matern_spacetime_categorical_local {GpGp}

Space-Time Matern covariance function with local random effects for categories

Description

From a matrix of locations and covariance parameters of the form (variance, spatial range, temporal range, smoothness, cat variance, cat spatial range, cat temporal range, cat smoothness, nugget), return the square matrix of all pairwise covariances. This is the covariance for the following model for data from cateogory k

Y_k(x_i,t_i) = Z_0(x_i,t_i) + Z_k(x_i,t_i) + e_i

where Z_0 is Matern with parameters (variance,spatial range,temporal range,smoothness) and Z_1,...,Z_K are independent Materns with parameters (cat variance, cat spatial range, cat temporal range, cat smoothness), and e_1, ..., e_n are independent normals with variance (variance * nugget)

Usage

matern_spacetime_categorical_local(covparms, locs)

d_matern_spacetime_categorical_local(covparms, locs)

Arguments

covparms

A vector with covariance parameters in the form (variance, spatial range, temporal range, smoothness, category, nugget)

locs

A matrix with n rows and d columns. Each row of locs gives a point in R^d.

Value

A matrix with n rows and n columns, with the i,j entry containing the covariance between observations at locs[i,] and locs[j,].

Functions

• d_matern_spacetime_categorical_local(): Derivatives of isotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, range, smoothness, category, nugget) = (\sigma^2,\alpha_1,\alpha_2,\nu,c^2,\tau^2), and the covariance function is parameterized as

d = ( || x - y ||^2/\alpha_1 + |s-t|^2/\alpha_2^2 )^{1/2}

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (d)^\nu K_\nu(d)

(x,s) and (y,t) are the space-time locations of a pair of observations. The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix. The category variance c^2 is added if two observation from same category NOTE: the nugget is \sigma^2 \tau^2 , not \tau^2 .