# Intelligent Identification of Trend Components in Singular Spectrum Analysis
## 7 Conclusions  
The paper presented a novel approach to automated identification by combining methods to improve separability and methods for automatic identification of decomposition components. The new separation-improvement method, EOSSA, was proposed and fully justified. Its modification with distance clustering ultimately proved to be the best in combination with the automatic identification of the trend in the case of finite-rank signals with precisely defined signal ranks.  
Using a numerical study, we provided recommendations for selecting parameters for automatic trend identification in both finite- and infinite-rank signals, and also described the strengths and weaknesses of the proposed methods.  
The methods exhibit good stability with respect to the choice of automatic identification parameters. In terms of stability to overestimation of the signal rank, the methods can be ranked as follows: the most stable method is autoSSA without improvement of separability, followed by autoFOSSA, which only improves contribution separability, then autoIOSSA + FOSSA, and the most unstable method is autoEOSSA. However, when the rank is determined correctly, the methods act in reverse order in terms of trend extraction accuracy. So, when the rank is slightly exceeded, autoEOSSA remains preferable, while the autoSSA method is not aided by its stability in exceeding the rank due to poor trend extraction caused by mixed components in the time series decomposition.  
For time series of infinite rank, a similar effect occurs if the concept of rank is replaced by the number of signal approximation components that are not mixed with the trend.  
It should be noted that although the EOSSA algorithm is formally designed for cases with roots of multiplicity one, meaning it is not suitable for polynomial trends, its implementation has proven to be effective in extracting polynomial trends.  
The application of automated SSA to real-world data, specifically the 'US Unemployment' data, demonstrates that the method effectively extracts an annual trend despite the presence of seasonal effects.  
Thus, the methods autoFOSSA, autoIOSSA + FOSSA, and autoEOSSA can be recommended for extracting trends in time series. These methods are based on SSA, which is nonparametric and robust to various forms of trends, periods of oscillations, and noise models. Due to its robustness in handling a moderate overestimation of time series ranks, the proposed approach can be considered general for cases when the parametric form of the trend is unknown. We expect that autoEOSSA will help extract the trend in cases where the trend is only approximately and locally governed by LRRs (therefore, the SSA analysis of moving and overlapping segments of a time series is considered [19; 20]) and forecast the trend using the technique described in [21].  
Automated decomposition of time series also requires methods for automated extraction of periodic components and estimation of signal ranks. In the future, it is expected to combine these methods with automatic trend extraction methods, as well as with the methods discussed in this paper to improve separability.  
Conceptualization, N.G. and A.S.; Formal analysis, N.G., P.D. and A.S.; Funding acquisition, N.G.; Investigation, N.G., P.D. and A.S.; Methodology, N.G., P.D. and A.S.; Project administration, N.G.; Resources, N.G.; Software, P.D. and A.S.; Supervision, N.G.; Validation, N.G., P.D. and A.S.; Visualization, N.G., P.D. and A.S.; Writing--original draft, N.G. and P.D.; Writing--review & editing, N.G., P.D. and A.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by RSF, project number 23-21-00222.  
Not applicable.  
The data presented in this study are openly available. The R-code for data generation and result replication can be found at [[https://zenodo.org/record/7982115](https://zenodo.org/record/7982115)]([https://zenodo.org/record/7982115](https://zenodo.org/record/7982115)), doi:10.5281/zenodo.7982115 (accessed on 18 July 2023).