Constructing a Hilbert dominant cycle and a maximum-entropy refinement

A constructed cycle reading

The historical workflow treats a dominant-cycle reading as a chain of constructed instruments rather than as a property read off the chart. Lag is chosen first. The series is then detrended, split into orthogonal components, gated by a signal-to-noise-ratio test, and only then passed to a maximum-entropy-spectrum estimate whose history length is set by the instantaneous frequency.

Lag as a design knob

A uniformly weighted average plotted at the right edge of its observation window produces observation-window-lag of half the window width. A linearly weighted average lags by one-third of that width.

On a constant-slope input the exponential-smoothing-constant equals 1 divided by lag plus 1, so a three-bar lag uses 0.25. A related mapping of 2 divided by period plus 1 follows because a simple average lags by about half its period. The constant is therefore fixed once a tolerable lag is chosen.

Reconstructing a linear trend without lateral delay

A zero-lag-average on a linear trend is a causal reconstruction. It adds the vertical gap between a lag-L average and a lag-2L average back onto the first average so the linear trend is recovered without lateral delay.

Choosing a four-bar lag hides cycle periods shorter than about four bars. A two-pass forward-then-backward average cancels lag only if future bars are available at the right edge.

In-phase and quadrature after the detrend

Spectral-analysis here means decomposing a detrended price series into frequency, amplitude, and phase by forming orthogonal in-phase and quadrature components. After the detrend, a Hilbert-transform all-pass construction synthesizes those traces from a single real sampled series so phase and amplitude can be read bar by bar.

The implemented transform carries about four bars of lag, which is a half-cycle on an eight-bar oscillation and can invert the apparent turning-point crossings. That lag is part of the instrument, not a later correction.

A decibel gate on the phasor

A phasor is applied only after the series is detrended. Signal amplitude is the phasor length formed from the squared in-phase and quadrature terms. Noise is defined as a smoothed bar range.

The signal-to-noise-ratio is reported in decibels: 0 dB when signal equals noise, and 6 dB when signal amplitude is twice the noise amplitude. The ratio is the test of whether a cycle reading is distinguishable from sampling uncertainty.

Accumulating phase into a period

Instantaneous cycle period is formed by taking the arctangent of quadrature over in-phase, differencing that phase from bar to bar, and summing the increments backward until they reach 360 degrees. A 10-bar cycle corresponds to 36 degrees of phase per bar.

The dominant-cycle is that prevailing oscillatory period after the bar count is smoothed into a usable cycle length. The Hilbert instantaneous frequency is presented as finer than a fast Fourier measurement.

Setting the spectral history length

A maximum-entropy-spectrum is then used to shrink the effective noise bandwidth. The instantaneous frequency measurement sets how much history is fed into that cycle estimate, so the data length of the spectral reading is not an independent free parameter.

When the period lengthens

On the assembled indicators, cycle readings are treated as usable only when the signal-to-noise-ratio exceeds 6 dB. A lengthening instantaneous period, illustrated peaking at 34 after a major low, is interpreted as departure from a cyclic turning point into a trend.