BRISC_bootstrap {BRISC}

Function for performing bootstrap with BRISC

Description

The function BRISC_bootstrap performs bootstrap to provide confidence intervals for parameters of univariate spatial regression models using outputs of BRISC_estimation. The details of the bootstrap method can be found in BRISC (Saha & Datta, 2018). The optimization is performed with C library of limited-memory BFGS libLBFGS: a library of Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), http://www.chokkan.org/software/liblbfgs/ (Naoaki Okazaki). For user convenience the soure codes of the package libLBFGS are provided in the package. 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_bootstrap(BRISC_Out, n_boot = 100, h = 1, n_omp = 1,
                init = "Initial", verbose = TRUE,
                nugget_status = 1)

Arguments

BRISC_Out

an object of class BRISC_Out, obtained as an output of
BRISC_estimation.

n_boot

number of bootstrap samples. Default value is 100.

h

number of core to be used in parallel computing setup for bootstrap samples. If h = 1, there is no parallelization. Default value is 1.

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.

init

keyword that specifies initialization scheme to be used. Supported keywords are: "Initial" and "Estimate" for initialization of parameter values for bootstrap samples with initial values used in BRISC_estimate and estimated values of parameters in BRISC_estimate respectively.

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.

nugget_status

if nugget_status = 0, tau.sq is fixed to 0, if nugget_status = 1 tau.sq is estimated. Default value is 1.

Value

A list comprising of the following:

boot.Theta

estimates of spatial covariance parameters corresponding to bootstrap samples.

boot.Beta

estimates of beta corresponding to bootstrap samples.

confidence.interval

confidence intervals corresponding to the parameters.

boot.time

time (in seconds) required to perform the bootstrapping after preprocessing data in R, reported using proc.time().

Author(s)

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

References

Saha, A., & Datta, A. (2018). BRISC: bootstrap for rapid inference on spatial covariances. Stat, e184, DOI: 10.1002/sta4.184.

Okazaki N. libLBFGS: a library of Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS),
http://www.chokkan.org/software/liblbfgs/ .

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

rmvn <- function(n, mu = 0, V = matrix(1)){
  p <- length(mu)
  if(any(is.na(match(dim(V),p))))
    stop("Dimension not right!")
  D <- chol(V)
  t(matrix(rnorm(n*p), ncol=p)%*%D + rep(mu,rep(n,p)))
}

set.seed(1)
n <- 300
coords <- cbind(runif(n,0,1), runif(n,0,1))

beta <- c(1,5)
x <- cbind(rnorm(n), rnorm(n))

sigma.sq = 1
phi = 5
tau.sq = 0.1

B <- as.matrix(beta)
D <- as.matrix(dist(coords))
R <- exp(-phi*D)
w <- rmvn(1, rep(0,n), sigma.sq*R)

y <- rnorm(n, x%*%B + w, sqrt(tau.sq))

estimation_result <- BRISC_estimation(coords, y, x)
bootstrap_result <- BRISC_bootstrap(estimation_result, n_boot = 10)