R: Compute Integral of Function Against Cumulative Distribution

stieltjes {spatstat} | R Documentation

## Compute Integral of Function Against Cumulative Distribution

### Description

Computes the Stieltjes integral of a function f with respect to a function M.

### Usage

    stieltjes(f, M, ...)

### Arguments

`f` | The integrand. A function in the R language.
`M` | The cumulative function against which `f` will be integrated. An object of class `"fv"` or `"stepfun"`.
`...` | Additional arguments passed to `f`.

### Details

This command computes the Stieltjes integral

I = integral f(x) dM(x)

of a real-valued function f(x) with respect to a nondecreasing function M(x).

One common use of the Stieltjes integral is to find the mean value of a random variable from its cumulative distribution function F(x). The mean value is the Stieltjes integral of f(x)=x with respect to F(x).

The argument `f` should be a `function` in the R language. It should accept a numeric vector argument `x` and should return a numeric vector of the same length.

The argument `M` should be either a step function (object of class `"stepfun"`) or a function value table (object of class `"fv"`, see `fv.object`). Objects of class `"stepfun"` are returned by `ecdf`, `ewcdf`, `spatialcdf` and other utilities. Objects of class `"fv"` are returned by the commands `Kest`, `Gest`, etc.

### Value

A list containing the value of the Stieltjes integral computed using each of the versions of the function `M`.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### See Also

`fv.object`, `Gest`

### Examples

      # estimate cdf of nearest neighbour distance in redwood data
      G <- Gest(redwood)
      # compute estimate of mean nearest neighbour distance
      stieltjes(function(x){x}, G)
      # estimated probability of a distance in the interval [0.1,0.2]
      stieltjes(function(x,a,b){ (x >= a) & (x <= b)}, G, a=0.1, b=0.2)

      # stepfun example
      H <- spatialcdf(bei.extra$elev, normalise=TRUE)
      stieltjes(function(x){x}, H)

[Package spatstat version 1.63-3 Index]