R: ZTNB estimator

ztnb.rSAC {preseqR} | R Documentation

## ZTNB estimator

### Description

`ztnb.rSAC` predicts the expected number of species represented at least r times in a random sample, based on the initial sample.

### Usage

    ztnb.rSAC(n, r=1, size=SIZE.INIT, mu=MU.INIT)

### Arguments

`n` | A two-column matrix. The first column is the frequency j = 1,2,…; 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.
`size` | A positive double, the initial value of the parameter `size` in the negative binomial distribution for the EM algorithm. Default value is 1.
`mu` | A positive double, the initial value of the parameter `mu` in the negative binomial distribution for the EM algorithm. Default value is 0.5.

### Details

The statistical assumption is that for each species the number of individuals in a sample follows a Poisson distribution. The Poisson rate `lambda` are numbers generated from a gamma distribution. So the random variable `X`, which is the number of species represented x (x > 0) times in the sample, follows a zero-truncated negative binomial distribution. The unknown parameters are estimated by the function `preseqR.ztnb.em` based on the initial sample. Using the estimated distribution, we calculate the expected number of species represented at least r times in a random sample. Details of the estimation procedure can be found in the supplement of Daley T. and Smith AD. (2013).

### 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.

### Author(s)

Chao Deng

### References

Daley, T., & Smith, A. D. (2013). Predicting the molecular complexity of sequencing libraries. Nature methods, 10(4), 325-327.

Deng C, Daley T & Smith AD (2015). Applications of species accumulation curves in large-scale biological data analysis. Quantitative Biology, 3(3), 135-144.

### See Also

`preseqR.ztnb.em`

### Examples

    ## load library
    library(preseqR)

    ## import data
    data(FisherButterfly)

    ## construct the estimator for SAC
    ztnb1 <- ztnb.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
    ztnb1(c(10, 20))

    ## construct the estimator for r-SAC
    ztnb2 <- ztnb.rSAC(FisherButterfly, r=2)
    ## The number of species represented at least twice in a sample,
    ## when the sample size is 50 or 100 times of the initial sample
    ztnb2(c(50, 100))

[Package preseqR version 4.0.0 Index]