Poisson lognormal | R Documentation |
Density and random generation for the Poisson lognormal distribution with parameters mu
and sig
.
dpoilog(n, mu, sig) rpoilog(S, mu, sig, nu=1, condS=FALSE, keep0=FALSE)
n |
vector of observed individuals for each species |
S |
number of species in the community |
mu |
mean of lognormal distribution |
sig |
standard deviation of lognormal distribution |
nu |
sampling intensity, defaults to 1 |
condS |
logical; if TRUE random deviates are conditional on S |
keep0 |
logical; if TRUE species with count 0 are included in the random deviates |
The following is written from the perspective of using the Poisson lognormal distribution to describe community structure (the distribution of species when sampling individuals from a community of several species).
Under the assumption of random sampling, the number of individuals sampled from a given
species with abundance y, say N, is Poisson distributed with mean \code{nu}\emph{y}
where the parameter nu
expresses the sampling intensity. If ln y is normally distributed
with mean mu
and standard deviation sig
among species, then the vector of individuals sampled
from all S
species then constitutes a sample from the Poisson lognormal distribution
with parameters (mu
+ ln nu
, sig
), where mu
and sig
are the mean and standard deviation of the log abundances. For nu
= 1, this is the Poisson
lognormal distribution with parameters (mu
,sig
) which may be written in the form
P(N=\code{n}; \code{mu},\code{sig}) = q(\code{n}; \code{mu},\code{sig}) = \int_-infty^infty g_\code{n}(\code{mu},\code{sig},u) phi(u) du,
where φ(u) is the standard normal distribution and
g_n(\code{mu},\code{sig},u) = exp(\code{mu} \code{sig} \code{n} + \code{mu} \code{n} + exp(-\code{mu} \code{sig} + \code{mu})) / \code{n}!
Since S
is usually unknown, we only consider the observed number of individuals for the observed species.
With a general sampling intensity nu
, the distribution of the number of individuals then follows the
zero-truncated Poisson lognormal distribution
q(\code{n}; \code{mu},\code{sig})/(1 - q(0; \code{mu},\code{sig}))
dpoilog
returns the density
rpoilog
returns random deviates
Vidar Grotan vidar.grotan@ntnu.no and Steinar Engen
Engen, S., R. Lande, T. Walla & P. J. DeVries. 2002. Analyzing spatial structure of communities using the two-dimensional Poisson lognormal species abundance model. American Naturalist 160: 60-73.
poilogMLE
for ML estimation
### plot density for given parameters barplot(dpoilog(n=0:20,mu=2,sig=1),names.arg=0:20) ### draw random deviates from a community of 50 species rpoilog(S=50,mu=2,sig=1) ### draw random deviates including zeros rpoilog(S=50,mu=2,sig=1,keep0=TRUE) ### draw random deviates with sampling intensity = 0.5 rpoilog(S=50,mu=2,sig=1,nu=0.5) ### how many species are likely to be observed ### (given S,mu,sig2 and nu)? hist(replicate(1000,length(rpoilog(S=30,mu=0,sig=3,nu=0.7)))) ### how many individuals are likely to be observed ### (given S,mu,sig2 and nu)? hist(replicate(1000,sum(rpoilog(S=30,mu=0,sig=3,nu=0.7))))
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