# Rational approximations of stationary Gaussian Matern random fields

Source:`R/fractional.operators.R`

`matern.operators.Rd`

`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"),
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".

- 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.

## See also

`fractional.operators()`

,
`spde.matern.operators()`

,
`matern.operators()`

## 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)
```