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1994issue C031-7

Window length for moving-window spectral forecasts

A moving-window spectral forecast advances a fixed lookback along ordered observations so each cycle can be measured, forecast, and recombined. Spectral density and frequency-domain variance show which periods must finish inside that lookback. The historical workflow then locked window length to a whole multiple of those periods and compared the choice on unused later observations.

  • Window length sets how many sampling intervals sit inside each lookback and therefore which periods can be observed whole.
  • Read spectral density and frequency-domain variance first so every cycle that carries a large share of non-trend variation can complete inside the sample.
  • Lock the window to a whole multiple of the dominant periods, then compare that choice on unused later observations.
  • The historical workflow recommended 80 days for daily sampling, 32 weeks for weekly sampling, and 48 months for monthly sampling, aligned to a recurring 16-period cycle and its whole multiples.
Entries in this reading3 entries
Current trackFourier analysis41 readingsSpectral analysis63 readingsDominant cycle detection180 readings

What window length controls

A moving-window spectral procedure treats ordered observations as a trend plus many cycles whose magnitude and phase may change independently. A fixed-length window is advanced point by point so each component can be recorded, forecast separately, and then recombined. Window length is the parameter that sets how many sampling intervals sit inside each lookback and therefore which periods can be observed whole.

Time-domain work indexes the series by sampling interval, while spectral and Fourier-style work indexes each sinusoid by frequency. A component that spans 80 days has a period of 80 days and a frequency of 1/80 cycles per day. Fourier analysis represents those oscillations by period and frequency so each component can be measured, allowed to evolve, forecast on its own, and then recombined.

Read spectral density first

Density analysis of a daily industrial-average sample flagged important cycles of 16, 40, and 80 days and preferred an 80-day window because that length is a whole multiple of both 16 and 40 and therefore contains all three. The non-repeating zero-frequency trend held 95.43% of density. When a forecasting model is built, that trend is removed by differencing daily changes and restored only at the last step.

Among repeating components, the 40-day cycle had the largest density at 0.76% and the 80-day cycle had 0.57%. Lengthening the window is described as increasing long-horizon capacity while still capturing shorter cycles that divide the chosen length.

Measure frequency-domain variance next

Frequency-domain analysis of variance measures how each cycle changes as the window slides. The 16-day component contributed 20.15% of total variation, the 40-day component 13.47%, and the 80-day dominant cycle 4.68%, so each needed to be represented. The dominant cycle is the repeating component that must fit inside the chosen window because it accounts for a large share of non-trend variation as the lookback slides.

After the trend and the six longest cycles, a remainder of 50.09% of variation was spread across 34 smaller components. The combined moving-window spectral representation left a grand-total accounting error of 2.4461%.

Compare unused later observations last

Window length was next judged by comparing holdout error across the 16-, 40-, and 80-day candidates already singled out by density and variance, using a model fit on the first 88 days of a 100-day 1991 sample and unused later observations. Combining density, variance, and holdout-error checks produced recommended window lengths of 80 days for daily sampling, 32 weeks for weekly sampling, and 48 months for monthly sampling, aligned to a recurring 16-period cycle and its whole multiples.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
10 of 26 in the Fourier analysis track
19941-2 pp.Next on Fourier analysisConstructing WAMI power and phase spectraWAMI is a three-parameter cycle operator whose frequency behavior is read from constructed power and phase curves, not from a single opaque oscillator reading.
All readings on this track · 26 readings
  1. 1982Critiquing Fourier, Elliott, and seasonal commodity models
  2. 1982Evaluating half-hour crossovers with cycle priors and stops
  3. 1984Building moving-window spectral cycle forecasts
  4. 1988Fourier cycles versus calendar seasonal regimes
  5. 1989Constructing analog cycle signatures from price lows
  6. 1990Curve fits fail the forecast test
  7. 1991Auditing moving-average length with Fourier spectra
  8. 1991Closed-form phase for dominant-cycle spectra
  9. 1993Evaluating windowed spectral forecasts of index turning points
  10. 1994Window length for moving-window spectral forecasts
  11. 1994Constructing WAMI power and phase spectra
  12. 1994Fourier-tuned momentum filter design
  13. 1994Aliasing, leakage, and dominant-cycle reads from sampled prices
  14. 1995Constructing indicator lookbacks from Fourier spectra
  15. 1996Constructing phase from a dominant-cycle sine and cosine pair
  16. 1999Sinusoidal pseudo-securities for indicator evaluation
  17. 2000Constructing dominant cycle length from Hilbert phase
  18. 2000Constructing adaptive channel breakouts from a measured cycle period
  19. 2005Embedding forex prices into a singular-spectrum forecast basis
  20. 2005Constructing recurrent forecasts from grouped spectral components
  21. 2007Constructing a dominant cycle from transformed Fourier spectra
  22. 2010Why cycle indicators fail when parameters shift
  23. 2019Constructing futures momentum as open-interest-scaled rate of change
  24. 2019Judging a dominant cycle and a trend filter by sample, lag and the move they keep
  25. 2020Constructing futures turnover time and urgency
  26. 2020Stock power ratio as a growth trend filter
By fieldCycles and spectral analysis129 readingsBy year1994135 readings
All 41 readings tagged Fourier analysis
Also on Fourier analysis5 readings
1984Building moving-window spectral cycle forecasts1989Constructing analog cycle signatures from price lows1991Auditing moving-average length with Fourier spectra1991Closed-form phase for dominant-cycle spectra1993Evaluating windowed spectral forecasts of index turning points