Whittaker

The Whittaker algorithm is part of the Estimate Gridded Data option in the Non-Parametric menu. Unlike the B-Spline, Bicubic+Akima, and Akima III algorithms, which pass exactly through every data point, Whittaker is a smoothing procedure: it produces a surface that balances closeness to the data against overall smoothness, with the trade-off controlled by the Lambda value.

Description

For each row and column of the data grid, the Whittaker algorithm balances two competing goals: staying close to the original values, and keeping the surface free of sudden bends (technically, minimizing the grid's second differences). The Lambda value sets this balance: a Lambda of 0 reproduces the data exactly, with no smoothing at all, while larger Lambda values increasingly suppress noise at the cost of following the data less closely. The smoothed grid is then fit with a bicubic spline, so the resulting surface and its partial-derivative surfaces remain smooth and continuous, consistent with the other spline-based algorithms in this option.

Choosing Lambda — GCV and rGCV

Rather than choosing a Lambda value by trial and error, the GCV button automatically selects a Lambda that minimizes prediction error, using Generalized Cross-Validation, which estimates how well the surface would predict each grid value if that value were withheld, without needing to refit once per point. TableCurve Studio searches 45 logarithmically spaced Lambda values between 1e-4 and 1e7 and keeps the one that minimizes the GCV score. The rGCV button applies a more robust form of the same criterion, biased toward additional smoothing; it is the better choice if a plain GCV result still looks noisy. Either button gives an excellent starting value rather than a final answer — after using one, check the partial-derivative surfaces as well as the surface itself, since residual noise often shows up in the derivatives before it is visible in the surface.

Order and Knots

The Order X/Y and Knots X/Y fields are shared with the other algorithms in this dialog, but are disabled for Whittaker. Its basis is always cubic, so the flexibility of the surface is controlled entirely by Lambda rather than by a knot count.

Statistics

Because Whittaker is a smoothing rather than an exact-interpolation algorithm (like B-Spl Var knots and NURBS), the dialog reports a squared sum of residuals (SSE) and a coefficient of determination (r2) so the fit can be compared against parametric results. With Lambda set to 0, the surface passes exactly through the data and SSE will be 0.

Extrapolation

Whittaker does not extrapolate. The surface is defined only within the bounds of the input grid.

Considerations

Whittaker smooths each dimension of the grid independently — first along X, then along Y — which is fast and effective for the great majority of gridded surfaces. This is the recommended algorithm whenever the input grid is known to contain measurement noise that an exact-interpolation algorithm (B-Spline, Bicubic+Akima, Akima III) would otherwise fit into the surface as spurious detail.

Author

E. T. Whittaker. The practical form used here, and the modern "Whittaker–Henderson" name, are due to Paul H. C. Eilers.

References

Whittaker, E. T. (1923). "On a new method of graduation." Proceedings of the Edinburgh Mathematical Society, 41, 63–75. Eilers, P. H. C. (2003). "A perfect smoother." Analytical Chemistry, 75(14), 3631–3636. Eilers, P. H. C. & Marx, B. D. (1996). "Flexible smoothing with B-splines and penalties." Statistical Science, 11(2), 89–121. Eilers, P. H. C., Currie, I. D. & Durbán, M. (2006). "Fast and compact smoothing on large multidimensional grids." Computational Statistics & Data Analysis, 50, 61–76.