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Calculate Smallest Eigenvalue for Power Exponential Correlation Matrices

Source: R/utils.R
sev_pexp.Rd

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.

Usage

sev_pexp(range_nu, range_rho, grid_len = 50, dmat)

Arguments

range_nu

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).

range_rho

A numeric vector of length 2, specifying the minimum and maximum values for the practical range parameter rho. rho must be positive.

grid_len

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.

dmat

A numeric matrix representing the distance matrix between locations. The distances should be non-negative.

Value

A tibble with three columns:

rho

The practical range parameter value.

nu

The power parameter value.

lambda

The smallest eigenvalue of the power exponential correlation matrix corresponding to the rho and nu pair.

Details

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.