k_t {quaxnat}

Dispersal kernels from spatial t distribution

Description

k_t computes the value, multiplied by N, of the dispersal kernel from Clark et al. (1999) that represents a multivariate t distribution.

Usage

k_t(x, par, N = 1, d = NCOL(x))

Arguments

x

Numeric matrix of positions x relative to the seed source, or vector of distances \left\|{x}\right\| to the seed source.

par

Numeric vector with two elements representing the log-transformed parameters a and b.

N

The multiplier N.

d

The spatial dimension.

Details

The dispersal kernel, i.e. spatial probability density of the location of a seed relative to its source, is here given by

k(x)={\Gamma ((b+d)/2) \over \pi ^{d/2}a^{d}\Gamma (b/2)} (1+{\left\|{x}\right\|^{2} \over a^{2}})^{-(b+d)/2},

which corresponds to a probability density of the distance given by

p(r)={2 \over a^{d}B(d/2,b/2)}r^{d-1} (1+{r^{2} \over a^{2}})^{-(b+d)/2},

where d is the spatial dimension, \left\|{\,}\right\| denotes the Euclidean norm and the normalizing constants involve the beta and gamma functions; see Clark et al. (1999) and Austerlitz et al. (2004) for the planar case (with a,b replaced by \sqrt{u},2p and a,2b-d, respectively). This means the position is a \over \sqrt{b} times a random vector having a standard d-variate t distribution with b degrees of freedom (a standard Gaussian vector divided by \sqrt{z/b}, where z is independent and chi-squared distributed with b degrees of freedom), and the squared distance is da^{2} \over b times a random variable having an F distribution with d and b degrees of freedom.

This results from the kernel being defined as a mixture of Gaussian kernels with an inverse variance having a gamma distribution with shape parameter b\over 2 and inverse scale parameter a^{2}\over 2, which for a=1 is a chi-squared distribution with b degrees of freedom.

The dispersal kernel always has its maximum at zero, and the distance has a fat-tailed distribution for all choices of b.

Value

Numeric vector of function values k(x) multiplied by N.

References

Clark, J.S., Silman, M., Kern, R., Macklin, E., HilleRisLambers, J. (1999). Seed dispersal near and far: patterns across temperate and tropical forests. Ecology 80, 1475–1494. doi:10.1890/0012-9658(1999)080[1475:SDNAFP]2.0.CO;2

Austerlitz, F., Dick, C.W., Dutech, C., Klein, E.K., Oddou-Muratorio, S., Smouse, P.E., Sork, V.L. (2004). Using genetic markers to estimate the pollen dispersal curve. Molecular Ecology 13, 937–954. doi:10.1111/j.1365-294X.2004.02100.x

Examples

k_t(2:5, par=c(0,0), d=2)