2007issue C011-6
Constructing a dominant cycle from transformed Fourier spectra
How a discrete Fourier periodogram is detrended, transformed, and collapsed into one dominant-cycle length, and when mixed periods, a long trend, or a weak signal make that length unusable as an indicator clock.
- Fixed-length oscillators and moving-average pairs are treated as theoretically mismatched when the cycle period is changing. Estimate a dominant-cycle first, then retune lengths to that period or a fraction of it.
- A relative-strength lookback equal to half of one cycle period is described as theoretically producing a full-amplitude swing.
- After a 40-bar high-pass-detrend and a six-element smoother, a resolution-transform of the Fourier periodogram is collapsed with a center-of-gravity-period on bins below a 3-decibel threshold.
- The transformed Fourier estimator did not separate equal-amplitude 20-bar and 24-bar cycles. Leave the clock unused when periods mix, the reading is longer than 50 bars, or signal-to-noise is weak.
Retune length after the cycle is estimated
Fixed-length oscillators and moving-average pairs are treated as theoretically mismatched when the market cycle period is changing.
The proposed construction first estimates a dominant-cycle, then retunes indicator lengths to that period or a fraction of it.
A relative-strength calculation is described as theoretically producing a full-amplitude swing when its lookback equals half of one cycle period.
Estimate frequency, amplitude, and phase together
Cycle measurement is framed as jointly estimating frequency, amplitude, and phase in a low signal-to-noise setting, while remaining responsive to shifting periods without injecting transient artifacts.
Spectral-resolution is the ability to separate nearby cycle periods. A narrow line can separate two nearby periods, whereas a broad line can merge those periods into one averaged reading.
An untransformed Fourier heat map stays diffuse
Fourier-analysis, as used here, is a discrete Fourier periodogram of ordered mid-price samples. It is the baseline spectrum that is later transformed and collapsed into a single period.
Spectral-analysis is the estimate of amplitude versus cycle period, including heat-map display and resolution checks that compare Fourier-type spectra with higher-resolution estimators.
An untransformed discrete Fourier heat map aligned to price, with the period axis scaled from 8 to 50 bars, is shown as too diffuse to isolate a usable cycle on the sample series.
Detrend before the transform
Before the transform, the series is detrended with a high-pass-detrend: a two-pole high-pass filter with a 40-bar cutoff, so long-cycle and trend energy do not leak into the spectrum.
A six-element finite-impulse-response smoother then removes two-bar, three-bar, and four-bar components.
Map the spectrum before you trust a sharp peak
A resolution-transform maps a Bartlett-style spectrum S in the unit interval to a MUSIC-style spectrum as 1 divided by 1 minus S.
After that mapping the two estimators are treated as having the same resolution even when visual inspection makes one look sharper.
Applying that mapping to a discrete Fourier estimate, with 20-decibel peak amplification, produced a spectrum from which a time-varying dominant-cycle could be read.
Collapse the bins without drawing the map
Power is accumulated for periods 8 through 50 over a 50-bar window and converted to a normalized decibel spectrum.
That spectrum is collapsed to one dominant-cycle length by a center-of-gravity-period, a weighted average of spectral bins below a 3-decibel threshold. The extraction does not depend on drawing the heat map.
The dominant-cycle is the single period extracted from the spectrum and then used as the construction length, or a fraction of that length, for other indicators.
Periods longer than 50 bars were treated as a trend regime where the cycle reading is of little help.
Name when the clock is unusable
On synthetic data with equal-amplitude 20-bar and 24-bar cycles, the transformed Fourier estimator failed to separate the pair, while a higher-resolution spectral estimator succeeded.
Remaining limits include a long data requirement, averaging of mixed periods inside the window, and doubtful accuracy in low signal-to-noise conditions.
Editorial: do not inherit the period into other models until you can point to mixed cycles, a reading longer than 50 bars treated as trend, or a weak signal as reasons to leave the clock unused.
All readings on this track · 26 readings
- 1982Critiquing Fourier, Elliott, and seasonal commodity models
- 1982Evaluating half-hour crossovers with cycle priors and stops
- 1984Building moving-window spectral cycle forecasts
- 1988Fourier cycles versus calendar seasonal regimes
- 1989Constructing analog cycle signatures from price lows
- 1990Curve fits fail the forecast test
- 1991Auditing moving-average length with Fourier spectra
- 1991Closed-form phase for dominant-cycle spectra
- 1993Evaluating windowed spectral forecasts of index turning points
- 1994Window length for moving-window spectral forecasts
- 1994Constructing WAMI power and phase spectra
- 1994Fourier-tuned momentum filter design
- 1994Aliasing, leakage, and dominant-cycle reads from sampled prices
- 1995Constructing indicator lookbacks from Fourier spectra
- 1996Constructing phase from a dominant-cycle sine and cosine pair
- 1999Sinusoidal pseudo-securities for indicator evaluation
- 2000Constructing dominant cycle length from Hilbert phase
- 2000Constructing adaptive channel breakouts from a measured cycle period
- 2005Embedding forex prices into a singular-spectrum forecast basis
- 2005Constructing recurrent forecasts from grouped spectral components
- 2007Constructing a dominant cycle from transformed Fourier spectra
- 2010Why cycle indicators fail when parameters shift
- 2019Constructing futures momentum as open-interest-scaled rate of change
- 2019Judging a dominant cycle and a trend filter by sample, lag and the move they keep
- 2020Constructing futures turnover time and urgency
- 2020Stock power ratio as a growth trend filter