Constructing market phase from dominant-cycle components

Phase sits on a 360-degree circuit

A completed cycle is treated as a 360-degree phase circuit, so a 10-bar cycle has a theoretical phase increment of 36 degrees per bar. Phase is the angular location inside that circuit, measured in degrees.

The phase routine requires a dominant-cycle length. That length is the one-period lookback for the sine and cosine correlations used to build phase. It may be a fixed swing count, a default of 15 bars, or a value remeasured on every bar. Spectral-analysis is the period-by-period strength display used to inspect or update the dominant-cycle length that the phase construction consumes.

Building phase from one dominant-cycle window

Dominant-cycle phase is constructed by correlating ordered prices over one full cycle length with sine and cosine. Those weights form the orthogonal-components: the real and imaginary sums over one dominant-cycle window.

Phase is then taken as the arctangent of the orthogonal-component ratio, after tangent-quadrant ambiguity is resolved. If the input series is a linear slope, the same one-cycle summation yields a phase of 180 degrees on an upslope and 0 degrees on a downslope.

Comparing observed and theoretical increment

Phase-rate-of-change is the bar-to-bar increment of measured phase, compared with the theoretical increment of 360 degrees divided by the dominant-cycle length. A near-zero observed phase increment, relative to that theoretical increment, is the construction used to flag trend-mode.

The supplied construction classifies a series as trend-mode when the measured phase-rate-of-change is less than 67 percent of the theoretical phase rate of the dominant-cycle. Cycle-mode is the complementary classification, in which measured phase advances near the theoretical dominant-cycle increment.

Two spectral peaks and the sine-wave-price-model

When two spectral peaks are present at once, measured phase can stop advancing uniformly, and the construction reads that stall as a shift from cycle-mode into trend-mode.

A sine-wave-price-model is a pair of traces equal to the sine of measured phase and the sine of that phase advanced by 45 degrees. The pair can cross only while phase is still changing.

Historical series illustration

On the September 1996 Treasury bond series, phase remaining near 0 degrees coincided with a February to March decline. Later uniform phase advance restored a sine-shaped overlay with successive crossings.