Given a distance matrix from calcVinEll, calculate a
stochastic matrix where one step movement probabilities follow a gamma density.
Arguments
- distMat
Distance matrix from
calcVinEll- shape
Shape parameter of
GammaDistdistribution- rate
Rate parameter of
GammaDistdistribution
Details
The distribution and density functions for the gamma kernel are given below: $$ F(x)=\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x) $$ $$ f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} $$ where \(\Gamma(\alpha)\) is the Gamma function, \(\gamma(\alpha,\beta x)\) is the lower incomplete gamma function, and \(\alpha,\beta\) are the shape and rate parameters, respectively.
Examples
# setup distance matrix
# two-column matrix with latitude/longitude, in degrees
latLong = cbind(runif(n = 5, min = 0, max = 90),
runif(n = 5, min = 0, max = 180))
# Vincenty Ellipsoid distance formula
distMat = calcVinEll(latLongs = latLong)
# calculate gamma distribution over distances
# shape and rate are just for example
kernMat = calcGammaKernel(distMat = distMat, shape = 1, rate = 1)