# Information Criteria for Signal Extraction Using Singular Spectrum Analysis: White and Red Noise
## 3 Modifications of Information Criteria for the Case of Hankel Noise
### White Noise
#### 4.2.1 Sum of Two Sinusoids
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\sigma\) & TRMAT & SVD & MDL \\ \hline \(\sigma_{1}\) & 0.983 & 0.990 & 0.969 \\ \(\sigma_{2}\) & 0.235 & 0.277 & 0.480 \\ \(\sigma_{3}\) & 0.984 & 0.988 & 0.966 \\ \(\sigma_{4}\) & 0.992 & 0.994 & 0.974 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example ’sine’: proportion of ranks coinciding with the individual optimal ranks. Figure 1: Example ’sine’: The dependence of criteria \(\text{IC}(k)\) on rank \(k\) for one realization of noise. At plateaus, both methods yield accurate estimates of the rank. During transition periods, both methods underestimate the rank, but the MDL method does so to a lesser extent. The graphs of RMSE versus noise level are presented in Figure 4. We will now explain the specifics of the graph depicted in this figure. First, as a baseline, we consider the RMSE at the maximum possible rank, that is, when the signal estimate is the entire original series and the RMSE is equal to the root of the mean squared time series values. All errors depicted in the plot are presented on a relative scale, wherein they are divided by the baseline RMSE. Accordingly, the value of 1 corresponds to a signal estimate that is equal to the original series. In a sense, this represents the most unfavorable scenario. Figure 4 also presents the optimal case, which corresponds to the rank associated with the lowest average error. The lines exhibit minimal divergence at the plateau, indicating stable rank detection, while diverging at the transition periods. It is evident that the error of both criteria exceeds the minimum error at the transitions. However, the error of the MDL criterion is slightly smaller than that of TRMAT, which is consistent with the results presented in Figure 3.  
Let us include MGN in the consideration. When the criterion MGN is applied, we obtain an MGN signal estimate and use this estimate for calculating the MSE of the signal estimate; in particular, for finding the optimal ranks. Figure 3: Example ‘sine’: Dependence of rank estimation on white noise level, TRMAT and MDL. Figure 2: Example ‘sine’: The dependence of criteria \(\mathrm{IC}(k)\) on rank \(k\) for one realization of noise (noise level \(\sigma_{1}\) on the left, \(\sigma_{3}\) on the right). Figure 5, which depicts the mean of the rank estimates as a function of noise level, shows that MGN overestimates the rank at plateaus and thus is much closer to the optimal rank at transitions than the other criteria, which estimate the rank accurately at plateaus and underestimate at transitions. It should be noted that the optimal ranks depicted in Figure 5 may differ from those presented in Figure 3. This discrepancy arises from the fact that, in the MGN criterion, the estimate of the signal is obtained through the MGN method, rather than the SSA. A comparison of the two figures reveals that the optimal MGN ranks do not exhibit a transition range of noise levels, whereas the optimal SSA ranks display noise levels that correspond to an intermediate rank of 3. Consequently, it is possible to calculate both minimal mean squared errors (MSE) using the MGN estimates and minimal MSEs using the SSA estimates. The former are, naturally, smaller, given that the MGN estimates are obtained by the least-squares method. As a consequence of rank overestimation, the resulting RMSE for the MGN-estimated rank is observed to be larger at plateaus than that of the minimal MGN error. Therefore, at plateaus, the MGN criterion provides the same level of accuracy (Figure 6) as TRMAT, with smaller errors occurring at transitions. Figure 6 also presents a hybrid scenario in which the criterion is TRMAT and the signal estimation is conducted using MGN. This combination results in the smallest errors, as TRMAT determines rank with greater precision and MGN generates a more precise estimate of the signal.