用自动微分法拟合材料光滑度参数
Fitting Matérn Smoothness Parameters Using Automatic Differentiation
摘要：
Matérn协方差函数在高斯过程应用于空间统计学及其他领域中是普遍存在的。也许最重要的原因是平滑参数$ν$完全控制了过程的均方可微性，这对估计量（如插值和预测）的行为具有重要意义。不幸的是，Matérn协方差函数关于$ν$的导数需要修改的第二类贝塞尔函数$\mathcal{K}_ν$关于$ν$的导数。虽然这些导数的封闭形式表达式确实存在，但它们的计算非常困难和昂贵。由于这个原因，许多软件包需要修复$ν$而不是估计它，并且所有试图提供估计$ν$的功能的现有软件包都使用$\partial_ν\mathcal{K}_ν$的有限差分估计。在这项工作中，我们介绍了一个新的实施$\mathcal{K}_ν$已被设计为通过自动微分（AD），并提供衍生物，其产生的衍生物是显着更快，更准确地比那些使用有限差分计算。我们提供了全面的测试速度和准确性，并表明我们的AD解决方案可用于建立准确的海森矩阵的二阶极大似然估计的设置，其中海森建立有限差分近似完全失败。
英文摘要：
The Matérn 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 $ν$ gives complete control over the mean-square differentiability of the process, which has significant implications for the behavior of estimated quantities such as interpolants and forecasts. Unfortunately, derivatives of the Matérn covariance function with respect to $ν$ require derivatives of the modified second-kind Bessel function $\mathcal{K}_ν$ with respect to $ν$. 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 $ν$ as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating $ν$ use finite difference estimates for $\partial_ν\mathcal{K}_ν$. In this work, we introduce a new implementation of $\mathcal{K}_ν$ that has been designed to provide derivatives via automatic differentiation (AD), and whose resulting derivatives are significantly faster and more accurate than those computed using finite differences. We provide comprehensive testing for both speed and accuracy and show that our AD solution can be used to build accurate Hessian matrices for second-order maximum likelihood estimation in settings where Hessians built with finite difference approximations completely fail.
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