Skip to main content
Track Fourier analysis
19 / 26
Search the archiveLibrary
Library

2005issue C101-4

Embedding forex prices into a singular-spectrum forecast basis

A one-dimensional price series is first embedded as overlapping delay windows, then read as a singular spectrum, then reduced to leading principal components. Fourier analysis is kept as a comparison of shapes, and reconstruction or forecasting is left to a later stage.

  • Treat equally spaced prices as system states and embed a finite run of N observations with a delay window no larger than the integer part of N divided by 2.
  • Form the trajectory matrix, then use spectral analysis of its singular values to decide where signal ends and a residual plateau begins.
  • Keep a leading block by spectrum truncation, project those axes as principal components, and use Fourier analysis only to check whether paired waveforms are ordinary harmonics.
  • Stop once the principal components are extracted. Reconstruction of the original series and any forecast step belong to a subsequent stage.
Entries in this reading3 entries
Principal component analysis1 readingCurrent trackFourier analysis41 readingsSpectral analysis63 readings

Start with embedding, not a forecast

A one-dimensional price series is treated as equally spaced system states. Embedding converts a finite run of those prices into overlapping delay windows that become the rows of a trajectory matrix.

The construction chooses an interval of N observations and a delay window m no larger than the integer part of N divided by 2. Spectral analysis, spectrum truncation, and principal component analysis all act on that matrix, so the matrix has to exist before any later forecast step is attempted.

The trajectory matrix

That embedding produces a trajectory matrix with m rows and N minus m plus 1 columns. Each column is one successive window, so the original series is rewritten as an m-dimensional representation built from overlapping segments.

The row count is the chosen window length. The column count is the number of successive windows.

Reading the singular spectrum

Spectral analysis of the trajectory matrix is obtained from its singular value decomposition. The square roots of the ordered diagonal covariance entries are the singular values. Lined up from largest to smallest, those values are the singular spectrum.

Gaps and slope changes in that list are used to decide which components belong to signal and which belong to a residual plateau.

Truncation instead of a change of axes

An empirical orthogonal function is a left or right singular vector of the trajectory matrix. Additive white noise is described as raising singular values without changing those functions.

Separating signal from residual is therefore a truncation of the singular spectrum rather than a change of axes. Spectrum truncation keeps a leading block of singular values as signal and sets the remainder to zero.

An hourly euro-dollar reading

In the constructed hourly euro-dollar example, N equals 200 and m equals 100. The first singular value stands apart from the rest and is identified with the main trend component.

The second and third singular values are separated from the remaining 97 by a smaller gap, so a leading block can be kept as signal and the rest set to zero. Changes in slope of the singular-value curve between the fifth and 20th values and between the 20th and 80th values are used as candidate truncation thresholds.

Principal components from selected functions

Principal component analysis is constructed by projecting the trajectory matrix onto selected empirical orthogonal functions. The first component maximizes variance. Later components are uncorrelated and maximize remaining variance.

An elementary pattern is a reconstructed component associated with one or more retained singular values. It is the language used later to describe trend, oscillation, or residual structure. This stage names the retained axes. It does not yet rebuild the series.

Fourier analysis as a comparison only

Fourier analysis is used as a comparison construction. Both approaches sum oscillatory pieces of different frequencies and amplitudes, but the singular vectors need not form sine-cosine pairs.

The comparison asks whether a retained pair looks like an ordinary harmonic or like a data-adaptive shape belonging to the embedded series.

Where this construction stops

This construction stops at the extracted principal components. Reconstruction of the original series and any later forecast step are treated as a subsequent stage.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
19 of 26 in the Fourier analysis track
20051-4 pp.Next on Fourier analysisConstructing recurrent forecasts from grouped spectral componentsChoose the delay window first. Lengthen it for smoothing, shorten it toward the leading-component count when searching for hidden periods, and set it to a multiple of a known period when separating periodic fluctuations. The window should begin and end at roughly the same level.
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 fieldTime series and forecasting58 readingsBy year2005167 readings
All 41 readings tagged Fourier analysis
Also on Fourier analysis5 readings
1982Building FFT spectra to size cycle filters1984Building moving-window spectral cycle forecasts1989Constructing analog cycle signatures from price lows1991Auditing moving-average length with Fourier spectra1991Closed-form phase for dominant-cycle spectra