# Information Criteria for Signal Extraction Using Singular Spectrum Analysis: White and Red Noise
### Low-Rank Approximation and SSA  
If the signal is a series of known rank \(r\), there are numerous methods for extracting it, including low-rank approximation [18; 19; 20; 21]. In particular, the paper [21] proposes an efficient MGN (Modified Gauss-Newton) method for finding the best low-rank approximation using the least-squares method. It is significant to note that in the case of Gaussian white noise, the least-squares approximation coincides with the maximum likelihood estimate. Alternative approaches include the Cadzow method, which consists of alternating projections to \(\Pi_{\mathcal{H}}\) and \(\Pi_{\mathcal{M}_{\mathcal{H}}}\) at each iteration. This method has been discussed in the literature, including in the following sources: [22; 23; 24]. In all versions, low-rank approximation methods are iterative, which makes even efficient methods quite time-consuming and not guaranteed to find the global minimum. Furthermore, it is always necessary to know the rank of the signal in advance. Let us describe the cases when low-rank approximation methods, with the choice of \(r\) equal to the signal rank, will prove ineffective. The first situation is when the noise level is too high. In this case, the approximation by a signal-rank series includes a significant portion of the noise in the result. In order to extract the signal more accurately, it is necessary to take \(r\) less than the signal rank. The second situation is when the signal is not exactly a low-rank series. This is usually the case for real-world time series. In this case, the low-rank approximation can perform poorly. The version of SSA for signal extraction is a single iteration of the Cadzow method and has a very efficient implementation [25]. Given that the method employs a single iteration, it is not constrained to the extraction of low-rank signals. Instead, it can be used to identify trends and periodic components in real-world time series, which can then be subjected to further analysis and forecasting. By employing the intermediate SSA result in the form of the singular value decomposition of the trajectory matrix, one can perform visual identification of the signal-related components of the decomposition. It is evident that this approach is not feasible when dealing with a vast amount of data. Consequently, techniques for automated component identification within SSA have been developed. For example, in [26], a method for automated trend identification has been proposed, wherein the number of components to be identified must be specified. Consequently, it is also necessary to set \(r\). The Problem Statement and Contribution  
In this study, we propose a novel approach to the problem of estimating the signal rank. Rather than determining the signal rank itself, our objective is to identify the optimal parameter \(r\) in the SSA algorithm that minimizes the mean square error (MSE) of the signal reconstruction. In the case of a low-rank signal and low noise, this approach will yield the same result as the conventional method of finding the signal rank. However, in situations where the signal is not exactly of low rank or the noise level is high, the signal rank may not provide the optimal \(r\). This study will be based on the methods of signal rank detection (model order selection). These methods can be divided into two types: those based on information criteria and those based on the properties of the SSA method (properties of the signal subspace). The information criteria that are currently available were not designed for use in the context of the SSA case; therefore, we propose modifications to them. Given that the original version of the information criteria was developed for the case of Gaussian white noise, we suggest an approach that would allow us to extend them to the case of red noise. Structure of the Paper  
Let us describe the structure of the paper. Section 2 describes known methods for estimating the model order \(r\). In Section 3, we propose an approach that is based on signal estimation by SSA.