View source: R/family.extremes.R

poisson.points | R Documentation |

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

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

`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 |

`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 |

`idensity` |
Optional initial value for the parameter.
A |

`imethod` |
An integer with value |

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.

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

,
`rrvglm`

and `vgam`

.

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.

T. W. Yee

`poissonff`

,
`maxwell`

,
`rayleigh`

.

```
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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