Adaptive averages driven by cycle-phase speed

Phase increment as the adaptive control

An exponential moving average is made adaptive by tying its smoothing coefficient to the measured bar-to-bar cycle-phase increment instead of to price volatility. The adaptive-moving-average is an exponential average whose smoothing coefficient is reset from measured cycle-phase speed instead of from a fixed length or from price volatility.

Cycle phase is the arctangent of the quadrature component divided by the in-phase component, and the phase rate of change is the difference of successive phase values. The phase-rate-of-change is the bar-to-bar increment of cycle phase, floored at 1 and used as the divisor of the FastLimit.

One cycle corresponds to 360 degrees of phase advance, so a 36-bar cycle changes about 10 degrees per bar and a 10-bar cycle changes about 36 degrees per bar. Any computed negative phase increment is reset to 1, both because phase is required to advance with time and because the arctangent wraps from +90 degrees back to -90 degrees every half cycle.

FastLimit, clipping, and the ratcheting hold

The adaptive coefficient equals a FastLimit of 0.5 divided by the phase rate of change, then is clipped between 0.05 and 0.5. A coefficient of 0.5 matches a four-bar exponential average.

Half-cycle wrap forces the coefficient to 0.5, so the average steps toward current price and then holds or trails slowly until the next wrap, creating a ratcheting sample-and-hold shape. The ratcheting-hold is a fast step toward current price when the coefficient hits its upper bound, followed by a slow hold until the next half-cycle wrap.

A half-coefficient follower and an order change

A follower average that applies half of the same coefficient to the first adaptive line stays synchronized but moves less, and the two lines change order only after a large shift in market direction. The following-adaptive-average is a second exponential average of the first adaptive line that uses half of that line's coefficient so the pair stays synchronized but the follower travels less.

A moving-average-crossover is a directional hypothesis formed when the phase-adaptive average and its half-coefficient follower change order after a large shift in market direction.

Bounding the measured dominant cycle

The dominant-cycle is the cycle period inferred from successive Hilbert phase readings and from a one-bar homodyne product of the in-phase and quadrature components. Dominant-cycle period from the one-bar homodyne product of the in-phase and quadrature components is bounded between 6 and 50, limited to 1.5 times and 0.67 times its previous value, then smoothed.

When price is in a trend the measured cycle tends to lengthen, so the fast-attack update occurs less often than when a shorter cycle is present.