Residual-impact construction of exponential smoothers

Two common calculation methods

Simple moving averages and exponentially smoothed moving averages are identified as the two calculation methods most often found among widely used trading systems.

A moving average is a finite-window average of ordered price, volume, or breadth observations whose contribution ends on the interval after they leave the window. Exponential smoothing is a recursive smoother that mixes the newest ordered observation with the prior output through a constant alpha so earlier observations decay geometrically and never vanish completely.

What leaves a window and what remains

An observation leaves an n-period simple or weighted moving average on the next interval, but the same observation never fully exits an exponentially smoothed average.

Two controls for the smoothing factor

The proposed construction selects the smoothing factor from two controls: the share of an observation's original impact to be used over a span, and the number of days in that span.

Residual impact is the remaining fraction of one observation after a chosen number of intervals. The leftover share after n intervals is (1-alpha) to the n.

How each update weights one observation

Each update is the newest price times alpha plus the previous smoother times one minus alpha. Unfolding that recursion shows the observation's weight on day n is alpha times (1-alpha) to the power n minus one.

The cumulative share of one price over n days is 1 minus (1-alpha) to the n, so the leftover share is (1-alpha) to the n.

A residual-and-span example

Using 95 percent of an observation in the first 10 days, a 5 percent residual, produces alpha equal to 0.25887 and matches the closed form 1 minus residual impact raised to 1 over days.

Length-only mappings

Conventional mappings treat a smoothing factor as equivalent to an n-day simple or weighted average by using either 1/n or 2/(n+1). Those shortcuts are a length-mapped alpha: a conversion of a window length n into alpha with either one over n or two over n plus one.

For a length-only shortcut, 2/(n+1) is preferred to 1/n. As the period lengthens, residuals under those two mappings approach about 36.79 percent and 13.534 percent, and many residual-and-days pairs can yield the same alpha.