gensphere: Generalized spherical distribution definition, density, simulation

Description

Define a generalized spherical distribution by specifying a contour function, a radial density function, a radial simulation function, and a value of the density at the origin. Once it is defined, compute density and simulate that distribution.

Usage

gensphere(cfunc, dradial, rradial, g0)
dgensphere(x, gs.dist)
rgensphere(n, gs.dist)

Arguments

cfunc

contour function object defined by cfunc.new, cfunc.add.term and cfunc.finish

dradial

a function to evaluate the density for the radial component of distribution

rradial

a function to simulate values of the radial distribution

g(0) = value of the multivariate density at the origin

(d x n) matrix of point where the density is to be evaluated. Columns x[,i] are vectors in d-space

gs.dist

a generalized spherical distribution, an object returned by function gensphere

number of values to generate

Value

gensphere returns an S3 object of class "gensphere.distribution" with components:
cfunca contour function defined with cfunc.new, etc.
dradiala function that evaluates the desnity of the radial component
rradiala function that simulates values of the radial component
g0g(0), the value of the multivariate density g(x) at the origin
dgensphere returns a numeric vector y that contains the value of the density of X: y[i]=g(x[,i]), i=1,...,n. Note that g(x) is the density of the vector X, whereas dradial is the denis of the univariate radial term R. rgensphere returns a (d x n) matrix of simulated values of X. Note that these values are an approximation to the distribution of X because the contour is approximated to a limited accuracy in cfund.finish. Here are plots of the density surface and simulated points generated by the examples below. html{
densitysurface.png

rgensphere.png
} latex{
densitysurface.png
{options: width=2.5in}
rgensphere.png
{options: width=2.5in} }

Details

A generalized spherical distribution is specified by calling function gensphere with the contour function (defined via function cfunc.new, cfunc.add.term and cfunc.finish), a function to compute the density of the radial term R, a runction to simulate from the radial term R, and g(0)=the value of the density at the origin. See the general representation of generalized spherical laws in gensphere-package. If the distribution is d dimensional and the radial term is a gamma distribution with shape=shape and scale=1,g(0)=0 if d < shape, g(0)=cfunc$norm.const if d=shape, $g(0)=\infty$ if d > shape. In general, $g(0)=\lim_{r \to 0^+} r^{1-d} dradial(r)$.

Examples

Run this code

# define a diamond shaped contour
cfunc1 <- cfunc.new(d=2)
#cfunc1 <- cfunc.add.term( cfunc1,"lp.norm",k=c(1,1))
cfunc1 <- cfunc.add.term( cfunc1,"gen.lp.norm",k=c(1,1,2,0,0,1))
cfunc1 <- cfunc.finish( cfunc1 )
cfunc1

# define a generalized spherical distribution
rradial <- function( n ) { rgamma( n, shape=2 ) }
dradial <- function( x ) { dgamma( x, shape=2 ) }
dist1 <- gensphere( cfunc1, dradial, rradial, g0=cfunc1$norm.const ) 
dist1

# calculate density at a few points
dgensphere( x=matrix( c(0,0, 0,1, 0,2), nrow=2, ncol=3), dist1 )

# calculate and plot density surface on a grid
xy.grid <- seq(-3,3,.1)
z <- gs.pdf2d.plot( dist1, xy.grid )
title3d("density surface")

# simulate values from the distribution
x <- rgensphere( 10000, dist1 )
plot(t(x),xlab="x",ylab="y",main="simulated points")