On Information About Covariance Parameters in Gaussian Matérn Random Fields

Abstract

The Matérn family of covariance functions is currently the most commonly used for the analysis of geostatistical data due to its ability to describe different smoothness behaviors. Yet, in many applications, the smoothness parameter is set at an arbitrary value. This practice is due partly to computational challenges faced when attempting to estimate all covariance parameters and partly to unqualified claims in the literature stating that geostatistical data have little or no information about the smoothness parameter. This work critically investigates this claim and shows it is not true in general. Specifically, it is shown that the information the data have about the correlation parameters varies substantially depending on the true model and sampling design and, in particular, the information about the smoothness parameter can be large, in some cases larger than the information about the range parameter. In light of these findings, we suggest to reassess the aforementioned practice and instead establish inferences from data-based estimates of both range and smoothness parameters, especially for strongly dependent non-smooth processes observed on irregular sampling designs. A data set of daily rainfall totals is used to motivate the discussion and gauge this common practice.

Appendix

Derivation of Identities ( 4.6 ) and ( 4.7 )

To derive (4.6), write the Matérn correlation function as $K_{\varvec{\varvec{\vartheta }}}(r) = c(\nu ) b(\vartheta )^{\nu } {\mathcal {K}}_{\nu }(b(\vartheta ))$, where $c(\nu ) := {2^{1 - \nu }}/{\Gamma (\nu )}$ and $b(\vartheta ) := {2r\sqrt{\nu }}/{\vartheta }$. Then by direct differentiation

$$\begin{aligned} \frac{\partial }{\partial \vartheta } K_{\varvec{\vartheta }}(r)&= c(\nu ) \Big ( \nu b(\vartheta )^{\nu - 1} b'(\vartheta ) {\mathcal {K}}_{\nu }(b(\vartheta )) + b(\vartheta )^{\nu } \frac{\partial }{\partial x} {\mathcal {K}}_{\nu }(x) \Big |_{x=b(\vartheta )} \!\! \cdot b'(\vartheta ) \Big ) \\&= c(\nu ) b(\vartheta )^{\nu - 1} b'(\vartheta ) \Big ( \nu {\mathcal {K}}_{\nu }(b(\vartheta )) -b(\vartheta )\Big [ {\mathcal {K}}_{\nu - 1}(b(\vartheta )) + \frac{\nu \vartheta }{2\sqrt{\nu } r} {\mathcal {K}}_{\nu }(b(\vartheta )) \Big ] \Big ) \\&= -\frac{c(\nu ) (2\sqrt{\nu }r)^{\nu }}{\vartheta ^{\nu + 1}} \Bigg ( \nu {\mathcal {K}}_{\nu }\Big ( \frac{2\sqrt{\nu }}{\vartheta } r \Big ) -\frac{2\sqrt{\nu }}{\vartheta } r {\mathcal {K}}_{\nu - 1}\Big ( \frac{2\sqrt{\nu }}{\vartheta } r \Big ) -\nu {\mathcal {K}}_{\nu }\Big ( \frac{2\sqrt{\nu }}{\vartheta } r \Big ) \Bigg ) \\&= \frac{4 \nu ^{\frac{\nu + 1}{2}} r^{\nu +1}}{\Gamma (\nu )\vartheta ^{\nu + 2}} {\mathcal {K}}_{\nu - 1}\Big ( \frac{2\sqrt{\nu }}{\vartheta } r \Big ) , \end{aligned}$$

where the second identity follows from (4.3).

To derive (4.7), write the Matérn correlation function as

$$\begin{aligned} K_{\varvec{\vartheta }}(r) = 2 e^{\nu \log ( 1/2)} (\Gamma (\nu ))^{-1} e^{\nu \log ( \frac{2r}{\vartheta } \sqrt{\nu })} {\mathcal {K}}_{\nu }\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big ) , \end{aligned}$$

so after direct differentiation we have

$$\begin{aligned} \frac{\partial }{\partial \nu } K_{\varvec{\vartheta }}(r) = \left( \frac{1}{2} + \log \left( \frac{\sqrt{\nu }}{\vartheta } r\right) - \psi (\nu )\right) K_{\varvec{\vartheta }}(r) + h(\nu ) \frac{\partial }{\partial \nu } {\mathcal {K}}_{\nu }\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big ) . \end{aligned}$$

(6.1)

Now, let $G(x, y) := {\mathcal {K}}_{x}(y)$. Then

$$\begin{aligned} \frac{\partial }{\partial \nu } {\mathcal {K}}_{\nu }\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big )&= \frac{\partial }{\partial \nu } G\Big (\nu , \frac{2r}{\vartheta } \sqrt{\nu }\Big ) \nonumber \\&= \frac{\partial }{\partial x} G(x, y) \Big |_{x=\nu , y=\frac{2r}{\vartheta } \sqrt{\nu }} +\; \frac{\partial }{\partial y} G(x, y) \Big |_{x=\nu , y=\frac{2r}{\vartheta } \sqrt{\nu }} \cdot \frac{r}{\vartheta \sqrt{\nu }} \nonumber \\&= \int _{0}^{\infty } t \sinh (\nu t) e^{-\frac{2r}{\vartheta } \sqrt{\nu } \cosh (t)} dt \nonumber \\&\quad -\frac{r}{\vartheta \sqrt{\nu }} \Bigg ( {\mathcal {K}}_{\nu - 1}\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big ) + \frac{\vartheta \sqrt{\nu }}{2r} {\mathcal {K}}_{\nu }\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big ) \Bigg ) \nonumber \\&= \int _{0}^{\infty } t \sinh (\nu t) e^{-\frac{2r}{\vartheta } \sqrt{\nu } \cosh (t)} dt -\frac{r}{\vartheta \sqrt{\nu }} {\mathcal {K}}_{\nu - 1}\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big ) -\frac{1}{2} {\mathcal {K}}_{\nu }\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big ) , \end{aligned}$$

(6.2)

where the third identity follows from (4.3) and (4.5). Finally, replacing (6.2) into (6.1), we get

$$\begin{aligned} \frac{\partial }{\partial \nu } K_{\varvec{\vartheta }}(r)&= \frac{1}{2}K_{\varvec{\vartheta }}(r) + \left( \log \left( \frac{\sqrt{\nu }}{\vartheta } r\right) - \psi (\nu )\right) K_{\varvec{\vartheta }}(r) + h(\nu ) \int _{0}^{\infty } t \sinh (\nu t) e^{-\frac{2r}{\vartheta } \sqrt{\nu } \cosh (t)} dt \\&\quad - h(\nu ) \frac{r}{\vartheta \sqrt{\nu }} {\mathcal {K}}_{\nu - 1}\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big ) - \frac{1}{2} \underbrace{h(\nu ) {\mathcal {K}}_{\nu }\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big )}_{K_{\varvec{\vartheta }}(r)} \\&= \left( \log \left( \frac{\sqrt{\nu }}{\vartheta } r\right) - \psi (\nu )\right) K_{\varvec{\vartheta }}(r) \\&\quad - h(\nu ) \Big (\frac{r}{\vartheta \sqrt{\nu }} {\mathcal {K}}_{\nu - 1}\Big ( \frac{2r}{\vartheta } \sqrt{\nu } \Big ) - \int _{0}^{\infty } t \sinh (\nu t) e^{-\frac{2r}{\vartheta } \sqrt{\nu } \cosh (t)} dt\Big ) . \end{aligned}$$