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2007issue C041-2

Construct a momentum difference from the dominant cycle

Differencing subtracts an earlier observation from the latest one and yields a signed series that regularly crosses through zero. On a teaching sine wave, a half-cycle interval keeps full amplitude and coincident turns, while shorter or longer lookbacks trade amplitude for lead or lag.

  • Differencing subtracts an earlier observation from the latest observation over a fixed interval and yields a signed series that regularly crosses through zero.
  • On a teaching 20-period sine wave, a 10-step half-cycle interval keeps full amplitude and coincident highs and lows.
  • A 5-step interval turns early with half the amplitude, a 15-step interval turns late with half the amplitude, and a full-period interval returns zeros.
  • A shorter lookback is more exposed to random false turns, a longer lookback can miss much of an extended move, and a packaged default used without inspecting cycles is a construction risk.
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Current trackDominant cycle detection119 readings

Differencing as a signed series

Differencing subtracts an observation from a fixed number of steps earlier from the latest observation and yields a signed series that regularly crosses through zero.

A single sine wave is only a teaching model for the differencing operation. Actual series combine several cycles, trend, and irregular fluctuation.

The teaching sine wave

The worked cyclical model is a sine wave with a period of 20 sampling steps, an amplitude of 10, and a central value of 100.

On that 20-period wave, a 10-step difference reproduces the original series with no amplitude loss and coincident highs and lows.

A 5-step difference on the same wave keeps a similar shape, halves the amplitude, and turns before the original series, which is also where a split between the indicator and the source path can appear.

A 15-step difference on the same wave halves the amplitude and turns after the original series.

The half-cycle interval

The interval that maximizes difference amplitude and aligns turns with the modeled cycle is one-half the cycle period. That half-cycle interval is the differencing lookback equal to one-half the period of the cycle under study.

Size the lookback from the dominant cycle, the repeating up-and-down component whose measured period sets the interval. Count period in sampling steps from any point to the next equivalent point, often between successive highs or lows.

Amplitude is the distance from a cycle's central value to an extreme high or low. Moving away from that half-period in either direction shrinks amplitude, and an interval equal to the full period returns zeros.

Leading and lagging differences

An interval shorter than half the cycle can turn early but, with reduced amplitude, is more exposed to random false turns. That earlier series is a leading difference.

An interval longer than half the cycle yet shorter than the full period lags and can miss much of an extended move. That later series is a lagging difference.

A packaged default interval used without inspecting the data for cycles is treated as a construction risk.

Teaching sine wave with a 20-day period

A constructed price model swinging from 90 to 110 around a center of 100, so amplitude is 10 and each high or low repeats 20 days later. Those parameters are stated in the article; the points follow that sine rather than any traded market. Half-cycle differencing is taken from this series.
A constructed price model swinging from 90 to 110 around a center of 100, so amplitude is 10 and each high or low repeats 20 days later. Those parameters are stated in the article; the points follow that sine rather than any traded market. Half-cycle differencing is taken from this series.1 day

The source uses this sine only to show how a differencing interval relates to cycle length, and warns that real prices also contain trend, other cycles, and noise.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
21 of 31 in the Dominant cycle detection track
20071-1 pp.Next on Dominant cycle detectionNaive dominant-cycle rules fail without crowd testsCyclical swings are treated as important in markets, yet the associated dips and peaks are described as difficult to detect.
All readings on this track · 31 readings
  1. 1982Cycle phase windows for chart signal filters
  2. 1987Constructing a cycle-scaled trend oscillator
  3. 1987Constructing a dominant-cycle grid from marked lows
  4. 1988Cycle lead from staggered exponential averages
  5. 1988Auditing the forty-month stock-price cycle
  6. 1989When long-wave dominant cycles cannot be disproved
  7. 1991Half-cycle average plot shift versus cycle attenuation
  8. 1991Half-cycle average contact as an amplitude-ratio test
  9. 1993Building a restoring-pull indicator from cycle frequency and volume
  10. 1995Regime filters for a dominant long wave
  11. 1995A cycle-tuned lead filter from bounded oscillators
  12. 1998Testable cycle rules instead of fear and greed
  13. 1999Nested Euro cycle timing as one checkable procedure
  14. 2002Constructing an instantaneous trendline from a dominant cycle
  15. 2002Half-cycle center of gravity oscillator from moving-average balance
  16. 2004Testing a locked forty-week cycle with a hold-or-sit-out rule
  17. 2005Nested timing bands for dominant-cycle confirmation
  18. 2005Dominant-cycle baselines versus policy-news narratives
  19. 2006Pairing a dominant-cycle horizon with trend and oscillators
  20. 2006A dominant-cycle split into a trend filter and residual Relative Strength Index
  21. 2007Construct a momentum difference from the dominant cycle
  22. 2007Naive dominant-cycle rules fail without crowd tests
  23. 2012Constructing a dominant-cycle forecast as a timing window
  24. 2012Open-parameter construction of dominant-cycle baselines
  25. 2013Using a second-term election to check a predeclared dominant-cycle forecast
  26. 2014Constructing a dominant-cycle forecast baseline
  27. 2014Quotient transform as an early-onset trend filter
  28. 2014Construct a trough-to-trough cycle map with the Detrended Price Oscillator
  29. 2015Dominant-cycle alignment before an earnings catalyst
  30. 2017Causal reverse exponential average for cycle and trend
  31. 2020Constructing a cycle-plus-trend oscillator from a one-wavelength chord
By fieldCycles and spectral analysis77 readingsBy year200773 readings
All 119 readings tagged Dominant cycle detection
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