In one sentence — Parabolic interpolation uses three already-known values to estimate the points between them. Hurst uses it to place filter outputs calculated at different intervals onto the same time grid.
What problem does it solve?
One filter may produce a value every five weeks and another every three. Before comparing, adding, or subtracting the two outputs, values are needed at the same instants.
In Appendix V of The Profit Magic of Stock Transaction Timing, Hurst constructs a local parabola from three consecutive points and evaluates it between the centre point and the next. He states that the book's filter results were thereby brought to a common one-week interval for charting.
How the three-point window works
1 — Fix the triple
The consecutive samples are S₋₁, S₀, and S₊₁, separated by a constant interval tₙ. Local time is set to zero at the centre sample S₀.
2 — Compute the parabola
The local form is S(t) = a₀ + a₁t + a₂t². In Hurst's symmetric case the coefficients are:
a₀ = S₀
a₁ = (S₊₁ − S₋₁)/(2tₙ)
a₂ = (S₊₁ + S₋₁ − 2S₀)/(2tₙ²)
Therefore:
S(t) = S₀ + a₁t + a₂t², with 0 < t < tₙ.
The new values belong to the interval already bounded by S₀ and S₊₁. They are not newly observed data: they are estimates produced by the curve.
3 — Slide one sample forward
After filling that segment, the window advances. The old S₊₁ becomes the new centre, another point enters on the right, and the next segment is calculated.
Adjacent parabolas share the value at their common node, so the sampled series remains continuous there. Their slopes, however, need not match when one parabola gives way to the next.
What “exact” means
Three points supply three equations for three coefficients. The parabola therefore passes exactly through S₋₁, S₀, and S₊₁: residual error at the nodes is zero by construction.
This does not show that the real shape between the nodes is parabolic and is not independent validation. General interpolation theory locates the error between the known points; it depends on the function's smoothness and on sample spacing.
Hurst derives the result through least-squares equations. In the specific three-point case, however, the system is exactly determined; no additional observations remain with which to measure the statistical adequacy of the parabola.
Interpolation, extrapolation, and forecasting
To estimate an instant between S₀ and S₊₁, S₊₁ must already be known. The intermediate value becomes available only after the sample closing the interval has arrived.
Extending the parabola beyond S₊₁ is extrapolation and is not part of the procedure described in Appendix V. Interpolation error is not a forecast error: the first concerns points inside an observed interval, while the second concerns future data not used in the calculation.
The method is therefore suited to historical reconstruction and numerical alignment of outputs. It does not make the latest unfinished segment available in real time while the right-hand point is still missing.
Limits and implementation choices
Series type. Hurst intends the technique for band-pass outputs with little high-frequency content. High-pass data, noise, or abrupt turns may require more points or a different method.
Sample density. The faster the dynamics are relative to tₙ, the less adequately three points describe what occurs between them. Reducing the spacing helps, but it does not turn interpolation into observation.
Boundaries. The central formula requires one point on the left and one on the right. Treatment of the first and last interval — omission, extension of the first or last triple, or another rule — must be declared by the implementation; Hurst does not specify a general boundary policy.
A reproducible check
The figure uses a synthetic sine with period 60, sampled every 5 units: twelve points per cycle. Applying Hurst's window and resampling every unit produces a maximum absolute error of about 0.009, or 0.9% of unit amplitude; error is zero at the original samples.
This is a deterministic check of the formula, not a general bound and not a measurement on prices. Changing frequency, noise, or distance between points changes the result.
Sources
- J. M. Hurst, The Profit Magic of Stock Transaction Timing, Prentice-Hall, 1970, Appendix V, pp. 212–214 — incompatible grids, sliding window, formula, and historical limits.
- NIST Digital Library of Mathematical Functions, §3.3 Interpolation — interpolation polynomials and the error term between nodes.
- SciPy, Extrapolation tips and tricks — the operational distinction between evaluation inside and extension outside the known domain.
Links
Ormsby numerical filters · Prony method · Spectral analysis · Hurst appendices