kernels: Reproducing kernels
In ppstat: Point Process Statistics

Description Usage Arguments Details Value Author(s) See Also Examples

Computation reproducing kernel evaluations.

1 2	sobolevKernel(s, r, t = 1, sub = NULL, constraint = 'boundary') gaussianKernel(s, r, t = 1, c = 1)

`s`	a `numeric`.
`r`	a `numeric`.
`t`	a `numeric`. Arguments are rescaled by `t`. Default value 1.
`c`	a `numeric`. The squared inverse scale parameter for the Gaussian kernel.
`sub`	a `numeric`. Determines if the kernel should be projected to a subspace. Default value `NULL` means no subspace. If 0, the kernel is projected to the 2-dimensional subspace spanned by the constant and linear functions, and if anything else, the kernel is projected to the orthogonal complement.
`constraint`	a `character`. The Sobolev space is a direct sum of a 2-dimensional space and a constraint space. This can either be the space of functions with boundary values 0, or the space of functions with initial value and derivative 0. The corresponding direct sum kernels differ. The default is 'boundary'.

The Sobolev kernel is the reproducing kernel for the (2,2) Sobolev space on [0, t]. The arguments s and r must be in the interval. If sub = 1 the kernel is projected to the subspace of functions being 0 in 0 and t

The Gaussian kernel is exp(-c(s-r)^2/t^2). Both arguments t and c are needed when used with ppKernel, where t is always the upper bound on the support.

A numeric. The value of the kernel in (s,r).

Niels Richard Hansen Niels.R.Hansen@math.ku.dk.

ppKernel

sobolevKernel(1:5, 1:5)
sobolevKernel(1:5, 1:5, t = 5)

gaussianKernel(1:5, 1:5)
gaussianKernel(1:5, 1:5, t = 5)
gaussianKernel(1:5, 1:5, t = 5, c = 2)