Mathematical methods for assessing the investment attractiveness of territories

. The article is devoted to the use of mathematical methods in assessing the investment attractiveness of territories. The work includes the selection of indicators that are the basis for comparison, their mathematical processing and generalization using such methods as normalization of indicators, correlation analysis, t-SNE visualization method, cluster analysis, principal components analysis, ranking method. Using these methods, it became possible to obtain the rating of municipalities on the example of one of the regions of the Southern Federal District of the Russian Federation - Rostov region. The rating is compiled separately for urban districts and municipal districts. The principal components analysis was used to study the relative importance of indicators, which eliminates the need to interview experts in the course of the research. The use of mathematical methods in assessing the investment attractiveness of municipalities made it possible to obtain a final assessment for each municipality, as well as to identify leaders and underperformers. This approach, which is notable for the availability of the information and less time-consuming calculations, is of interest both to local governments of municipalities and regional authorities. This methodology can be recommended to private investors for the selection of investment objects and the assessment of investment risks.


Introduction
The development of territories involves the successful identification and solution of demographic, economic, social and politico-legal problems.This contributes to a positive reputation and improved investment climate, which can attract more investments to increase economic stability.
Investment attractiveness is a complex parameter, which includes various indicators characterizing the object of investment from different positions for investor's decisionmaking.Such categories as investment climate, investment potential, investment activity, investment risks are directly interconnected with the concept of investment attractiveness.
The investment climate is a much broader concept than investment attractiveness.The investment climate is an interval characteristic of a territory, determined in dynamics, over a number of years.Assessment of investment attractiveness may vary from year to year, so in order to obtain an objective rating it is necessary to examine the region's indicators for a long period of time.
The assessment of the investment climate, and as a result, the investment attractiveness, can be carried out at different levels of detail.It is possible to assess individual enterprises or areas of the economy.It is possible to assess the investment attractiveness of the territory -country, region, or municipality.In the framework of this study, there were investigated approaches to the analysis of investment attractiveness of territories within the countrynamely, municipal formations.Nevertheless, similar approaches can be applied to the assessment of investment attractiveness of regions, largely due to the absence of interregional boundaries.
When choosing indicators for assessing investment attractiveness, it is important to determine the goal that needs to be achieved in the course of the assessment.It may depend on who is the subject of investment activity: while for a private investor it may be important to choose a territory for investment (one of many), or to identify a group of such territories, the authorities may be interested in identifying the specifics in the activities of each region, ranking entities for a more efficient allocation of resources.In the latter case, choosing the best alternative is not enough.
For any investor (both private and state) it is important to study the investment potential of the territory and possible risks.Private investors (foreign and domestic) will not invest their own funds in those objects that are highly risky and will not provide a decent return, but will choose the most promising ones.At the same time, public authorities must make decisions in the interests of the people living in these territories -including less attractive locations.It is the authorities -municipal, regional, or federal -that are interested in the equal development of the territory of the corresponding level -municipality, region, or country.
The purpose of this study is to estimate the investment attractiveness of municipalities by creating an integral assessment, using mathematical methods.
When assessing the investment attractiveness, the most important characteristics are the investment potential of the territory and investment risks.
The investment potential of a territory includes all the means and resources available to the territorial object to achieve certain financial results with their efficient use.The investment potential of a territory consists of nine groups of indicators: production; labor; financial; consumer; institutional; infrastructure; natural resources; tourism; innovation.
Investment risks represent potential problems and limitations that an investor may encounter during the implementation of an investment project.The following groups of risks are distinguished: financial; social; management; economic; environmental; criminal.
The most commonly used methods of assessing the investment attractiveness of regions.The best known are methodology of the National Rating Agency (NRA), methodology of the rating agency "Expert", methodology of the Institute of Economics of the Russian Academy of Sciences and other techniques [1].
There is no universal assessment methodology that would be identical for all municipalities.Each region assesses the attractiveness of municipalities independently.Some regions, for example, the Rostov Region, use methods which evaluate the activities of their authorities in this field.
A significant contribution to the development of theoretical and practical foundations for assessing the investment attractiveness of territories is made by international authors who have developed and applied methods that allow to obtain an integral assessment of territories [2][3][4][5].
Various mathematical methods can be used to summarize the data and obtain an integral assessment.One of the methods used to solve the problem of finding the optimal region for opening a supermarket was the construction of a weighted Voronoi diagram [6].In this diagram, the weights change depending on the values of the indicators.
Regions can be ranked based on the normalized values of the attributes [7].Each attribute is mapped on a scale from 0 to 1, depending on its position relative to the best/worst region for each individual indicator.The final ranking is a function of the Euclidean distance between the object and the best or worst vector.
Regression models can also be used to estimate the investment potential [8].In this model, all indicators are first normalized, and then regression coefficients are selected based on historical data.
Another approach used to assess the attractiveness of regions is the application of multiattribute utility theory [9].It uses both indicator analysis, and an additional step of expert verification to clarify the relative importance of the indicators.
Fuzzy logic methods are also used to assess the investment attractiveness of regions [10].For each attribute an expert assigns a verbal assessment (from "excellent" to "very bad"), which is then converted into a numerical value from 0 to 1 using characteristic functions.
The above mathematical methods take into account the relative importance of the indicators in different ways.A group of experts often determines the significance of indicators, but their evaluations can vary greatly.Other methods assign equal importance to all indicators, which can lead to incorrect results.Therefore, there is a need to develop methods to assess investment attractiveness that don't depend on expert assessments and take into account the relative importance of indicators.

Materials and methods
To assess the investment attractiveness of municipalities, the right choice of indicators, which should have a quantitative form of presentation, is of great importance.On the one hand, the evaluation indicators should be sufficient to be able to consider the characteristic under study from different sides, on the other hand, a large number of evaluated parameters significantly complicates the calculations and increases the probability of errors.
In the course of the research, we selected the indicators that are the basis for comparison (Tables 1 and 2).The preference was given to the indicators that are available in the public domain, we used the "Database of indicators of municipalities", available on the website of the Federal State Statistics Service.
The municipalities of Rostov Region, a subject of the Russian Federation, which is part of the Southern Federal District, were chosen as the objects of the study.The population of the region is 4.4 million people.(5th place in the Russian Federation), the area is 100.8 thousand square kilometers.(33rd largest in Russia).Rostov region includes 12 urban districts and 43 municipal districts.Shipped goods of own production, works and services performed by own forces (without small businesses), thousand rubles. 1.2 Sales of non-produced goods (without small businesses), thousand rubles. 1.3 Agricultural production, thousand rubles. 1.4 Cattle and poultry produced (sold) for slaughter (live weight), quintals 1.5 Number of business entities, units. 1.6 Volume of extrabudgetary investment in fixed capital per capita (per capita rubles) The volume of work performed by type of economic activity "construction" per capita (the number of received notifications of the completion of construction) 2.
Labor potential The average number of employees of organizations of municipal form of ownership, people.

2.2
Average number of employees of organizations (data for 2016) The volume of paid services for the population per capita, thousand rubles The cost of fixed assets of commercial organizations of municipal ownership per capita, thousand rubles

5.2
The total area of land in the municipality, hectares The entire sown area, hectares 5.5 The number of cattle, heads (units) 6.

Infrastructural potential
6.1 Length of public roads of local importance, km 6.2 The number of educational institutions of secondary vocational and higher education per 1,000 people 7.

Institutional potential
7.1 Share of the average number of employees of SMEs (small and medium enterprises), percent Transformation and processing of information was carried out using mathematical methods such as normalization of indicators, correlation analysis, t-SNE visualization method, cluster analysis, method of principal components, ranking method.Let us consider the specifics of applying these methods at each of the stages of information processing.
Stage 1: Normalization of the indicators.
To improve representativeness, the raw data were scaled from 0 to 1 using the following formula [11]: Step 2: Visualization.To reduce the data dimensionality, the t-SNE method is used [12].Stage 3: Cluster analysis.To check the efficiency of the ranking, all municipal entities are divided into clusters.The hierarchical clustering algorithm [13] was used to group entities into clusters.The Euclidean norm L2 (2) is used to estimate the distance between objects: (, ) = √∑ (  −   ) 2

𝑖
(2) Step 4: Correlation analysis.To exclude the possibility of mutual correlation of indicators, a Pearson correlation matrix was used [14]: Here:   ,   -values of the i-th and j-th indicator,   ̅ ,   ̅average values of the i-th and j-th indicators

N -total number of indicators
Step 5: Determine the significance of individual indicators using the principal components analysis.When compiling the rating, different indicators may have different significance.Mathematically it is reflected in the value of dispersion, which shows the degree to which objects differ in this indicator.To account for the probability of mutual dependence and ensure the accuracy of the ranking result, data on the population and the area of the municipality were added to the dataset.
To obtain the significance of the indicators, taking into account their partial mutual correlation, we used the principal components analysis method [15].This method uses the number of components in the vector space, to determine how the original data will be transformed.
The components obtained as a result of the method will differ in their contribution to the variance.The normalized table of indicators was divided into two parts (cities and municipal districts separately), and then the method of principal components with the number of components equal to 12 was applied to each of them (based on the number of cities).
Step 6. Ranking.The final ranking was calculated using the following formula: () =   ̂ *   *   *   (5) where:   ̂− the normalized value of the indicator,calculated according to the (1),   -coefficient of significance of the indicator, obtained as a result of the principal component analysis,   = 1 for indicators to be ranked in descending order ("the more, the better"); -1 for indicators to be ranked in descending order ("the less, the better").
− correction factor (equal to 0.5 for indicators from the risk group).

Results
An application of mathematical methods allowed us to obtain the following results.Stage 1: All the values used to assess the investment attractiveness of municipalities have been normalized, as previously mentioned in Section 2.
Stage 2. The visualization of the t-SNE transformation in Figure 1 shows that cities and municipal districts are represented in a distinct area in space.This suggests that the data is representative of other important indicators for ranking.Step 4: To perform a correlation analysis, a correlation matrix was constructed (Figure 3), which allowed a number of generalizations to be made: -indicators 2.2, 3.  Correlation analysis allowed us to exclude from the ranking such indicators as: -Local budget income/expences per capita (3.1, 3.2); -Share of profitable organizations in the total number of organizations (3.5); -Area of perennial plantations (5.3), which have a high correlation coefficient with other parameters.
Step 5. Determination of the significance of the indicators using the principal components analysis.The basis vectors obtained as a result of the application of the principal components are shown in Figure 4.The color represents the absolute value of the index projection on the basis vector of the component.
The principal component method allowed us to determine the significance of each indicator for ranking (Table 3).We can see that the significance of the indicators for urban districts and municipal districts is different.For urban districts, the most important indicators are the value of fixed assets per capita, the average number of employees of organizations and the average per capita income per month (the significance coefficients are 7.76, 6.50, 6.46, respectively).The most important indicators for municipal districts are the production of cattle and poultry for slaughter, the number of cattle, total land area of the municipal formation per capita (the significance coefficients are 5.58, 4.97, 4.82).Step 6.The rating was constructed according to formula 2. Figure 5 shows the results of the rating by urban districts and Figure 6 -by municipal districts.The columns 1-10 in the table denote the arithmetic mean between the values of indicators within a group: thus, column 1 shows all indicators of group 1 (Industrial potential), column 2 -all indicators of group 2 (Labor potential), and so on.
For the urban districts, the leader is Rostov-on-Don with an integral coefficient equal to 1.684.It leads in five blocks of nine -production, labor, consumer, infrastructure potential, in two blocks it is close to the best indicators.The outsiders are the cities of Gukovo, Novoshakhtinsk and Zverevo, whose integral indicators are several times less than the leader.
Among the municipal districts, Aksai is in the first position with an integral coefficient of 1.037.The Milutinsky and Zimovnikovsky districts were also in the top three, with their indicators only slightly inferior to the best and equal to 1.008 and 0.967.It is noteworthy that the difference between the final values of municipal districts is much smaller than that between cities; the maximum value for districts exceeds the minimum value by 2.4 times, and by 5.1 times for urban districts.
The following conclusions can also be drawn from the rating: -In this rating, the position occupied by each urban district corresponds to the results of the cluster analysis: Rostov-on-Don is placed first, followed by all cities in the first cluster, followed by all cities in the second cluster.This allows us to conclude that their division into clusters reflects the actual economic situation.In addition, these methods can be used each separately: if researchers are interested only in the order of objects, the cluster analysis will be sufficient; obtaining quantitative differences involves ranking.
-Rostov-on-Don is ahead of the other cities by a significant margin.This once again emphasizes its special status as the capital of the region.-For districts, the property of ordering is not preserved: they may be higher or lower in the ranking, regardless of whether they fall into one or another cluster.This can be explained by the fact that the number of regions is larger, and the probability of getting into the wrong cluster is higher than for the list of 12 cities.

Discussion
The materials of this study show the advantages of using mathematical methods to assess the investment attractiveness of municipalities, which are aimed at reducing labor and time costs.Combining the methods allowed us to obtain an integral assessment for each municipality of the region, separately for urban and municipal districts, as there are significant differences between the two clusters.
The mathematical methods used in the study allowed us to divide municipalities into clusters (cluster analysis), identify the closeness of the relationship between the selected characteristics (correlation analysis), determine the degree of their importance for the investment rating of territories based only on the quantitative characteristics (principal component method) and build the final rating of urban districts and municipal areas (ranking method).It should be noted that for municipalities, some indicators were absent or presented for earlier periods, limiting the possibility of their use in the study, however, more data is presented at the regional level.

Conclusion
The investment attractiveness of a territorial entity is one of the indicators on the basis of which decisions about investing in a particular municipality are made.The formation of a favorable investment climate is one of the main tasks of the authorities and local government, which have a direct interest in it.
The combination of mathematical methods used in this study allowed us to identify the main regularities in the formation of the investment attractiveness of municipalities, to identify leaders and outsiders, as well as to identify ways to improve the investment climate of territories and to improve the financing of subordinate territories.In turn, the growth of investment attractiveness of urban districts and municipal areas will increase the attractiveness of the region as a whole, which will be reflected in the results of the regional rating of the subject of the Federation.

Fig. 1 .
Fig. 1.The result of performing the t-SNE transformation to the normalized indices.The circles represent municipal districts, the triangles represent urban districts.Stage 3: Cluster analysis, based on the results of stage 2, was conducted separately in the context of urban districts and municipal areas.Figure2shows the dendrograms for urban districts and municipal areas.
Figure2shows the dendrograms for urban districts and municipal areas.

Fig. 3 .
Fig. 3. Correlation matrix of the indicators used.The color of the cell shows the absolute value of the matrix element -darker for negative values, lighter for positive values.

Fig. 4 .
Fig. 4. Principal components for urban and municipal districts.The darker the cell of the diagram, the greater the absolute value of the component.

Fig. 5 .
Fig. 5. Rating results by city.The most significant indicators in the columns are highlighted in gray.

Table 1 .
Indicators used to assess the investment potential of municipalities of the Rostov region.

Table 2 .
Indicators for assessing investment risks.

Table 3 .
Significance of the first 10 principal components of the Rostov region municipalities.