## 4 Discussion  
The three tests performed aimed at comparing DMDct vs csDMD in terms of both dynamic mode shape reconstruction and fully-sampled dataset reconstruction based on a sparsely-sampled dataset. While csDMD is developed to reconstruct the mode shapes, DMDct reconstructs the fully-sampled dataset. To use csDMD for fully-sampled dataset reconstruction, the spDMD method was incorporated to find the optimal amplitudes of DMD modes. The first test showed both methods reconstructed the mode shapes almost identical to the ones resulting from applying exact DMD on the fully-sampled dataset even though a very small set of basis vectors was used. Both methods resulted in RMSE of 0.002 in reconstructing the fully-sampled dataset. Figure 8: Investigating the effect of changing various parameters on per-snapshot PVNR values of DMDct and csDMD reconstructions of \(u\) velocity component of the wake behind a cylinder at Reynolds number \(\text{Re}=100\). Blue and red, respectively, represent DMDct and csDMD. The solid lines correspond to the nominal values. (**a**) The effect of variation of noise standard deviation. (**b**) The effect of variation of sampling ratio. (**c**) The effect of changing the number of basis vectors. (**d**) The effect changing the number of dynamic modes. The lines corresponding to 10 dynamic modes are overlapped by the lines corresponding to 5 modes most of the times. These results show neither method outperforms the other in dealing with the test dataset which has a few dynamic modes. The second test consisted of four cases. In the first case, where there is no gap in the data and the samples are noise-free, csDMD reconstruction shows some glitches, especially in the first snapshot, whereas the DMDct reconstruction has much fewer glitches (Figure 5a). The glitches reduce as the time goes on which is probably due to the high decay rate of the corresponding modes. As depicted in Figure 4b, the amplitudes of about half of the modes reduce to 10% or less of their initial values after 20 snapshots which means the corresponding modes die out quickly. In the third and fourth cases, where there is a rectangular gap in the data, DMDct has resulted in less reconstruction error than csDMD in both noisy and noise-free cases. Obviously, the RMSE values are higher compared to those of the no-gap case. Visually comparing, both methods were able to fill the rectangular gap but DMDct seems to have resulted in a smoother and more consistent filling than csDMD. This is also confirmed numerically for the first sampling mask through the RMSE values listed in row \"_inside_\" of Table 1.  
As the statistical analysis showed, the RMSE values of DMDct reconstruction are significantly lower than those of csDMD. The post hoc analysis also showed the noise has no significant effect on the error of DMDct. This means DMDct is robust with respect to the noise. The glitches in the noisy reconstruction of csDMD seem to be less than the noise-free case, which is probably due to the smaller number of DMD modes taken (5 vs. 25) and the fewer basis vectors used (20 vs. 41). It is also seen DMDct has resulted in more reconstruction error for the noisy cases than the noise-free cases which is as expected, but the reconstruction errors of csDMD for the noisy cases are less than those of the noise-free cases, which indicates csDMD is more sensitive to the number of mode shapes and basis vectors than DMDct. As stated earlier, the noisy reconstructions were performed using a fewer number of DMD modes and basis functions than the noise-free ones. Comparing the RMSE values in Figure 6 reveals DMDct resulted in less changes in the RMSE values compared to csDMD. In addition, the standard deviation of DMDct results is much lower than csDMD's according to the error bars in Figure 6. Thus, DMDct is more robust than csDMD. The third test showed the effect of changing the values of various parameters on the PVNR values of DMDct and csDMD reconstructions. The first parameter to investigate was the standard deviation of the random Gaussian noise.