Basis for Harmonic Functions
Evaluates a basis for the harmonic polynomials in x and y of degree less than or equal to n.
harmonic(x, y, n)
x |
Vector of x coordinates |
y |
Vector of y coordinates |
n |
Maximum degree of polynomial |
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.
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.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.