Rational approximations of stationary Gaussian Matern random fields

matern.operators is used for computing a rational SPDE approximation of a stationary Gaussian random fields on $R^d$ with a Matern covariance function $$C(h) = \frac{\sigma^2}{2^{\nu-1}\Gamma(\nu)} (\kappa h)^\nu K_\nu(\kappa h)$$

Usage

matern.operators(
  kappa = NULL,
  tau = NULL,
  alpha = NULL,
  sigma = NULL,
  range = NULL,
  nu = NULL,
  G = NULL,
  C = NULL,
  d = NULL,
  mesh = NULL,
  graph = NULL,
  range_mesh = NULL,
  loc_mesh = NULL,
  m = 1,
  type = c("covariance", "operator"),
  parameterization = c("spde", "matern"),
  compute_higher_order = FALSE,
  return_block_list = FALSE,
  type_rational_approximation = c("chebfun", "brasil", "chebfunLB"),
  fem_mesh_matrices = NULL,
  compute_logdet = FALSE
)

Arguments

kappa: Parameter kappa of the SPDE representation. If NULL, the range parameter will be used. If the range is also NULL, a starting value based on the mesh will be supplied.
tau: Parameter tau of the SPDE representation. If both sigma and tau are NULL, a starting value based on the mesh will be supplied.
alpha: Parameter alpha of the SPDE representation. If alpha is NULL, a starting value will be supplied.
sigma: Standard deviation of the covariance function. Used if parameterization is matern. If NULL, tau will be used. If tau is also NULL, a starting value based on the mesh will be supplied.
range: Range parameter of the covariance function. Used if parameterization is matern. If range is NULL, a starting value based on the mesh will be supplied.
nu: Shape parameter of the covariance function. Used if parameterization is matern. If NULL, a starting value will be supplied.
G: The stiffness matrix of a finite element discretization of the domain of interest. Does not need to be given if either mesh or graph is supplied.
C: The mass matrix of a finite element discretization of the domain of interest. Does not need to be given if either mesh or graph is supplied.
d: The dimension of the domain. Does not need to be given if either mesh or graph is provided.
mesh: An optional fmesher mesh. Replaces d, C and G.
graph: An optional metric_graph object. Replaces d, C and G.
range_mesh: The range of the mesh. Will be used to provide starting values for the parameters. Will be used if mesh and graph are NULL, and if one of the parameters (kappa or tau for spde parameterization, or sigma or range for matern parameterization) are not provided.
loc_mesh: The mesh locations used to construct the matrices C and G. This option should be provided if one wants to use the rspde_lme() function and will not provide neither graph nor mesh. Only works for 1d data. Does not work for metric graphs. For metric graphs you should supply the graph using the graph argument.
m: The order of the rational approximation, which needs to be a positive integer. The default value is 1.
type: The type of the rational approximation. The options are "covariance" and "operator". The default is "covariance".
parameterization: Which parameterization to use? matern uses range, std. deviation and nu (smoothness). spde uses kappa, tau and alpha. The default is spde.
compute_higher_order: Logical. Should the higher order finite element matrices be computed?
return_block_list: Logical. For type = "covariance", should the block parts of the precision matrix be returned separately as a list?
type_rational_approximation: Which type of rational approximation should be used? The current types are "chebfun", "brasil" or "chebfunLB".
fem_mesh_matrices: A list containing FEM-related matrices. The list should contain elements c0, g1, g2, g3, etc.
compute_logdet: Should log determinants be computed while building the model? (For covariance-based models)

Value

If type is "covariance", then matern.operators

returns an object of class "CBrSPDEobj". This object is a list containing the following quantities:

C: The mass lumped mass matrix.
Ci: The inverse of C.
GCi: The stiffness matrix G times Ci
Gk: The stiffness matrix G along with the higher-order FEM-related matrices G2, G3, etc.
fem_mesh_matrices: A list containing the mass lumped mass matrix, the stiffness matrix and the higher-order FEM-related matrices.
m: The order of the rational approximation.
alpha: The fractional power of the precision operator.
type: String indicating the type of approximation.
d: The dimension of the domain.
nu: Shape parameter of the covariance function.
kappa: Range parameter of the covariance function
tau: Scale parameter of the covariance function.
sigma: Standard deviation of the covariance function.
type: String indicating the type of approximation.

If type is "operator", then matern.operators

returns an object of class "rSPDEobj". This object contains the quantities listed in the output of fractional.operators(), the G matrix, the dimension of the domain, as well as the parameters of the covariance function.

Details

If type is "covariance", we use the covariance-based rational approximation of the fractional operator. In the SPDE approach, we model $u$ as the solution of the following SPDE: $$L^{\alpha/2}(\tau u) = \mathcal{W},$$ where $L = -\Delta +\kappa^2 I$ and $\mathcal{W}$ is the standard Gaussian white noise. The covariance operator of $u$ is given by $L^{-\alpha}$. Now, let $L_h$ be a finite-element approximation of $L$. We can use a rational approximation of order $m$ on $L_h^{-\alpha}$ to obtain the following approximation: $$L_{h,m}^{-\alpha} = L_h^{-m_\alpha} p(L_h^{-1})q(L_h^{-1})^{-1},$$ where $m_\alpha = \lfloor \alpha\rfloor$, $p$ and $q$ are polynomials arising from such rational approximation. From this approximation we construct an approximate precision matrix for $u$.

If type is "operator", the approximation is based on a rational approximation of the fractional operator $(\kappa^2 -\Delta)^\beta$, where $\beta = (\nu + d/2)/2$. This results in an approximate model of the form $$P_l u(s) = P_r W,$$ where $P_j = p_j(L)$ are non-fractional operators defined in terms of polynomials $p_j$ for $j=l,r$. The order of $p_r$ is given by m and the order of $p_l$ is $m + m_\beta$ where $m_\beta$ is the integer part of $\beta$ if $\beta>1$ and $m_\beta = 1$ otherwise.

The discrete approximation can be written as $u = P_r x$ where $x \sim N(0,Q^{-1})$ and $Q = P_l^T C^{-1} P_l$. Note that the matrices $P_r$ and $Q$ may be be ill-conditioned for $m>1$. In this case, the methods in operator.operations() should be used for operations involving the matrices, since these methods are more numerically stable.

Examples

# Compute the covariance-based rational approximation of a
# Gaussian process with a Matern covariance function on R
kappa <- 10
sigma <- 1
nu <- 0.8
range <- sqrt(8 * nu) / kappa

# create mass and stiffness matrices for a FEM discretization
nobs <- 101
x <- seq(from = 0, to = 1, length.out = 101)
fem <- rSPDE.fem1d(x)

# compute rational approximation of covariance function at 0.5
op_cov <- matern.operators(
  loc_mesh = x, nu = nu,
  range = range, sigma = sigma, d = 1, m = 2,
  parameterization = "matern"
)

v <- t(rSPDE.A1d(x, 0.5))
# Compute the precision matrix
Q <- op_cov$Q
# A matrix here is the identity matrix
A <- Diagonal(nobs)
# We need to concatenate 3 A's since we are doing a covariance-based rational
# approximation of order 2
Abar <- cbind(A, A, A)
w <- rbind(v, v, v)
# The approximate covariance function:
c_cov.approx <- (Abar) %*% solve(Q, w)
c.true <- folded.matern.covariance.1d(
  rep(0.5, length(x)),
  abs(x), kappa, nu, sigma
)

# plot the result and compare with the true Matern covariance
plot(x, c.true,
  type = "l", ylab = "C(h)",
  xlab = "h", main = "Matern covariance and rational approximations"
)
lines(x, c_cov.approx, col = 2)



# Compute the operator-based rational approximation of a Gaussian
# process with a Matern covariance function on R
kappa <- 10
sigma <- 1
nu <- 0.8
range <- sqrt(8 * nu) / kappa

# create mass and stiffness matrices for a FEM discretization
x <- seq(from = 0, to = 1, length.out = 101)
fem <- rSPDE.fem1d(x)

# compute rational approximation of covariance function at 0.5
op <- matern.operators(
  range = range, sigma = sigma, nu = nu,
  loc_mesh = x, d = 1,
  type = "operator",
  parameterization = "matern"
)

v <- t(rSPDE.A1d(x, 0.5))
c.approx <- Sigma.mult(op, v)
c.true <- folded.matern.covariance.1d(
  rep(0.5, length(x)),
  abs(x), kappa, nu, sigma
)

# plot the result and compare with the true Matern covariance
plot(x, c.true,
  type = "l", ylab = "C(h)",
  xlab = "h", main = "Matern covariance and rational approximation"
)
lines(x, c.approx, col = 2)