Create the sparse triangular matrix U for multivariate Vecchia models

This creates the sparse triangular matrix U for multivariate Vecchia models. This matrix can be used to estimate the likelihood or transform the data to be iid. This function is a multivariate version of createU.

Usage

createUMultivariate(vec.approx, params, cov_func = NULL)

Arguments

vec.approx: Object returned by vecchia_Mspecify.
params: Vector of covariance parameters. See create.param.sequence or the examples below for details about the format of this vector.
cov_func: The function used to compute the covariance between two observations. Defaults to a Matern model.

Value

A list containing the sparse upper trianguler U, plus additional objects required for other functions.

Details

This function will be much slower if a non-default cov_func is specified. More importantly, there is no guarantee that the resulting covariance matrices will be positive definite. We recommend using the default (Matern) covariance function unless you know exactly what you are doing. See Apanasovich et al. (2012) for a description of how the cross-covariances are computed.

References

Apanasovich, T.V., Genton, M.G. and Sun, Y. "A valid Matérn class of cross-covariance functions for multivariate random fields with any number of components", Journal of the American Statistical Association (2012) 107(497):180-193.
Katzfuss, M., and Guinness, J. "A general framework for Vecchia approximations of Gaussian processes", Statistical Science (2021) 36(1):124-141.

Examples

data(soil)
soil <- soil[!is.na(soil[,5]),] # remove rows with NA's
locs <- as.matrix(soil[,1:2])
locsm <- list()
locsm[[1]] <- locsm[[2]] <- locs
soil.va <- vecchia_Mspecify(locsm, m=10)

pseq <- create.param.sequence(2)
# Initialize the vector of covariance parameters
params <- rep(NA, pseq[5,2])
# Sigma parameters:
params[pseq[1,1]:pseq[1,2]] <- c(100, 80)
# Scale parameters:
params[pseq[2,1]:pseq[2,2]] <- c(60, 50)
# Smoothness parameters:
params[pseq[3,1]:pseq[3,2]] <- c(0.5, 0.5)
# Nuggets:
params[pseq[4,1]:pseq[4,2]] <- c(30, 30)
# Correlation:
params[pseq[5,1]:pseq[5,2]] <- -0.9

soil.u <- createUMultivariate(soil.va, params)