Spectral whipsaw limits for a price-crossing average

The lookback is already licensed

Once a tradable cycle has already set the simple-moving-average length, the remaining modeled hurdle is whether a shorter residual cycle can displace the price-path crossover times. The dominant-cycle is that selected spectral peak, and it licenses the averaging length of the crossover rule.

Spectral-analysis is a transform of ordered prices that separates peaks above a noise cutoff from shorter residuals treated as contaminants. Spectral peaks below a noise-based cutoff are classified as whipsaw cycles, and the modeled contaminant is the peak immediately below the selected tradable cycle.

The modeled average is formed by applying a rectangular window to a sine-style price path. The moving-average-crossover flips long when the average sits below the path and short when the average sits above it. That flip is against the price path rather than against another constructed series.

A timing window, not a cost test

The whipsaw-cycle is that next-lower peak, modeled as magnitude B. B-over-A is the ratio of whipsaw-cycle magnitude to tradable-cycle magnitude and is used as the contamination budget.

A setup is scored useless when a nonzero whipsaw magnitude moves any crossover outside a window of plus or minus 10 percent of the tradable-cycle wavelength centered on the zero-whipsaw crossings. That band is the whipsaw-window. Commissions and slippage are not part of that test.

On a 500-day tradable cycle with a 250-day average, the longest interval that stays outside those 10 percent windows captures 43.5 percent of the cycle magnitude.

The residual budget does not travel with trend

With phase and linear trend set to zero, maximum allowable B-over-A ratios were independent of tradable-cycle wavelength. A reference point is a B-over-A of 0.63 at a signal-to-noise-ratio of 2 when the average length is half the tradable cycle. Signal-to-noise-ratio is tradable-cycle strength relative to the noise band around the price path in the numerical model.

An illustrated spectrum meets the 0.63 B-over-A reference at a 0.588 magnitude ratio under a 0.381 noise reading and fails it at 0.778 under a 0.78 noise reading. Most other examined series sat above the tabulated maxima.

Repeating the same 10 percent window with a linear-trend-ratio of 0.005 reduces the allowable B-over-A relative to the zero-trend case. Linear-trend-ratio is the slope of an added linear drift divided by tradable-cycle magnitude, written b/A.

False crossings and a noise-band gate

Zero whipsaw magnitude produces four crossings per cycle. With a 500-day tradable cycle, a 10-day residual, and a 250-day average, false crossings rise as B-over-A approaches 0.63 at a signal-to-noise-ratio of 2 or 0.18 at a signal-to-noise-ratio of 0.5.

A noise-band-filter requires the average to clear both edges of the noise band before a flip is accepted. Forcing the average to clear both edges of the noise band stops a half-cycle average near a signal-to-noise-ratio of 0.43. A filter four noise levels wide would pass only signals near twice the noise level, while quarter-cycle and tenth-cycle averages stop at higher signal-to-noise-ratio.