Recency weighting in simple, linear, and exponential moving averages

Lookback length and the target bar

A moving average is a lookback smoother on ordered price, volume, or breadth observations that reduces short-lived fluctuations over a defined sampling interval. It is specified by two construction choices: how many sampling periods enter the lookback, and which observation the average is computed for.

Lookback length is the number of sampling periods included in the average, stated in the same periodicity as the series. Simple, linear, and exponential constructions all average a window of observations to damp short-lived fluctuations and isolate an underlying trend, but they distribute weight across that window differently.

Equal-weight average

In an equal-weight average, each observation in the lookback receives the same fraction of the total weight. In a simple moving average each observation in an N-period lookback receives equal weight 1/N, so an older change of a given size offsets a newer change of the same size.

Because a long equal-weight average still gives full weight to its oldest observation, the next update can be anticipated once a large old print is about to leave a 200-period window.

Linear-weight average

A weighted moving average assigns larger influence to more recent observations than to older ones inside the same lookback. A linear-weight average scales observations by integers from one for the oldest bar to the lookback length for the newest bar, then divides by the sum of those integers.

A linear weighted moving average multiplies the oldest observation by 1 and the newest by N, then divides by the sum of the integers from 1 through N rather than by N. For a five-period linear average the newest observation is scaled by 5 and the oldest by 1, and the weighted sum is divided by 15.

Exponential-weight average

An exponential-weight average is a recursive weighted average that applies a single newest-bar weight to the latest observation and the residual weight to the previous average. An exponential moving average uses the same newest-observation weight as the matching linear average, then applies the leftover weight to the previous exponential value instead of assigning a distinct weight to every older bar.

The newest-bar weight of an exponential moving average can be computed as 2 divided by one more than the lookback length. In the five-period case that shared newest-bar weight is 0.33, so the update mixes 0.33 times the latest observation with the complementary weight on the prior exponential average.