From aliasing and dilation to a roofed stochastic swing

The archive construction begins with the sampling interval, then a roofing preprocessor, then a stochastic oscillator computed on that roofed series and smoothed once. A confirmation swing on that stochastic is then counted against the half-swing of an assumed dominant cycle.

Aliasing below the Nyquist period

Daily bars remain one sample per session even if the high and low are averaged in. Aliasing-noise is spurious short-period energy created because those bars are discrete samples, not a continuous path, so cycles near and below the two-bar Nyquist period cannot be recovered. Aliasing several octaves below that two-bar Nyquist period swamps four-bar and eight-bar cycle estimates.

Inefficient averages

Simple and exponential averages reduce aliasing only by lengthening the window, which adds lag, so they are inefficient stopband tools for sampled price series. With both critical periods set to 10 bars, a 10-bar exponential average shows only 13 dB of rejection at the Nyquist frequency. A two-pole smoother, converted from a two-pole analog prototype, is built to nearly remove that aliasing with less lag.

Spectral dilation

Cyclic amplitude grows about 6 dB per octave of period. That spectral dilation lets longer waves dominate raw oscillator output. First-order oscillators such as the stochastic, momentum, relative-strength, moving-average-convergence, and commodity-channel constructions roll off at the same 6 dB per octave and can at best flatten that dilated spectrum.

A second-order high-pass

A first-order high-pass leaves a non-zero mean in a long uptrend because longer cycles leak. A second-order high-pass at 12 dB per octave exceeds the 6 dB dilation slope and recenters the oscillator near zero with less leak-induced lag.

Roofing then one stochastic smooth

Stacking the second-order high-pass with the two-pole smoother yields a roofing filter. The stacked preprocessor is what later oscillators see: a band-limited, near-zero-mean series. A following stochastic is the current roofing value minus its lookback low, divided by the lookback range, then smoothed once instead of with stacked conventional smoothers.

Confirmation delay versus the half-swing

Confirmation-style stochastic swing rules buy after a cross back above 20 and sell short after a cross back below 80. About two bars of smoother lag, one bar of next-bar entry, and about three bars of confirmation total about eight bars of delay. That delay is set against an assumed 20-bar dominant cycle whose half-swing is 10 bars.

Early crosses and the swing-setup

Over about 2,500 daily samples spanning 10 years, the roofing-filtered stochastic's value density is nearly that of a sine wave. That historical density motivates acting when the line crosses under 20 or over 80 before the extremum, replacing confirmation delay with about two bars of net lag versus price turning points.

A swing-setup procedure pairs a smoothed, zero-mean cycle line that stays nearly in step with price peaks and troughs with a lagged stochastic-like momentum line. A candidate long setup is a cycle trough in the last few days while momentum is still declining or at a minimum.