Who this entry is for — The complete recipe for measuring a stock's spectrum by hand, as the book gives it (in Lanczos's simple variant): paper, trigonometric tables and patience. Today a computer does it in a flash — but understanding it step by step is the best way to know what a spectrum really means.
Source: J. M. Hurst, The Profit Magic of Stock Transaction Timing, Prentice-Hall, 1970 — Chapter 11, §§ How to Do Fourier Analysis → The Kind of Results You Can Expect (Lanczos method, pp. 171–175).
Prerequisites
Spectral analysis — what period, amplitude, phase and angular frequency are.
1 · Assemble the data
Three rules: data equi-spaced in time (daily, weekly, monthly…); one representative value per interval, always by the same criterion (usually the close); an odd number of datums, in chronological order. The more data, the finer the spectrum's resolution.
2 · The two sequences
Mark the central datum of the series and build two new sequences from there — every further operation works on these, not on the prices:
- Sequence 1 (for cosines) — first element: the middle number; then the sum of pairs equidistant from the centre, widening out to the ends; the last element (first + last price) is divided by 2.
- Sequence 2 (for sines) — same, except: first element zero; differences instead of sums (always the later minus the earlier); last element set to zero.
3 · The frequencies
With m datums, compute Z = π / ((m−1)/2). The analysis's angular frequencies are: 0 (the value without oscillation), then Z, 2Z, 3Z… up to ((m−1)/2)·Z — each divided by the data spacing. Each converts back to a period via T = 2π/ω. Convert everything to radians/year (from weekly: ×52) to avoid confusion.
4 · The amplitudes
For each frequency, two amplitudes:
- cosine (from Sequence 1): for the k-th frequency, the sum of each element times cos(k·j·Z) — j being the element's index — divided by (m−1)/2. For ω = 0: the sum of the whole sequence, divided by (m−1)/2.
- sine (from Sequence 2): identical, with sines instead of cosines; the first sinusoidal amplitude is zero.
5 · The composite spectrum
For each frequency, the composite amplitude is √(a² + b²) of the two. Plot the amplitude-frequency (or amplitude-period) pairs — and your Fourier analysis is complete: a spectrum like Fig. A I-1's.
The book's test bench — This analysis was carried out on 2,300 weekly closes of the DJ 30: the results, with interpretation and correlations with the other spectral methods, are the heart of Appendix I.
The warning worth the chapter
Warning — Draw a ruled line, tabulate its points, and Fourier will return "a set of oscillating components which can be made to approximate the original line as closely as you desire". The analysis found the line's spectrum — but you made the line with a ruler, not by summing waves. Fourier is the starting point when you suspect hidden periodicities; to know whether the generating process really put them there, the other techniques are needed. For prices, the answer is the book's thesis: about 23% of price motion is not an artificial spectrum but an intrinsic process — the case is developed in the Appendices.
Summary card
| Step | Operation |
|---|---|
| 1 | m odd datums, equi-spaced, same price criterion |
| 2 | Two sequences from the centre: sums (cosines) and differences (sines) |
| 3 | Z = π/((m−1)/2); frequencies 0, Z, 2Z, … /(spacing) |
| 4 | Cosine and sine amplitudes per frequency |
| 5 | Composite √(a²+b²) → amplitude-period spectrum |
| ⚠ | A spectrum always exists; predictability only if the process is truly oscillatory |
Links
- Spectral analysis — Ch. 11's framework
- The Ormsby filters — the next step: isolating what Fourier found
- Appendix I — the DJIA's real spectrum (2,300 weeks)
- The nominal cycles — what the spectrum confirms
- Hurst tradition — chapter index