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 = "loge",
               idensity = NULL, imethod = 1)

Arguments

ostatistic

Order statistic. A single positive value, usually an integer. For example, the value 5 means the response are the distances of the fifth nearest value to that point (usually over many planes or volumes). Non-integers are allowed because the valu

dimension

The value 2 or 3; 2 meaning a plane and 3 meaning a volume.

link

Parameter link function applied to the (positive) density parameter, called $\lambda$ below. See Links for more choices.

idensity

Optional initial value for the parameter. A NULL value means a value is obtained internally. Use this argument if convergence failure occurs.

imethod

An integer with value 1 or 2 which specifies the initialization method for $\lambda$. If failure to converge occurs try another value and/or else specify a value for idensity.

Value

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

Warning

Convergence may be slow if the initial values are far from the solution. This often corresponds to the situation when the response values are all close to zero, i.e., there is a high density of points.

Formulae such as the means have not been fully checked.

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)

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