A three-part workshop for adaptive-filter length search

Three pieces, one construction

Editorial framing: read this archive construction as a three-part workshop. The first part is a moving-average view of how noise is folded into a mean. The second is an exponential-smoothing recursion used as the live output. The third is an adaptive-filter length search that shrinks a shared window until a median and that smoother finally agree.

Editorial note: the archive facts describe a historical workflow. They do not present a trading rule.

A moving average folds noise into a mean

An averaging filter reduces noise by folding it into a summed series so that, given enough samples, the noise component tends toward a near-zero mean. Here the moving average is that same-length window mean. It is the conceptual noise-folding baseline, and it is the averaging counterpart in the replacement example that follows.

A median keeps a single spike from moving the output

A median filter reduces impulsive spikes by ranking the window and emitting the middle value, so one extreme observation does not change the output. The median is the middle ranked value in an odd-length window, which is why a single extreme sample does not move that output.

In a ten-sample window of ones, both the average and the median equal 1. After one sample is replaced by 10, the average becomes 1.9 while the median remains 1.

Exponential smoothing is the live output

The live output is exponential smoothing: a recursive average that blends the newest observation with the prior output through a coefficient that can be reset whenever the selected length changes.

That coefficient is alpha, the exponential-smoothing weight obtained by dividing 2 by one plus the selected length.

The length search shortens a shared odd window

The length search begins with a relatively long odd window so the median sits at the exact center, then shortens that window by 2 on each pass. At each candidate length, a same-length median of a pre-smoothed series is compared with an exponential-smoothing output of that series, and the absolute percentage difference of those two outputs is recorded.

The search keeps the shortest length whose absolute percentage difference falls below a chosen threshold, then converts that length into an exponential-smoothing coefficient of 2 divided by one plus the length. The threshold is that absolute percentage gap between the same-length median and exponential-smoothing outputs.

Illustrated defaults and a changing memory

The illustrated construction pre-smooths the high-low midpoint with a four-sample weighted average, starts the search at length 39, applies a default threshold of 0.002, and floors length at 3.

Because that coefficient can be recomputed with each new sample, the final adaptive-filter output changes its effective memory as conditions change. An adaptive filter is a smoother whose effective lookback, and therefore its smoothing weight, is recomputed from incoming observations instead of being held fixed.

With the threshold set to 0.002, the plotted adaptive-filter output is described as adjusting quickly on larger price moves and remaining comparatively quiet through a congestion interval.