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

The Original Article was published on 19 September 2019

## 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. 1.

Definition of c(xy) 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. 2.

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

3. 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. 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.

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Correspondence to David L. Miller.

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