## 2 Materials and Methods
### Formalization and Mathematical Statement of Problem  
Section 2.1 describes how text mining isolated a cluster of texts mentioning attitude to the subway line project from the entire social media data array. Text mining also split the entire cluster into subclusters based on attitude: negative, positive, or neutral. Filtered and clustered data helped word the problem.  
A project is planned that may affect the interests of residents in a specific, limited area. The problem is to build a mathematical model to assess the utility of this project on social media data only without any additional research such as population surveying. The utility is defined herein as the residents' positive attitude to the project.  
Let:  
1. A be the subpopulation with a negative attitude to the project;
2. B be the subpopulation with a positive attitude to the project;
3. C be the subpopulation with a neutral attitude to the project.  
Social attitude to the project could be assessed by analyzing the size of these subpopulations in dynamics. However, such data is not available. Therefore, it is proposed to estimate such subpopulations by means of digital footprint. The models are based on the assumption that the community consists of three subpopulations A, B, C, holding a negative, positive, or neutral attitude to the project. Let N(A), N(B), and N(C) be the sizes of the sets A, B, C, and D be the sum of the disjoint sets D = A + B + C. Subjects may move from one group to another as a result of communication or an event. Therefore, the percentage of each subpopulation may change over time. Figure 2 shows the flowchart of the modeled objects. Let D1 be the set of social media actors whose posts and comments contain one of the three possible attitudes to the project. Assume that the set D is mapped to the set D1,  
\[\text{F: D}\to\text{D1}.\]  
Assume that the mapping F is such that the percentage of negative/positive/neutral groups in D1 fully matches that in D.  
The objective is to use digital data of the actors in D1 to model the dynamics of the residents' attitude to the project.  
Mathematical Statement of Problem. Let \(X(t)\), \(P(t)\), \(Z(t)\) be the number of posts at the time t that express negative, positive, or neutral attitude to the project, respectively; let \(S(t)\) = \(X(t)+P(t)+Z(t)\), \(t\) = 1, 2, \(\ldots\), \(n\).  
Consider society as a dynamic system. Let the state of the dynamic system at the time \(t\) be described by the vector \(Y(t)\) = \((X(t)\), \(P(t)\), \(Z(t))\). The objective is to use observations of \(X(t)\), \(P(t)\), \(Z(t)\) over the period t = 1, 2, \(\ldots\), n to build a mathematical model to describe the behavior of the dynamic system.  
The behavior dynamics of \(X(t)\), \(P(t)\), \(Z(t)\) contains important data: the ratio of these figures; for this reason, the figures are normalized for modeling. The normalization rule is as follows: \(x=X/S;p=P/S;z=Z/S\), then the follows holds: \(x+p+z=1\), \(y=(x,p,z)\), \(yeD\subset R^{3}\).  
To make a managerial decision on the basis of the built mathematical models, introduce the preferential attitude and its numerical representation. Let \(\chi\) be a non-empty set. Interpret the elements of this set as a possible choice of attitude to the project: \(\bar{x}\), \(\bar{p}\), \(\bar{z}\) (negative, neutral, positive). Numerical representation of the preferential attitude \(\succ\) is such function \(V\colon\chi\to R\) that  
\[y\succ x\Leftrightarrow V(y)>V(x).\]  
The function \(V(\cdot)\) can be defined in different ways, e.g., as a function of the first differences of a time series.