## 5 Numerical Examples
### Model Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **LSEs** & **MSE** & **AIC** & **PRR** & **PP** & **SAE** & \(\mathbf{R^{2}}\) \\ \hline GOM & \(\hat{\alpha}=197.387\), \(\hat{b}=0.399\) & 80.678 & 184.331 & 0.170 & 0.101 & 104.403 & 0.939 \\ DSM & \(\hat{\alpha}=192.528\), \(\hat{b}=0.882\) & 232.628 & 331.857 & 1.291 & 0.333 & 142.544 & 0.823 \\ ISM & \(\hat{\alpha}=197.354\), \(\hat{b}=0.399\) & 86.440 & 186.334 & 0.171 & 0.101 & 104.370 & 0.939 \\  & \(\hat{\beta}=0.00001\) & & & & & & \\ YIDM & \(\hat{\alpha}=182.934\), \(\hat{b}=0.464\) & 78.837 & 157.825 & 0.128 & 0.087 & 100.617 & 0.944 \\  & \(\hat{\alpha}=0.0071\) & & & & & & \\ PNZM & \(\hat{\alpha}=183.124\), \(\hat{b}=0.463\) & 84.902 & 159.873 & 0.128 & 0.087 & 100.608 & 0.944 \\  & \(\hat{\alpha}=195.990\), \(\hat{b}=0.3987\) & 100.989 & 190.332 & 0.172 & 0.102 & 104.354 & 0.939 \\  & \(\hat{\alpha}=1000.00\), \(\hat{\beta}=0.00\), \(\hat{\alpha}=1.390\) & & & & & \\ DPM & \(\hat{\alpha}=26124.0\), \(\hat{\gamma}=0.0044\) & 769.282 & 480.341 & 0.415 & 0.712 & 334.128 & 0.494 \\  & \(\hat{\alpha}=0.0053\), \(\hat{b}=0.774\), & & & & & & \\  & \(\hat{\alpha}=181.0\), \(\hat{\beta}=38.6\), \(\hat{\rm N}=204.1\) & 72.283 & 158.933 & 0.052 & 0.048 & 103.196 & 0.956 \\  & \(\hat{\alpha}=0.028\), \(\hat{b}=0.210\) & & & & & & \\
3 PFDM & \(\hat{\alpha}=9.924\), \(\hat{\beta}=0.005\) & 81.090 & 163.797 & 0.073 & 0.061 & 106.341 & 0.951 \\  & \(\hat{\rm N}=206.387\) & & & & & & \\ New & \(\hat{\alpha}=0.008\), \(\hat{\alpha}=0.275\), & 60.623 & 151.156 & 0.043 & 0.041 & 98.705 & 0.960 \\ Model & \(\hat{\beta}=0.001\), \(\hat{\rm N}=207.873\) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model parameter estimation and comparison criteria from MRS data set (DS2). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **LSEs** & **MSE** & **AIC** & **PRR** & **PP** & **SAE** & \(\mathbf{R^{2}}\) \\ \hline GOM & \(\hat{\alpha}=136.050\), \(\hat{b}=0.138\) & 33.822 & 121.878 & 0.479 & 0.262 & 118.530 & 0.972 \\ DSM & \(\hat{\alpha}=124.665\), \(\hat{b}=0.356\) & 134.582 & 210.287 & 12.787 & 1.181 & 239.335 & 0.889 \\ ISM & \(\hat{\alpha}=136.050\), \(\hat{b}=0.138\) & 35.363 & 123.878 & 0.479 & 0.262 & 118.532 & 0.972 \\  & \(\hat{\beta}=0.0001\) & & & & & & \\ YIDM & \(\hat{\alpha}=81.252\), \(\hat{b}=0.340\) & 9.435 & 116.403 & 0.035 & 0.031 & 60.842 & 0.993 \\  & \(\hat{\alpha}=0.033\) & 9.888 & 118.388 & 0.037 & 0.032 & 60.877 & 0.993 \\ PNZM & \(\hat{\alpha}=0.01\), \(\hat{b}=0.138\) & & & & & & \\  & \(\hat{\alpha}=800.0\), \(\hat{b}=0.00\), \(\hat{c}=136.04\) & 38.995 & 127.878 & 0.479 & 0.262 & 118.530 & 0.972 \\  & \(\hat{\alpha}=28650\), \(\hat{\beta}=0.003\) & & & & & & \\  & \(\hat{\alpha}=0.000\), \(\hat{\alpha}=71.8\) & 274.911 & 382.143 & 0.857 & 3.568 & 304.212 & 0.792 \\ TCM & \(\hat{\alpha}=0.00033\), \(\hat{b}=0.734\), & & & & & & \\  & \(\hat{\alpha}=0.29\), \(\hat{\beta}=0.002\), \(\hat{\rm N}=427\) & 7.640 & 116.932 & 0.019 & 0.019 & 47.304 & 0.995 \\  & \(\hat{\alpha}=1.696\), \(\hat{\rm N}=0.001\) & & & & & & \\
3 PFDM & \(\hat{\alpha}=1.696\), \(\hat{\rm N}=1.574\) & 17.827 & 119.523 & 0.137 & 0.100 & 81.313 & 0.987 \\ New & \(\hat{\alpha}=0.277\), \(\hat{\alpha}=0.328\) & & & & & & \\ Model & \(\hat{\beta}=17.839\), \(\hat{\rm N}=228.909\) & 7.361 & 114.982 & 0.022 & 0.022 & 47.869 & 0.994 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Model parameter estimation and comparison criteria from RTC&CS data set (DS1). Least-squares estimate (LSE); mean squared error; Akaike’s information criterion (AIC); predictive ratio risk (PRR); predictive power (PP), sum absolute error (SAE), correlation index of the regression curve equation (\(\text{R}^{2}\) ).