## 3 Results
### Variation of Parameters  
As the third test, a dataset representing the 2-D velocity field for the wake behind a cylinder at Reynolds number Re = 100 taken from [23] was used. The size of the mesh grid is \(449\times 199\). The dataset consists of 151 snapshots with regular time intervals of 0.2 s. Random Gaussian noise with a known standard deviation was added to both components. Two rectangular gaps were made \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Type of Gap**} & \multirow{2}{*}{**Region**} & \multicolumn{3}{c}{**Noise-Free**} & \multicolumn{3}{c}{**Noisy**} \\ \cline{3-7}  & & **csDMD** & **DMDct** & **Dataset** & **csDMD** & **DMDct** \\ \hline \multirow{2}{*}{None} & \multirow{2}{*}{whole} & \multirow{2}{*}{0.291} & \multirow{2}{*}{0.130} & 0.250 & 0.182 & 0.119 \\  & & & & (19.0 dB) & (21.8 dB) & (25.5 dB) \\ \hline \multirow{4}{*}{Rectangular} & inside & 0.408 & 0.199 & - & 0.231 & 0.185 \\  & outside & 0.316 & 0.142 & - & 0.168 & 0.126 \\ \cline{1-1}  & whole & 0.334 & 0.154 & 0.250 & 0.181 & 0.138 \\ \cline{1-1}  & & & (19.1 dB) & (21.8 dB) & (24.2 dB) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The summary of the error metrics for the first random sampling mask. The numbers given are RMSE values with the PVNR values in dB shown inside parenthesis when applicable. In all cases studied, DMDct resulted in lower reconstruction error than csDMD. Figure 6: The mean RMSE values of the two methods for the four test cases of the Duffing dataset. Each test case consisted of twenty different random sampling masks. The error bars show the standard deviations. A three-factor analysis of variance was conducted to determine whether the reconstruction error significantly changed with the three factors method, noise, gap, and their interaction. Tukey post hoc analysis showed the reconstruction error of DMDct was significantly lower than the error of csDMD in all cases (all \(p<0.001\)). in the dataset. The size of the first gap was \(60\times 70\) with the bottom-left and top-right corners at \((270,115)\) and \((329,184)\), respectively. The size of the second gap was \(46\times 46\) with the bottom-left corner at \((97,44)\) and the top-right corner at \((142,89)\). The aim of this test was to investigate the effect of changing various parameters on the quality of reconstruction. The parameters are noise standard deviation, sampling ratio, number of basis vectors, and number of dynamic modes. The nominal values of the parameters were chosen as noise standard deviation of 0.25, 2% sampling, \(s_{x}=67,s_{y}=30\), and five dynamic modes (according to SVHT). Although both \(u\) and \(v\) velocity components were used for analysis, only the results corresponding to the \(u\) component are presented here. Figure 7 shows four sample snapshots of the reference noise-free \(u\) velocity components, the reference with noise added, the random sample, reconstructions of csDMD and DMDct, and reconstruction errors for the nominal values of the parameters. Figure 8 shows the effects of the variation of parameters on PVNR values of csDMD and DMDct reconstructions per snapshot. In all sub-figures, the blue and red curves correspond to DMDct and csDMD results, respectively. The solid lines represent the results based on the nominal values. Figure 8a shows the effect of changing noise standard deviation. As the noise standard deviation increases, the PVNR values drop but in all snapshots, DMDct results in higher PVNR values than csDMD. The effect of changing the sampling ratio is shown in Figure 8b. As expected, increasing the sampling ratio results in higher PVNR values. Figure 8c shows the effect of taking different numbers of basis vectors. As the number of basis vectors increases, the PVNR values drop slightly. Finally, the effect of  
Figure 7: Four sample snapshots of reconstructing the noisy \(u\) velocity component of the wake behind a cylinder at Reynolds number \(\text{Re}=100\). The circular hollow represents the cylinder.