Purpose of the research. Let’s assume that the dynamics of the state of some object is being investigated. Its state is described by a system of specified indicators. Among them, some may be a linear combination of other indicators. The aim of any forecasting procedure is to solve two problems: first, to estimate the expected forecast value, and second, to estimate the confidence interval for possible other forecast values. The prediction procedure is multidimensional. Since the indicators describe the same object, in addition to explicit dependencies, there may be hidden dependencies among them. The principal component analysis effectively takes into account the variation of data in the system of the studied indicators. Therefore, it is desirable to use this method in the forecasting procedure. The results of forecasting would be more adequate if it were possible to implement different forecasting strategies. But this will require a modification of the traditional principal component analysis. Therefore, this is the main aim of this study. A related aim is to investigate the possibility of solving the second forecasting problem, which is more complex than the first one.
Materials and research methods. When estimating the confidence interval, it is necessary to specify the procedure for estimating the expected forecast value. At the same time, it would be useful to use the methods of multidimensional time series. Usually, different time series models use the concept of time lag. Their number and weight significance in the model may be different. In this study, we propose a time series model based on the exponential smoothing method. The prediction procedure is multidimensional. It will rely on the rule of agreed upon data change. Therefore, the algorithm for predictive evaluation of a particular indicator is presented in a form that will be convenient for building and practical use of this rule in the future. The principal component analysis should take into account the weights of the indicator values. This is necessary for the implementation of various strategies for estimating the boundaries of the forecast values interval. The proposed standardization of weighted data promotes to the implementation of the main theorem of factor analysis. This ensures the construction of an orthonormal basis in the factor area. At the same time, it was not necessary to build an iterative algorithm, which is typical for such studies.
Results. For the test data set, comparative calculations were performed using the traditional and weighted principal component analysis. It shows that the main characteristics of the component analysis are preserved. One of the indicators under consideration clearly depends on the others. Therefore, both methods show that the number of factors is less than the number of indicators. All indicators have a good relationship with the factors. In the traditional method, the dependent indicator is included in the first main component. In the modified method, this indicator is better related to the second component.
Conclusion. It was shown that the elements of the factor matrix corresponding to the forecast time can be expressed as weighted averages of the previous factor values. This will allow us to estimate the limits of the confidence interval for each individual indicator, as well as for the complex indicator of the entire system. This takes into account both the consistency of data changes and the forecasting strategy.

The purpose of the work is to study the foundations of a general risk theory. To form a single formal concept of risk, a number of definitions of the term “risk” found in the literature have been analyzed. There is a certain redundancy in the number of entities involved in the definition of this term. The necessary attributes of the genesis of this concept have been identified. Based on the analysis, using the instrumentation of modern algebra, a new formal, mathematically strict definition of risk is built. In fact, the paper proposes a new relational theory of risk, attracting only two entities to define the concept of “risk”: set and order of preference, inducing on this set the minimum structure of a semilattice or family of semilattices.
The paper also describes an approach for studying in theoretical-risk setting problems in which there is no risk, in the traditional sense, in which the preference relation does not induce a semilattice and/or is a preorder. It is shown that in this case, when identifying a suitable equivalence relation on a set of outcomes, the problem can be reduced to a classical theoretical-risk setting. The second part of the paper contains an example of the direct use of a new relational risk theory in the study of nonnumerical economies. The problem of analysis of the situation of confrontation between two technologically unequal countries in game-theoretical staging is considered. We are talking about a grandiose space program - the organization of manned flights to Mars. The scales of the object of confrontation, the impossibility of quantitative assessments of the consequences of the implementation of scenarios of this project, make any quantitative assessments impossible at the stage of preliminary analysis. Therefore, only expert estimates of preferences at multiple outcomes can be used as initial data for confrontation analysis. Under these conditions, the emergence of certain risks of the project implementation for both players was demonstrated. During the analysis of the example illustrating the application of the new relational risk theory, a number of optimality principles considered in game theory were extended to the case when only partial orders are given on a set of game outcomes. As the methodological basis of the research we used the achievements of modern algebra, in particular the theory of relational systems, as well as the concepts and methods of game theory, such as representing the game in normal form, selecting dominant and eliminating dominant strategies, choosing solutions from many cautious strategies, as well as from the set of Nash Equilibria.
The main result of the paper is the substantiation of relational risk theory, the formation of its conceptual base, the demonstration of the constructive nature of the theory on the example of solving a specific problem of risk analysis in an economic system described in terms of non-numerical characteristics. The material presented in the article is of interest to researchers in the field of risk theory and game theory, as well as to practitioners engaged in socio-economic and political forecasting in conditions of lack of information.

The article presents in detail the methodological approaches to calculating the integrated assessment of brand strength, as well as the results of testing the system of indicators and the mathematical model on the example of a comprehensive assessment of the power of the regional brand “The real Vologda product”. The developed methodology as a whole is the result of research within the framework of the implementation of a state scientific grant from the government of the Vologda Oblast. In addition to the system of indicators, built on the basis of the results of generalization of theoretical materials and practical experience of domestic and foreign authors on this problem and the use of modeling techniques, a questionnaire for researching consumer attitudes towards brand product is presented, a methodology for presenting the results of respondents’ answers to questions in a quantitative coefficient form is described. A mathematical model has been developed to calculate the integrated indicator for assessing brand strength based on the survey results, which includes information received both from the consumer and from outside experts. The scaling of the possible values of this estimate has been performed. The matrix of indicators for assessing brand strength is graphically presented, which allows you to give both a general and a detailed qualitative characteristic of brand strength based on the results of using the model, to determine the strengths and weaknesses of the brand, and to reveal its potential.
Purpose of the research. The purpose of this research was to develop a comprehensive methodology for assessing brand strength and its approbation on the example of the regional product brand “The real Vologda product”.
Materials and methods. The research was carried out on the basis of data from a questionnaire survey of consumers and experts, construction of a mathematical model, including coefficient values of indicators reflecting the level of indicators that allow assessing brand strength in the complex. Mathematical and statistical methods were used as the basis for transferring the results of the respondents’ answers into the coefficient form, their standardization and generalization. A graphical method was used to represent the matrix of indicators for assessing brand strength.
Results. The studied domestic and foreign methods for assessing the power of a brand, with all their obvious positive aspects and completeness, have a weak side – they are qualitative from the standpoint of characteristics of the strength, potential and status of a brand, they only allow a description of these properties. As a result of the study, it was possible to generalize the existing experience, develop a system of brand strength indicators, draw up a questionnaire for a consumer survey that allows collecting information to measure the values of these indicators, build a mathematical model for calculating an integrated brand strength assessment, develop a methodology for the qualitative characteristics of the results of calculating this assessment based on the matrix. Approbation of the methodology made it possible to obtain a comprehensive quantitative assessment of the power of the regional product brand “The Real Vologda product”, to formulate recommendations that were taken into account by the copyright holder of the brand – Budgetary Institution of the Vologda Region “Vologda Information and Consulting Center of the Agro-Industrial Complex” when elaborating its development strategy.
Conclusion. The results of assessing the influence of the indicators’ contribution to the integrated indicator of the brand strength “The Real Vologda product” showed that its strength and potential are mostly due to the factors such as physical availability and quality of products, highly rated by consumers, as well as demand from the buyer. At the same time, according to the experts, these factors also received very high weighted coefficients.

In recent years, Russia has taken a serious step towards innovative development, to a large extent by providing systematic incentives for innovation and technological development of sectors of the economy. Today, innovation, research, development and their implementation are an important part of the political ambitions of most developed and developing countries. Global spending on innovation continues to grow, and the share of business in it is increasing. The relevance of this topic is also due to the acceleration of the processes of digital transformation of the Russian and international economy, which is why the need to create conditions for innovation, information and communication technologies is increasing. Based on this, it is important for the Government of the Russian Federation to form a national innovation system that promotes the innovative activity of domestic enterprises and increases the scientific orientation of research activities of organizations. Developed countries achieve competitive advantages through the introduction of innovative technologies, a highly qualified workforce with creative thinking, efficient use of financial resources, and strong R&D. Identifying new market opportunities is easy for highly innovative companies. Currently, not only the emergence of new high-tech industries, but also the development of traditional ones is based on information and scientific potential. The development of science is closely related to the growing flow of information, which is both the result and the basis for the development of science, the results of which are subsequently embodied in economic development.
The purpose of the article is to identify the main trends in the development of innovative processes in the Russian Federation. The active creation and modernization of technologies, introduction of various technological projects is one of the driving forces of the innovation process. In recent years, there has been a strengthening of innovative development in various sectors of the Russian economy, and a significant role in this development is played by the systemic stimulation of innovation and technological development.
Materials and methods. The methodological basis of the study consists of works on regional statistics, mathematical modeling, and econometrics. Graphic and tabular methods of visualization of research results, statistical methods of analysis of economic phenomena were used. To solve the tasks of the study, application packages SPSS, Statistica were used.
Results. The creation of new, unique advanced production technologies is an essential element of the innovative development of the country’s economy. The development of industrial design is one of the main stimulating factors for the innovative development of Russia. The increase in the number of issued patents for inventions with a high degree of probability predetermines an increase in the number of manufacturing technologies, which is the basis for progress in the field of manufacturing design. However, the main challenge remains the issue of financing.
Conclusion. The article contains an analysis of the dynamics of the main indicators characterizing the innovative activity of organizations and provides forecast estimates of the indicator based on trend models.

The purpose of this paper is to develop a comprehensive approach to evaluating digitalization of interaction of the state and citizens in the sphere of providing state services. The objectives of the study include the analysis of current Russian and foreign approaches to assessing the digitalization of interaction between the state and citizens; development and testing of a composite indicator methodology for conducting such an assessment in the provision of public services.
Materials and methods. When preparing the paper, the authors studied Russian and foreign theoretical and practical approaches to evaluating digitalization of interaction between the state and citizens. The approbation of the digitalization index of interaction between the state and citizens proposed in the article was carried out on the basis of data from a selective federal statistical observation of the Federal State Statistics Service on the use of information technologies, information and communication networks by the population, as well as the data from the Ministry of Economic Development of the Russian Federation. To achieve the objective, theoretical methods (comparative analysis, synthesis, and generalization), modelling, and correlation analysis were used.
Results. Based on the analysis of foreign and Russian approaches, their advantages and limitations, the article substantiates the methodology for calculating the digitalization index of interaction between the state and citizens, taking into account the immediate results (the availability of state and municipal services in the electronic form), intermediate results (the demand for the electronic form of service provision) and the final results of digital transformation in the provision of public services (satisfaction of citizens with the quality of services provided in the electronic form), as well as the risks of digital interaction (the level of digital trust of citizens in the interaction with the state). Approbation of the digitalization index of interaction between the state and citizens in the regional context was carried out. To assess the applicability of the proposed index, the correlation dependence of its values and the values of the index components on the level of education of the population, employment, income and urbanization, as well as differences in the availability of information and communication technologies infrastructure was investigated. At the same time, the level of digitalization correlation of interaction between the state and citizens with social factors is higher than with infrastructural ones. This means that to solve the problem of digital inequality in Russian regions, it is not enough to provide universal access to the Internet; factors of human capital development also play an important role in promoting digital channels of interaction.
Conclusion. Assessing the digitalization of interaction between the state and citizens requires taking into account various indicators that reflect the immediate, intermediate and final results of such an interaction, as well as the risks associated with the transition to “digit”. In terms of the provision of public services for such an assessment, it is advisable to use the digitalization index of interaction between the state and citizens. Further promising areas of research in the field of assessing the digitalization of interaction between the state and citizens are related to the study of the influence of citizens’ trust in government bodies on the level of digitalization of interaction between the state and citizens, as well as taking into account the interaction of citizens and the state within the framework of other types of state functions (development and implementation of state policy, regulation, control and supervisory activities). The implementation of these directions will require the expansion of the currently available information base.

China, being the country with the largest population, inevitably influences the processes taking place in the world economy. In terms of GDP, the Chinese economy has been the first economy in the world over the past several years, being among the most important economic indicators such as GDP, industrial production, and export of goods in leading positions. Under the influence of the pandemic crisis, a deep recession occurred in industrially developed countries, where the reproduction system is based on the “TNC economy”, with the exception of China, which managed to transfer the economy into a state of reproductive expansion, while maintaining the positive dynamics of the country’s economic development on an annualized basis. The aim of the paper was to analyze data that gives an idea of the potential of the Chinese economy. The main indicators characterizing the development and dynamics of the Chinese economy are considered.
Materials and methods. In the course of the work, a dynamic, structural analysis of analytical and statistical information was used; methods of analytical, logical, system analysis were used.
Results. The paper examines certain macroeconomic indicators that characterize the state and development of the Chinese economy, taking into account the achievements in the context of the global economic crisis caused by the COVID-19 pandemic. The GDP of China was analyzed, calculated in national currency or PPP.
Conclusion. The country was faced with a difficult international situation, difficult tasks in search of reform, development and stability within the country and, in particular, with the heavy blows of the COVID-19 epidemic. Steps taken to stabilize employment, financial transactions, foreign trade, foreign investment, domestic investment and market expectations, as well as ensuring fully guaranteed employment, basic living needs, functioning of market actors, food and energy security, stability of production and supply chains, helped to restore the economy making China the only major economy with positive economic growth in the world in 2020.