Fitting Matérn Smoothness Parameters Using Automatic Differentiation

The Mat\'ern covariance function is ubiquitous in the application of Gaussian processes to spatial statistics and beyond. Perhaps the most important reason for this is that the smoothness parameter $\nu$ gives complete control over the mean-square differentiability of the process, which has significant implications about the behavior of estimated quantities such as interpolants and forecasts. Unfortunately, derivatives of the Mat\'ern covariance function with respect to $\nu$ require derivatives of the modified second-kind Bessel function $\mathcal{K}_\nu$ with respect to $\nu$. While closed form expressions of these derivatives do exist, they are prohibitively difficult and expensive to compute. For this reason, many software packages require fixing $\nu$ as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating $\nu$ use finite difference estimates for $\partial_\nu \mathcal{K}_\nu$. In this work, we introduce a new implementation of $\mathcal{K}_\nu$ that has been designed to provide derivatives via automatic differentiation, and whose resulting derivatives are significantly faster and more accurate than using finite differences. We provide comprehensive testing for both speed and accuracy and provide a motivating demonstration in which maximum likelihood estimation via second order optimization using finite difference approximations for derivatives with respect to $\nu$ gives completely incorrect answers.