Construct a set of local basis functions

Construct a set of local basis functions based on pre-specified location and scale parameters.

Usage

local_basis(
  manifold = sphere(),
  loc = matrix(c(1, 0), nrow = 1),
  scale = 1,
  type = c("bisquare", "Gaussian", "exp", "Matern32"),
  res = 1,
  regular = FALSE
)

radial_basis(
  manifold = sphere(),
  loc = matrix(c(1, 0), nrow = 1),
  scale = 1,
  type = c("bisquare", "Gaussian", "exp", "Matern32")
)

Arguments

manifold

object of class manifold, for example, sphere

loc

a matrix of size n by dimensions(manifold) indicating centres of basis functions

scale

vector of length n containing the scale parameters of the basis functions; see details

type

either "bisquare", "Gaussian", "exp", or "Matern32"

res

vector of length n containing the resolutions of the basis functions

regular

logical indicating if the basis functions (of each resolution) are in a regular grid

Details

This functions lays out local basis functions in a domain of interest based on pre-specified location and scale parameters. If type is “bisquare”, then $$\phi(u) = \left(1- \left(\frac{\| u \|}{R}\right)^2\right)^2 I(\|u\| < R),$$ and scale is given by \(R\), the range of support of the bisquare function. If type is “Gaussian”, then $$\phi(u) = \exp\left(-\frac{\|u \|^2}{2\sigma^2}\right),$$ and scale is given by \(\sigma\), the standard deviation. If type is “exp”, then $$\phi(u) = \exp\left(-\frac{\|u\|}{ \tau}\right),$$ and scale is given by \(\tau\), the e-folding length. If type is “Matern32”, then $$\phi(u) = \left(1 + \frac{\sqrt{3}\|u\|}{\kappa}\right)\exp\left(-\frac{\sqrt{3}\| u \|}{\kappa}\right),$$ and scale is given by \(\kappa\), the function's scale.

See also

auto_basis for constructing basis functions automatically, and show_basis for visualising basis functions.

Examples

library(ggplot2)
G <-  local_basis(manifold = real_line(),
                   loc=matrix(1:10,10,1),
                   scale=rep(2,10),
                   type="bisquare")
if (FALSE) show_basis(G) # \dontrun{}