R: MacArthur's Broken-stick distribution

dbs {sads} | R Documentation

## MacArthur's Broken-stick distribution

### Description

Density, distribution function, quantile function and random generation for the Broken-stick distribution with parameters `N` and `S`.

### Usage

    dbs( x, N, S, log = FALSE )
    pbs( q, N, S, lower.tail = TRUE, log.p = FALSE )
    qbs( p, N, S, lower.tail = TRUE, log.p = FALSE )
    rbs( n, N, S )
    drbs( x, N, S, log = FALSE )
    prbs( q, N, S, lower.tail = TRUE, log.p = FALSE )
    qrbs( p, N, S, lower.tail = TRUE, log.p = FALSE )
    rrbs( n, N, S)

### Arguments

`x` | vector of (non-negative integer) quantiles. In the context of species abundance distributions, this is a vector of abundances (for `dbs`) or abundance ranks (for `drbs`) of species in a sample.
`q` | vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundances (for `dbs`) or abundance ranks (for `drbs`) of species in a sample.
`n` | number of random values to return.
`p` | vector of probabilities.
`N` | positive integer 0 < N < Inf, sample size. In the context of species abundance distributions, the sum of abundances of individuals in a sample.
`S` | positive integer 0 < S < Inf, number of elements in a collection. In the context of species abundance distributions, the number of species in a sample.
`log, log.p` | logical; if TRUE, probabilities p are given as log(p).
`lower.tail` | logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

### Details

The Broken-stick distribution was proposed as a model for the expected abundance of elements in a collection:

n(i) = N/S (sum(from k=i to S) 1/k)

where n(i) is the abundance in the i-th most abundant element (MacArthur 1960, May 1975). Hence the probability (or expected proportion of occurrences) in the i-th element is

p(i) = n(i)/N = (sum(from k=i to S) 1/k) / S

`[dpq]rbs` stands for "rank-abundance Broken-stick" and return probabilities and quantiles based on the expression above, for p(i). Therefore, `[dpq]rbs` can be used as a rank-abundance model for species' ranks in a sample or in a biological community see `fitrad`.

The probability density for a given abundance value in the Broken-stick model is given by

p(x) = (1 - x/N)^(S-2) (S - 1)/N

Where x is the abundance of a given element in the collection (May 1975). `[dpq]bs` return probabilities and quantiles according to the expression above for p(x). Therefore, `[dpq]bs` can be used as a species abundance model see `fitsad`.

### Value

`dbs` gives the (log) density and `pbs` gives the (log) distribution function of abundances, and `qbs` gives the corresponding quantile function. `drbs` gives the (log) density and `prbs` gives the (log) distribution function of ranks, and `qrbs` gives the corresponding quantile function.

### Author(s)

Paulo I Prado prado@ib.usp.br and Murilo Dantas Miranda.

### References

MacArthur, R.H. 1960. On the relative abundance of species. Am Nat 94:25–36.

May, R.M. 1975. Patterns of Species Abundance and Diversity. In Cody, M.L. and Diamond, J.M. (Eds) Ecology and Evolution of Communities. Harvard University Press. pp 81–120.

### See Also

`fitbs` and `fitrbs` to fit the Broken-stick distribution as a abundance (SAD) and rank-abundance (RAD) model.

### Examples

    x <- 1:25
    PDF <- drbs(x=x, N=100, S=25)
    CDF <- prbs(q=x, N=100, S=25)
    par(mfrow=c(1,2))
    plot(x,CDF, ylab="Cumulative Probability", type="b",
         main="Broken-stick rank distribution, CDF")
    plot(x,PDF, ylab="Probability", type="h",
         main="Broken-stick rank distribution, PDF")
    par(mfrow=c(1,1))

    ## quantile is the inverse of CDF
    all.equal( qrbs( CDF, N=100, S=25), x) # should be TRUE

[Package sads version 0.4.2 Index]