Exponential Moving average
An exponential moving average (EMA), sometimes also called an exponentially weighted moving average (EWMA), applies weighting factors which decrease exponentially. The weighting for each older data point decreases exponentially, giving much more importance to recent observations while still not discarding older observations entirely. The graph at right shows an example of the weight decrease.
The degree of weighing decrease is expressed as a constant smoothing factor α, a number between 0 and 1. α may be expressed as a percentage, so a smoothing factor of 10% is equivalent to α=0.1. Alternatively, α may be expressed in terms of N time periods, where α=2/(N + 1). For example, N=19 is equivalent to α=0.1.
The observation at a time period t is designated Yt, and the value of the EMA at any time period t is designated St. S1 is undefined. S2 may be initialized in a number of different ways, most commonly by setting S2 to Y1, though other techniques exist, such as setting S2 to an average of the first 4 or 5 observations. The prominence of the S2 initialization's effect on the resultant moving average depends on α; smaller α values make the choice of S2 relatively more important than larger α values, since a higher α discounts older observations faster.
The formula for calculating the EMA at time periods t≥2 is
This formulation is according to Hunter (1986); an alternate approach by Roberts (1959) uses Yt in place of Yt-1:
This formula can also be expressed in technical analysis terms as follows, showing how the EMA steps towards the latest data point, but only by a proportion of the difference (each time):
Expanding out EMA yesterday each time results in the following power series, showing how the weighting factor on each data point p1, p2, etc, decrease exponentially:
In theory this is an infinite sum, but because 1-α is less than 1, the terms become smaller and smaller, and can be ignored once small enough. The denominator approaches 1/α, and that value can be used instead of adding up the powers, provided one is using enough terms that the omitted portion is negligible.
The N periods in an N-day EMA only specify the α factor. N is not a stopping point for the calculation in the way it is in an SMA or WMA. The first N data points in an EMA represent about 86% of the total weight in the calculation.
The power formula above gives a starting value for a particular day, after which the successive days formula shown first can be applied.
The question of how far back to go for an initial value depends, in the worst case, on the data. If there are huge p price values in old data then they'll have an effect on the total even if their weighting is very small. If one assumes prices don't vary too wildly then just the weighting can be considered.
For example, to have 99.9% of the weight,
terms should be used. Since log(1 - α) approaches -2/(N + 1) as N increases, this simplifies to approximately
for this example (99.9% weight).