sads (version 0.2.2)

dbs: MacArthur's Broken-stick distribution

Description

Density, distribution function and quantile function 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 )
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 )

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.

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].

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.

Details

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

$$n(i) = \frac{N}{S} \sum_{k=i}^S 1/k$$

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

$$p(i) = \frac{n(i)}{S} = S^{-1}\sum_{k=i}^S 1/k$$

[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

$$p(x) = \frac{S-1}{N} \left( 1 - \frac{x}{N} \right)^{S-2}$$

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.

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