assist (version 2.0)

lspline: Calculate Reproducing Kernels for Some L-splines

Description

Return a matrix evaluating reproducing kernels for some L-splines at observed points.

Usage

lspline(x,y=x, type="exp", ...)

Arguments

x

a numeric vector on which reproducing kerenls are evaluated.

y

an optional vector, specifying the second argument of reproducing kernels. Default is x.

type

a string indicating the type of L-splines. Available options are "exp", "logit","sine", "sine1", and "linSinCos". Default is "exp".

...

other arguments needed.

Value

• a matrix with the numbers of row and column equal to the lengths of x and y respectively. The [i, j] element is the reproducing kernel evaluated at (x[i], y[j]).

Details

Denote L as the differential oprator, $H_0$ as the null (kernel) space. The available kernels correspond to the following L:

• exp:$L=rD+D^2$,$H_0=span{1,exp(-rx)}$.$r>0$, default to be 1;
• logit:$L=D-1/(1+e^t)$,$H_0=span{e^t/(1+e^t)}$;
• sine0:$L=D^2+(2\pi)^2$,$H_0=span{sin(2\pi x),cos(2\pi x)}$;
• sine1:$L=D(D^2+(2\pi)^2)$,$H_0=span{1, sin(2\pi x),cos(2\pi x)}$;
• linSinCos:$L=D^4+D^2$,$H_0=spac{1, x, sin(x), cos(x)}$.

References

Wahba, G. (1990). Spline Models for Observational Data. SIAM, Vol. 59. Heckman, N and Ramsay, J. O. (2000). Penalised regression with model-based penalties. To appear in Canadian Journal of Statisitcs.

See Also

ssr

Examples

x<- seq(0,1, len=20)
lspline(x, type="exp", r=1.5)