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For geostatistics problems, I am used to working with the following parametrization for the Matérn covariance function.

For a stationary and isotropic Gaussian random field $X(\boldsymbol{s})$, $\boldsymbol{s} \in \mathbb{R}^d$, we have

\begin{align*} \text{Cov}\left[X(\boldsymbol{s}_1), X(\boldsymbol{s}_2)\right] &= r(||h||), \text{ such that } ||h|| \text{ is the Euclidian distance between } \boldsymbol{s}_1 \text{ and } \boldsymbol{s}_2\\ &=\frac{\sigma^2}{2^{\nu - 1}\Gamma(\nu)} (\kappa \cdot ||h||)^{\nu} K_{\nu}(\kappa \cdot ||h||) \end{align*} where $K_{\nu}(\cdot)$ is a modified Bessel functions of the $2^{\text{nd}}$ order. Additionally,

  • $r(0) = \sigma^2$ is the variance.
  • $\nu > 0$ is the smoothness parameter (a Gaussian process with this covariance function is $\lceil\nu\rceil - 1$ differentiable).
  • $\kappa > 0$ determines the practical correlation range; in particular, $\rho = \frac{\sqrt{8\nu}}{\kappa}$ is the distance at which the spatial correlation is close to $0.1$.

However, using the RandomFields in R, according to its documentation (p.337), the RMmatern() uses the following parameterizationn: \begin{align*} r(||h||) = \frac{1}{2^{\nu-1}\Gamma(\nu)}(\sqrt{2\nu}\cdot||h||)^{\nu}K_{\nu}(\sqrt{2\nu}\cdot||h||), \end{align*} where

  • $\nu > 0$ is said to be the smoothness parameter (no problem here).

Moreover, the function has the following arguments var, scale, ....

Regarding this parametrization, I have two questions:

  1. What does the scale parameter mean? How can I relate it to my original parameterization?
  2. Is the parameter var the same as $\sigma^2$ in my original parameterization?
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There's a hyperlink in the explanation to the documentation for RMmodel, where var and scale parameters are explained. If $r=\phi(h)$ is the correlation model, the covariance model is $$C = \mathtt{var}\times\phi(h/\mathtt{scale})$$ That is, var is $\sigma^2$ and scale is $\sqrt{2\nu}/\kappa$

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    $\begingroup$ The general principle is OK but the expression for scale is ambiguous since it depends on the parameterisation, the OP having two parameterisations involving a parameter $\nu$. This is for the first parameterisation. $\endgroup$ – Yves Apr 15 at 7:09

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