# thinplate: Thin-Plate Spline Basis and Penalty In npreg: Nonparametric Regression via Smoothing Splines

## Description

Generate the smoothing spline basis and penalty matrix for a thin-plate spline.

## Usage

 1 2 3 basis.tps(x, knots, m = 2, rk = TRUE, intercept = FALSE, ridge = FALSE) penalty.tps(x, m = 2, rk = TRUE)

## Arguments

 x Predictor variables (basis) or spline knots (penalty). Numeric or integer vector of length n, or matrix of dimension n by p. knots Spline knots. Numeric or integer vector of length r, or matrix of dimension r by p. m Penalty order. "m=1" for linear thin-plate spline, "m=2" for cubic, and "m=3" for quintic. Must satisfy 2m > p. rk If true (default), the reproducing kernel parameterization is used. Otherwise, the standard thin-plate basis is returned. intercept If TRUE, the first column of the basis will be a column of ones. ridge If TRUE, the basis matrix is post-multiplied by the inverse square root of the penalty matrix. Only applicable if rk = TRUE. See Note and Examples.

## Details

Generates a basis function or penalty matrix used to fit linear, cubic, and quintic thin-plate splines.

The basis function matrix has the form

X = [X_0, X_1]

where the matrix X_0 is of dimension n by M-1 (plus 1 if an intercept is included) where M = {p+m-1 \choose p}, and X_1 is a matrix of dimension n by r.

The X_0 matrix contains the "parametric part" of the basis, which includes polynomial functions of the columns of x up to degree m-1 (and potentially interactions).

The matrix X_1 contains the "nonparametric part" of the basis.

If rk = TRUE, the matrix X_1 consists of the reproducing kernel function

ρ(x, y) = (I - P_x) (I - P_y) E(|x - y|)

evaluated at all combinations of x and knots. Note that P_x and P_y are projection operators, |.| denotes the Euclidean distance, and the TPS semi-kernel is defined as

E(z) = α z^{2m-p} \log(z)

if p is even and

E(z) = β z^{2m-p}

otherwise, where α and β are positive constants (see References).

If rk = FALSE, the matrix X_1 contains the TPS semi-kernel E(.) evaluated at all combinations of x and knots. Note: the TPS semi-kernel is not positive (semi-)definite, but the projection is.

If rk = TRUE, the penalty matrix consists of the reproducing kernel function

ρ(x, y) = (I - P_x) (I - P_y) E(|x - y|)

evaluated at all combinations of x. If rk = FALSE, the penalty matrix contains the TPS semi-kernel E(.) evaluated at all combinations of x.

## Value

Basis: Matrix of dimension c(length(x), df) where df = nrow(as.matrix(knots)) + choose(p + m - 1, p) - !intercept and p = ncol(as.matrix(x)).

Penalty: Matrix of dimension c(r, r) where r = nrow(as.matrix(x)) is the number of knots.

## Note

The inputs x and knots must have the same dimension.

If rk = TRUE and ridge = TRUE, the penalty matrix is the identity matrix.

## Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

## References

Gu, C. (2013). Smoothing Spline ANOVA Models. 2nd Ed. New York, NY: Springer-Verlag. doi: 10.1007/978-1-4614-5369-7

Helwig, N. E. (2017). Regression with ordered predictors via ordinal smoothing splines. Frontiers in Applied Mathematics and Statistics, 3(15), 1-13. doi: 10.3389/fams.2017.00015

Helwig, N. E., & Ma, P. (2015). Fast and stable multiple smoothing parameter selection in smoothing spline analysis of variance models with large samples. Journal of Computational and Graphical Statistics, 24(3), 715-732. doi: 10.1080/10618600.2014.926819