# measureVariation

##### 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`

.

##### 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 \(\mu\) is a signed measure,
it can be represented as
$$\mu = \mu_{+} - \mu_{-}$$
where \(\mu_{+}\) and \(\mu_{-}\)
are *nonnegative* measures called the positive and negative
parts of \(\mu\).
In a nutshell, the positive part of \(\mu\)
consists of all positive contributions or increments,
and the negative part consists of all negative contributions
multiplied by `-1`

.

The variation \(|\mu|\) is defined by $$\mu = \mu_{+} + \mu_{-}$$ 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.

##### References

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

##### See Also

##### Examples

```
# NOT RUN {
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)
# }
```

*Documentation reproduced from package spatstat, version 1.57-1, License: GPL (>= 2)*