# tpaps

Thin-plate smoothing spline

## Description

is the stform of a thin-plate smoothing spline `st`

= tpaps(`x`

,`y`

)*f* for the given data sites `x(:,j)`

and the given data values `y(:,j)`

. The `x(:,j)`

must be distinct points in the plane, the values can be scalars, vectors, matrices, even ND-arrays, and there must be exactly as many values as there are sites.

The thin-plate smoothing spline *f* is the unique minimizer of the weighted sum

$$pE(f)+(1-p)R(f)$$

with *E*(*f*) the error measure

$$E(f)={\displaystyle \sum _{j}{\left|y(:,j)-f\left(x(:,j)\right)\right|}^{2}}$$

and *R*(*f*) the roughness measure

$$R(f)={{\displaystyle \int (\left|{D}_{1}{D}_{1}f\right|}}^{2}+2{\left|{D}_{1}{D}_{2}f\right|}^{2}+{\left|{D}_{2}{D}_{2}f\right|}^{2})$$

Here, the integral is taken over all of *R*^{2}, |*z*|^{2} denotes the sum of squares of all the entries of *z*, and *D _{i}f* denotes the partial derivative of

*f*with respect to its

*i*-th argument, hence the integrand involves second partial derivatives of

*f*. The function chooses the smoothing parameter

`p`

so that `(1-p)/p`

equals the average of the diagonal entries of the matrix `A`

, with `A + (1-p)/p*eye(n)`

the coefficient matrix of the linear system for the `n`

coefficients of the smoothing spline to be determined. This ensures staying in between the two extremes of interpolation (when `p`

is close to `1`

and the coefficient matrix is essentially `A`

) and complete smoothing (when `p`

is close to `0`

and the coefficient matrix is essentially a multiple of the identity matrix). This serves as a good first guess for `p`

.

also inputs the `st`

= tpaps(`x`

,`y`

,`p`

)*smoothing parameter*, `p`

, a number between 0 and 1. As the smoothing parameter varies from 0 to 1, the smoothing spline varies, from the least-squares approximation to the data by a linear polynomial when `p`

is `0`

, to the thin-plate spline interpolant to the data when `p`

is `1`

.

`[...,`

also returns the value of the smoothing parameter used in the final spline result whether or not you specify `P`

] = tpaps(...) `p`

. This syntax is useful for experimentation in which you can start with `[pp,P] = tpaps(x,y)`

and obtain a reasonable first guess for `p`

.

## Examples

## Input Arguments

## Output Arguments

## Limitations

The determination of the smoothing spline involves the solution of a linear system with as many unknowns as there are data points. Since the matrix of this linear system is full, the solving can take a long time even if, as is the case here, an iterative scheme is used when there are more than 728 data points. The convergence speed of that iteration is strongly influenced by `p`

, and is slower the larger `p`

is. So, for large problems, use interpolation, i.e., `p`

equal to 1, only if you can afford the time.

## Version History

**Introduced in R2006b**