. Weight values are decreased exponentially depending on how far the observation is from the time moment t. That is why this technique is named exponential average.

Exponential smoothing can be described as a filter where the members of the source series are input as a stream, and the output forms values of exponential averages. The smoothed series St has the same mean of distribution as X, but smaller variance.

When the α value is large, the variance of the smoothed series does not much differ from the variance of the series X. The smaller is α, the more reduced is the variance of the smoothed series (that is, oscillations of the source series are suppressed).

Exponential average can be used to make short-term forecasts. In this case, the source series is assumed to be described by the model:

where:

at. Mean series level that varies in time.
errt. Random non-autocorrelated deviations with zero mean.

Given below is the forecasting model:

where:

. Forecast made at the moment T for т steps forward.
. Estimate a_T.

The estimate for parameter of the model aT is exponential average of the series ST. Thus, the forecast model has all properties of exponential average. For example, if a recurrence formula is brought to the following form:

and regard St-1 as a one-step-forward forecast, the value (Xt-1 - St-1) is the error of this forecast, and the new forecast St is obtained by correcting the previous forecast taking into account its error. This describes adaptation.

Using simple exponential smoothing, more complex models have been developed to smooth time series that include cyclic seasonal oscillations and/or tend to increase.

Like simple exponential smoothing, this model is used to smooth a source series based on the growth effect (linear, exponential or damped) and the seasonality effect (additive or multiplicative).

Given below is the general form of the recurrence formula of exponential smoothing:

where the factors d1 and d2 depend on the selected smoothing model. For example, after the simple exponential smoothing described above, d1 = Xt, d2 = St-1.