, in order to balance a measure of goodness of fit of
The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where
is defined to be the unique minimizer, in the Sobolev space
of fitted values, the sum-of-squares part of the spline criterion is fixed.
, and the minimizer is a natural cubic spline that interpolates the points
This interpolating spline is a linear operator, and can be written in the form where
As a result, the roughness penalty has the form where the elements of A are
The basis functions, and hence the matrix A, depend on the configuration of the predictor variables
Δ is an (n-2)×n matrix of second differences with elements:
De Boor's approach exploits the same idea, of finding a balance between having a smooth curve and being close to the given data.
is a parameter called smooth factor and belongs to the interval
are the quantities controlling the extent of smoothing (they represent the weight
converges to the "natural" spline interpolant to the given data.
converges to a straight line (the smoothest curve).
is a task of trial and error, a redundant constant
meets the following condition: The algorithm described by de Boor starts with
means the solution is the "natural" spline interpolant.
means we obtain a smoother curve by getting farther from the given data.
There are two main classes of method for generalizing from smoothing with respect to a scalar
The first approach simply generalizes the spline smoothing penalty to the multidimensional setting.
we might use the Thin plate spline penalty and find the
minimizing The thin plate spline approach can be generalized to smoothing with respect to more than two dimensions and to other orders of differentiation in the penalty.
[1] As the dimension increases there are some restrictions on the smallest order of differential that can be used,[1] but actually Duchon's original paper,[9] gives slightly more complicated penalties that can avoid this restriction.
The thin plate splines are isotropic, meaning that if we rotate the
co-ordinate system the estimate will not change, but also that we are assuming that the same level of smoothing is appropriate in all directions.
This is often considered reasonable when smoothing with respect to spatial location, but in many other cases isotropy is not an appropriate assumption and can lead to sensitivity to apparently arbitrary choices of measurement units.
For example, if smoothing with respect to distance and time an isotropic smoother will give different results if distance is measure in metres and time in seconds, to what will occur if we change the units to centimetres and hours.
The second class of generalizations to multi-dimensional smoothing deals directly with this scale invariance issue using tensor product spline constructions.
Smoothing splines are related to, but distinct from: Source code for spline smoothing can be found in the examples from Carl de Boor's book A Practical Guide to Splines.
The updated sources are available also on Carl de Boor's official site [1].