gensphere (version 1.3)

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

g0

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

x

(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

n

number of values to generate

Value

gensphere returns an S3 object of class "gensphere.distribution" with components:

cfunc

a contour function defined with cfunc.new, etc.

dradial

a function that evaluates the desnity of the radial component

rradial

a function that simulates values of the radial component

g0

g(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

latex

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, \( g(0)=\lim_{r \to 0^+} r^{1-d} dradial(r)\).

See Also

gensphere-package, cfunc.new

Examples

# NOT RUN {
# define a diamond shaped contour
cfunc1 <- cfunc.new(d=2)
cfunc1 <- cfunc.add.term( cfunc1,"gen.lp.norm",k=c(1,1,2,0,0,1))
cfunc1 <- cfunc.finish( cfunc1 )
cfunc1

# }
# NOT RUN {
# 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 )

# simulate values from the distribution
x <- rgensphere( 10000, dist1 )

# calculate and plot density surface on a grid
xy.grid <- seq(-3,3,.1)
if( interactive() ) {
  z <- gs.pdf2d.plot( dist1, xy.grid )
  title3d("density surface")
  plot(t(x),xlab="x",ylab="y",main="simulated points")
} 

# }