Constructing channel-normalized Fisher reversal signals

Occupancy comes before a crossover

Market prices are described as lacking a Gaussian probability density, contrary to the common modeling story that about 68 percent of samples fall within one standard deviation of the mean. A probability-density-function records the relative frequency with which a price or oscillator occupies each value inside a stated range. The gaussian-assumption says samples cluster in a bell curve around a mean, with most observations near the center rather than at the extremes.

Why a moving-average cross arrives late

In a two-value square-wave sketch each level has a 50 percent occupancy, so a moving-average cross would register a move only after the series had already switched to the opposite value. A moving-average is a lagging location baseline. A simple price cross of that baseline is used here as the default detector that fails when a series spends most of its time at extremes.

A sinewave occupancy built from 2000 samples concentrates near the maxima and minima, producing a cycle density closer to a square wave than to a bell curve.

A bounded mid-price map and a tail stretch

Channel-normalization rescales a mid-price series to the highest and lowest values of a lookback window, then centers and stretches it into the open interval from -1 to 1. The fisher-transform is a logarithmic map applied to a series already confined inside that open interval so values near the bounds are amplified and the output occupancy becomes closer to a bell curve.

The transform is defined as one-half the natural logarithm of (1 + x) divided by (1 - x), with the input x required to stay strictly inside the open interval from -1 to 1. Near the mean the map has roughly unit gain when the absolute input is below 0.5, while values approaching either bound are strongly amplified, turning a cyclic input density into a nearly Gaussian output density.

The working recipe

The working recipe rescales mid-price, computed as high plus low divided by 2, inside an adjustable 10-bar high-low channel, applies an exponential smoother with alpha 0.33, clamps absolute values of 0.99 to 0.999, then adds a one-half one-bar recursion after the logarithm.

Comparison layers on the same hypothesis

The transformed series plotted against its one-bar delay is the crossover construction used to mark cyclic turning points. That delayed series is the one-bar-trigger: the transformed series delayed by one observation and used as a crossover counterpart.

A similarly scaled MACD overlay is introduced as a conventional comparison whose turning points are described as rounded and lagging relative to the tail-stretched series. MACD is a conventional momentum overlay kept on a comparable scale so those rounded, delayed turns can be compared with the tail-stretched reversal series.

Because the channel length is 10 bars, ten times the rate of change of the transformed series is crossed with the series itself as a second construction for locating major turns. That period-matched multiple is the amplified-rate-of-change, crossed back through the series to mark larger swings.

The same logarithmic stretch is also specified after a series is already normalized, including by a relative-strength-index construction, so peak swings become relatively rare events used to mark reversals. The relative-strength-index is a bounded oscillator that already lives on a normalized scale and can receive the same logarithmic stretch used on channel-normalized mid-price.

Editorial: the moving-average cross, the MACD overlay, and the relative-strength-index input are comparison layers that can each be falsified against the same turning-point hypothesis.