# Information Criteria for Signal Extraction Using Singular Spectrum Analysis: White and Red Noise
###### Abstract  
In singular spectrum analysis, which is applied to signal extraction, it is of critical importance to select the number of components correctly in order to accurately estimate the signal. In the case of a low-rank signal, there is a challenge in estimating the signal rank, which is equivalent to selecting the model order. Information criteria are commonly employed to address these issues. However, singular spectrum analysis is not aimed at the exact low-rank approximation of the signal. This makes it an adaptive, fast, and flexible approach. Conventional information criteria are not directly applicable in this context. The paper examines both subspace-based and information criteria, proposing modifications suited to the Hankel structure of trajectory matrices employed in singular spectrum analysis. These modifications are initially developed for white noise, and a version for red noise is also proposed. In the numerical comparisons, a number of scenarios are considered, including the case of signals that are approximated by low-rank signals. This is the most similar to the case of real-world time series. The criteria are compared with each other and with the optimal rank choice that minimizes the signal estimation error. The results of numerical experiments demonstrate that for low-rank signals and noise levels within a region of stable rank detection, the proposed modifications yield accurate estimates of the optimal rank for both white and red noise cases. The method that considers the Hankel structure of the trajectory matrices appears to be a superior approach in many instances. Reasonable model orders are obtained for real-world time series. It is recommended that a transformation be applied to stabilize the variance before estimating the rank.  
title: _Algorithms_ **2024**, 17, 395. [[https://doi.org/10.3390/a17090395](https://doi.org/10.3390/a17090395)]([https://doi.org/10.3390/a17090395](https://doi.org/10.3390/a17090395))**  
title: **Copyright**: \(\copyright\) 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/)]([https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/))).  
title: _Algorithms_ **2024**, 17, 395. [[https://doi.org/10.3390/a17090395](https://doi.org/10.3390/a17090395)]([https://doi.org/10.3390/a17090395](https://doi.org/10.3390/a17090395))**  
title: **Copyright**: \(\copyright\) 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/)]([https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/))).  
title: _Algorithms_ **2024**, 17, 395. [[https://doi.org/10.3390/a17090395](https://doi.org/10.3390/a17090395)]([https://doi.org/10.3390/a17090395](https://doi.org/10.3390/a17090395))**  
title: **Copyright**: \(\copyright\) 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/)]([https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/))).  
title: _Algorithms_ **2024**, 17, 395.  
signal estimation is required. Signal estimation is performed by grouping the leading \(r\) components. As a result, it is of great importance to know \(r\), which is referred to as the signal rank or signal model order. The following section will describe the SSA algorithm for solving a particular signal estimation problem.