# A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model
## 3 Simulated Data Analysis
### Five-DOF System
Figure 5: Monte Carlo simulation results for the 5 th identified natural frequency (**a**) and damping ratio (**b**): exact quantity (solid black line), LSCF estimates (blue dots) and MLE-PMM estimates after 10 iterations (red dots). Figure 3: Element(1,1) of the FRF matrix contaminated with 10% noise: exact (black line) and noisy (red line) FRF, noise standard deviation (green line) and exact natural frequencies (vertical lines) (**a**); and stabilization diagram (with ‘\(\circ\)’ denoting a pole with stable frequency and damping and ‘\(+\)’ a pole with stable frequency only) constructed with the LSCF estimator by identifying common denominator polynomial models with order ranging from 1 to 50 (**b**). Analyzing these figures, one verifies that, while no changes can be visually perceived for the estimated natural frequencies with regard to their exact counterparts, a significant improvement in accuracy of the damping ratio estimates after the optimization with MLE-PMM is clearly visible. This conclusion is corroborated by the results presented in Table 4 where the results obtained from the Monte Carlo simulations are compared in terms of mean relative bias error. One of the main advantages of the MLE-PMM approach herein proposed with regard to the optimization technique described in Reference [19] is that the former takes into account not only the FRFs but also the noise information as primary data and, therefore, apart from the modal parameter estimates, it also predicts the uncertainties on theses estimates. In order to illustrate this capability, the standard deviations of the natural frequencies and damping ratios predicted after 10 iterations of the proposed MLE-PMM for the 3 rd and 5 th modes in each Monte Carlo run are compared to the respective sample standard deviations in Figures 6 and 7. In these figures, the results obtained with the proposed MLE-PMM are also compared to those from the MLE-MM described in Reference [14]. Similarly to the MLE-PMM, the optimization of the pLSCF estimates with the MLE-MM was also performed with 10 iterations in each Monte Carlo run. It is clear from Figures 6 and 7 that the standard deviations provided by the proposed MLE-PMM are in very good agreement both with their corresponding sample standard deviations and with standard deviation obtained with MLE-MM. The estimates for the damping ratios obtained after 10 iterations of the MLE-PMM in each Monte Carlo run are plotted as a function of the corresponding natural frequency estimates in Figure 8. Also, in this figure, the 'classical' 95% confidence ellipses calculated both with the sample covariance and the mean covariance computed from the estimates obtained with Equation (26) over 5000 Monte Carlo runs are shown. It is clearly seen from such figure that not only the shape of the uncertainty regions defined by the damping ratio and natural frequency estimates but also the estimated 95% confidence ellipses are in good agreement with the corresponding sample covariance ellipses. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**LSCF Estimates**} & \multicolumn{3}{c}{**MLE-PMM Estimates (10 Iterations)**} \\ \cline{2-9}
**Mode** & \(\mu_{f_{\pi}}\) & **Rel.Bias** & \(\mu_{g_{\pi}}\) & **Rel.Bias** & \(\mu_{f_{\pi}}\) & **Rel.Bias** & \(\mu_{g_{\pi}}\) & **Rel.Bias** \\  & **(Hi2)** & **(\% \(\times\)103)** & **(\%)** & **(\% \(\times\)102)** & **(Hz)** & **(\% \(\times\)103)** & **(\%)** & **(\% \(\times\)102)** \\ \hline
1 & 26.0603 & 1.129 & 2.0449 & 2.246 & 26.0600 & 0.152 & 1.9998 & 0.011 \\
2 & 36.8404 & 1.131 & 2.0317 & 1.585 & 36.8398 & 0.425 & 1.9999 & 0.004 \\
3 & 51.4715 & 2.927 & 2.0685 & 3.423 & 51.4702 & 0.337 & 2.0005 & 0.025 \\
4 & 56.2125 & 4.378 & 2.0274 & 1.370 & 56.2100 & 0.042 & 1.9999 & 0.004 \\
5 & 62.6006 & 0.934 & 2.0275 & 1.373 & 62.6001 & 0.216 & 2.0000 & 0.002 \\ \hline \hline \end{tabular}
\end{table}