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2002issue C021-4

Low-lag trendline from elliptic and dominant-cycle notches

A trendline is the residual after an elliptic lowpass and two notches cancel the measured dominant cycle with less lag than a matching simple average. Failure of a four-bar weighted price to recross that residual for more than half a dominant cycle is treated as trend mode.

  • Averaging 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.
  • An elliptic lowpass, a fixed 10-bar notch, and an adaptive notch at the dominant-cycle period spend 6.7 bars of lag at a 21-bar cycle, versus 10 bars for the matching simple average.
  • When the measured cycle lengthens toward 40 bars, raising the adaptive-notch alpha to 0.9 keeps cascade lag near 8.2 bars, versus about 20 bars for a 40-bar simple average.
  • A four-bar weighted price that fails to recross the residual for more than half a dominant cycle is treated as trend mode, because a cycle mode would recross each half-cycle.
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Current trackMaximum entropy spectrum analysis30 readingsDominant cycle detection119 readingsTrend filter137 readings

What the residual represents

Averaging price over the measured dominant-cycle length cancels that cycle. The residual of a trend-plus-cycle series is treated as the trend even though smaller secondary cycles remain.

The dominant cycle is the bar-by-bar cycle length that sets the variable averaging window or the adaptive notch period so the primary oscillation can be removed.

Lag of a matching simple average

A simple average whose length equals a 21-bar dominant cycle lags price by 10 bars. That lag comes from the relation of cycle length minus 1, divided by 2.

Elliptic lowpass and a leftover delay band

A three-pole elliptic lowpass with 0.8 dB passband ripple and 30 dB stopband attenuation, and with the passband set at normalized frequency 0.22 (a nine-bar period), notches a five-bar cycle and attenuates shorter cycles by 30 dB or more.

That elliptic stage has low-frequency group delay under three bars, but delay near a 10-bar cycle is large because that band is not attenuated.

Fixed and adaptive notches

A 10-bar notch with alpha 0.6, chosen so the upper 3 dB point is 44 percent above the notch frequency, removes the large-delay band. The composite lowpass-plus-notch then has 4.2 bars of low-frequency lag.

A second notch tuned to the measured dominant cycle uses alpha 0.8. At a 21-bar cycle it adds 2.5 bars of lag, for a cascade total of 6.7 bars versus 10 bars for the matching simple average. The period of that adaptive notch is the continuously measured dominant-cycle length supplied by a maximum-entropy spectral estimate.

When the measured cycle lengthens toward 40 bars, raising alpha to 0.9 keeps the tunable-notch delay near four bars and the cascade near 8.2 bars, versus about 20 bars for a 40-bar simple average.

Half-cycle recross timing

A four-bar weighted price smoother with one-bar lag is compared with the residual trendline. Failure to recross for more than half a dominant cycle is treated as trend mode because a cycle mode would recross each half-cycle.

Implemented cascade

The implemented cascade is an elliptic lowpass, then a fixed 10-bar notch, then an adaptive notch at the dominant-cycle period from a maximum-entropy spectral estimate, plotted against a 4-3-2-1 weighted price smoother.

That elliptic lowpass plus the fixed and adaptive notches is the trend filter whose residual is the constructed trendline.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
22 of 28 in the Maximum entropy spectrum analysis track
20041-4 pp.Next on Maximum entropy spectrum analysisSpectral peaks are mode diagnostics, not forecastsConventional Fourier transforms were judged unsuitable for market cycle measurement because the series do not remain stationary long enough to support a reliable estimate.
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 year200284 readings
All 30 readings tagged Maximum entropy spectrum analysis
Also on Maximum entropy spectrum analysis5 readings
2000Switch trend and cycle indicators after a half-cycle dwell test2000Constructing a dominant-cycle squelch trend filter1984How to construct a maximum-entropy cycle model1984Constructing maximum-entropy spectra for dominant-cycle forecasts1985Constructing period-locked half-cycle and full-cycle averages