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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to David L. Miller.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Miller, D.L., Glennie, R. & Seaton, A.E. Correction to: Understanding the Stochastic Partial Differential Equation Approach to Smoothing. JABES 25, 276 (2020). https://doi.org/10.1007/s13253-020-00383-6

Download citation