https://github.com/cran/spatstat
Tip revision: 43e2374b0fee50899d3e1a4de55422c1ec521d04 authored by Adrian Baddeley on 29 January 2018, 18:45:23 UTC
version 1.55-0
version 1.55-0
Tip revision: 43e2374
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}},
\code{\link{polynom}}
}
\examples{
# inhomogeneous point pattern
X <- unmark(longleaf)
\testonly{
# smaller dataset
X <- X[seq(1,npoints(X), 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 * (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)
}
\author{
\spatstatAuthors.
}
\keyword{spatial}
\keyword{models}