B-Spline

The B-Spline algorithm is part of the Estimate Gridded Data option in the Non-Parametric menu.

Author

Carl de Boor
IMSL

References

Carl de Boor, "A Practical Guide to Splines", Springer-Velag, 1978, p332-346, also SPLI2D p347-349.
IMSL Library, BS2IN

Description

This option offers the standard two-dimensional tensor product B-Spline. The orders can be set separately for the X and Y directions. The default is a bicubic (order 3 in both X and Y). Orders two through five are available. The algorithm does not offer extrapolation.

Interpolation

A biquadratic order 2/2 spline offers a smooth surface and continuous but not smooth first partial derivatives.
A bicubic order 3/3 spline offers a smooth surface, smooth first partial derivatives, and continuous second derivatives.
A biquartic order 4/4 spline offers a smooth surface, smooth first and second partial derivatives, and continuous third derivatives.
A biquintic order 5/5 spline offers a smooth surface, smooth first, second, and third partial derivatives, and continuous fourth derivatives.

The splines exactly interpolate the data. The x and y knots will equal the number of x and y mesh values. Because of the oscillations which can occur with higher order splines, a bicubic spline is recommended unless a smooth second or higher derivative is needed.

Extrapolation

The algorithm offers no extrapolation. Points outside the rectangular bounds of the data are mapped to the bounds and evaluated there. A Bicubic+Akima procedure is offered for those instances where a spline with extrapolation is needed.

Estimated Partial Derivatives

The algorithm computes exact partial derivatives. These are used for graphing the partial derivative surfaces and for partial derivatives computed in the Evaluation procedure.

Estimated Volumes

The algorithm computes exact integrals within the bounds of the data. If any of the integration limits are outside the bounds, the spline is used to interpolate a uniform grid of 10,000 points. A B-spline is fitted to this generated data and this second B-spline is then integrated. The precision error reported in the Evaluation procedure will be the fractional difference with a similar 2,500 node integration. Note that such integrations will involve values that have been forced to the bounds. A repeat evaluation with the same limits will use a numeric double integration procedure. A very fast double Gaussian quadrature procedure is first attempted to 1E-5 precision. If this is unsuccessful, a double adaptive quadrature procedure is then used. This second evaluation is offered as a verification of accuracy.

Considerations

The Bicubic+Akima algorithm offers all of the benefits of the standard B-spline procedure with the addition of the very stable Akima III extrapolant.

Algorithm Adjustments

This algorithm has two user adjustments, the order of the spline in the X and Y directions. Orders 2 through 5 are available.