# 'INLA' implementation of Whittle-Matérn fields for metric graphs

Source:`R/graph_spde.R`

`graph_spde.Rd`

This function creates an 'INLA' object that can be used in 'INLA' or 'inlabru' to fit Whittle-Matérn fields on metric graphs.

## Usage

```
graph_spde(
graph_object,
alpha = 1,
directional = FALSE,
stationary_endpoints = "all",
parameterization = c("matern", "spde"),
start_range = NULL,
prior_range = NULL,
start_kappa = NULL,
prior_kappa = NULL,
start_sigma = NULL,
prior_sigma = NULL,
start_tau = NULL,
prior_tau = NULL,
shared_lib = "detect",
debug = FALSE
)
```

## Arguments

- graph_object
A

`metric_graph`

object.- alpha
The order of the SPDE.

- directional
Should a directional model be used? Currently only implemented for

`alpha=1`

.- stationary_endpoints
Which vertices of degree 1 should contain stationary boundary conditions? Set to "all" for all vertices of degree 1, "none" for none of the vertices of degree 1, or pass the indices of the vertices of degree 1 for which stationary conditions are desired.

- parameterization
Which parameterization to be used? The options are 'matern' (sigma and range) and 'spde' (sigma and kappa).

- start_range
Starting value for range parameter.

- prior_range
a

`list`

containing the elements`meanlog`

and`sdlog`

, that is, the mean and standard deviation of the range parameter on the log scale. Will not be used if prior.kappa is non-null.- start_kappa
Starting value for kappa.

- prior_kappa
a

`list`

containing the elements`meanlog`

and`sdlog`

, that is, the mean and standard deviation of kappa on the log scale.- start_sigma
Starting value for sigma.

- prior_sigma
a

`list`

containing the elements`meanlog`

and`sdlog`

, that is, the mean and standard deviation of sigma on the log scale.- start_tau
Starting value for tau.

- prior_tau
a

`list`

containing the elements`meanlog`

and`sdlog`

, that is, the mean and standard deviation of tau on the log scale.Which shared lib to use for the cgeneric implementation? If "detect", it will check if the shared lib exists locally, in which case it will use it. Otherwise it will use 'INLA's shared library. If 'INLA', it will use the shared lib from 'INLA's installation. If 'rSPDE', then it will use the local installation of the rSPDE package (does not work if your installation is from CRAN). Otherwise, you can directly supply the path of the .so (or .dll) file.

- debug
Should debug be displayed?

## Details

This function is used to construct a Matern SPDE model on a metric graph. The latent field \(u\) is the solution of the SPDE $$(\kappa^2 - \Delta)^\alpha u = \sigma W,$$ where \(W\) is Gaussian white noise on the metric graph. This model implements exactly the cases in which \(\alpha = 1\) or \(\alpha = 2\). For a finite element approximation for general \(\alpha\) we refer the reader to the 'rSPDE' package and to the Whittle–Matérn fields with general smoothness vignette.

We also have the alternative parameterization \(\rho = \frac{\sqrt{8(\alpha-0.5)}}{\kappa}\), which can be interpreted as a range parameter.

Let \(\kappa_0\) and \(\sigma_0\) be the starting values for \(\kappa\) and \(\sigma\), we write \(\sigma = \exp\{\theta_1\}\) and \(\kappa = \exp\{\theta_2\}\). We assume priors on \(\theta_1\) and \(\theta_2\) to be normally distributed with mean, respectively, \(\log(\sigma_0)\) and \(\log(\kappa_0)\), and variance 10. Similarly, if we let \(\rho_0\) be the starting value for \(\rho\), then we write \(\rho = \exp\{\theta_2\}\) and assume a normal prior for \(\theta_2\), with mean \(\log(\rho_0)\) and variance 10.