BRISC_correlation {BRISC}

Function for simulating correlated data with BRISC

Description

The function BRISC_correlation creates correlated data (known structure) using Nearest Neighbor Gaussian Processes (NNGP). BRISC_correlation uses the sparse Cholesky representation of Vecchia’s likelihood developed in Datta et al., 2016. Some code blocks are borrowed from the R package: spNNGP: Spatial Regression Models for Large Datasets using Nearest Neighbor Gaussian Processes https://CRAN.R-project.org/package=spNNGP .

Usage

BRISC_correlation(coords, sim, sigma.sq = 1, tau.sq = 0, phi = 1,
                  nu = 1.5, n.neighbors = NULL, n_omp = 1,
                  cov.model = "exponential",
                  search.type = "tree", stabilization = NULL,
                  verbose = TRUE, tol = 12)

Arguments

coords

an n \times 2 matrix of the observation coordinates in R^2 (e.g., easting and northing).

sim

an n \times k matrix of the k many n \times 1 vectors from which the correlated data are calculated (see Details below).

sigma.sq

value of sigma square. Default value is 1.

tau.sq

value of tau square. Default value is 0.1.

phi

value of phi. Default value is 1.

nu

value of nu, only required for matern covariance model. Default value is 1.5.

n.neighbors

number of neighbors used in the NNGP. Default value is max(100, n -1). We suggest a high value of n.neighbors for lower value of phi.

n_omp

number of threads to be used, value can be more than 1 if source code is compiled with OpenMP support. Default is 1.

cov.model

keyword that specifies the covariance function to be used in modelling the spatial dependence structure among the observations. Supported keywords are: "exponential", "matern", "spherical", and "gaussian" for exponential, Matern, spherical and Gaussian covariance function respectively. Default value is "exponential".

search.type

keyword that specifies type of nearest neighbor search algorithm to be used. Supported keywords are: "brute", "tree" and "cb".
"brute" and "tree" provide the same result, though "tree" should be faster. "cb" implements fast code book search described in Ra and Kim (1993) modified for NNGP. If locations do not have identical coordinate values on the axis used for the nearest neighbor determination, then "cb" and "brute" should produce identical neighbor sets. However, if there are identical coordinate values on the axis used for nearest neighbor determination, then "cb" and "brute" might produce different, but equally valid neighbor sets, e.g., if data are on a grid. Default value is "tree".

stabilization

when we use a very smooth covarince model (lower values of phi for spherical and Gaussian covariance and low phi and high nu for Matern covarinace) in absence of a non-negligble nugget, the correlation process may fail due to computational instability. If stabilization = TRUE, performs stabilization by setting tau.sq = max{\code{tau .sq}, \code{sigma.sq} * 1e-06}. Default value is TRUE for cov.model = "expoenential" and FALSE otherwise.

verbose

if TRUE, model specifications along with information regarding OpenMP support and progress of the algorithm is printed to the screen. Otherwise, nothing is printed to the screen. Default value is TRUE.

tol

the input observation coordinates are rounded to this many places after the decimal. The default value is 12.

Details

Denote g be the input sim. Let \Sigma be the precision matrix associated with the covariance model determined by the cov.model and model parameters. Then BRISC_correlation calculates h, where h is given as follows:

S ^{-0.5} h = g

where, S ^{-0.5} is a sparse approximation of the cholesky factor \Sigma ^{-0.5} of the precision matrix \Sigma ^{-1}, obtained from NNGP.

Value

A list comprising of the following:

coords

the matrix coords.

n.neighbors

the used value of n.neighbors.

cov.model

the used covariance model.

Theta

parameters of covarinace model; accounts for stabilization.

input.data

the matrix sim.

output.data

the output matrix h in Details.

time

time (in seconds) required after preprocessing data in R,
reported using, proc.time().

Author(s)

Arkajyoti Saha [e-mail],
Abhirup Datta [e-mail]

References

Datta, A., S. Banerjee, A.O. Finley, and A.E. Gelfand. (2016) Hierarchical Nearest-Neighbor Gaussian process models for large geostatistical datasets. Journal of the American Statistical Association, 111:800-812.

Andrew Finley, Abhirup Datta and Sudipto Banerjee (2017). spNNGP: Spatial Regression Models for Large Datasets using Nearest Neighbor Gaussian Processes. R package version 0.1.1. https://CRAN.R-project.org/package=spNNGP

Examples

set.seed(1)
n <- 1000
coords <- cbind(runif(n,0,1), runif(n,0,1))

sigma.sq = 1
phi = 1

set.seed(1)
sim <- matrix(rnorm(3*n),n, 3)
correlation_result <- BRISC_correlation(coords, sigma.sq = sigma.sq,
                                        phi = phi, sim = sim)