sads (version 0.2.2)

dgs: Geometric series distribution

Description

Density, distribution function and quantile function for the Geometric Series distribution, with parameter k.

Usage

dgs( x, k, S, log = FALSE )
pgs( q, k, S, lower.tail = TRUE, log.p = FALSE )
qgs( p, k, 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 abundance ranks of species in a sample.

k

positive real, 0 < k < 1; geometric series coefficient; the ratio between the abundances of i-th and (i+1)-th species.

q

vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundance ranks of species in a sample.

p

vector of probabilities.

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

• dgs gives the (log) density and pgs gives the (log) distribution function of ranks, and qgs gives the corresponding quantile function.

Details

The Geometric series distribution gives the probability (or expected proportion of occurrences) of the i-th most abundant element in a collection: $$p(i) = C k (1-k)^{i-1}$$ where C is a normalization constant which makes the summation of p(i) over S equals to one: $$C = \frac{1}{1 - (1-k)^S}$$ where S is the number of species in the sample. Therefore, [dpq]gs can be used as rank-abundance model for species ranks in a sample or biological community see fitrad-class.

References

Doi, H. and Mori, T. 2012. The discovery of species-abundance distribution in an ecological community. Oikos 122: 179--182. 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

fitgs, fitrad to fit the Geometric series as a rank-abundance model.

Examples

x <- 1:25
PDF <- dgs(x=x, k=0.1, S=25)
CDF <- pgs(q=x, k=0.1, S=25)
par(mfrow=c(1,2))
plot(x,CDF, ylab="Cumulative Probability", type="b",
     main="Geometric series distribution, CDF")
plot(x,PDF, ylab="Probability, log-scale", type="h",
     main="Geometric series distribution, PDF", log="y")
par(mfrow=c(1,1))

## quantile is the inverse of CDF
all.equal(qgs(CDF, k=0.1, S=25), x)