Correction to: Understanding the Stochastic Partial Differential Equation Approach to Smoothing

1 Correction to: JABES https://doi.org/10.1007/s13253-019-00377-z

Unfortunately, in the original publication of the article, several definitional mistakes crept into the manuscript during writing. A complete, corrected version of the manuscript can be found at https://arxiv.org/abs/2001.07623. The results of the paper are unchanged, but the errors and their corrections are listed below.

• 1. Definition of c(x, y) in Sect. 2.3. The Matérn formulation should be

  $$\begin{aligned} c(x, y) = \dfrac{2^{1-\nu }}{(4\pi )^{d/2}\kappa ^{2\nu }\tau ^2\Gamma (\nu + d/2)} (\kappa \Vert x - y\Vert )^{\nu }K_{\nu }(\kappa \Vert x - y\Vert ) \end{aligned}$$

• 2. At end of second paragraph in Sect. 2.3 should be $\alpha = \nu + d/2$.

• 3. In Sect. 3.1, the thin plate penalties should have squared integrands, i.e., $J(\varvec{\beta }, \lambda ) = \lambda \int \left( \partial ^2 f / \partial x^2\right) ^2 + 2\left( \partial ^2 f / \partial x \partial y\right) ^2 + \left( \partial ^2 f / \partial y^2\right) ^2 \mathrm {d}x\mathrm {d}y$.

• 4. In Sect. 3.3, in paragraph 2, the expression should be $\langle Df, Df \rangle = \tau ^2(\kappa ^4 \langle f, f \rangle + 2\kappa ^2 \langle \nabla f, \nabla f \rangle + \langle \Delta f, \Delta f \rangle )$. Following on from that, the definitions of $\varvec{G}_1, \varvec{G}_2$ should be $\langle \nabla \psi _i, \nabla \psi _j \rangle$ and $\langle \Delta \psi _i, \Delta \psi _j \rangle$, respectively.