spatstat.explore (version 3.2-5)

stieltjes: 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, ...)

Value

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

Author

Adrian Baddeley , Rolf Turner and Ege Rubak .

Details

This command computes the Stieltjes integral $$I = \int f(x) dM(x)$$ 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.

See Also

fv.object.

Use integral.fv to integrate a function object of class "fv".

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)