Correction: Souza et al. A Regularized Mixture of Linear Experts for Quality Prediction in Multimode and Multiphase Industrial Processes. Appl. Sci. 2021, 11, 2040

1. Text Correction

We, the authors, wish to make the following corrections to our paper [1]. We have found that Equations (16), (21) and (22) were stated wrong in the manuscript. These equations relate to the BIC criteria for parameters and model selection.

In Equation (16), the BIC for the selection of ${\lambda }_p^e$ for the expert p should be replaced by the correct format

$${\mathrm{BIC}}^e({\lambda }_p^e,\alpha )=n_p^e\log \left(\sum _{i=1}^n{\gamma }_{pi}^t{\left(y_i-{\mathbf{x}}_i^T{\mathit{\theta }}_p^{t+1}\right)}^2\right)+\log \left(n_p^e\right){\psi }_p^e,$$

Similarly, in Equation (21), the BIC for the selection of ${\lambda }_p^g$ for the gate p, should be replaced by the correct format

$${\mathrm{BIC}}^g({\lambda }_p^g,\alpha )=n_p^g\log \left(\sum _{i=1}^n{r}_{pi}{\left(z_{pi}-{\mathbf{x}}_i^T{\mathit{v}}_p^k\right)}^2\right)+\log \left(n_p^g\right){\psi }_p^g,$$

The BIC criteria for the model selection, stated in Equation (22) should be replaced by the correct format

$$\mathrm{BIC}\left(\mathsf{\Omega }\right)=m\log \left(\frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{m}\right)+\log \left(m\right)\sum _{p=1}^P\left({\psi }_p^e+{\psi }_p^g\right).$$

We also state that there is no impact on the experimental results, as we ran the experiments with the correct equations

The authors apologize for any inconvenience caused and state that the scientific conclusions are unaffected. The original article has been updated.