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Correction to: Multivariate Gaussian and Student-t process regression for multi-output prediction

The Original Article was published on 31 December 2019

Correction to: Neural Computing and Applications https://doi.org/10.1007/s00521-019-04687-8

Unfortunately, Theorem 3 was published with incorrect equations in the online publication of the article.

The correct theorem should read as follows:

Theorem 3

(Marginalization and conditional distribution) Let\( X\sim{\mathcal{M}\mathcal{N}}_{n,d} (M,\varSigma ,\Omega ) \)and partition\( X,M,\Sigma \)and\( \Omega \)as

where\( n_{1} ,n_{2} ,d_{1} ,d_{2} \)is the column or row length of the corresponding vector or matrix. Then,

  1. 1.

    \( X_{1r} \sim{\mathcal{M}\mathcal{N}}_{{n_{1} ,d}} \left( {M_{1r} ,\Sigma _{11} ,\Omega } \right) \),

    $$ X_{2r} |X_{1r} \sim{\mathcal{M}\mathcal{N}}_{{n_{2} ,d}} \left( {M_{2r} +\Sigma _{21}\Sigma _{11}^{ - 1} (X_{1r} - M_{1r} ),\Sigma _{22 \cdot 1} ,\Omega } \right); $$
  2. 2.

    \( X_{1c} \sim{\mathcal{M}\mathcal{N}}_{{n,d_{1} }} \left( {M_{1c} ,\Sigma ,\Omega _{11} } \right) \),

    $$ X_{2c} |X_{1c} \sim{\mathcal{M}\mathcal{N}}_{{n,d_{2} }} \left( {M_{2c} + (X_{1c} - M_{1c} )\Omega _{11}^{ - 1}\Omega _{12} ,\Sigma ,\Omega _{22 \cdot 1} } \right), \, $$

where\( \Sigma _{22 \cdot 1} \)and\( \Omega _{22 \cdot 1} \)are the Schur complement [30] of\( \Sigma _{11} \)and\( \Omega _{11} \), respectively,

$$ \Sigma _{22 \cdot 1} =\Sigma _{22} -\Sigma _{21}\Sigma _{11}^{ - 1}\Sigma _{12} ,\quad\Omega _{22 \cdot 1} =\Omega _{22} -\Omega _{21}\Omega _{11}^{ - 1}\Omega _{12} . \, $$

The original article has been updated accordingly.

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Correspondence to Zexun Chen.

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Chen, Z., Wang, B. & Gorban, A.N. Correction to: Multivariate Gaussian and Student-t process regression for multi-output prediction. Neural Comput & Applic 32, 11963 (2020). https://doi.org/10.1007/s00521-020-04774-1

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