https://github.com/cran/spatstat
Tip revision: adc4d1a73ce0bc27b15a9ab4b12d1a52f68a4a55 authored by Adrian Baddeley on 19 March 2010, 07:31:32 UTC
version 1.18-1
version 1.18-1
Tip revision: adc4d1a
harmonic.Rd
\name{harmonic}
\alias{harmonic}
\title{Basis for Harmonic Functions}
\description{
Evaluates a basis for the harmonic polynomials in \eqn{x} and \eqn{y}
of degree less than or equal to \eqn{n}.
}
\usage{
harmonic(x, y, n)
}
\arguments{
\item{x}{
Vector of \eqn{x} coordinates
}
\item{y}{
Vector of \eqn{y} coordinates
}
\item{n}{
Maximum degree of polynomial
}
}
\value{
A data frame with \code{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.
}
\details{
This function computes a basis for the harmonic polynomials
in two variables \eqn{x} and \eqn{y} up to a given degree \eqn{n}
and evaluates them at given \eqn{x,y} locations.
It can be used in model formulas (for example in
the model-fitting functions
\code{\link{lm},\link{glm},\link{gam}} and \code{\link{ppm}}) to specify a
linear predictor which is a harmonic function.
A function \eqn{f(x,y)} is harmonic if
\deqn{\frac{\partial^2}{\partial x^2} f
+ \frac{\partial^2}{\partial y^2}f = 0.}{
(d/dx)^2 f + (d/dy)^2 f = 0.}
The harmonic polynomials of degree less than or equal to
\eqn{n} have a basis consisting of \eqn{2 n} functions.
This function was implemented on a suggestion of P. McCullagh
for fitting nonstationary spatial trend to point process models.
}
\seealso{
\code{\link{ppm}}
}
\examples{
data(longleaf)
X <- unmark(longleaf)
# inhomogeneous point pattern
\testonly{
# smaller dataset
longleaf <- longleaf[seq(1,longleaf$n, by=50)]
}
# fit Poisson point process with log-cubic intensity
fit.3 <- ppm(X, ~ polynom(x,y,3), Poisson())
# fit Poisson process with log-cubic-harmonic intensity
fit.h <- ppm(X, ~ harmonic(x,y,3), Poisson())
# Likelihood ratio test
lrts <- 2 * (fit.3$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)
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{models}