Wave counting as a sampling frame for head and shoulders

Count similar four-swing shapes

An exact replay of the current swing is treated as too scarce to estimate a future bias. Wave counting therefore uses similar four-swing length classes that can be found often enough to count.

Wave counting means grouping successive price swings by the relative length of the last four legs so similar shapes can be counted instead of hunting for an exact replay.

32 M and W classes

The relative lengths of the prior four swings define 32 classes, divided into 16 M-shaped outlines and 16 W-shaped outlines. Each mw-class is one of those 32 four-swing shapes.

Turning points are ranked from the highest extreme downward so a four-swing shape can be coded independently of calendar order. That turning-point rank numbers swing extremes from highest to lowest so the same shape receives a stable code such as 2-3-1.

A percent-filter is a minimum swing size used to mark turning points before a four-swing class is assigned.

Head and shoulders as a named subset

Head-and-shoulders bottoms occupy six M classes and head-and-shoulders tops occupy six W classes. In this evaluation the familiar three-trough or three-peak outline is not one pattern but twelve named M-class bottoms and W-class tops inside the four-swing catalog.

The pattern is evaluated as a subset of the wave-count catalog rather than as a single formation.

A chance check on the later split

A chi-square test is specified as the check that a historical up-versus-down split after a similar swing class is not merely chance. It is a textbook check on whether an advance-versus-decline split after a wave class is large enough to reject a chance explanation.

What the historical workflow recorded

The evaluation isolated industrial-average swings with a 5% filter over 90 years and recorded whether the average was higher 26 weeks later and one year later after each matched M or W class.

Five tabulated formations did not produce significant directional probabilities, and the W11 class was too rare to evaluate.