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Hybrid non-stationary Whittle-Matern SPDE model with non-zero mean

Source: R/hybrid.spde.R
hybrid.spde.Rd

hybrid.spde constructs a finite element/rational SPDE approximation of the model $$Y(s) = \beta L^{-\alpha/2} X(s) + \tau^{-1} L^{-\alpha/2} W(s),$$ where \(L = \kappa(s)^2 - \Delta\) is the (possibly non-stationary) Whittle-Matern differential operator, \(\tau\) is a positive scale that affects only the random component, \(X\) is a (matrix of) covariate field(s) supplied at the mesh nodes, \(\beta\) is a vector of regression coefficients and \(W\) is Gaussian white noise.

Usage

hybrid.spde(
  X = NULL,
  beta_X = NULL,
  kappa = NULL,
  tau = NULL,
  kappa_mu = NULL,
  theta = NULL,
  B.tau = NULL,
  B.kappa = NULL,
  B.sigma = NULL,
  B.range = NULL,
  alpha = NULL,
  nu = NULL,
  parameterization = c("spde", "matern"),
  G = NULL,
  C = NULL,
  d = NULL,
  graph = NULL,
  mesh = NULL,
  range_mesh = NULL,
  loc_mesh = NULL,
  m = 1,
  type = c("covariance", "operator"),
  type_rational_approximation = c("brasil", "chebfun", "chebfunLB"),
  check_stationarity = TRUE
)

Arguments

X

Numeric vector or matrix containing the covariate field(s) evaluated at the mesh nodes. If a matrix is supplied, each column is treated as a separate covariate field. The number of rows must equal the number of mesh nodes of the underlying FEM discretization.

beta_X

Numeric vector of regression coefficients corresponding to the columns of X. Defaults to a vector of zeros of length ncol(X).

kappa, tau, theta, B.tau, B.kappa, B.sigma, B.range, alpha, nu, parameterization, G, C, d, graph, mesh, range_mesh, loc_mesh, m, type, type_rational_approximation, check_stationarity

Arguments passed to spde.matern.operators(). The default check_stationarity = TRUE allows the underlying covariance model to be returned as a stationary matern.operators() object whenever tau and kappa are constant after the log-linear regression on theta is applied.

kappa_mu

Optional. If NULL (the default), the operator \(L_\mu = \kappa^2 - \Delta\) used for the deterministic mean \(\mu = \beta L_\mu^{-\alpha/2} X\) shares the range parameter kappa with the random component. If a positive scalar (or vector of length n_mesh for non-stationary models) is supplied, \(L_\mu = \kappa_\mu^2 - \Delta\) is used instead.

Value

hybrid.spde returns an object of class "hybrid_spde" which also inherits from the class of the object returned by spde.matern.operators() (spde_matern_operator plus either rSPDEobj or CBrSPDEobj depending on type). The object stores the covariate matrix X, the regression coefficients beta_X, the cached operator object op_mean used to evaluate \(L^{-\alpha/2}\), and the evaluated mean function mu at the mesh nodes.

See also

spde.matern.operators(), simulate.hybrid_spde(), predict.hybrid_spde()

Examples

set.seed(1)
x <- seq(from = 0, to = 1, length.out = 51)
kappa <- 10
tau <- 0.5
alpha <- 1.3
X <- cbind(sin(2 * pi * x), cos(2 * pi * x))
beta_X <- c(1.5, -0.5)
op <- hybrid.spde(
  X = X, beta_X = beta_X,
  kappa = kappa, tau = tau, alpha = alpha,
  loc_mesh = x, d = 1, parameterization = "spde"
)
Y <- simulate(op, nsim = 1)