# On the Quality of Deep Representations for Kepler Light Curves Using Variational Auto-Encoders
## 3 Experimental Setup and Results
### Results
#### 3.3.2 Quality Evaluation  
The reconstruction and smoothing scores for different methods and configurations are presented in Table 2. As expected, passband methods offer a very good denoising behavior, but fail to properly reconstruct the signal. For example, if we vary the filter on the passband, by specifically narrowing the pass frequency, the reconstruction error increases and becomes smoother (high Auto-C and low Diff-M). The same effect occurs for the moving average method when we increase the window size. Compared to the passband, the moving average method presents smaller reconstruction errors but producing a rougher time series (low Auto-C). The Mandel-Agol fit has a smaller reconstruction error compared to the passband and moving average methods, which is because it performs a specific and specialized problem fit (transit light curves). The simulation function does not consider factors of smoothness, as it models the ideal behavior of the transit objects. Nevertheless, the auto-correlation is still strong as in the previous methods, with the smallest difference on consecutive values. It is clear that the learning-based methods (auto-encoders), due to data learning, have the lowest reconstruction errors only comparable with a moving average with a window size of 3. Furthermore, the reconstructed data are still significantly smoother than those of the original time series (_denoising effect_). Between them, variational proposals have a greater auto-correlation (i.e. smoother) than their deterministic counterpart \(\text{RAE}_{t}\). Except for the Mandel-Agol fit and the narrowest passband, they have the smallest difference on consecutive values (Diff-M), showing smoother transitions in the local behavior of the time series. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Reconstruction**} & \multicolumn{2}{c}{**Denoising**} & \multicolumn{2}{c}{**Residual Noise**} \\
**Method** & **Time** & **RMSE \(\downarrow\)** & **MAE \(\downarrow\)** & **Auto-C \(\uparrow\)** & **Diff-M \(\downarrow\)** & **Spectral-H \(\uparrow\)** \\ \hline \multirow{2}{*}{\(\text{RAE}_{t}\)} & \(\times\) & 0.630 & 0.448 & 0.429 & 0.206 & 0.889 \\  & ✓ & 0.680 & 0.475 & 0.502 & 0.125 & 0.895 \\ \hline \multirow{2}{*}{\(\text{VRAE}_{t}\)} & \(\times\) & 0.689 & 0.480 & 0.559 & 0.074 & 0.900 \\  & ✓ & 0.688 & 0.484 & 0.594 & 0.068 & 0.901 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reconstruction and denoising results with the associated metrics. In both, compared methods are shown if the time information is used (as delta intervals \(\delta\)). \(\uparrow\) symbol means that a higher value is better on that task, while \(\downarrow\) means that a lower value is better. If we take a look at the spectral entropy of the residual, we clearly identify that learning-based methods exhibit the best scores. This means that these methods really focus on learning the time series patterns to perform the reconstruction instead of removing structure for achieving high smoothness as in the generic denoising methods presented. Please note that the spectral entropy has a reference value of 0.840 (i.e., on the original light curve) and its maximum value is 1.000 (i.e., a uniform distribution of frequencies). Figure 5 shows a few examples of reconstructed light curves for a better understanding of the previous results. The RAE\({}_{t}\) approach tends to keep the noise on the edges of the time series (in order to perform a good reconstruction), while our proposal (S-VRAE\({}_{t}\)) generates a smoother transit to the extent of being comparable with the M-A transit model, removing more high-frequency noises, as can be seen in the figure. The VRAE\({}_{t}\) model is not shown because visually it is quite similar to S-VRAE\({}_{t}\).