## 2 Materials and Methods
In addition, as analysis of the costs of innovation activity in Volgograd region showed, SSA does not always effectively separate the weakly expressed trend and medium-term (long-term) fluctuations, as a result of which one component must be divided into two mathematical models. To facilitate and partially automate the expert evaluation in our algorithm, we propose to use the toolkit of fuzzy sets, which will be implemented in the phasifier and the database of fuzzy rules. This toolkit, based on fuzzy inference rules, will allow you to determine the dynamics model for a particular component (or propose several of the most appropriate models for experts to choose from), as well as to make preliminary conclusions about the possible economic content of specific components on the basis of connected databases and/or knowledge bases. It should be noted that, in [6], a tool for identifying the types of dynamics (trend and cyclic component) on the basis of modeling is proposed for similar problems; however, this approach does not allow expanding the set of rules and including, for example, rules for trend changes or rules for analysis of cyclic fluctuations, which are not sinusoidal. Another approach, effective from the point of view of extracting the cyclic component, is a combination of SSA (MSSA) and Fisher filter methods [20]. For example, [21] suggests using SSA as a preprocessing method to increase the efficiency of detecting hidden periodicities in the phenomenon under study. On the other hand, in their paper [18], the authors use logic close to that proposed by the authors in the algorithm, according to which \"Fisher g test <is used> to select the principal components to be aggregated in the reconstruction step\". However, even in this approach, there are difficulties with the unclear economic content of individual economic cycles reconstructed on the basis of [PERSON]'s g test, as well as with the impossibility of changing the model used and combining, for example, several approaches to model the cyclicality of different components. Based on this review, we can conclude that the task of finding a way to extend the toolkit for recognition of SSA series components is relevant, and the toolkit for phasing and applying fuzzy inference rules, which is easily extensible and customizable, is presumably effective enough to solve this problem. The fuzzifier converts the indicators extracted by SSA (in particular, the values of the most significant criteria) into fuzzy linguistic variables belonging to fuzzy sets Aj (individual for each indicator). As a rule, these variables have an affiliation function (AF), for example, of the triangular form  
\[\mu_{A}(s)=\left\{ \begin{array}{cc}1-\frac{|s-c|}{d}&\forall s\in[c-d,c+d]\\ 0&\forall s\
otin[c-d,c+d]\end{array} \right., \tag{7}\]  
where \(c\) is the center point of the triangle; \(d\) is the width of the triangle. The peculiarity of this transformation is that we can construct fuzzy estimates for any criterion, and, if necessary, make fuzzy estimates based on expert reasoning. After the transformation, formation of fuzzy conclusions about the possible type of dynamics of the phenomenon defined by the SSA component under study takes place. For this purpose, additive convolution is used, which is performed on the basis of the rules of the form:  
\[\text{if S1 is A11 and S2 is A21, then Z is B,} \tag{8}\]  
where S1 is a fuzzy value for one indicator, S2 for another indicator, and so on; Z is a fuzzy value for the resulting component type. For example, \"if Spearman criterion value is high and L value obtained by Foster-Stewart method is high, then presence of trend with positive first derivative is very high\". The set of rules contained in the database of fuzzy rules and/or knowledge base, if necessary, can be adjusted, which is an important distinguishing feature of this algorithm, giving it high flexibility.