preseqR (version 4.0.0)

ztp.rSAC: ZTP estimator

Description

ztp.rSAC predicts the expected number of species represented at least \(r\) times in a random sample, based on the initial sample.

Usage

ztp.rSAC(n, r=1)

Arguments

n

A two-column matrix. The first column is the frequency \(j = 1,2,\dots\); and the second column is \(N_j\), the number of species with each species represented exactly \(j\) times in the initial sample. The first column must be sorted in an ascending order.

r

A positive integer. Default is 1.

Value

The estimator for the \(r\)-SAC. The input of the estimator is a vector of sampling efforts \(t\), i.e., the relative sample sizes comparing with the initial sample. For example, \(t = 2\) means a random sample that is twice the size of the initial sample.

Details

The statistical assumption is that for each species the number of individuals in a sample follows a Poisson distribution. The Poisson rate lambda is the same among all species. So the random variable X, which is the number of species represented x (x > 0) times, follows a zero-truncated Poisson distribution. The unknown parameters are estimated by Cohen (1960). Based on the estimated distribution, we calculate the expected number of species in a random sample.

References

Cohen, A. Clifford. "Estimating the parameter in a conditional Poisson distribution." Biometrics 16, no. 2 (1960): 203-211.

Examples

# NOT RUN {
## load library
library(preseqR)

## import data
data(FisherButterfly)

## construct the estimator for SAC
ztp1 <- ztp.rSAC(FisherButterfly, r=1)
## The number of species represented at least once in a sample, 
## when the sample size is 10 or 20 times of the initial sample
ztp1(c(10, 20))

## construct the estimator for r-SAC
ztp2 <- ztp.rSAC(FisherButterfly, r=2)
## The number of species represented at least once in a sample, 
## when the sample size is 10 or 20 times of the initial sample
ztp2(c(50, 100))
# }