RFGLS_estimate_spatial {RandomForestsGLS}

Function for estimation in spatial data with RF-GLS

Description

The function RFGLS_estimate_spatial fits univariate non-linear spatial regression models for spatial data using RF-GLS in Saha et al. 2020. RFGLS_estimate_spatial uses the sparse Cholesky representation of Vecchia’s likelihood (Vecchia, 1988) developed in Datta et al., 2016 and Saha & Datta, 2018. The fitted Random Forest (RF) model is used later for prediction via the RFGLS_predict and RFGLS_predict_spatial.

Some code blocks are borrowed from the R packages: spNNGP: Spatial Regression Models for Large Datasets using Nearest Neighbor Gaussian Process
https://CRAN.R-project.org/package=spNNGP and randomForest: Breiman and Cutler's Random Forests for Classification and Regression
https://CRAN.R-project.org/package=randomForest .

Usage

RFGLS_estimate_spatial(coords, y, X, Xtest = NULL,
                       nrnodes = NULL, nthsize = 20,
                       mtry = 1, pinv_choice = 1,
                       n_omp = 1, ntree = 50, h = 1,
                       sigma.sq = 1, tau.sq = 0.1,
                       phi = 5, nu = 0.5,
                       n.neighbors = 15,
                       cov.model = "exponential",
                       search.type = "tree",
                       param_estimate = FALSE,
                       verbose = FALSE)

Arguments

coords

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

y

an n length vector of response at the observed coordinates.

X

an n \times p matrix of the covariates in the observation coordinates.

Xtest

an ntest \times p matrix of covariates for prediction locations. Its Structure should be identical (including intercept) with that of covariates provided for estimation purpose in X. If NULL, will use X as Xtest. Default value is NULL.

nrnodes

the maximum number of nodes a tree can have. Default choice leads to the deepest tree contigent on nthsize. For significantly large n, one needs to bound it for growing shallow trees which trades off efficiency for computation time.

nthsize

minimum size of leaf nodes. We recommend not setting this value too small, as that will lead to very deep trees that takes a lot of time to be built and can produce unstable estimaes. Default value is 20.

mtry

number of variables randomly sampled at each partition as a candidate split direction. We recommend using the value p/3 where p is the number of variables in X. Default value is 1.

pinv_choice

dictates the choice of method for obtaining the pseudoinverse involved in the cost function and node representative evaluation. if pinv_choice = 0, SVD is used (slower but more stable), if pinv_choice = 1, orthogonal decomposition (faster, may produce unstable results if nthsize is too low) is used. 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.

ntree

number of trees to be grown. This value should not be too small. Default value is 50.

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.

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 5.

nu

value of nu, only required for matern covariance model. Default value is 0.5.

n.neighbors

number of neighbors used in the NNGP. Default value is 15.

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: "tree" and "brute". Both of them provide the same result, though "tree" should be faster. Default value is "tree".

param_estimate

if TRUE, using the residuals obtained from fitting a classical RF with default options and nodesize = nthsize, will estimate the coefficeints corresponding to cov.model from BRISC_estimate with the deafult options. Default value is FALSE.

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 FALSE.

Value

A list comprising:

P_matrix

an n \times ntree matrix of zero indexed resamples. t-th column denote the n resamples used in the t-th tree.

predicted_matrix

an ntest \times ntree matrix of predictions. t-th column denote the predictions at ntest datapoints obtained from the t-th tree.

predicted

preducted values at the ntest prediction points. Average (rowMeans) of the treewise predctions in predicted_matrix,

X

the matrix X.

y

the vector y.

RFGLS_Object

object required for prediction.

Author(s)

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

References

Saha, A., Basu, S., & Datta, A. (2020). Random Forests for dependent data. arXiv preprint arXiv:2007.15421.

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

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

Andy Liaw, and Matthew Wiener (2015). randomForest: Breiman and Cutler's Random Forests for Classification and Regression. R package version 4.6-14.
https://CRAN.R-project.org/package=randomForest

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 <- 200
coords <- cbind(runif(n,0,1), runif(n,0,1))

set.seed(2)
x <- as.matrix(rnorm(n),n,1)

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

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

y <- rnorm(n, 10*sin(pi * x) + w, sqrt(tau.sq))

estimation_result <- RFGLS_estimate_spatial(coords, y, x, ntree = 10)