assist (version 3.1.9)

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", ...)

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]).

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.

Author

Chunlei Ke and Yuedong Wang

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

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