Algorithm 2 Monte Carlo sampling for regression analysis.

for$k=1:N$do ${\overline{x}}_{\mathrm{sampled}}=\overline{x}+{\overline{\sigma }}_x;R\sim \mathcal{N}(0,1)$ ${\overline{y}}_{\mathrm{sampled}}=\overline{y}+{\overline{\sigma }}_y;R\sim \mathcal{N}(0,1)$ $\widehat{{\mathit{P}}_{\mathit{k}}}=\underset{{\mathit{P}}_{\mathit{k}}}{arg\; min}S\left({\mathit{P}}_{\mathit{k}}\right),$ ${\widehat{y}}_k=f({\overline{x}}_{\mathrm{sampled}},\widehat{{\mathit{P}}_{\mathit{k}}})$ end for $co{v}_{\mathit{P}}=cov\left({\mathit{P}}_{\mathbf{k}(\mathbf{1}:\mathbf{N})}\right)$ ${\overline{\sigma }}_{\mathit{P}}=\sqrt{\mathrm{diag}(co{v}_{\mathit{P})}}$ ${\overline{\sigma }}_{\widehat{y}}=\frac{{\sum }_{k=1}^N({\widehat{y}}_k-\overline{y}{)}^2}{N-1}$