Data Analysis: How to Forecast With Exponential Smoothing

All moving-average schemes have a number of problems.

• They are painful to evaluate. For each point, the calculation has to be performed from scratch. It is not possible to evaluate weighted moving averages by updating a previous result.
  
  
• Moving averages can never be extended to the true edge of the available data set, because of the finite width of the averaging window. This is especially problematic because often it is precisely the behavior at the leading edge of a data set that we are most interested in.
  
  
• Similarly, moving averages are not defined outside the range of the existing data set. As a consequence, they are of no use in forecasting.

Fortunately, there exists a very simple calculational scheme that avoids all of these problems. It is called exponential smoothing or Holt–Winters method. There are various forms of exponential smoothing: single exponential smoothing for series that have neither trend nor seasonality, double exponential smoothing for series exhibiting a trend but no seasonality, and triple exponential smoothing for series with both trend and seasonality. The term “Holt–Winters method” is sometimes reserved for triple exponential smoothing alone.

All exponential smoothing methods work by updating the result from the previous time step using the new information contained in the data of the current time step. They do so by “mixing” the new information with the old one, and the relative weight of old and new information is controlled by an adjustable mixing parameter. The various methods differ in terms of the number of quantities they track and the corresponding number of mixing parameters.

The recurrence relation for single exponential smoothing is particularly simple:

si = αxi + (1 – α)si – 1

with 0 ≤ α ≤ 1

Here s_i is the smoothed value at time step i, and x_i is the actual (unsmoothed) data at that time step. You can see how s_i is a mixture of the raw data and the previous smoothed value s_i–1. The mixing parameter α can be chosen anywhere between 0 and 1, and it controls the balance between new and old information: as α approaches 1, we retain only the current data point (i.e., the series is not smoothed at all); as α approaches 0, we retain only the smoothed past (i.e., the curve is totally flat).

Why is this method called “exponential” smoothing? To see this, simply expand the recurrence relation:



What this shows is that in exponential smoothing, all previous observations contribute to the smoothed value, but their contribution is suppressed by increasing powers of the parameter α. That observations further in the past are suppressed multiplicatively is characteristic of exponential behavior. In a way, exponential smoothing is like a floating average with infinite memory but with exponentially falling weights. (Also observe that the sum of the weights, Σ_j α(1 – α)^j, equals 1 as required by virtue of the geometric series Σ_i qⁱ = 1/(1 – q) for q < 1.)

The results of the simple exponential smoothing procedure can be extended beyond the end of the data set and thereby used to make a forecast. The forecast is extremely simple:

x_i+h = s_i

where s_i is the last calculated value. In other words, single exponential smoothing yields a forecast that is absolutely flat for all times.