sads (version 0.6.3)

dgs: Geometric series distribution

Description

Density, distribution function, quantile function and random generation 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 )
rgs( n, k, S )

Value

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

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.

n

number of random values to return.

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].

Author

Paulo I Prado and Murilo Dantas Miranda.

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)