# Intelligent Identification of Trend Components in Singular Spectrum Analysis
## 6 Numerical Investigation
### Comparison of Accuracy  
Let us compare the results of the methods for all examples described in Section 6.1. The following general parameters were used:  
* Threshold \(T_{0}=0.2\).
* Threshold \(T_{1}=0.5\).
* Maximum number of iterations for IOSSA method \(n_{iter}=10\).  
As before, the comparison is performed using the same 1000 generated time series. In Equations (6)-(9) below, the rows with significantly smallest mean MSE errors are in bold.  
Example (6).  
This example corresponds to the accurate separability of the trend from the periodic components. Take the frequency threshold \(\omega_{0}=\frac{1}{24}\) and the number of components equal to the signal rank, \(t=r=4\). Table 3 contains the results on the accuracy of the trend extraction.  
All methods showed approximately the same errors. When the trend is accurately separated from the periodic, the best method to extract the trend with the selected parameters is autoSSA, since the method is the fastest and the most stable.  
Example (7).  
This example corresponds to no weak separability of the trend from the periodic components. Take the frequency threshold \(\omega_{0}=\frac{1}{40}\) and the number of components equal to the signal rank, \(t=r=5\). Table 4 contains the results on the accuracy of the trend extraction.  
The autoEOSSA method is the most accurate one. The autoIOSSA + FOSSA method also shows small errors, although it reaches the maximum number of iterations. The methods autoFOSSA and autoSSA do not improve weak separability, so they show large errors. Example (8).  
This example contains the trend of infinite rank and corresponds to more or less accurate weak separability and no contribution separability of the trend from the periodic components. Take the frequency threshold \(\omega_{0}=\frac{1}{24}\) and large number of components equal to \(t=r=12\). Table 5 contains the results on the accuracy of the trend extraction.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **trend\_mse** & **iterations\_num** \\ \hline autoIOSSA + FOSSA & mean: 0.2573, med: 0.1766 & mean: 10, med: 10 \\ autoEOSSA & **mean: 0.1593, med: 0.1183** & mean: 1, med: 1 \\ autoSSA & mean: 1.0696, med: 1.0510 & mean: 1, med: 1 \\ autoFOSSA & mean: 1.9922, med: 1.9942 & mean: 1, med: 1. \\ \hline \hline \end{tabular}
\end{table}
Table 4: MSE of trend estimates. Example (7).  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **trend\_mse** & **iterations\_num** \\ \hline autoIOSSA + FOSSA & **mean: 0.068, med: 0.059** & mean: 2, med: 2 \\ autoEOSSA & **mean: 0.068, med: 0.059** & mean: 1, med: 1 \\ autoSSA & **mean: 0.068, med: 0.059** & mean: 1, med: 1 \\ autoFOSSA & **mean: 0.068, med: 0.059** & mean: 1, med: 1 \\ autoFOSSA & **mean: 0.068, med: 0.059** & mean: 1, med: 1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: MSE of trend estimates. Example (6).  
Despite a slightly inaccurate weak separability, the autoFOSSA method is the most accurate in extracting the trend. The errors of the autoIOSSA + FOSSA method are significantly larger than those of the autoEOSSA method.  
Example (9).  
In this example, the trend is a polynomial of degree 2, which produces a signal root of multiplicity 3. There is neither weak nor contribution separability. Take the frequency threshold \(\omega_{0}=\frac{1}{40}\) and set the number of components equal to the signal rank, \(t=r=5\). Table 6 contains the results on the accuracy of the trend extraction. The only method that accurately extract the trend is autoEOSSA.