Maximum Likelihood Estimation for Bivariate Poisson Lognormal Distribution

Description

bipoilogMLE fits the bivariate Poisson lognormal distribution to data

Usage

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bipoilogMLE(n1, n2 = NULL, 
            startVals = c(mu1=1, mu2=1, sig1=2, sig2=2, rho=0.5), 
            nboot = 0, zTrunc = TRUE, file = NULL, 
            method = "BFGS", control = list(maxit=1000))

Arguments

n1

a vector or a matrix with two columns of pairwise counts of observed individuals for each species

n2

if n1 is not given as a matrix, a vector of counts with same ordering of species as in argument n1

startVals

starting values of parameters

nboot

number of parametric bootstraps, defaults to zero

zTrunc

logical; if TRUE (default) the zero-truncated distribution is fitted

file

text file to hold copies of bootstrap estimates

method

method to use during optimization, see details

control

a list of control parameters for the optimization routine, see details

Details

The function estimates the parameters mu1, sig1, mu2, sig2 and rho. In cases of incomplete sampling the estimates of mu1 and mu2 will be confounded with the sampling intensities (see rbipoilog). Assuming sampling intensities ν1 and ν2, the estimates of the means are \code{mu1}+ \ln nu1 and \code{mu2}+ \ln nu2. Parameters sig1, sig2 and rho can be estimated without any knowledge of sampling intensities. The parameters must be given starting values for the optimization procedure (default starting values are used if starting values are not specified in the function call).

A zero-truncated distribution (see dbipoilog) is assumed by default (zTrunc = TRUE). In cases where the number of zeros is known the zTrunc argument should be set to FALSE.

The function uses the optimization procedures in optim to obtain the maximum likelihood estimate. The method and control arguments are passed to optim, see the help page for this function for additional methods and control parameters.

The approximate fraction of species revealed by each sample is estimated as 1 minus the zero term of the univariate Poisson lognormal distribution: 1-q(0;\code{mu1},\code{sig1}) and 1-q(0;\code{mu2},\code{sig2}).

Parametric bootstrapping could be time consuming for large data sets. If argument file is specified, e.g. file =C:\\myboots.txt’, the matrix with bootstrap estimates are copied into a tab-seperated text-file providing extra backup. Bootstrapping is done by simulating new sets of observations conditioned on the observed number of species (see rbipoilog).

Value

par

Maximum likelihood estimates of the parameters

p

Approximate fraction of species revealed by the samples for sample 1 and 2 respectively

logLval

Log likelihood of the data given the estimated parameters

gof

Godness of fit measure obtained by checking the rank of logLval against logLval's obtained during the bootstrap procedure, (gof < 0.05) or (gof > 0.95) indicates lack of fit

boot

A data frame containing the bootstrap replicates of parameters and logLval

Author(s)

Vidar Grøtan vidar.grotan@bio.ntnu.no, Steinar Engen

References

Engen, S., R. Lande, T. Walla and P. J. DeVries. 2002. Analyzing spatial structure of communities using the two-dimensional Poisson lognormal species abundance model. American Naturalist 160, 60-73.

See Also

optim, dbipoilog, rbipoilog

Examples

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## simulate observations
xy  <- rbipoilog(S=30,mu1=1,mu2=1,sig1=2,sig2=2,rho=0.5)

## obtain estimates of parameters
est <- bipoilogMLE(xy)

## similar, but now with bootstrapping 
## Not run: est <- bipoilogMLE(xy,nboot=10)

## change start values and request tracing information 
## from optimization procedure
est <- bipoilogMLE(xy,startVals=c(2,2,4,4,0.3),
       control=list(maxit=1000,trace=1, REPORT=1))

## effect of sampling intensity 
xy  <- rbipoilog(S=100,mu1=1,mu2=1,sig1=2,sig2=2,rho=0.5,nu1=0.5,nu2=0.5)
est <- bipoilogMLE(xy)
## the expected estimates of mu1 and mu2 are now 1-log(0.5) = 0.3 (approximately)