Estimating the density parameter of the distances from a fixed point to the u-th nearest point, in a plane or volume.
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
The value 2 or 3; 2 meaning a plane and 3 meaning a volume.
Parameter link function applied to the (positive) density parameter,
called lambda below.
Optional initial value for the parameter.
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,.... Here, u corresponds to the
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
The object is used by modelling functions such as
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
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)
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