Paired tests for moving-average lag and smoothness

What the paired scores measure

A moving average is a lookback smoother that turns ordered prices into a delayed, less noisy companion series over a stated sampling interval. Exponential smoothing is a recursive average that weights the newest observation more than older ones and can be run once or stacked on its own output. A triangular moving average is a moving average whose weights rise to the middle of the window and then fall, emphasizing the center of the lookback.

A multi-pass exponential average is a short exponential average applied to its own output several times, used as a low-noise counterpart to a longer single-pass exponential average. The archive paired an 11-period exponential moving average with a 3-period exponential average applied five times.

Linear-input-lag on a ramp and a tent

The two exponential averages were treated as sharing a five-day lag when the input was linear. Linear-input-lag is delay measured when the input is a straight ramp, which can understate delay on peaked or oscillatory paths.

On a constructed tent path that rose for 10 days and then fell for 10 days, the 11-period exponential average peaked 3 days after the crest and the five-times-smoothed 3-period average peaked 4 days after the crest. Phase-lag is how many bars a smoother's turning point trails the same turn in the raw series.

Smoothness on a noisy flat line

On a flat line with added random noise, average absolute distance from that line was used as a smoothness score. Smoothness here is how little leftover oscillation remains after filtering, scored as distance from a quiet baseline or as how rarely the one-period change crosses zero.

The single 11-period exponential average reduced that distance more than the five-times-smoothed 3-period average while also showing less delay. In that lag-matched noise test, the single exponential average was judged to smooth more than simple, weighted, triangular, and double-exponential moving averages.

A crossing count on daily closes

On more than two years of one equity's daily closes, each smoother's one-period rate of change was used to count zero-line crossings as a noise proxy. That crossing-interval test found the five-times-smoothed 3-period average crossed zero about half as often as the 11-period exponential average.

Extra delay as the counterpart of lower noise

A reply held that both smoothers still share five days of lag on linear input, that extra delay for the five-times-smoothed average appears when the input has spectral content such as a sine or sawtooth wave, and that this extra delay is the counterpart of lower noise.