This function computes the smallest eigenvalue of a correlation
matrix derived from the power exponential correlation function. It
evaluates this across a grid of values for the power parameter (nu) and
the practical range parameter (rho), based on a provided distance
matrix.
sev_pexp(range_nu, range_rho, grid_len = 50, dmat)A tibble with three columns:
The practical range parameter value.
The power parameter value.
The smallest eigenvalue of the power exponential correlation
matrix corresponding to the rho and nu pair.
A numeric vector of length 2, specifying the minimum and
maximum values for the power parameter nu. nu typically ranges between
0 and 2 (e.g., nu = 1 for exponential, nu = 2 for Gaussian).
A numeric vector of length 2, specifying the minimum and
maximum values for the practical range parameter rho. rho must be
positive.
An integer specifying the number of points to create for
both nu and rho sequences. The total number of grid combinations will
be grid_len^2. Default is 50.
A numeric matrix representing the distance matrix between locations. The distances should be non-negative.
The practical range rho is defined here as the distance at which
the correlation is 0.1. The internal scale parameter phi is calculated
as phi = rho / (log(10)^(1/nu)). The power exponential correlation
function is assumed to be of the form C(h) = exp(-(h/phi)^nu), where h is
distance. The function smile:::pexp_cov is used internally to compute
the covariance/correlation matrix with a sill of 1.
The function first creates a grid of nu and rho parameters. For
each pair of (rho, nu) in the grid: 1. It calculates the scale
parameter phi for the power exponential correlation function, where phi = rho / (log(10)^(1/nu)). This definition implies that the correlation is
0.1 at the distance rho. 2. It computes the power exponential
correlation matrix using smile:::pexp_cov(dists = dmat, sill = 1, range = phi, smooth = nu). Note the use of an internal function from the smile
package. 3. It calculates the eigenvalues of this correlation matrix.
4. The minimum eigenvalue is extracted. The final output is a tibble
containing all parameter combinations and their corresponding minimum
eigenvalues.