poisson.points: Poisson-points-on-a-plane/volume Distances Distribution

Description

Estimating the density parameter of the distances from a fixed point to the u-th nearest point, in a plane or volume.

Usage

poisson.points(ostatistic, dimension = 2, link = "loglink",
               idensity = NULL, imethod = 1)

Arguments

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm,

rrvglm

and vgam.

Details

Suppose the number of points in any region of area \(A\) of the plane is a Poisson random variable with mean \(\lambda A\) (i.e., \(\lambda\) is the density of the points). Given a fixed point \(P\), define \(D_1\), \(D_2\),... to be the distance to the nearest point to \(P\), second nearest to \(P\), etc. This VGAM family function estimates \(\lambda\) since the probability density function for \(D_u\) is easily derived, \(u=1,2,\ldots\). Here, \(u\) corresponds to the argument ostatistic.

Similarly, suppose the number of points in any volume \(V\) is a Poisson random variable with mean \(\lambda V\) where, once again, \(\lambda\) is the density of the points. This VGAM family function estimates \(\lambda\) by specifying the argument ostatistic and using dimension = 3.

The mean of \(D_u\) is returned as the fitted values. Newton-Raphson is the same as Fisher-scoring.

Examples

Run this code

pdata <- data.frame(y = rgamma(10, shape = exp(-1)))  # Not proper data!
ostat <- 2
fit <- vglm(y ~ 1, poisson.points(ostat, 2), data = pdata,
            trace = TRUE, crit = "coef")
fit <- vglm(y ~ 1, poisson.points(ostat, 3), data = pdata,
            trace = TRUE, crit = "coef")  # Slow convergence?
fit <- vglm(y ~ 1, poisson.points(ostat, 3, idensi = 1), data = pdata,
            trace = TRUE, crit = "coef")
head(fitted(fit))
with(pdata, mean(y))
coef(fit, matrix = TRUE)
Coef(fit)