R: Predicting generalized sample coverage

preseqR.sample.cov {preseqR} | R Documentation

## Predicting generalized sample coverage

### Description

`preseqR.sample.cov` predicts the probability of observing a species represented at least r times in a random sample.

### Usage

      preseqR.sample.cov(n, r=1, mt=20)

### 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.
`mt` | A positive integer constraining possible rational function approximations. Default is 20.

### Details

Suppose a sample is given and one more individual is randomly drawn from the population. `preseqR.sample.cov` estimates the probability of the species, which represents the individual, has been observed at least r times in the sample. When r = 1, the probability is called the sample coverage.

Let N_j be the number of species represented exactly j times in a sample. The probability of observing a species represented at least r times in the sample is estimated as ∑_{j=r+1}^∞ jN_j / ∑_{j=1}^∞ jN_j. The theory is described by Mao and Lindsay (2002). For a random sample where N_j is unknown, a modified rational function approximation is first used to predict the value of N_j. Then the estimates are substituted to obtain an estimator for the probability of observing a species represented at least r times in the sample.

This function is the fast version of `preseqR.sample.cov.bootstrap`. The function does not provide the confidence interval. To obtain the confidence interval along with the estimates, one should use the function `preseqR.sample.cov.bootstrap`.

### Value

The estimator for the probability of observing a species represented at least r times in a random sample. 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

Good, I. J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika, 40(3-4), 237-264.

Mao, C. X. and Lindsay, B. G. (2002). A Poisson model for the coverage problem with a genomic application. Biometrika, 89(3), 669-682.

Deng, C., Daley, T., Calabrese, P., Ren, J., & Smith, A.D. (2016). Estimating the number of species to attain sufficient representation in a random sample. arXiv preprint arXiv:1607.02804v3.

### Examples

    ## load library
    library(preseqR)

    ## import data
    data(FisherButterfly)

    ## construct the estimator for the sample coverage
    estimator1 <- preseqR.sample.cov(FisherButterfly, r=1)
    ## Given a sample that is 10 times or 20 times the size of an initial samples,
    ## suppose one randomly draws one more individual from the population. The
    ## value of the function is the probability that the representing species
    ## has been observed in the sample
    estimator1(c(10, 20))

    ## construct the estimator
    estimator2 <- preseqR.sample.cov(FisherButterfly, r=2)
    ## the probability a species represented at least twice when the sample size
    ## is 50 times or 100 times of the initial sample
    estimator2(c(50, 100))

[Package preseqR version 4.0.0 Index]