R: Positive and Negative Parts, and Variation, of a Measure

measureVariation {spatstat.core} | R Documentation

## Positive and Negative Parts, and Variation, of a Measure

### Description

Given a measure `A` (object of class `"msr"`) these functions find the positive part, negative part and variation of `A`.

### Usage

    measurePositive(x)
    measureNegative(x)
    measureVariation(x)
    totalVariation(x)

### Arguments

`x` | A measure (object of class `"msr"`).

### Details

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.

### Value

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.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au.

### References

Halmos, P.R. (1950) Measure Theory. Van Nostrand.

### See Also

`msr`, `with.msr`, `split.msr`, `measureDiscrete`

### 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)

[Package spatstat.core version 2.3-0 Index]