The study presents a comprehensive exploration of canonical correlation analysis applied within advanced information systems for assessing complex interrelations between economic growth indicators and environmental security factors. Leveraging robust computational methodologies and integrated information systems, this research utilizes canonical correlation analysis to quantitatively evaluate relationships between two sets of multidimensional variables: indicators representing economic wellbeing-including GDP per capita, gross fixed capital formation, value-added industrial production, and household expenditures-and variables reflecting environmental threats such as carbon dioxide and greenhouse gas emissions, and natural resource depletion. Data sourced from the World Bank for 136 countries for the year 2020 served as the empirical foundation of the study. The employed computational information system facilitated advanced preprocessing and normalization procedures, essential for ensuring analytical accuracy given substantial variability across datasets. Statistical computations were executed within a structured digital environment, leveraging computational efficiency to identify canonical variables demonstrating maximal correlation. Results indicated a significant canonical correlation coefficient (r = 0.9762), underscoring the robustness of the identified relationships. Further analytical interpretation using Pearson's pairwise correlation confirmed the validity and significance of these variables within constructed canonical sets. The presented findings reaffirm previous scholarly insights into economic-environmental interdependencies and reinforce the pivotal role of computational analysis supported by sophisticated information systems in elucidating complex socio-economic phenomena. This methodological approach proves indispensable for strategic policy formulation aimed at balancing economic advancement and environmental sustainability, contributing to the broader discourse on achieving sustainable development goals through innovative computational and analytical techniques.
In the context of rapid globalization processes, the internationalization of economies, and the innovative development of production, humanity faces an increasing number of challenges associated with the extensive use of natural resource potential. This leads to its uncontrolled depletion, resulting in significant social stratification within society due to uneven access to resources, a decline in biodiversity, a reduction in the productivity of geoecosystems, and environmental deterioration. These factors disrupt the balance of ecosystems, leading to the emergence of threats to ecological security, which in turn contribute to declines in national health, a decrease in life expectancy, and the exacerbation of social conflicts.
Such disturbances necessitate a shift in the trajectory of modern society’s development and the formulation of strategic prospects for global development, thereby creating more favorable conditions for living and reproduction. The solution to this problem is the transition to balanced, sustainable development, which combines the even development of economic, ecological, and social aspects of societal development and determines the global priorities for social growth [1]. The implementation of the concept of sustainable development involves the use of a model of economic growth, in which urgent issues of the livelihood of society will be effectively resolved based on the perception of nature and its resources as one of the highest values and the priority of preserving the natural environment will prevail over the criterion of economic efficiency. Such a strategy is based on dynamic balance, which avoids irreversible environmental changes and does not threaten humanity’s long-term existence.
The interaction between the economic and ecological components occurs within the environmental and economic system. The economic system formed by humans develops exceptionally quickly and is often a destabilizing factor in the environmental system, disrupting its overall balance. In this regard, there is a pressing need to study the relationship between indicators of economic development and factors that pose threats to environmental safety to identify their causes and consequences and to determine the nature of the interaction between these indicators. The application of a scientific approach and modern analytical tools, including economic and mathematical modeling, to data analysis will contribute to the development of strategic decisions necessary for reducing ecological and economic tension, preventing the catastrophic consequences of the global ecological crisis, and mutually beneficial integration of economic growth and environmental protection.
Sustainable development should be considered in the context of the interaction between two
alternative problems: the preservation of ecosystems and the provision of favorable conditions for
the stable economic development of society. The deepening of the climate crisis continues to cause
significant economic damage, illustrating the essential relationship between the economy and
ecology [
Problems related to the interaction between the economy and ecology in the context of
achieving sustainable development goals have been explored in the research of a wide range of
scientists. One of the dominant research areas in this field is assessing the relationship between
agricultural development, GDP growth, and waste generation. Agriculture is a branch of the
economy that exerts a significant anthropogenic impact on the environment and utilizes available
natural resources to a considerable extent. The use of econometric models is quite common in
studying these issues. Using Ukraine’s data as an example, Zomchak et al. [
Using the example of Pakistan, Sajjad et al. [
Studies [
The study results reflect the heterogeneity of environmental security degree for different territories, which is determined by the economic development level, economic specialization, natural and climatic conditions of economic activity, and geographical location of territories. The analysis of the presented studies reveals that various analytical tools are currently employed to investigate the mutual influence of economic and ecological systems. At the same time, reasonably thorough results were obtained, illustrating the presence of close relationships between indicators of economic development and environmental threats. At the same time, it is worth noting that environmental threats are often assessed based on single indicators, particularly the volume of carbon dioxide emissions. Our research aims to determine the degree of correlation between a set of indicators of economic development and a set of indicators that reflect environmental threats.
For this, we used the method of canonical correlations. Studies [
Canonical correlation analysis is a method of multivariate data analysis. This is the most general form of correlation analysis, which allows you to examine the relationship between two sets of variables. In contrast, factor analysis is used to establish relationships within a single set of variables. The primary purpose of the method is to identify combinations of initial and resulting indicators for which the correlation between them is the strongest. The problem of reducing the number of indicators included in the model can also be solved by eliminating those that are insignificant. To some extent, canonical analysis combines methods from correlation, regression, and factor analysis.
Let the resulting indicators Y1, Y2, ..., Yk be influenced by factors X1, X2, ..., Xn. The measurement is carried out on m objects. It is logical to assume that there are fewer resulting indicators than factors, k ≤ m. The output matrix of observations is usually written in the form:
Let’s write the linear combinations of the initial indicators in the form of canonical variables U and V:
The density of the connection between the canonical variables U and V is determined by the canonical correlation coefficient:
The r values depend on which linear combinations form the initial indicators, that is, on the values of the parameters ai and bj. The canonical correlations method is aimed to find such a combination of indicators for which the value of the canonical correlation coefficient will be the largest by the modulus.
The calculation of the canonical correlation coefficient, as follows from formula (3), is based on the extended correlation matrix of indicators: where RXX is the correlation matrix between indicators Xi and Xj, i, j = 1..n.
RYY is the correlation matrix between indicators Yi and Yj, i, j = 1..k.
RXY is the correlation matrix between indicators Xi and Yj, i = 1..n, j = 1..k.
RYX = RTXY.
Let’s write the canonical variables in matrix form: Then, we get the following expression for the canonical correlation coefficient: , (5) (6) Assume that the canonical variables are standardized; that is, their mean values are equal to zero, and their root mean square deviations are equal to one. The expression (6) is simplified and takes the form:
Finding the maximum of function (7) is a problem of finding a conditional extremum and can be solved using the method of Lagrange multipliers. As a result, we get the equation:
It follows from this equation that the problem of finding the coefficients of the vector B is reduced to the task of finding the eigenvectors of the matrix:
Note that all eigenvalues of in this case will be positive: β = λ2. Analogous transformations can give an expression for finding the coefficients of the matrix A as eigenvectors of some matrix similar to (9). However, this is not necessary: we can find the matrix A from the expression: (7) (8) (9) (10)
We began the search for the parameters of the linear combination of indicators by considering the coefficients of the vector B, as initially, the assumption was made that the number of resulting indicators would be less than the number of factors. Otherwise, the search for parameters must begin with vector A.
Since AT S XX A= B T S YY B = 1, then λ= AT S XY B = r . Since the goal of the method is to find combinations with the largest value of the canonical correlation coefficient, among all eigenvalues, the largest should be selected for calculations. However, for each eigenvalue, it is possible to construct its pair of canonical variables U and V.
Let us consider the application of the method of canonical correlations in the context of the
research tasks at hand. A critical stage is the choice of indicators. The authors’ considerations
regarding macroeconomic indicators of economic growth and indicators of environmental threats
and the results of the analysis of the results of other researchers [
X1—Agriculture, forestry, and fishing, value added (constant 2015 US$).
X2—Industry (including construction), value added (constant 2015 US$).
X3—GDP per capita (constant 2015 US$).
X4—Households and NPISHs Final consumption expenditure (constant 2015 US$). X5—Gross fixed capital formation (constant 2015 US$).
X6—Exports of goods and services (constant 2015 US$).
As indicators characterizing environmental threats, we will single out the following: Y1—CO2 emissions (kt).
Y2—Total greenhouse gas emissions (kt of CO2 equivalent).
Y3—Adjusted savings: natural resources depletion (% of GNI).
The study was conducted for countries worldwide, based on data from the World Bank [
Y3 indicators selected for the study (and not only for these). Therefore, we limit our analysis to data from 2020. The conditions that determined the technique of collecting the necessary data are that, firstly, the canonical analysis gives reasonable results under the condition of a large sample that contains at least 100 elements. Secondly, it is recommended that the sample size be 15–20 times larger than the number of indicators selected for the study. These conditions determined both the number of indicators selected for the research and the size of the sample.
The values of the extended correlation matrix calculated for the selected set of indicators are shown in Table 1.
β2
β3 0.9530 0.5437
0.0564
Eigenvectors 0.8107 –0.5630 0.5854 0.0000 0.7798 0.2738
0.1603 –0.2220
0.9618
The first eigenvalue is β1 = 0.9530. Accordingly, the first canonical correlation coefficient is r1 = 0.9762. The corresponding coefficients of the canonical variable V are the components of the first eigenvector. The values of the parameters of the canonical variable U, calculated according to formula 10, are recorded in Table 3. In this case, χ2emp=511.03, χ2kr(0.05; 18)=28.87. Therefore, the hypothesis that a relationship exists between the selected sets of indicators is accepted. Expressions for the canonical variables U and V have a form: We check the significance of the canonical correlation coefficients using the Wilks λ-criterion. The corresponding function is approximated by the distribution χ2 with n⋅k degrees of freedom:
The analysis of the obtained dependencies reveals that the canonical variable U is most influenced by indicators X2 is Industry (including construction), value added, and X5 is Gross fixed capital formation, and for variable V, the indicator Y1 is CO2 emissions. Let’s pay attention to the fact that the indicator X3 is GDP per capita has a relatively insignificant effect on the canonical variable U, and the indicator Y3 is Adjusted savings: natural resources depletion, in this case, does not affect the corresponding canonical variable V.
Let’s analyze the value of Pearson’s pairwise correlation coefficients between the initial indicators of each group and the corresponding canonical variables. They are interpreted as canonical factor loadings, reflecting the weight of each output indicator in the corresponding canonical variable. The corresponding values are given in Table 4. Its analysis confirms the earlier conclusion about the importance of the influence of each of the initial indicators on the related canonical variables. Canonical correlations do not contain information about what part of the variance each canonical root explains in the studied indicators. However, it is possible to calculate the proportion of the original indicators’ variance explained by the corresponding canonical variable as the average value of the squares of the corresponding pairwise correlation coefficients. In this case, for each of the canonical variables, this share is 66%. (12) (13) Please note that the obtained results were significantly affected by sample heterogeneity, for which the calculations were performed. So, for the X2 indicator, the largest value exceeds the smallest by 50,000 times, for the X5 indicator—by 40,000 times, and for the Y1 indicator—by 36,000 times. The same heterogeneity occurs for normalized values. At the same time, the calculations allow us to conclude that there is a connection between the development of the world’s economic system and global environmental threats. This negatively affects progress in achieving the goals of sustainable development. This conclusion is consistent with the results obtained by other scientists, including for different sets of indicators. The research results can be utilized as analytical information for policy review about long-term, sustainable economic development.
Please note that the obtained results were significantly affected by sample heterogeneity, for which the calculations were performed. So, for the X2 indicator, the largest value exceeds the smallest by 50,000 times, for the X5 indicator—by 40,000 times, and for the Y1 indicator—by 36,000 times. The same heterogeneity occurs for normalized values. At the same time, the calculations allow us to conclude that there is a connection between the development of the world’s economic system and global environmental threats. This negatively affects progress in achieving the goals of sustainable development. This conclusion is consistent with the results obtained by other scientists, including for different sets of indicators. The research results can be used as analytical information for policy review regarding long-term sustainable economic development.
Currently, various analytical tools are used to solve the task. We employed the method of canonical correlations, which enables us to assess the relationship between two sets of indicators: one representing economic growth and well-being, and the other representing environmental threats. The study was conducted for 136 countries using World Bank data from 2020. For analysis, six indicators were selected for the first group and three for the second. Several indicators were identified due to certain limitations inherent in the chosen method. The calculations revealed a reasonably high correlation between the constructed canonical variables, with a value of the canonical correlation coefficient of r = 0.9762. Criterion verification confirmed the significance of the calculated indicator. At the same time, it was established that the following indicators influence the canonical variable of economic development: Industry (including construction) and Gross fixed capital formation, and the canonical variable of environmental threats—CO2 emissions. Calculating Pearson’s pairwise correlation coefficients between the original indicators and canonical variables confirmed the importance of the selected indicators. The research results can be utilized as analytical information for reviewing policies related to achieving sustainable development goals, which involves an effective combination of economic development and environmental protection, resulting in mutually beneficial outcomes for both environmental and economic benefits. The direction of further research is to assess the relationship between indicators of economic development and environmental threats for certain groups of countries, grouped, in particular, by the level of GDP per capita or by clustering according to a selected set of indicators. Declaration on Generative AI While preparing this work, the authors used the AI programs Grammarly Pro to correct text grammar and Strike Plagiarism to search for possible plagiarism. After using this tool, the authors reviewed and edited the content as needed and took full responsibility for the publication’s content. [1] Unated Nations. Department of Economic and Social Affairs. Sustainable Development. URL: https://sdgs.un.org/goals