Exponential Smoothing - The Exponential Moving Average

The Exponential Moving Average

Exponential smoothing was first suggested by Robert Goodell Brown in 1956, and then expanded by Charles C. Holt in 1957. The formulation below, which is the one commonly used, is attributed to Brown and is known as "Brown's simple exponential smoothing".

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

where α is the smoothing factor, and 0 < α < 1. In other words, the smoothed statistic s_t is a simple weighted average of the previous observation x_t-1 and the previous smoothed statistic s_t−1. The term smoothing factor applied to α here is something of a misnomer, as larger values of α actually reduce the level of smoothing, and in the limiting case with α = 1 the output series is just the same as the original series (with lag of one time unit). Simple exponential smoothing is easily applied, and it produces a smoothed statistic as soon as two observations are available.

Values of α close to one have less of a smoothing effect and give greater weight to recent changes in the data, while values of α closer to zero have a greater smoothing effect and are less responsive to recent changes. There is no formally correct procedure for choosing α. Sometimes the statistician's judgment is used to choose an appropriate factor. Alternatively, a statistical technique may be used to optimize the value of α. For example, the method of least squares might be used to determine the value of α for which the sum of the quantities (s_n-1 − x_n-1)2 is minimized.

Unlike some other smoothing methods, this technique does not require any minimum number of observations to be made before it begins to produce results. In practice, however, a "good average" will not be achieved until several samples have been averaged together; for example, a constant signal will take approximately 3/α stages to reach 95% of the actual value. To accurately reconstruct the original signal without information loss all stages of the exponential moving average must also be available, because older samples decay in weight exponentially. This is in contrast to a simple moving average, in which some samples can be skipped without as much loss of information due to the constant weighting of samples within the average. If a known number of samples will be missed, one can adjust a weighted average for this as well, by giving equal weight to the new sample and all those to be skipped.

This simple form of exponential smoothing is also known as an exponentially weighted moving average (EWMA). Technically it can also be classified as an Autoregressive integrated moving average (ARIMA) (0,1,1) model with no constant term.