SpatialExtremes (version 2.0-0)

rbpspline: Fits a penalized spline with radial basis functions to data

Description

Fits a penalized spline with radial basis functions to data.

Usage

rbpspline(y, x, knots, degree, penalty = "gcv", ...)

Arguments

y

The response vector.

x

A vector/matrix giving the values of the predictor variable(s). If x is a matrix, each row corresponds to one observation.

knots

A vector givint the coordinates of the knots.

degree

The degree of the penalized smoothing spline.

penalty

A numeric giving the penalty coefficient for the penalization term. Alternatively, it could be either 'cv' or 'gcv' to choose the penalty using the (generalized) cross-validation criterion.

...

Additional options to be passed to the cv or gcv function.

Value

• An object of class pspline.

Details

The penalized spline with radial basis is defined by:

$$f(x) = \beta_0 + \beta_1 x + \ldots + \beta_{m-1} x^{m-1} + \sum_{k=0}^{K-1} \beta_{m+k} || x - \kappa_k ||^{2m - 1}$$ where $\beta_i$ are the coefficients to be estimated, $\kappa_i$ are the coordinates of the i-th knot and $m = \frac{d+1}{2}$ where $d$ corresponds to the degree of the spline.

The fitting criterion is to minimize $$||y - X\beta||^2 + \lambda^{2m-1} \beta^T K \beta$$ where $\lambda$ is the penalty coefficient and $K$ the penalty matrix.

References

Ruppert, D. Wand, M.P. and Carrol, R.J. (2003) Semiparametric Regression Cambridge Series in Statistical and Probabilistic Mathematics.

See Also

cv, gcv

Examples

n <- 200
x <- runif(n)
fun <- function(x) sin(3 * pi * x)
y <- fun(x) + rnorm(n, 0, sqrt(0.4))
knots <- quantile(x, prob = 1:(n/4) / (n/4 + 1))
fitted <- rbpspline(y, x, knots = knots, degree = 3)
fitted

plot(x, y)
lines(fitted, col = 2)