measureVariation.Rd
\name{measureVariation}
\alias{measureVariation}
\alias{measurePositive}
\alias{measureNegative}
\alias{totalVariation}
\title{
Positive and Negative Parts, and Variation, of a Measure
}
\description{
Given a measure \code{A} (object of class \code{"msr"})
these functions find the positive part, negative part and variation
of \code{A}.
}
\usage{
measurePositive(x)
measureNegative(x)
measureVariation(x)
totalVariation(x)
}
\arguments{
\item{x}{
A measure (object of class \code{"msr"}).
}
}
\details{
The functions \code{measurePositive} and \code{measureNegative}
return the positive and negative parts of the measure,
and \code{measureVariation} returns the variation (sum of positive and
negative parts). The function \code{totalVariation} returns the total
variation norm.
If \eqn{\mu} is a signed measure,
it can be represented as
\deqn{\mu = \mu_{+} - \mu_{-}}{\mu = \mu[+] - \mu[-]}
where \eqn{\mu_{+}}{\mu[+]} and \eqn{\mu_{-}}{\mu[-]}
are \emph{nonnegative} measures called the positive and negative
parts of \eqn{\mu}.
In a nutshell, the positive part of \eqn{\mu}
consists of all positive contributions or increments,
and the negative part consists of all negative contributions
multiplied by \code{-1}.
The variation \eqn{|\mu|} is defined by
\deqn{\mu = \mu_{+} + \mu_{-}}{\mu = \mu[+] + \mu[-]}
and is also a nonnegative measure.
The total variation norm is the integral of the variation.
}
\value{
The result of \code{measurePositive}, \code{measureNegative}
and \code{measureVariation} is another measure (object of class \code{"msr"})
on the same spatial domain.
The result of \code{totalVariation} is a non-negative number.
}
\references{
Halmos, P.R. (1950) \emph{Measure Theory}. Van Nostrand.
}
\author{
\adrian.
}
\seealso{
\code{\link{msr}}, \code{\link{with.msr}}, \code{\link{split.msr}}
}
\examples{
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
measurePositive(rp)
measureNegative(rp)
measureVariation(rp)
# total variation norm
totalVariation(rp)
}
\keyword{spatial}
\keyword{math}