# 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
#### 4.2.2 Logarithmic Signal  
As an example of a signal that is not low-rank, consider the signal in the form of a logarithmic series:  
\[s_{n}=\log n. \tag{14}\]  
As the signal is not low-rank, it is not possible to determine a proper rank; however, the selection of an appropriate model order can be discussed. Figure 7 depicts the mean estimated ranks and the optimal ranks (model orders) for the smallest root of the average mean squared errors (MSEs) over 1000 realizations. As the noise level increases, the optimal rank values decrease from four to zero. The plateaus are relatively short, and there are significantly more transition regions than in the previous example involving a finite-rank signal. Figure 8 illustrates the dependence of the root mean square error (RMSE) on the noise level.  
In this example, we will consider the same noise levels as previously described in Section 4.2.1. In contrast with the previous example, the noise level \(\sigma_{1}\) falls on the transition period between ranks 3 and 2, \(\sigma_{2}\) is almost on the plateau (rank 1), \(\sigma_{3}\) lies exactly on the plateau (rank 1), and for \(\sigma_{4}\), the signal is not detected.  
Table 2 correlates with this description of noise levels. In the first row, the accuracy of the criteria is generally less precise; in the second row, it is more precise and in the third and fourth rows, it is highly precise. It can be seen that the TRMAT method produces the best result, providing a good match of ranks on transition periods as well.  
Figure 6: Example ‘sine’: Dependence of relative RMSE on white noise level, TRMAT and MGN.  
In the preceding example, the TRMAT and MDL criteria exhibited comparable accuracy, with MDL demonstrating a slight advantage at transition sections in the noise levels. In the case of the considered signal that is not low-rank, the MDL criterion exhibited instability, while the TRMAT criterion demonstrated a notable advantage. These observations are illustrated in Table 2 and Figure 8.  
\begin{table}
\begin{tabular}{c c c c} \hline \(\sigma\) & **TRMAT** & **SVD** & **MDL** \\ \hline \(\sigma_{1}\) & 0.744 & 0.214 & 0.206 \\ \(\sigma_{2}\) & 0.751 & 0.603 & 0.589 \\ \(\sigma_{3}\) & 0.983 & 0.983 & 0.976 \\ \(\sigma_{4}\) & 0.997 & 0.998 & 0.978 \\ \hline \end{tabular}
\end{table}
Table 2: Example ‘log’: proportion of ranks coinciding with the individual optimal ranks.  
Figure 8: Example ‘log’: Dependence of relative RMSE on white noise level, TRMAT and MDL.  
Figure 7: Example ‘log’: Dependence of rank estimation on white noise level, TRMAT and MDL.  
The MGN criterion is now to be incorporated into the comparison (see Figures 9 and 10). One can see that the error lines for TRMAT and MGN are intertwined, indicating that there is no clear advantage of one criterion over the other. In this instance, the combination of the TRMAT criterion for rank detection and the MGN signal estimation with the obtained rank does not result in any improvement and thus is not depicted in the graph.