Weight construction beyond the N-day label

A weighted average, not a named window

An exponential moving average is a weighted average. Each observation is scaled by a weight between zero and one, and those weights always sum to one. A simple moving average is the equal-weight case of that construction, giving every included observation the same weight.

Exponential-smoothing is a recursive forecast construction. It assigns a leading-weight to the newest observation and scales prior output by a retention-factor so older observations decay with age. A moving-average is an equal-weight average over a stated lookback. In the conventional exponential construction it can also serve as the opening block from which new weight is drawn.

How each new observation takes weight

The conventional recursive step forms the new output as the newest observation times a leading-weight K plus the previous output times the complementary factor 1 minus K. That complementary share is the retention-factor. It scales previous output and therefore how slowly older observations fade.

Unlike an N-day simple average that uses exactly N observations and drops the oldest, an exponential construction can start from a single observation and retains every observation it has acquired.

Where the leading-weight is drawn from

With K equal to 0.2, a simple-average initializer must cover at least five earlier observations if no leftover observation is to outweigh the newest observation. Two algebraic choices that meet the initializer bound K greater than 1 over N plus 1 are K equal to 2 over N plus 1, described as the more common mapping, and K equal to 1 over N.

How long the decay profile needs to settle

The weight-distribution is the full set of observation weights after the recursion is expanded. Early profiles are distorted and only later approach a smooth decaying shape. After enough observations for a smooth conventional profile, the most recent N observations carry between 86 percent and 89 percent of total weight. Under the conventional N parameterization, the weight profile generally needs about 2.3 times N observations to lie within 1 percent of its limiting values.

A modified construction and a window reading

A modified construction can start from the first observation without an initializing simple average, take incremental weight from every preceding weight, and set any retention-factor between zero and one rather than only rates implied by integer day counts. One way to make the N-day name refer to a window is to solve the retention-factor so a chosen cumulative share, illustrated at 95 percent, falls on the most recent N days.