CYCLEPEDIAThe trading encyclopedia
IT
James Marsden Hurst 1924—2005

Chapter 12.5

Appendix V — Parabolic interpolation (Hurst)

One parabola per triple of points: the technique the book uses to bring filter outputs to a common interval — exact formula, sliding-window procedure, limits.

On this page

Who this entry is for — Data spacing is a design parameter of numerical filters: two filters built on different spacings produce outputs that cannot be compared directly. This is the glue: "for the charts in this book, all filter results have been interpolated in this manner down to a common interval of one week".

Source: J. M. Hurst, The Profit Magic of Stock Transaction Timing, Prentice-Hall, 1970 — Appendix V, Parabolic Interpolation (pp. 212–214).

Prerequisites

The Ormsby filters — where the series to reconcile come from.

The problem and the solution

One filter outputs every 5 weeks, another every 3: to sum, difference or compare them you need values at the same instants. The solution: fit a least-square-error parabola through each consecutive triple of output points, and solve it for the in-between values on a common grid.

With t = 0 at the central point S₀ of the triple (S₋₁, S₀, S₊₁) at spacing tₙ, the book's derivation gives:

S(t) = S₀ + [(S₊₁ − S₋₁)/(2tₙ)]·t + [(S₊₁ + S₋₁ − 2S₀)/(2tₙ²)]·t²

— and the parabola through three points passes through them exactly (the derivation shows a₀ = S₀). The procedure is a sliding window: interpolate the segment between the central point and the next, then slide one point forward and repeat, until the data are exhausted.

HURST 1970 · APPENDIX V Parabolic interpolation at work Appendix V: from one spacing to another without losing the curve CYCLEPEDIA DIAGRAM — EMICICLO FILTER OUTPUT AT SPACING 5 → COMMON GRID AT SPACING 1 filter points (spacing 5) resampled to 1 true curve FORMULA S(t) = a₀ + a₁t + a₂t² MEASURED ERROR ≈0.9% at 12 pts/cycle Every chart in the book passes through here: each filter brought back to the week.
The technique at work: a filter output at spacing 5 (gold points) brought to spacing 1 (teal) — with one local parabola highlighted in violet and the true curve dashed.
Tap the parabola and the curve

The limits, declared — The technique is meant for band-pass outputs with little high-frequency content: on a sine sampled at 12 points per cycle the error stays below 1% (measured: ≈0.9%), consistent with Ch. 11's rule of ≥6–7 points per cycle. High-pass or otherwise non-smooth data may require more points in the fit — or other forms of curve fitting.

Summary card

Element Value
Purpose Filter outputs at different spacings → common grid
Formula S(t) = S₀ + a₁t + a₂t² per triple, exact on the three points
Procedure Sliding window, one segment at a time
In the book Everything interpolated to one week
Limits Smooth (band-pass) series; ≈0.9% error at 12 points/cycle
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