Fourier analysis according to Hurst

How does Fourier analysis work according to Hurst?

In Chapter 11 of The Profit Magic of Stock Transaction Timing, Hurst presents a procedure attributed to Cornelius Lanczos. It starts from prices observed at regular intervals, separates each frequency into a cosine component and a sine component, and then combines the pair into one amplitude.

It is a change of perspective. The original series says when price rose or fell; the spectrum says how much each frequency contributes to reconstructing the analysed segment.

The data contract

Hurst requires equally spaced data: for example, one close per day or per week, always selected by the same criterion. He uses an odd number m of observations in chronological order, so one datum lies exactly at the centre.

A longer window can distinguish more closely spaced frequencies. A higher sampling rate instead allows faster oscillations to be observed. Merely adding points without disclosing interval, missing data, and transformations does not automatically improve the analysis.

The historical procedure in four steps

1. Open the series around its centre. The first sequence contains the middle value followed by sums of equidistant pairs; its final sum is halved. The second starts at zero and uses “later minus earlier” differences; its final element is also zero.
2. Build the grid. Let q = (m−1)/2; Hurst defines Z = π/q. Angular frequencies are ωₖ = kZ/Δt, with k from 0 to q and Δt the interval between observations.
3. Compute the two coordinates. Cosine-weighted sums produce aₖ; sine-weighted sums produce bₖ. They are two views of the same frequency, a quarter-cycle apart.
4. Combine the amplitude. For each frequency, Aₖ = √(aₖ² + bₖ²). Plotting Aₖ against frequency, angular frequency, or period produces the spectrum.

This centred form, including its special treatment of the endpoints, is the manual recipe reported in the book. Modern software can calculate the same family of components with a DFT or FFT, but sign, normalisation, and coefficient ordering must always be disclosed.

A minimal example

With m = 9 weekly observations, q = 4 and Z = π/4. The procedure examines 0, π/4, π/2, 3π/4, and π radians per week. Excluding zero, the corresponding periods are 8, 4, about 2.67, and 2 weeks.

The final point, π radians per interval, is the Nyquist limit: with one datum per week, an oscillation faster than one cycle every two weeks cannot be distinguished correctly. A higher frequency may reappear under a false identity, a phenomenon called aliasing.

Amplitude, phase, and reconstruction

The pair (aₖ, bₖ) determines the amplitude and phase of component k. Amplitude is the vector's length; phase is its angle. The exact sign of a phase formula depends on the sine, cosine, and transform convention, so it should not be copied between software packages without checking their documentation.

Summing the components can reconstruct the window very accurately. This is a mathematical check of the decomposition on the same data, not a forecast error. Extracting coefficients and judging them on that same segment is in-sample analysis.

Resolution and window distortions

In modern notation, a window of N observations spaced by Δt has a typical frequency spacing of Δf = 1/(NΔt). A longer window makes the grid finer; more frequent sampling moves the Nyquist limit, but these are different improvements.

If the window does not contain a whole number of oscillations, energy spreads across nearby bins: this is spectral leakage. Mean, trend, missing values, window choice, and normalisation can change the peaks. These decisions are part of the result and must be reproducible.

What the book reports for the DJIA

Chapter 11 refers to 2,300 weekly closes of the Dow Jones Industrial Average. Appendix I instead specifies 2,229 observations from 29 April 1921 to 25 June 1965 and a resolution of 0.568 radians per year. The discrepancy is preserved: “2,300” appears to be a rounded description of the project, whereas the Appendix supplies the precise count and interval.

Hurst also warns that almost any shape has a spectrum: even points sampled from a ruled line can be reconstructed with oscillating components. The spectrum does not prove that the originating process actually summed those waves. If that condition does not hold, he writes, simple extrapolation of periodicities cannot support prediction.

How to use it rigorously today

The series, interval, data treatment, window, and frequencies to inspect are declared first. Peaks are then compared across subperiods and with complementary estimators. If forecasting is the objective, coefficients and rules are frozen on past data and measured on later data never used to select them.

Fourier is therefore an excellent tool for description and hypothesis formation. Moving from a good reconstruction to a predictive rule requires a separate out-of-sample test.

Sources

• J. M. Hurst, The Profit Magic of Stock Transaction Timing, Prentice-Hall, 1970, Chapter 11, pp. 171–175; Appendix I, pp. 188–200 — Lanczos procedure, DJIA counts, and the warning about extrapolation.
• Google Books bibliographic record for the 1970 Prentice-Hall edition.
• NIST/SEMATECH, Spectral Plot — Engineering Statistics Handbook — dominant frequencies and spectrum interpretation.
• NumPy, Discrete Fourier Transform — modern DFT conventions, frequency, amplitude, and coefficient ordering.

Connections

• Spectral analysis according to Hurst — the broader framework and inferential limits
• Ormsby numerical filters — isolating candidate bands
• Appendix I: DJIA spectrum — the book's historical application
• DJIA decomposition — reconstruction compared with forecasting
• Hurst tradition — complete historical path