R: Basis for Harmonic Functions

harmonic {spatstat.core} | R Documentation

## 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

`x` | Vector of x coordinates
`y` | Vector of y coordinates
`n` | Maximum degree of polynomial

### 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.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### See Also

`ppm`, `polynom`

### Examples

       # inhomogeneous point pattern
       X <- unmark(longleaf)

       # 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 * (logLik(fit.3) - logLik(fit.h))
       df <- with(coords(X),
                  ncol(polynom(x,y,3)) - ncol(harmonic(x,y,3)))
       pval <- 1 - pchisq(lrts, df=df)

[Package spatstat.core version 2.0-0 Index]