# poisson.points: Poisson-points-on-a-plane/volume Distances Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2``` ```poisson.points(ostatistic, dimension = 2, link = "loglink", 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 value 1.5 coincides with `maxwell` when `dimension = 2`. Note: if `ostatistic = 1` and `dimension = 2` then this VGAM family function coincides with `rayleigh`. `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`.

## 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,.... 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.

## 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.

## Author(s)

T. W. Yee

`poissonff`, `maxwell`, `rayleigh`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```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) ```