Basis for Harmonic Functions

Description

Evaluates a basis for the harmonic polynomials in x and y of degree less than or equal to n.

Usage

   harmonic(x, y, n)

Arguments

Details

This function computes a basis for the harmonic polynomials in two variables x and y up to a given degree n and evaluates them at given x,y locations. It can be used in model formulas (for example in the model-fitting functions lm,glm,gam and ppm) to specify a linear predictor which is a harmonic function.

A function f(x,y) is harmonic if

(d/dx)^2 f + (d/dy)^2 f = 0.

The harmonic polynomials of degree less than or equal to n have a basis consisting of 2 n functions.

This function was implemented on a suggestion of P. McCullagh for fitting nonstationary spatial trend to point process models.

Value

A data frame with 2 * n columns giving the values of the basis functions at the coordinates. Each column is labelled by an algebraic expression for the corresponding basis function.

Author(s)

Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf

See Also

ppm

Examples

   data(longleaf)
   X <- unmark(longleaf)
   # inhomogeneous point pattern
   

   # fit Poisson point process with log-cubic intensity
   fit.3 <- ppm(X, ~ polynom(x,y,3), Poisson())

   # fit Poisson process with log-cubic-harmonic intensity
   fit.h <- ppm(X, ~ harmonic(x,y,3), Poisson())

   # Likelihood ratio test
   lrts <- 2 * (fit.3$maxlogpl - fit.h$maxlogpl)
   x <- X$x
   y <- X$y
   df <- ncol(polynom(x,y,3)) - ncol(harmonic(x,y,3))
   pval <- 1 - pchisq(lrts, df=df)