Positive and Negative Parts, and Variation, of a Measure
Given a measure A (object of class "msr")
these functions find the positive part, negative part and variation
of A.
measurePositive(x) measureNegative(x) measureVariation(x) totalVariation(x)
x |
A measure (object of class |
The functions measurePositive and measureNegative
return the positive and negative parts of the measure,
and measureVariation returns the variation (sum of positive and
negative parts). The function totalVariation returns the total
variation norm.
If μ is a signed measure, it can be represented as
μ = μ[+] - μ[-]
where μ[+] and μ[-]
are nonnegative measures called the positive and negative
parts of μ.
In a nutshell, the positive part of μ
consists of all positive contributions or increments,
and the negative part consists of all negative contributions
multiplied by -1.
The variation |μ| is defined by
μ = μ[+] + μ[-]
and is also a nonnegative measure.
The total variation norm is the integral of the variation.
The result of measurePositive, measureNegative
and measureVariation is another measure (object of class "msr")
on the same spatial domain.
The result of totalVariation is a non-negative number.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au.
Halmos, P.R. (1950) Measure Theory. Van Nostrand.
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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.