Oscillator distribution shape for mean-reversion filters

Measure the density first

A trend-filter, in this construction, is a forecast-style mean-reversion filter. Extreme readings are treated as low-probability departures from a measured center rather than as trend-following signals.

The usable extremes depend on a measured probability-density. Spectral-analysis, two-bar differencing, smoother scaling, and a Gaussianizing transform are applied to change that shape before any trade rule is written.

How the probability-density is accumulated

A mean-reversion filter can be constructed by assuming a zero-mean oscillator, placing each bar’s reading into 30 bins below zero and 30 bins above zero over a range of about -3 to +3, accumulating those counts through the sample, and exporting the histogram on the last bar as an empirical probability-density.

What a scaled RSI density looked like

On daily SPY from January 1, 2008 through December 31, 2017, a default-settings RSI scaled from -1 to +1 produced a nearly uniform density with an upside tilt, so the chance of an “overbought” reading exceeded the chance of a reading near the mean by more than a factor of two.

Editorial: TradersWeek does not treat familiar RSI 20/80 style thresholds as already valid filter events on that shape, because the measured density did not concentrate probability at the center.

Spectral-analysis and the zeros-oscillator

Spectral-analysis inspects an oscillator’s transfer response and the fractal amplitude slope of price so the construction can whiten the spectrum, cancel DC and Nyquist artifacts, and leave a usable probability shape.

The zeros-oscillator is a two-bar close-to-close difference whose transfer response is zero for a constant price, the DC term, and zero at the Nyquist period of two samples. That response removes a simple one-bar difference’s extra 6 dB of high-frequency noise and reduces aliasing in the oscillator output.

Because market cycle amplitudes rise statistically at about +6 dB per octave while the zeros-oscillator rolls off at about -6 dB per octave, that two-bar difference is constructed to whiten the price spectrum rather than leave a 1/f-type amplitude slope in the filter output.

Building a deviation-scaled-oscillator

A deviation-scaled-oscillator is built by sending the zeros series through a two-pole SuperSmoother whose critical period is half the input length, with coefficients computed on the first bar, then dividing the smoothed output by the RMS of that output over the input period so the plotted series is in local standard deviations.

The same 60-bin, -3-to-+3 accumulation used on RSI can be appended after RMS scaling so the constructor can see whether the scaled series concentrates probability at the center.

Fisher-gaussianizing and the filter event

A nonlinear Fisher transform, used here as fisher-gaussianizing, can convert a wide class of waveforms toward a near-Gaussian bell. Applying that construction, RocketRSI with swings limited to about plus or minus three standard deviations, to the same 10-year SPY sample yielded a bell-shaped density instead of the nearly uniform RSI histogram.

The same 60-bin accumulation can also be appended after the Fisher step, so the constructor can compare whether the raw deviation-scaled-oscillator or the Fisher-mapped series actually concentrates probability at the center.

Under a near-Gaussian oscillator, a cross of about -2 can be read as leaving a region that occupies only about 2.5% of the measured density. That measured share is the construction rationale for treating the crossing as a mean-reversion filter event rather than as an RSI 20/80 threshold.