mcmc_mra: Bayesian Multi-resolution Spatial Regression
In BayesMRA: Bayesian Multi-Resolution Gaussian Process Approximations

this function runs Markov Chain Monte Carlo to estimate the Bayesian multi-resolution spatial regression model.

mcmc_mra(
  y,
  X,
  locs,
  params,
  priors = NULL,
  M = 4,
  n_neighbors = 68,
  n_coarse_grid = 10,
  n_padding = 5L,
  n_cores = 1L,
  inits = NULL,
  config = NULL,
  verbose = FALSE,
  use_spam = TRUE,
  n_chain = 1
)

`y`	is a n vector of Gaussian data.
`X`	is a n x p matrix of fixed effects (like latitude, elevation, etc)
`locs`	is a n x 2 matrix of observation locations.
`params`	is a list of parameter settings. The list `params` must contain the following values: `n_adapt`: A positive integer number of adaptive MCMC iterations. `n_mcmc`: A positive integer number of total MCMC iterations post adaptation. `n_thin`: A positive integer number of MCMC iterations per saved sample. `n_message`: A positive integer number of frequency of iterations to output a progress message. For example, `n_message = 50` outputs progress messages every 50 iterations.
`priors`	is the list of prior settings.
`M`	The number of resolutions.
`n_neighbors`	The expected number of neighbors for each interior basis function. This determines the basis radius parameter.
`n_coarse_grid`	The number of basis functions in one direction (e.g. `n_coarse_grid = 10` results in a 10x10 course grid which is further extended by the number of additional padding basis functions given by `n_padding`.
`n_padding`	The number of additional boundary points to add on each boundary. For example, n_padding = 5 will add 5 boundary knots to the both the left and right side of the grid).
`n_cores`	is the number of cores for parallel computation using openMP.
`inits`	is the list of initial values if the user wishes to specify initial values. If these values are not specified, then the initial values will be randomly sampled from the prior.
`config`	is the list of configuration values if the user wishes to specify initial values. If these values are not specified, then default a configuration will be used.
`verbose`	Should verbose output be printed? Typically this is only useful for troubleshooting.
`use_spam`	is a boolean flag to determine whether the output is a list of spam matrix objects (`use_spam = TRUE`) or a an n x n sparse Matrix of class "dgCMatrix" `use_spam = FALSE` (see spam and Matrix packages for details).
`n_chain`	is the MCMC chain id. The default is 1.

set.seed(111)
## genereate the locations
locs <- matrix(runif(20), 10, 2)
## generate some covariates and regression coefficients
X <- cbind(1, matrix(rnorm(30), 10, 3))
beta <- rnorm(ncol(X))

## simulate the MRA process
M <- 2
MRA <- mra_wendland_2d(locs, M = M, n_coarse_grid = 4)
W <- do.call(cbind, MRA$W)

n_dims   <- rep(NA, length(MRA$W))
dims_idx <- NULL
for (i in 1:M) {
    n_dims[i] <- ncol(MRA$W[[i]])
    dims_idx  <- c(dims_idx, rep(i, n_dims[i]))
}
## set up the process precision matrices
Q_alpha <- make_Q_alpha_2d(sqrt(n_dims), c(0.9, 0.8))
Q_alpha_tau2 <- make_Q_alpha_tau2(Q_alpha, tau2 = c(2, 4))

## add in constraints so each resolution has random effects that sum to 0
A_constraint <- sapply(1:M, function(i){
    tmp = rep(0, sum(n_dims))
    tmp[dims_idx == i] <- 1
    return(tmp)
})
a_constraint <- rep(0, M)
alpha <- as.vector(spam::rmvnorm.prec.const(
    n = 1,
    mu = rep(0, nrow(W)),
    Q = Q_alpha_tau2,
    A = t(A_constraint),
    a = a_constraint))
## define the data
y <-  as.vector(X %*% beta + W %*% alpha + rnorm(10))

## define the params for MCMC fitting
params <- list(
    n_mcmc    = 5,
    n_adapt   = 5,
    n_thin    = 1,
    n_message = 5)

## define the model priors
priors <- list(
    alpha_tau2   = 1,
    beta_tau2    = 1,
    alpha_sigma2 = 1,
    beta_sigma2  = 1,
    mu_beta      = rep(0, ncol(X)),
    Sigma_beta   = 5 * diag(ncol(X)))

## Fit the MRA model using MCMC
out     <- mcmc_mra(
    y             = y,
    X             = X,
    locs          = locs,
    params        = params,
    priors        = priors,
    M             = 2,
    n_coarse_grid = 4,
    n_cores       = 1L,
    verbose       = FALSE
)