Constructing a cosine-correlation cycle and phasor trend filter

Building the cosine-correlation series

A cycle-mode series can be built by correlating ordered prices with a cosine wave over one full wavelength of a chosen period. The output stays in phase with a matching cyclic component.

Editorial interpretation: the chosen cosine period is the Dominant cycle detection assumption, and the cosine and sine correlations are the Spectral analysis construction being proved on the bench.

Proving the detector on a known waveform

On a known 20-bar sinusoid, correlation with a 20-bar cosine overlays the input so tightly that detuning the input to 21 bars is used to show the two series are distinct.

With the cosine period held at 20 bars, stretching the input to 30 bars produces a leading phase error under 45 degrees. Compressing the input to 15 bars produces a lagging phase error under 45 degrees. The construction does not collapse when the assumed dominant cycle is wrong.

An orthogonal companion and a line Trend filter

An orthogonal companion series is formed by correlating the same input with a negative sine of that period. That companion places a 90-degree lead, so its zero crossings mark the peaks and valleys of the cosine-correlation output.

A companion Trend filter is constructed by correlating prices with a straight line of positive slope. That line correlation is intended for trend-mode conditions, just as the cosine correlation is intended for cyclic, mean-reverting conditions.

Using the correlations on real prices

On real prices the synthetic input period can be set to zero so the correlations use the price series itself. A 14-bar cosine period was chosen to bias the output toward a leading phase error.

Recovering phase from the two components

Phase is recovered as the arctangent of the cosine-correlation component over the sine-correlation component, with quadrant correction, a wrap from -180 to +180 degrees, and a constraint that the angle is not allowed to step backward.

Reusing a stalled phasor as market state

The sine-based rate series fails when price enters a trend. That cycle-mode breakdown is reused as a rapid way to classify the market as having switched into trend mode.

A market-state variable is 0 while phase keeps rotating, +1 when the angle freezes at or above zero, and -1 when it freezes below zero. Phase rates under 9 degrees per bar, equivalent to a 40-bar or longer dominant cycle, are classified as trend rather than cycle.

Because the phasor is assembled from two independent orthogonal inputs, the construction is described as having a 6 dB signal-to-noise advantage over a single conventional line indicator.