# Information Criteria for Signal Extraction Using Singular Spectrum Analysis: White and Red Noise
## 5 Real-Life Time Series  
Consider ten real-world time series, they are described at https://ssa-with-r-book. github.io/datasets.html (accessed on 10 August 2024):  
* wines (from 1 to 6), Australian wine sales (Total, Drywhite, Fortified, Red, Rose, Sparkling), monthly, length 174;
* co2, Atmospheric concentrations of CO\({}_{2}\), monthly, length 468;
* dwarf, Time variation of the intensity of the variable white dwarf star, every 10 s, length 618;
* oilprod, Crude oil and natural gas plant liquids production, monthly, length 300;
* tererings, Tree ring indices, annual, length 669.  
We take the window length for each series to be half the length of the series.  
Table 3 presents the model order estimates by the specified criteria in three versions: for the white noise case and with the estimation of the parameter \(\phi\) of the autoregressive process AR(1), without checking for stationarity (labeled \"no\") and with checking for stationarity. The last column contains the ranks that were estimated visually. It is not always possible to determine the number of components associated with a given signal. In such cases, a number of values are provided. In general, the estimates of the model orders produced by the various criteria are consistent with the visual estimates. As can be observed, the MGN criterion produces an overestimate of the rank in the majority of cases, while the TRMAT criterion produces an underestimate in several instances. The stationarity check has a negligible impact on the outcome. The results are, for the most part, satisfactory. In particular, the signal rank for the series \"tree rings\" is indeed equal to 1, as the series is similar to a random process with a non-zero constant mean.  
Since the criteria were developed for the case of stationary noise, we will assess whether the performance of the criteria improves after the variance is stabilized. Table 4 illustrates the outcomes of the criteria applied to the series after the Box-Cox transformation [38] with the automatically estimated parameter \(\lambda\), the value of which is provided in the last column. It can be observed that the results have been equalized, and the effects of over-ranking by the MGN criterion and under-ranking by the TRMAT criterion have been mitigated. The  
\begin{table}
\begin{tabular}{l r r r r r r r r} \hline \hline
**ts** & **MGN** & **TRMAT** & **MDL** & \begin{tabular}{c} **TRMAT** \\ **ar1 (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} **Visual Est.** \\ \end{tabular} \\ \hline wine1 & 14 & 13 & 13 & 14 & 13 & 14 & 13 & 13 \\ wine2 & 12 & 11 & 11 & 11 & 11 & 11 & 11 & 11–14 \\ wine3 & 15 & 11 & 11 & 7 & 11 & 7 & 11 & 11 \\ wine4 & 13 & 5 & 10 & 6 & 10 & 6 & 10 & 14 \\ wine5 & 22 & 2 & 11 & 1 & 12 & 1 & 13 & 12 \\ wine6 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\ co2 & 11 & 12 & 21 & 6 & 6 & 6 & 6 & 6, 15, 25 \\ oilprod & 22 & 21 & 21 & 16 & 16 & 16 & 16 & 16+ \\ dwarf & 10 & 15 & 15 & 11 & 14 & 11 & 14 & 12–14 \\ tererings & 5 & 1 & 3 & 1 & 1 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Original series: Ranks estimated by the criteria and visual rank estimation.  
effect of the stationarity test is entirely absent. Therefore, although the SSA method does not require stationarity, it is recommended to perform a variance stabilizing transformation for more accurate and stable rank estimation.