## 3 Results
The ARIMA model showed the best results, so we will compare them with the results obtained by the presented algorithm. As can be seen from Figure 3, the model created with the help of the presented algorithm interpolates quite accurately the value of RF GDP in the middle of the series, deviating significantly from the real values in the periods of crises (which is generally characteristic of statistical models). Even in these periods, in general, the model proved to be more efficient than ARIMA. At the same time, both models have high enough accuracy; if we evaluate them on the basis of the proportion of the explained variance \(\mathbb{R}^{2}\), it is estimated as 0.973 for ARIMA and 0.985 for the model created before the change in fuzzy recognition rules. After creating rules that account for the variation in the amplitude of the oscillations and the complex nature of the Juglar cycle, the \(\mathbb{R}^{2}\) estimate rose to 0.992. Moreover, even the model of the first three components provided an \(\mathbb{R}^{2}\) value of 0.976, which shows interpolation better than the ARIMA method. The forecast error for 2021 for the created model was +2.2% compared to \(-3.4\%\) for ARIMA, but the accuracy of this forecast is not high also because of the COVID-19 crisis (as mentioned above, crisis phenomena reduce the accuracy of the model). Figure 2: Modeling results for the 2 nd (**a**) and 3 rd (**b**) components extracted by the SSA method. We also compared the results of our algorithm with the results of the SSA + LSTM algorithm, which was studied in [8; 9]. In those cases, SSA was also used as the method of preliminary dataset analysis and the result obtained was used to train the LSTM network. We used TensorFlow [23] as an end-to-end open-source platform for machine learning and scikit-learn [24] to build and use a network. The result of this algorithm was worse both than ARIMA and the method proposed. SSA + LSTM had an R\({}^{2}\) value of 0.960, but, in our case, we had a small dataset, so the LSTM network lacked enough data to train. Therefore, we cannot conclude that the SSA + LSTM algorithm is worse than ours, and the effectiveness of them is not constant but depends on inner components. Therefore, the SSA + LTSM effectiveness depends on hidden element number and input window size, and our algorithm effectiveness highly depends on fuzzy rules database content. Hence, both algorithms can progress to improve their effectiveness, so we cannot affirm which is the best. When analyzing the dynamics of the number of researchers in the Russian Federation (you can see the data at Supplementary Materials, Table_S1.csv and Table_S2.csv), we also obtain six components of time series using SSA (Table_S3.csv). After modeling them (Table_S4.csv, Table_S5.csv, Table_S6.csv, Table_S7.csv, Table_S8.csv, Table_S9.csv) we can assert, that the accuracy of the model (Table_S10.csv) was much lower than in the analysis of GDP. This is quite expected, since the number of researchers refers to indicators of innovation activity, which is characterized by high quantitative and qualitative uncertainty. Obvious fuzzy rules quite effectively distinguished cyclic components and assumed the presence of a quadratic trend. Difficulties appeared in analysis of the second component, where a sharp change in the trend is modeled (from growth in the number of researchers in 2000-2012 to a sharp decrease in 2012-2020). The proposed rules based on the difference in moving averages with different periods showed a possible trend change, but the algorithm highlighted as more preferable the autocorrelation model with lag 1. Theoretically, this could have been changed by adjusting the rules to reduce the preference of the autocorrelation model as a whole, but this would have required a readjustment of the entire algorithm, so a rule tracking a single change in the trend was introduced. This situation confirmed that the question of the optimal approach to trend change tracking using the proposed algorithm remains open.