# Information Criteria for Signal Extraction Using Singular Spectrum Analysis: White and Red Noise
## 6 Summary and Discussion  
In this paper, we considered a variety of criteria for determining the model order (signal rank) in SSA, including non-conventional cases, namely, signals that are not low-rank or mixed with noise. The criteria considered were ESTER, SAMOS, MGN, MDL, SVD, and TRMAT (the latter two were proposed by us). The ESTER and SAMOS criteria appeared to be unsuitable for determining model order in the majority of the considered cases. While the information criteria can be employed, none of them have a comprehensive theoretical justification for application to the considered problem statement. It is also important to note that this justification is unlikely to be obtained due to the overly general formulation of the problem, which allows for the possibility of signals that are not low-rank and have a high noise level. An exception is the MGN criterion, which is theoretically justified in the case of a low-rank unmixed signal in the presence of white noise. This is due to the fact that it numerically searches for the maximum likelihood estimate (MLE) of the signal. The MGN method, which is used in the MGN criterion for signal estimation, is computationally expensive, while it is implemented as computationally fast as possible. The used optimization method is local, as are many others, and can converge to a local extremum, leading to an excessive estimation of the standard error and thus an overestimation of the rank by the MGN criterion. In general, an overestimation may not be a significant issue, as the higher the number of decomposition components, the smaller their contribution. However, the high computational cost represents a significant obstacle to the application of the MGN criterion, even in the case of a low-rank signal. In the case of SSA, when the singular value decomposition is performed once and the signal estimation is generally not a low-rank series, three variants, MDL, SVD, and TRMAT, were considered in the BIC version, since the penalty from the AIC criterion leads to a significant overestimation of the signal rank. In this case, the MSE signal estimation error (an estimate of \(\sigma^{2}\)) used in the information criteria is not based on the time series decomposition after diagonal averaging, but on the estimated noise matrix before hankelization. In our proposed version of TRMAT, the Hankel structure of the trajectory matrix is accounted for using weights. Numerical studies have demonstrated that for relatively simple cases, such as a noisy sum of sinusoids with a low noise level, the methods yield approximately the same results. A slight advantage of the MDL method can be observed in transition regions where the optimal order of the model changes. In the case of a signal that is not low-rank, such as a \begin{table}
\begin{tabular}{l r r r r r r r r} \hline \hline
**ts** & **MGN** & **TRMAT** & **MDL** & \begin{tabular}{c} **TRMAT** \\ **AR (no)** \\ \end{tabular} & \begin{tabular}{c} **MDL** \\ **AR (no)** \\ \end{tabular} & \begin{tabular}{c} **TRMAT** \\ **AR** \\ \end{tabular} & \begin{tabular}{c} **MDL** \\ **AR** \\ \end{tabular} & \
\begin{tabular}{c} **MDL** \\ **AR** \\ \end{tabular} &
\begin{tabular}{c} **\lambda** \\ **AR** \\ \end{tabular} \\ \hline wine1 & 14 & 13 & 13 & 14 & 13 & 14 & 13 & 0.35 \\ wine2 & 12 & 11 & 11 & 11 & 11 & 11 & 0.38 \\ wine3 & 15 & 11 & 11 & 7 & 11 & 7 & 11 & 0.19 \\ wine4 & 14 & 10 & 12 & 6 & 12 & 6 & 12 & 0.51 \\ wine5 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & \(-\)0.24 \\ wine6 & 12 & 9 & 9 & 9 & 9 & 9 & 9 & \(-\)0.27 \\ co2 & 21 & 12 & 21 & 6 & 6 & 6 & 6 & \(-\)0.03 \\ oilprod & 18 & 21 & 21 & 16 & 16 & 16 & 16 & \(-\)0.33 \\ dwarf & - & - & - & - & - & - & - & - \\ treerings & - & - & - & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Series after the Cox-Box transformation: Rank values estimated by the criteria and estimates of the transformation parameters. logarithmic signal, the proposed TRMAT criterion is preferred. In both cases, the MGN criterion is comparable to TRMAT.