Correction to: Scientific Reports https://doi.org/10.1038/s41598-025-09625-4, published online 3 July 2025 The original version of this Article contained errors in Eqs. 4, 5, 6, 11, 13, 14, 15, 16, 17, 21, 22 and 23, where these were partially not rendered correctly. As a result, Equation 4, $$\begin{aligned} \mathcalligra{\scriptstyle L}\hspace{000.1cm}_P(x;\theta ,\mathcalligra{\scriptstyle P}\hspace{000.1cm})=-\log_{EMPTY}{EMPTY}p(y=k|x,\varTheta ,\mathcalligra{\scriptstyle P}\hspace{000.1cm}).