Constructing a deviation-scaled adaptive moving average

Two stages of construction

A deviation-scaled-moving-average is assembled in two stages. First, an exponential baseline is locked to a user-chosen critical-period. Second, a whitened two-bar residual, expressed in its own rms-scale units, raises or lowers the mixing weight from that baseline.

An exponential mix and its alpha

An exponential moving average is built as a convex mix of the current close and the previous filter value. The mixing weight is alpha, which lies between zero and one. Smaller alpha increases smoothing. Larger alpha reduces smoothing and gives more weight to the latest close.

Why common adaptive modifiers add lag

Adaptive averages change that alpha from a separately measured market condition. Two common constructions derive the modifier from an RSI-like statistic or from net price change over a window divided by the sum of bar-to-bar changes. Both estimate the modifier over multiple bars, which adds computational lag.

A critical-period baseline

The user period is treated as the filter critical-period, the cycle length at which passed signal power is halved. The baseline exponential alpha is set to five divided by that period. When the oscillator’s scaled amplitude deviation equals one, the adaptive alpha matches an exponential average with the same critical-period.

A zeros-oscillator and spectrum-whitening

The scaling oscillator is a two-bar close-to-close difference. This zeros-oscillator has a transfer response of zero for an unchanging series, which supplies a nominal zero mean, and it rolls off at six decibels per octave. Fractal market amplitudes are described as rising at the same rate, so the difference is used for spectrum-whitening.

The same two-bar difference also has a transfer-response zero at the Nyquist period, the shortest cycle that sampled data can represent. That zero is intended to cancel the six-decibel noise gain of a one-bar difference and to lessen aliasing in the oscillator.

A two-pole residual and rms-scale

The oscillator is passed through a two-pole smoother whose critical-period is half the user period so the residual stays responsive. The smoother coefficients are computed only on the first bar. Because the input is nominally zero-mean, the filtered series is treated as zero-mean as well.

Local scale is the rms-scale of that filtered oscillator over the user period, the root-mean-square of the residual. Dividing the filtered series by that rms-scale expresses it in standard-deviation units. Small local deviation then yields a small effective alpha and heavier smoothing. Large deviation raises alpha and lets the filter follow price more quickly.

Working alpha and the adaptive series

The working alpha is the absolute scaled value times five divided by the period. The adaptive series is the usual exponential recursion that mixes the current close with the previous adaptive value using that working alpha. Tracking speed is retuned by changing the single period input that sets both the baseline alpha and the rms-scale window.