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2000issue C121-7

Phasor displays for dominant-cycle construction

A Hilbert transform converts a rotating phasor into inphase and quadrature components so a two-axis track can show when a single dominant cycle still holds and when leftover components warp the path. Entry rules then key off defined inphase axis crossings.

  • Build the display from phasor length and phase angle as two orthogonal series, the inphase and quadrature components extracted by a Hilbert transform.
  • A clean clockwise rotation supports one dominant cycle; uneven sample spacing flags a frequency change, and radius changes usually indicate a second cycle.
  • Shorter leftovers appear as incomplete loops near the origin and longer leftovers pull the trajectory off center, without an extra filter that would add lag.
  • Sell when the inphase component is at a maximum and buy when it is at a minimum, after allowing for the lag inherited from the Hilbert computation.
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Current trackMaximum entropy spectrum analysis30 readingsDominant cycle detection119 readingsRule-based entry174 readings

What a phasor records

A cycle can be constructed as a rotating phasor. When the tip is recorded at a uniform sampling rate, it traces a sinewave, and the angle uniquely locates the current position in that waveform.

How the Hilbert pair is built

The display is built by converting phasor length and phase angle into two orthogonal series, the inphase and quadrature components. A Hilbert transform extracts that pair from an analytic waveform.

The inphase component is the horizontal Hilbert output. The quadrature component is the vertical Hilbert output, offset by a quarter cycle from the inphase series so the pair can draw a rotating trajectory.

Reading a single-cycle path

On a perfect single-cycle path the track rotates clockwise. Period can be estimated by counting samples in one quadrant and multiplying by four. Uneven spacing flags a frequency change. Radius changes usually indicate a second cycle.

Leftover components as path distortions

The phasor is drawn as if only the dominant cycle is present, so subordinate components appear as path distortions. Shorter leftovers show as incomplete loops near the origin. Longer leftovers produce an off-center trajectory. Those leftovers are read from the path itself, without an extra filter that would add lag.

When the path enters trend mode

When the dominant-cycle phase stops advancing, the same plot identifies a shift into trend mode. In the illustrated bond example that shift occurred about 17 bars before the end of the sample around point 24.

In trend mode the trajectory no longer completes a regular rotation, and trend-following takes precedence over cycle entries.

March 1996 T-bond futures daily closes

Weekly closes read from the daily March 1996 T-bond candlesticks show a grind from about 111 to a 122 high, then a late-February break. A trader sees the September dip-and-recovery and the winter top that the article’s phasor windows isolate. Values come from the plotted bars, to the nearest quarter-point, except the 1 March 1996 close of 116.03 printed in the chart header.
Weekly closes read from the daily March 1996 T-bond candlesticks show a grind from about 111 to a 122 high, then a late-February break. A trader sees the September dip-and-recovery and the winter top that the article’s phasor windows isolate. Values come from the plotted bars, to the nearest quarter-point, except the 1 March 1996 close of 116.03 printed in the chart header.US 96H March 1996 T-bond futures · Daily · 1995-08-14T00:00:00.000Z to 1996-03-01T00:00:00.000Z

Digitized from the candlestick raster at weekly Friday samples; quarter-point resolution matches the one-point grid. The 1 March 1996 close is the header print, not a visual estimate. Yellow shading on this frame covers September through early November 1995.

Quadrant crossings as entry rules

A rule-based entry sells when the inphase component is at a maximum, the phasor crossing from quadrant I to IV. It buys when the inphase component is at a minimum, the crossing from quadrant III to II.

Lag inherited by those rules

Those entry rules inherit computation lag: a one-bar lag from a four-bar weighted moving average, a three-bar lag from detrending at the filter center, and a three-bar lag from the final inphase calculation.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
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20021-4 pp.Next on Maximum entropy spectrum analysisLow-lag trendline from elliptic and dominant-cycle notchesAveraging price over the measured dominant-cycle length cancels that cycle, so the residual of a trend-plus-cycle series is treated as the trend even though smaller secondary cycles remain.
All readings on this track · 28 readings
  1. 1984Constructing maximum-entropy spectra for dominant-cycle forecasts
  2. 1984How to construct a maximum-entropy cycle model
  3. 1984Constructing a maximum-entropy forecast from a chosen lookback
  4. 1985Constructing period-locked half-cycle and full-cycle averages
  5. 1986Why Fourier windows limit dominant-cycle resolution
  6. 1987Assembling short-lookback maximum-entropy cycle forecasts
  7. 1988Why a fitted dominant cycle is not a forecast
  8. 1989Evaluating commodity cycle personalities with spectral histograms
  9. 1989Evaluating next-session cycle forecasts with stops
  10. 1989Constructing cycle-aged volatility trailing stops
  11. 1990A channel signal-to-noise gate for dominant-cycle forecasts
  12. 1990Year-over-year dominant cycle personality audit
  13. 1991Cyclic entry from a locked dominant-cycle phase
  14. 1992Stationarity states on synchronized futures spectral contours
  15. 1997Hidden horizon assumptions in dominant-cycle readings
  16. 1997When market cycles are absent more than present
  17. 1997A spectral estimator that retunes indicators to the measured cycle
  18. 2000Constructing a Hilbert dominant cycle and a maximum-entropy refinement
  19. 2000Switch trend and cycle indicators after a half-cycle dwell test
  20. 2000Constructing a dominant-cycle squelch trend filter
  21. 2000Phasor displays for dominant-cycle construction
  22. 2002Low-lag trendline from elliptic and dominant-cycle notches
  23. 2004Spectral peaks are mode diagnostics, not forecasts
  24. 2004Compressive last-stage oscillator construction
  25. 2013Constructing trend failure curves from qualified-trend transitions
  26. 2014Lookback range, a two-lag smoother, and next-bar fills
  27. 2014Constructing a MESA stochastic with roofing and SuperSmoother filters
  28. 2016Constructing spectral heatmaps for dominant market cycles
By fieldCycles and spectral analysis77 readingsBy year200059 readings
All 30 readings tagged Maximum entropy spectrum analysis
Also on Maximum entropy spectrum analysis5 readings
1991Cyclic entry from a locked dominant-cycle phase1984How to construct a maximum-entropy cycle model1984Constructing maximum-entropy spectra for dominant-cycle forecasts1985Constructing period-locked half-cycle and full-cycle averages1986Why Fourier windows limit dominant-cycle resolution