Thin {assist}

Calculate Reproducing Kernels for Thin Plate Splines

Description

Return a matrix evaluating reproducing kernels for thin plate splines at observed points.

Usage

tp.pseudo(s, u=s, order=2)
tp(s, u=s, order=2)
tp.linear(s, u=s)

Arguments

s

a list or matrix of observations. One component, if a list, and one column, if a matrix, contains observations on one variable. If a list, all components must be of the same length.

u

a list or matrix of observations. If a list, all components must be of the same length. The number of componets of the list, or the number of column of the matrix must be the same as that for s. Default is s.

order

an optional integer specifying the order of the thin plate spline. Default is 2. Let d be the dimension of s (and u). Then order must satisfy 2*order-d>0.

Details

The pseudo kernel, which is conditional definite positive instead of definite positive, is easy to calculate, while the true reproducing kernel is complicated. Pseudo Kernels are enough to compute spline estimates, but to calcualte Bayesian confidnece intervals, the true kernel is required. For the special case of d=2 and order=2, the function tp.linear computes evaluations of the reproducing kernel of the space spanned by linear basis.

Value

a matrix with the numbers of row and column equal to the common length of componets or the number of row of s and t respectively. The [i, j] element is the pseudo, true, or linear reproducing kernel evaluated at the ith element of s and jth element of u.

Author(s)

Chunlei Ke [e-mail] and Yuedong Wang [e-mail]

References

Wahba, G. (1990). Spline Models for Observational Data. SIAM, Vol. 59.

Gu, C. and Wahba, G (1993). Smoothing Spline ANOVA with component-wise Bayesian confidence intervals. Journal of Computational and Graphical Statistics 55, 353–368.

See Also

ssr, cubic

Examples

data(acid)
## Not run: tp.pseudo(list(acid$x1, acid$x2))
## Not run: tp.pseud0(list(acid$x1, acid$x2), order=3)