Fitting Matérn smoothness parameters using automatic differentiation
Fitting Matérn smoothness parameters using automatic differentiation
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使用自动微分拟合 Matérn 平滑度参数
DOI：
10.1007/s11222-022-10127-w
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发表时间：
2022
影响因子：
2.2
通讯作者：
Michael L. Stein
中科院分区：
文献类型：
--
作者：
Christopher Geoga;Oana Marin;Michel Schanen;Michael L. Stein
Matérn协方差函数在空间统计和空间统计的高斯过程应用中是普遍存在的。也许最重要的原因是，光滑度参数$$\nu$$ν完全控制了过程的均方可微性，这对估计量的行为(如插值法和预测值)具有重要影响。不幸的是，Matérn协方差函数关于$$\nu$$ν的导数需要修改的第二类贝塞尔函数$${\mathcal{K}}_{\nu}$$Kν关于$$\nu$$ν的导数。虽然这些导数确实存在封闭形式的表达式，但它们的计算难度和成本都高得令人望而却步。因此，许多软件包需要修复$$\nu$$ν而不是估算它，并且尝试提供估算$$\nu$$ν功能的所有现有软件包都使用对$$\Partial_\nu{\Mathcal{K}}_{\nu}$$∂νKν的有限差分估算。在这项工作中，我们介绍了一个新的实现$${\Mathcal{K}}_{\nu}$$Kν，它被设计成通过自动微分(AD)来提供导数，并且其导数比使用有限差分计算的导数更快和更准确。我们对速度和精度进行了全面的测试，并表明我们的AD解决方案可以用来构建准确的海森矩阵，用于在用有限差分近似构建海森矩阵完全失败的情况下进行二阶最大似然估计。
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 $$\nu $$ 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 $$\nu $$ ν require derivatives of the modified second-kind Bessel function $${\mathcal {K}}_{\nu }$$ K ν with respect to $$\nu $$ 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 $$\nu $$ ν as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating $$\nu $$ ν use finite difference estimates for $$\partial _\nu {\mathcal {K}}_{\nu }$$ ∂ ν K ν . In this work, we introduce a new implementation of $${\mathcal {K}}_{\nu }$$ 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.
DOI：
10.1007/s13253-018-00348-w
发表时间：
2019-09-01
影响因子：
1.4
作者：
Heaton, Matthew J.;Datta, Abhirup;Zammit-Mangion, Andrew
通讯作者：
Zammit-Mangion, Andrew