Constructing symmetric market-breadth ratio accumulators

Construction starts with scale

Market-breadth is participation measured from advancing versus declining issues and from up volume versus down volume, used to turn a repeatable chart condition into a testable hypothesis. The first construction choice is how those counts are combined.

A ratio is one series divided by another. When the two inputs share similar ranges, the quotient tends to oscillate around 1.0. A conventional advance-decline line is a cumulative difference between advancing and declining issues, not a ratio of those two counts.

What the combined ratio encodes

In a ratio that divides the advance/decline split by the up-volume/down-volume split, readings above 1.0 correspond to relatively more down volume in declining issues, and readings below 1.0 correspond to the opposite mix.

Asymmetry pulls the moving average

Because that raw ratio can rise without an upper bound but cannot fall below zero, a 39-day exponential moving average of the unadjusted series is pulled toward values above 1.0. A moving average is a lookback average of ordered price, volume, or breadth observations that supplies an explicit quantitative baseline over a defined sampling interval.

The unity-symmetry-transform

Mapping each reading below 1.0 to its negative inverse, so that 0.5 becomes -2, centers the series on zero and evens the spike distribution before further smoothing or accumulation. That step is the unity-symmetry-transform: it maps each reading below 1.0 to its negative inverse so magnitudes on either side of unity are comparable.

What still drifted after the transform

After that transform, a 39-day exponential moving average remained above zero through a 15% decline in the Dow Jones Industrial Average, and accumulating the transformed readings produced a series that kept rising except for a limited setback around October 1987. A ratio-accumulator is a running sum of transformed ratio readings intended to show directional persistence rather than a single-session imbalance.

An accumulator built from the advance/decline ratio alone did not show that persistent upward drift, which isolated the volume terms as the remaining source of bias.

Equal-weight adjustment of volume

Scaling the up-volume inputs down by 20% flipped the accumulated series from a rising slope to a falling slope. A 7.5% reduction was the adjustment chosen for the January 1984 to February 1989 window.

A crossover on a two-sided series

After equal-weight adjustment, a 19-day exponential moving average of the advance/decline ratio oscillated near +1.0 to -1.0, and crossings between a 9-day and a 40-day exponential moving average were plotted as trading signals. A moving-average-crossover is a signal formed when a shorter moving average of an adjusted breadth series crosses a longer one, converting a repeatable chart condition into a falsifiable trade hypothesis.

Range-normalization for comparison

Recalculating each series as 100 times (value minus minimum) divided by (maximum minus minimum) placed a raw advance-decline accumulation and an adjusted advance/decline-ratio series on a shared 0-100 scale so their paths could be compared. That recasting is range-normalization: each series becomes a percent of its own minimum-to-maximum span.