Correction to: Scientific Reports https://doi.org/10.1038/s41598-021-92661-7, published online 05 July 2021


The original version of this Article contained errors in Equation 37, where \({n}_{o}\) was incorrectly given as \(n\).

$${{\varvec{R}}{\varvec{M}}{\varvec{S}}{\varvec{E}}}_{{\varvec{c}}{\varvec{o}}{\varvec{n}}{\varvec{t}}{\varvec{r}}{\varvec{o}}{\varvec{l}}\boldsymbol{ }}=\frac{1}{{\varvec{M}}}\sum_{{\varvec{M}}}\sum_{{\varvec{i}}=1:{\varvec{n}}}\frac{{({{\varvec{d}}}_{{\varvec{i}}}\boldsymbol{ }-{{{\varvec{g}}}_{2}({\varvec{h}}}_{{\varvec{i}}}^{{\varvec{s}}}))}^{2}}{{\varvec{n}}}$$

now reads:


$${{\varvec{R}}{\varvec{M}}{\varvec{S}}{\varvec{E}}}_{{\varvec{c}}{\varvec{o}}{\varvec{n}}{\varvec{t}}{\varvec{r}}{\varvec{o}}{\varvec{l}}\boldsymbol{ }}=\frac{1}{{\varvec{M}}}\sum_{{\varvec{M}}}\sum_{{\varvec{i}}=1:{n}_{o}}\frac{{({{\varvec{d}}}_{{\varvec{i}}}\boldsymbol{ }-{{{\varvec{g}}}_{2}({\varvec{h}}}_{{\varvec{i}}}^{{\varvec{s}}}))}^{2}}{{n}_{o}}$$

As a result, in the sentence preceding Equation 37,


“Given a test sample size of M data points, the prediction error was calculated in terms of the root mean squared error (RMSE) between the desired output \({d}_{i}\) and the predicted output \({g(h}_{i}^{s}),\) \(1<i<n,\) where \(n\) is the number of neurons in the output layer.”


now reads:


“Given a test sample size of M data points, the prediction error was calculated in terms of the root mean squared error (RMSE) between the desired output \({d}_{i}\) and the predicted output \({g(h}_{i}^{s}),\) \(1<i<{n}_{o},\) where \({n}_{o}\) is the number of neurons in the output layer.”


Furthermore, Equation 40 contained errors


$${\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}}_{{\varvec{k}}}=\frac{1}{{\varvec{M}}}\sum_{{\varvec{M}}}\sum_{{\varvec{i}}=1:{\varvec{I}}}\frac{{({{\varvec{d}}}_{{\varvec{i}}}\boldsymbol{ }-{{\varvec{V}}}_{{\varvec{j}}}^{{\varvec{k}}})}^{2}}{{\varvec{I}}}$$

now reads:


$${\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}}_{{\varvec{k}}}=\frac{1}{{\varvec{M}}}\sum_{{\varvec{M}}}\sum_{{\varvec{i}}=1:{n}_{o}}\frac{{({{\varvec{d}}}_{{\varvec{i}}}\boldsymbol{ }-{{{\varvec{g}}}_{2}({\varvec{h}}}_{{\varvec{i}}}^{{\varvec{s}}}))}^{2}}{{n}_{o}}$$

As a result, in the sentence preceding Equation 40,


“The root mean squared error obtained by shutting off \({k}^{th}\) neuron (\({RMSE}_{k})\) was calculated as below”


now reads:


“The root mean squared error obtained by shutting off \({k}^{th}\) neuron (\({RMSE}_{k})\) was estimated as below by using \({V}_{j}^{k}\) to estimate the new \({h}_{i}^{s}\) (see Eq. 10).”


The original Article has been corrected.