# 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  
Let us start with a simple example, a signal in the form of the sum of two sinusoids:  
\[s_{n}=10\sin(2\pi n/20)+\sin(2\pi n/10). \tag{13}\]  
The rank of the signal (13) is 4 [Example 5.2] in [3]. Since deterministic signals are asymptotically separated from noise [Section 6.1.3] in [3], the optimal value of \(r\) will be 4. However, as the noise level increases, the second sinusoid will start to mix with the noise and after some period of uncertainty, the optimal rank will become equal to 2, the rank of one sinusoid. It is clear that for any signal, as the noise increases, at some noise level the optimal rank becomes zero, i.e., the best estimate of the signal is the zero series. We will consider 20 values of \(\sigma\) from 0.01 to 100 with equal logarithmic steps. Separately, we will focus on four noise levels, \(\sigma_{1}=0.1128838\) (optimal rank 4), \(\sigma_{2}=1.274275\) (transition period from 4 to 2), \(\sigma_{3}=5.45559478\) (optimal rank 2) and \(\sigma_{4}=61.58482111\) (optimal rank 0). We begin by examining the proportion of matches between the estimated ranks and the individual optimal ranks (which yield the smallest RMSE of the signal estimates for the given series). In comparison to the methods SVD, TRMAT, and MDL, the methods ESTER and SAMOS demonstrate the most favorable outcome, with a proportion of matches reaching 0.998 (out of 1000 trials) at a low noise level (\(\sigma=0.01\)) and window length \(L=50\). This outcome aligns with the findings presented in the studies [32; 33]. However, as the noise level increases, the methods lose efficacy, yielding near-zero matches. Figure 1 illustrates the dependence of standardized criterion values on rank. For comparability, the criteria were standardized (i.e., the mean was subtracted and the values were divided by the standard deviation depicted in the graph) as the criterion scales may be incomparable. To more effectively illustrate the impact, we have selected the window length \(L=40\) as a multiple of both sinusoids' periods and the standard deviation \(\sigma=\sigma_{1}\). It is readily apparent that the ESTER and SAMOS criteria lead to an erroneous determination of the rank in this instance, particularly the ESTER method, with a maximum at the separability point \(r=2\). In contrast, the considered information criteria, such as TRMAT, effectively resolve the rank to be four. Thereby, in the following sections, we will not consider the ESTER and SAMOS criteria, particularly given that they are not applicable in the absence of a signal. Thus, in what follows, we will examine the criteria SVD, TRMAT, and MDL in greater detail. As illustrated in Table 1, for noise levels corresponding to the stable rank detection (i.e., all except the level \(\sigma_{2}\)), the TRMAT, SVD, and MDL methods consistently yield optimal rank estimates. For each of the aforementioned methods, the proportion of matches with optimal ranks is nearly one. However, at the noise level \(\sigma_{2}\), the TRMAT and SVD methods were unsuccessful, with only a small proportion of matches being identified. In comparison, the MDL method demonstrated better performance, with a success rate of 0.48, as opposed to 0.235 and 0.277 for the TRMAT and SVD methods, respectively. Figure 2 illustrates the distinction in the behavior of the criteria for the two noise levels, namely, \(\sigma_{1}\) and \(\sigma_{3}\). As previously, the criterion values have also been standardized. Figure 3 depicts the mean estimated ranks and the optimal ranks for the smallest root of average MSEs over 1000 realizations for the TRMAT and MDL methods (SVD has been omitted due to its similar behavior to TRMAT). It can be observed that the optimal rank graph as a function of noise levels contains plateaus with identical rank values and transition periods.