Modeling the Size of Protection Zones of Cultural Heritage Sites Based on Factors of the Historical and Cultural Assessment of Lands

: This article addresses the global issue of preserving cultural heritage, which is associated, among other things, with the lack of provision for boundaries of protection zones for cultural heritage sites. This paper analyzes the worldwide experience in the ﬁeld of establishing protection zones for cultural heritage sites, identiﬁes the issues of preserving cultural heritage in Russia, as well as imperfections in the management of lands containing cultural heritage sites. To improve the management of such lands, a method of historical and cultural assessment of territories has been developed, which, on the basis of evaluative factors, makes it possible to characterize the historical and cultural value of a territory. In this article, based on this list of factors, a method for mathematical substantiation of the boundaries of protection zones of cultural heritage sites has been developed; this method includes seven stages. Using mathematical and statistical methods, models were obtained to determine the area of the protection (buffer) zone and the development control zone, the analysis of which revealed differentiation between the factors of historical and cultural assessment that characterize the urban environment, depending on the type of zones. The interpretation of the obtained models and the ﬁelds of their use are given. In further studies, this method will be modiﬁed using the historical and cultural assessment of territories.


Introduction
One of the main principles of the UNESCO Convention is the preservation of cultural heritage. The need to strengthen the protection and sustainability of cultural and natural world heritage is also laid down in the Sustainable Development Goals (SDGs) adopted by the United Nations (UN) Member States in 2015 [1]. At present, in the context of active human economic activity in urbanized areas as well as the impact of negative environmental and urban planning factors, this issue is becoming more and more relevant [2,3]. In their study, G. Chen, J. Shi, Y. Xia, and K. Furuya [4] revealed the influence of urbanization on the attractiveness of historical landscapes for tourism and on the sustainable development of territories of historical landscaping. C. Udeaja, C. Trillo, K.G.B. Awuah, B.C.N. Makore et al. [5] conducted a study of the city of Surat with the involvement of experts and proved that urbanization processes led to the destruction of valuable historical and cultural monuments, as well as the loss of intangible cultural heritage.
Considering the inevitability of the renovation and urbanization of historical territories, which is required to maintain favorable environment for city residents, it is necessary to develop a historical urban environment management procedure.
According to L.A. Soares, M.D. Valente, A. Yago Sampaio Brito, V. Furtado [6], J. Whitehand, and K. Gu [7], unprecedented rates of urbanization require justified decisions to preserve the "urban fragment", which has a historical and social value, by establishing buffer zones. Modern mechanisms for the use of lands populated with cultural heritag falling into the zones of their influence, according to L.V. Averina and I.S. Myam economically and legally imperfect [21].
Immovable cultural heritage sites act as mode-forming objects that impos restrictions on the use of urban areas [22]. These restrictions are considered in th of land use regulation, which is associated both with creating conditions for su development of territories, ensuring the comfort of living of the population and in its wellbeing, as well as preserving the environment, reducing the negative impa environment and humans, and preserving cultural heritage sites. Effective and use of territories makes it possible to maintain land and resource potential and t sustainable development, which the modern world is seeking to achieve [18,23,2 According to the Federal Law dated 25,06,2002 No. 73-FZ "On Cultural Herit (Historical and Cultural Monuments) of the Peoples of the Russian Federation" t the safety of a real estate object's value from the point of view of architecture culture, or science, the boundaries of the protection zones are established withi torical and landscape environment. The current legislation of the zone of prot cultural heritage sites covers three types of zones, which differ by imposed res protection zones, urban development control zones and business activities cont (DCZ&BACZ), and zones of protected natural landscapes. Within the boundari protection zone of the cultural heritage site, a special land tenure mode shall be which provides for the prohibition of new construction and the restriction of e Modern mechanisms for the use of lands populated with cultural heritage sites or falling into the zones of their influence, according to L.V. Averina and I.S. Myamina, are economically and legally imperfect [21].
Immovable cultural heritage sites act as mode-forming objects that impose special restrictions on the use of urban areas [22]. These restrictions are considered in the process of land use regulation, which is associated both with creating conditions for sustainable development of territories, ensuring the comfort of living of the population and increasing its wellbeing, as well as preserving the environment, reducing the negative impact on the environment and humans, and preserving cultural heritage sites. Effective and rational use of territories makes it possible to maintain land and resource potential and to ensure sustainable development, which the modern world is seeking to achieve [18,23,24].
According to the Federal Law dated 25,06,2002 No. 73-FZ "On Cultural Heritage Sites (Historical and Cultural Monuments) of the Peoples of the Russian Federation" to ensure the safety of a real estate object's value from the point of view of architecture, history, culture, or science, the boundaries of the protection zones are established within its historical and landscape environment. The current legislation of the zone of protection of cultural heritage sites covers three types of zones, which differ by imposed restrictions: protection zones, urban development control zones and business activities control zones (DCZ&BACZ), and zones of protected natural landscapes. Within the boundaries of the protection zone of the cultural heritage site, a special land tenure mode shall be ensured, which provides for the prohibition of new construction and the restriction of economic activities-for example, improvement of the adjacent territory or placement of special memorial plaques. The DCZ&BACZ mode restricts the conduct of land management, engineering, and other economic works, but allows construction in historical areas in compliance with the norms for the preservation of valuable historical buildings. The protected Land 2021, 10, 1201 4 of 20 natural landscape zone enables restrictions, such as establishing a ban on the construction and reconstruction of buildings to preserve and restore the natural landscape [25,26]. Currently, less than 15% of cultural heritage sites have approved boundaries for protection zones, and no more than 5% of the established boundaries have been entered into the Unified State Register [27] (Figure 2).
OR PEER REVIEW 4 of zones, and no more than 5% of the established boundaries have been entered into th Unified State Register [27] (Figure 2). This circumstance leads to negative consequences for the state of cultural heritag sites. The lack of established land tenure modes leads to uncertainty and the possibility o urban planning and construction errors in the future [10,13,24]. Within the boundaries o the protection zones, construction and reconstruction are prohibited; the developmen and economic activity control zones restrict new construction and the scope of reconstru tion, which is a key point in the preservation of cultural heritage sites. According to th annual State Report on the State of Culture issued in 2020, about 10% of all cultural heri age sites are in an unsatisfactory condition, while protection zones have not been ap proved for almost half of them. Due to these circumstances, we can talk about noncom pliance with the protection legislation and the problem of impending threats to the con dition and value of cultural heritage sites, which leads to the irrational use of limited te ritorial resources [28,29].
The approval of the boundaries for protection zones relies upon the basis of borde designs that are developed by scientific and design organizations using deep archiva historical, cultural, and other research. The field of historical and cultural research is di tinguished by the fact that it is based on qualitative characteristics, as well as on archiva This circumstance leads to negative consequences for the state of cultural heritage sites. The lack of established land tenure modes leads to uncertainty and the possibility of urban planning and construction errors in the future [10,13,24]. Within the boundaries of the protection zones, construction and reconstruction are prohibited; the development and economic activity control zones restrict new construction and the scope of reconstruction, which is a key point in the preservation of cultural heritage sites. According to the annual State Report on the State of Culture issued in 2020, about 10% of all cultural heritage sites are in an unsatisfactory condition, while protection zones have not been approved for almost half of them. Due to these circumstances, we can talk about noncompliance with the protection legislation and the problem of impending threats to the condition and value of cultural heritage sites, which leads to the irrational use of limited territorial resources [28,29].
The approval of the boundaries for protection zones relies upon the basis of border designs that are developed by scientific and design organizations using deep archival, historical, cultural, and other research. The field of historical and cultural research is distinguished by the fact that it is based on qualitative characteristics, as well as on archival, historical, and cultural textual and graphic descriptions. This ensures the reliability and completeness of the substantiation of design boundaries for protection zones; however, these studies are rather time consuming and unsupported by objective numerical data [26,27].
One of the tools for solving the problem of establishing optimal sizes of protection zones for cultural heritage sites and modernizing the process of designing their boundaries can be the use of mathematical and statistical methods as a stage of project boundaries for modeling the required area of protection zones for cultural heritage sites. This will allow the collection of objective data on the current condition of the territory and predicting, based on the existing situation, the dimensional values for other objects. Moreover, models of zones of protection for cultural heritage sites will help make the process of developing the designs of the boundaries of protection zones more efficient and reduce their cost [28].
In Russia, the problems of the legal regime of territories occupied by cultural heritage sites and adjacent territories are studied by such scientists as A.A. Varlamov, R.A. Kirillov [30], L.V. Averina, I.S. Myamina [21], N.N. Melnikov [25], and M.S. Trofimova [27]. The work of Yu. Kurashov [26], as well as earlier works of the authors of this article [28,31], is concerned with assessing the lands encumbered with cultural heritage objects to regulate nature management.
Since it is necessary to maintain favorable conditions for the inhabitants of the city, the renovation of valuable historical quarters is inevitable. Due to these circumstances, the most optimal solution is the adaptation of the site to modern conditions with the parallel establishment of protection zone boundaries for the cultural heritage site; without these boundaries, the site becomes poorly protected from the influence of ongoing economic activities in the adjacent territory [10,14,15]. In this regard, it becomes necessary to find a balance between the stringent land tenure system in historical cities, which extends to the territories within the boundaries of the protection zones of cultural heritage sites, and the ability of these sites to adapt in the conditions of constant urban environment transformation. The latter is expressed in the most careful but effective possible use of these facilities, without stagnation in the use and development of the natural resource potential of the territories.
Since the main lever of influence on the modes of use of territories adjacent to cultural heritage sites are cultural heritage protection zones (buffer zones), improving the development and use of urban areas is associated with the establishment of their modes and boundaries [32,33].
Based on the above statement, the study pursued the goal of improving the mechanisms for preserving cultural heritage sites in terms of managing encumbered land resources. To achieve this goal, the following tasks were solved: first, by examining modern trends, the existing problems in the field of land use regulation within the boundaries of cultural heritage protection zones were identified; second, using economic and statistical methods, models of the size of the protection zones of cultural heritage sites were built; third, the differentiation of the historical and cultural assessment factors according to the types of protection zones of cultural heritage sites were proved.

Research Area
The study object is Orenburg, and the territory of historical settlement of regional significance is the regional center of the Orenburg region ( Figure 3).
Orenburg, or "Steppe Capital", was founded in 1743 at the intersection of the Ural and Sakmara rivers as a fortress to protect Russian borders, and later it became a center of trade between Russia and Central Asia. Within the boundaries of Orenburg, there are 478 monuments of architecture, urban planning, and history ( Figure 4).  Orenburg, or "Steppe Capital", was founded in 1743 at the intersection of t and Sakmara rivers as a fortress to protect Russian borders, and later it became of trade between Russia and Central Asia. Within the boundaries of Orenburg, t 478 monuments of architecture, urban planning, and history ( Figure 4).  Orenburg, or "Steppe Capital", was founded in 1743 at the intersection of the Ural and Sakmara rivers as a fortress to protect Russian borders, and later it became a center of trade between Russia and Central Asia. Within the boundaries of Orenburg, there are 478 monuments of architecture, urban planning, and history ( Figure 4).   The names of many famous people are associated with Orenburg: the first cosmonaut, Yu. A. Gagarin, who studied at the First Chkalovsk Higher Aviation School; P.I. Rychkov, scientist and the first associate member of the St. Petersburg Academy of Sciences; writer, ethnographer, and lexicographer V.I. Dal; poet and writer A.S. Pushkin; writer L.N. Tolstoy; Ukrainian poet and artist T.G. Shevchenko; poet Musa Jalil; musician and conductor M.L. Rostropovich; and many others lived and worked here. The long history of the city is reflected in its layout; the historical core is an image of a typical Russian provincial town with predominant buildings of the 18th and 19th centuries. The population of the city as of 2021 is more than 560,000 people.
Justification of the boundaries of the study area is based on the data from the project of the Historical and Cultural Baseline Plan, the master plan of the city of Orenburg.

Simulation Steps
Modeling the area of the buffer zone and DCZ&BACZ of cultural heritage sites was carried out using the city of Orenburg as an example. The cartographic base was created using Geographical Information System (GIS) MapInfo Professional 19.0, while statistical data processing was carried out using Microsoft Excel 2010 and Gretl software solutions.
The initial research data are information on the area of the established boundaries of protection zones in Orenburg, obtained from regulatory legal acts on the approval of boundaries and acts of state examination of projects of boundaries of cultural heritage site protection zones, as well as statistical information from the public cadastral map of Rosreestr (Russian State Register), UISIS, and the Unified State Register of Cultural Heritage Sites. All information was obtained from official sources that are publicly available on the Internet.
The analysis of sources showed that in December 2020, in Orenburg, the boundaries of protection zones and development control zones for 25 cultural heritage sites were established, while the boundaries of the protected natural landscape zone were established for one object only (the landmark "Zauralnaya Roscha"). The process of establishing the boundaries of such zones is rather slow. Thus, a question arises: why it is impossible to analyze the size of the boundaries of the zones directly? The fact is that the configuration and size of the boundaries of the zones of protection for cultural heritage sites are very different and do not lend themselves to systematic analysis [34].
Modeling the size of the protection zones and DSC&BACZ was carried out using mathematical and statistical methods in the sequence shown in Figure 5. provincial town with predominant buildings of the 18 th and 19th centuries. The population of the city as of 2021 is more than 560,000 people. Justification of the boundaries of the study area is based on the data from the project of the Historical and Cultural Baseline Plan, the master plan of the city of Orenburg.

Simulation Steps
Modeling the area of the buffer zone and DCZ&BACZ of cultural heritage sites was carried out using the city of Orenburg as an example. The cartographic base was created using Geographical Information System (GIS) MapInfo Professional 19.0, while statistical data processing was carried out using Microsoft Excel 2010 and Gretl software solutions.
The initial research data are information on the area of the established boundaries of protection zones in Orenburg, obtained from regulatory legal acts on the approval of boundaries and acts of state examination of projects of boundaries of cultural heritage site protection zones, as well as statistical information from the public cadastral map of Rosreestr (Russian State Register), UISIS, and the Unified State Register of Cultural Heritage Sites. All information was obtained from official sources that are publicly available on the Internet.
The analysis of sources showed that in December 2020, in Orenburg, the boundaries of protection zones and development control zones for 25 cultural heritage sites were established, while the boundaries of the protected natural landscape zone were established for one object only (the landmark "Zauralnaya Roscha"). The process of establishing the boundaries of such zones is rather slow. Thus, a question arises: why it is impossible to analyze the size of the boundaries of the zones directly? The fact is that the configuration and size of the boundaries of the zones of protection for cultural heritage sites are very different and do not lend themselves to systematic analysis [34].
Modeling the size of the protection zones and DSC&BACZ was carried out using mathematical and statistical methods in the sequence shown in Figure 5.   The first stage of research includes the formation of a list of factors of historical and cultural assessment of lands that characterize the features of the urban environment ( Figure 6) [31]. The initial substantiation of their composition was carried out through the analysis of scientific and regulatory literature [34,35], then they were tested for multicollinearity to confirm the possibility of using a linear regression method [36,37].
To identify the correlation between the factors, the method of inflation factors was used. The Variance Inflation Factor (VIF) shows how many times the variance of the regression coefficient increases due to the correlation of the regressors X1,…, Xk compared to the variance of this coefficient if the regressors were uncorrelated. The Variance Inflation Factor is calculated by Formula (1): where Rj 2 is the coefficient of determination of the j-th regressor Xj for all other regressors, and j = 1,…, k, k is the number of model factors.
It is believed that if the Variance Inflation Factor VIF > 10, then this regressor leads to multicollinearity [38].
The factor inflation method makes it possible to correctly identify multicollinearity, but its disadvantage is that when several factors are collinear, this method cannot be used to identify which of them should be excluded from the model [38,39]. In mathematical statistics, there are several methods for eliminating correlating factors, such as t-statistics, Belsley's method, analysis of the matrix of partial correlation coefficients of dependent correlating variables, and the resulting indicator. The simplest and most effective way is to identify which of the correlating factors has a greater correlation coefficient with the independent variable, that is, to assess the connection degree of one feature with one factor excluding the influence of all other factors.
Calculations to identify multicollinearity and the construction of correlated matrices are presented in earlier studies by the authors [31].
The next stage of modeling is the assignment of values to the estimated factors, that is, carrying out a historical and cultural assessment of the territory in accordance with the previously developed methodology [39]. This method is based on the expression of The initial substantiation of their composition was carried out through the analysis of scientific and regulatory literature [34,35], then they were tested for multicollinearity to confirm the possibility of using a linear regression method [36,37].
To identify the correlation between the factors, the method of inflation factors was used. The Variance Inflation Factor (VIF) shows how many times the variance of the regression coefficient increases due to the correlation of the regressors X1, . . . , Xk compared to the variance of this coefficient if the regressors were uncorrelated. The Variance Inflation Factor is calculated by Formula (1): where R j 2 is the coefficient of determination of the j-th regressor Xj for all other regressors, and j = 1, . . . , k, k is the number of model factors.
It is believed that if the Variance Inflation Factor VIF > 10, then this regressor leads to multicollinearity [38].
The factor inflation method makes it possible to correctly identify multicollinearity, but its disadvantage is that when several factors are collinear, this method cannot be used to identify which of them should be excluded from the model [38,39]. In mathematical statistics, there are several methods for eliminating correlating factors, such as t-statistics, Belsley's method, analysis of the matrix of partial correlation coefficients of dependent correlating variables, and the resulting indicator. The simplest and most effective way is to identify which of the correlating factors has a greater correlation coefficient with the independent variable, that is, to assess the connection degree of one feature with one factor excluding the influence of all other factors.
Calculations to identify multicollinearity and the construction of correlated matrices are presented in earlier studies by the authors [31].
The next stage of modeling is the assignment of values to the estimated factors, that is, carrying out a historical and cultural assessment of the territory in accordance with the previously developed methodology [39]. This method is based on the expression of historical and cultural features of the territories in the form of semantic values of evaluative factors with their subsequent transformation into a quantitative value using the fuzzy sets theory. A scoring scale has been developed for each factor. For example, the urban planning factor takes into account the impact of the site on the surrounding buildings and compositional relations (Table 1). Negatively affects the formation of the historical and cultural environment of the territory, creates discord in the environment, violates the architectural scene of the urban environment by its appearance (cladding, building facade style), height (number of stories), functions performed (garage or outbuilding near a valuable site)

Long-term construction on Sovetskaya Street
The fourth stage is associated with the use of the least-squares method for developing mathematical models for the area of the protection zone and the DCZ&BACZ [8,10]. Since each type of protection zone is characterized by and assigned with various restrictions and established modes of land plot use, the modeling is performed individually for each type of zone.
At this stage, the hypothesis about the linear form of the multiple regression equation was verified and was put forward because of the absence of multicollinearity between the factors [39,40]. The creation of the multiple regression equation was carried out on the basis of the obtained quantitative characteristics of the area of the studied protection zones (the resulting indicator) and on the values of the factors (regressors) by sequentially including the factors in the model and systematically checking the quality of the resulting equations until all the quality indicators of the constructed model stop improving, while all criteria for evaluating the model will represent satisfactory requirements [41][42][43].
The original model includes a list of all evaluative factors. The significant factors were selected by the method of sequential exclusion of factors by the value of t-statistics. The method operates so that, after constructing the regression equation and assessing the significance of all regression coefficients, the factor for which the coefficient is insignificant and has the smallest value of t-statistics is excluded from the model. A new multiple regression equation is then obtained, and the significance of all remaining regression coefficients is evaluated again. If there are insignificant ones among them, then again, the factor with the smallest value of the t-criterion is excluded. The process of excluding factors stops when all the regression coefficients are significant, i.e., the values of t obs > t crit [43].
The verification of the obtained models for compliance with the quality criteria was carried out on the basis of the analysis of statistical characteristics, including the coefficient of determination, t-statistics, Fisher's coefficient, the Durbin-Watson test, the Kolmogorov-Smirnov test, the Breusch-Godfrey test, and graphic methods [38,43].
The coefficient of determination is calculated to assess the quality of fitting the linear function of the model. It characterizes the proportion of the variance of the zone area, explained by regression in the overall regression of the final area (2) [44]: where y i is the observed values of the zone area, y is the average value according to the observed data,ŷ i is the model values of the area of the protection zone of cultural heritage sites built according to the estimated parameters.
Fisher's F test makes it possible to assess the significance of the correlation index, and, consequently, the entire regression equation. If the actual value of the F-criterion is greater than the tabular value, then the relationship between the factors and the area of the protection zone is reliable and the equation of the dependence of the area on the factors of historical and cultural assessment fully reflects this relationship. If the actual value of the F-criterion is less than the tabular value, then the relationship between them is random.
The observed value of the Fisher's coefficient F obs is calculated by Formula (3): where R is the coefficient of the correlation of factors and the area of the protection zone; The significance of each parameter is evaluated using the Student's t-test by comparing its values with the tabular value of t tabl at a given level of significance and degrees of freedom v = n-k, where n is the number of evaluated sites and k is the number of factors of historical and cultural assessment. If the actual value of the Student's t-test is greater than the tabular value, then the model parameters (factors included in the model) are significant.
In addition to evaluating the model itself, an analysis of the remainders (the difference between the model and actual values of the area of protection zones) is carried out. In order to exclude a situation in which the model does not include factors that have a significant effect on the area of the protection zones, which leads to autocorrelation of remainders, it is necessary to calculate the Durbin-Watson model using Formula (4) [45]: where e t is the remainders of the equation of the dependence of the zone area on the factors of historical and cultural assessment.
The algorithm for detecting autocorrelation of remainders based on the Durbin-Watson model is as follows. The hypothesis H 0 , the absence of autocorrelation of remainders, is put forward. Alternative hypotheses H 1 and H 1 have positive or negative autocorrelation in the remainders. Further, using special tables, the critical values of the Durbin-Watson model, dt and du, are determined for a given number of observations N = 25 (number of sites), the number of independent variables of the model M = 4 (factors of historical and cultural assessment in models), and significance level α = 0.95. According to these values, the numerical interval [0.4] is divided into five segments [43].
If the observed value of the Durbin-Watson model is less than the critical value 4-d u , that is d obs < 4-d u , then the main hypothesis of the absence of first-order autocorrelation between the remainders of the regression model is accepted.
To determine the normality of the distribution of the sample of predicted values of the area of protection zones, the Kolmogorov-Smirnov test is used [44]. We test hypothesis H 0 : F 1 (x) = F 2 (x) so that the data is described by the same distribution function. To do this, it is necessary to find the largest deviation according to Formula (5): where n 1 , n 2 are the frequencies for the predicted and actual values of the protection zone, respectively; n 1 *, n 2 * are the accumulated frequencies for the predicted and actual values of the area of protection zones of cultural heritage sites, respectively.
To conclude that the test criterion under consideration is similar between the predicted and actual values of the protection zone, it is necessary to compare the experimental value of the criterion with its critical value determined from a special table based on the significance level α = 0.05. As a null hypothesis, the statement that the compared groups differ insignificantly from each other in the level of assimilation is accepted. In this case, the null hypothesis shall be accepted if the observed value of the criterion does not exceed its critical value λ emp < λ theor .
The sixth stage of this process is the analysis and interpretation of the coefficients and the type of the obtained models. At the last stage, conclusions were made about the differences between the lists of factors that affect different types of protection zones.
To interpret the area of the protection zone, the average coefficient of elasticity is calculated, which shows how much the percentage will change on average and the resulting factor, when the dependent factor changes by 1% (6): where x, y are model values of the regressions and indicator [43,44].

Model of the Size of the Protection Zone of a Cultural Heritage Site
The indicators characterizing the quality of the model of the Protection Zone (PZ) area, expressed by linear multiple regression, turned out to be unsatisfactory (Table 2).
By developing various types of models of the protection zone area and analyzing their characteristics (Figure 7), it was found that, in terms of indicators, the most optimal was the multiplicative function, as shown in Figure 8.
The complex multiplicative function of the security zone area model has the following form (7): where y is the model of the protection zone of the cultural heritage site, x F1 is the value of the urban factor, x F3 is the value of the historical factor, x F12 is the value of the educational factor, and x F13 is the value of the degree of uniqueness of the building and structures. By developing various types of models of the protection zone area and analyzing their characteristics (Figure 7), it was found that, in terms of indicators, the most optimal was the multiplicative function, as shown in Figure 8.  The complex multiplicative function of the security zone area model has the following form (7): where y is the model of the protection zone of the cultural heritage site, xF1 is the value of the urban factor, xF3 is the value of the historical factor, xF12 is the value of the educational factor, and xF13 is the value of the degree of uniqueness of the building and structures. By developing various types of models of the protection zone area and analyzing their characteristics (Figure 7), it was found that, in terms of indicators, the most optimal was the multiplicative function, as shown in Figure 8.  The complex multiplicative function of the security zone area model has the following form (7): where y is the model of the protection zone of the cultural heritage site, xF1 is the value of the urban factor, xF3 is the value of the historical factor, xF12 is the value of the educational factor, and xF13 is the value of the degree of uniqueness of the building and structures. In order to estimate the parameters of the developed model by the least-squares method, this function was linearized by introducing the following substitutions (8) [38,43]: The parameters of the nonlinear regression model for the area of the security zone are presented in Table 3. Table 3. Quality indicators of the security zone area model.

Quality Criterion for the Developed Model Actual Value Critical Value Condition of Compliance Conclusions
Determination coefficient R 2 0.710 Chadok scale R 2 > 0.7; high correlation.
The condition is met. The model sufficiently describes the initial data.
Fisher's coefficient F 12.212 3.072 Ftabl < Fobs The condition is met. The regression equation is significant.
Values of the Student's t-criterion for the coefficients of the security zone area model To interpret the simulation results, we will use the calculations of the elasticity coefficients. Elasticity coefficients show, by percentage, how much change the value of the area of the protection zone will occur with an increase in the value of the factor by 1% [41]. The calculated values of the elasticity coefficients are presented in Table 4. Table 4. Elasticity coefficients of a function.

Factor
The Value of the

Coefficient of Elasticity E Interpretation
Urban factor −4.531 The area of the protection zone decreases by 4.53%, with an increase in the importance of the object in the urban planning environment (from a discordant object to an urban planning dominant) by 1%.
Historical factor 0.960 The area of the protection zone increases by 0.96%, with an increase in the historical value of the object by 1%.
Factor "Possession of educational, patriotic, spiritual, and aesthetic value" −0.010 The area of the protection zone decreases by 0.01% with an increase in the value of the educational and spiritual value of the object by 1%.
Factor "The degree of uniqueness of the technical construction of a structure, the latest building materials and technologies, a constructive solution" −0.333 The area of the protection zone is reduced by 0.33%, with an increase in the degree of uniqueness of technical construction and design solutions by 1%.
The analysis of the results presented in Table 3 describes the negative current trend in the field of establishing boundaries of protection zones in Orenburg's historical settlement of regional significance. The bottom line is that the higher the value indicators of the cultural heritage site, the smaller the established size of the protection zone. This phenomenon can be explained by the desire to reduce the area of land with stringent modes of restrictions on land use due to the presence of a cultural heritage site.

Model of the Size of DCZ&BACZ of a Cultural Heritage Site
To characterize the area of DCZ&BACZ for the cultural heritage site, a multiple linear regression model was built with criteria meeting the requirements of statistical characteristics. Its parameters are presented in Table 5. Linear regression for the dependence of the DCZ&BACZ area of the cultural heritage site and estimated factors has the form (9): y = 95,222 + 27,243x F4 + 37,967x F7 − 51,977x F8 − 19,328x F9 (9) where y is the DCZ&BACZ area, m 2 , x F4 is the value of the cultural factor, x F7 is the value of the protective factor, x F9 is the value of the construction and technical factor, x F2 is the value of the landscape and environmental factor, and x F8 is the value of the functional factor.
Analysis of the resulting model allows the interpretation as follows: First, the coefficients at x F4 , x F7 show an increase in the area of the DCZ&BACZ of the cultural heritage site by 37,967.35 m 2 with an increase in the degree of the object's influence on the public, cultural, and educational life of the city by one point; Second, the coefficients at x F8 , x F9 show a decrease in the area of the DCZ&BACZ of the cultural heritage site by 51,977.99 m 2 and 19,328.88 m 2 , respectively, with an increase in the degree of adaptation of the object to modern conditions and with an increase in the uniqueness of technical construction by one point.
For the resulting models, certain limitations should be stipulated. Due to the fact that the development of historical cities of regional and federal significance differs due to the originally laid down historical characteristics, and considering that in cities of federal significance, as a rule, united protection zones are established, the developed models are applicable only for the historical city of Orenburg.
Analyzing the obtained models, it can be concluded that the dependences of the buffer zone areas and DCZ&BACZ on the assessment factors that are used in the process of historical and cultural assessment of lands have acceptable quality indicators and they can be objectively interpreted (Table 6). Table 6. Differentiation of the factors of historical and cultural assessment of lands in the process of establishing the area of protection zones for cultural heritage sites, depending on the type of zone.

Factor
Characteristics of the Urban Environment Protection Territory DCZ&BACZ

Cultural
Characteristics of the cultural and social orientation of the building Figure 11. Polypseudorotaxanes inter-or intra-crosslink with each other via hydrogen bonding. Table 2. Ten percent 10KPEGMA and 10% P-10K solutions with 10% α-CD.

Construction and technical
Assigning the object to architectural style Figure 11. Polypseudorotaxanes inter-or intra-crosslink with each other via hydrogen bonding. Table 2. Ten percent 10KPEGMA and 10% P-10K solutions with 10% α-CD.

Functional
The degree of adaptation in modern conditions Figure 11. Polypseudorotaxanes inter-or intra-crosslink with each other via hydrogen bonding. Table 2. Ten percent 10KPEGMA and 10% P-10K solutions with 10% α-CD.         The symbol means that the factor is included in the model and affects the size of the zone, and the symbol O means that the factor is not included in the model and does not affect the establishment of the boundaries of this zone type.

Sample
The resulting models (7), (9) were tested on seven cultural heritage sites of regional significance for which no zones of protection of cultural heritage sites have been approved (Table 7). The obtained values of the area of protection zones can be used to determine the configuration and dimensions when graphically superimposed around the site by uniform distribution around the site. This also considers the existing buildings and the historical and cultural value of the adjacent territory. It is proposed to distribute the areas of the zones along the rectangle shape's perimeter from the boundaries of the site itself (Figure 9).
In further studies, this method will be modified using the historical and cultural assessment of territories.
The obtained values of the area of protection zones can be used to determine the configuration and dimensions when graphically superimposed around the site by uniform distribution around the site. This also considers the existing buildings and the historical and cultural value of the adjacent territory. It is proposed to distribute the areas of the zones along the rectangle shape's perimeter from the boundaries of the site itself (Figure 9). In further studies, this method will be modified using the historical and cultural assessment of territories.

Discussion
The use of economic and mathematical methods to determine the boundaries of zones for the protection of cultural heritage objects plays a special role in ensuring the rational use of historical territories, since it mathematically assesses the spatial and qualitative characteristics of construction that can affect the object and its zone. A. Khamis and N.K. Kamarudin [45] confirm the advantage of multiple regression analysis in the land

Discussion
The use of economic and mathematical methods to determine the boundaries of zones for the protection of cultural heritage objects plays a special role in ensuring the rational use of historical territories, since it mathematically assesses the spatial and qualitative characteristics of construction that can affect the object and its zone. A. Khamis and N.K. Kamarudin [45] confirm the advantage of multiple regression analysis in the land assessment process using traditional methods. S. Gnat [46] proved that even when working with small data sets, accurate evaluation results can be expected thanks to regression analysis algorithms.
The identification of qualitative factors affecting the historical environment and the establishment of protective zones is a complex and multistage process. In further studies, it is planned to apply the results to develop a methodology for historical and cultural assessment of a territory based on the use of fuzzy sets in combination with machine learning.
The study results correlate with the findings of M.B. Schlee [8] that historical background and spatial relationships influence the establishment of buffer zones.
In their study [47], A. Versaci and A. Cardici note that the protection zone of the monuments shall be 500 m from the boundaries of the site (experience of France). In contrast to this study, the proposed method and the developed models make it possible to approach each site individually, assessing its uniqueness and value.
T. Colleta [10] considers buffer zones as overlapping territories, the boundaries of which take development into account. This condition is also proposed to be observed in further developments. O. Martin and G. Piatti consider the buffer zone as the area around the site, the cadastral plot, and the level of protection. Therefore, it is very important to accurately determine its meaning and configuration [48]. K. Conradin and T. Hammer [49] propose, in addition to buffer zones, to allocate transition zones, which are less strict in nature, but contribute to the development of territories.
In addition, the obtained models of protected zones will be used to study the influence of the regimes of zones with special conditions for the use of territories on the value of real estate objects.

Conclusions
In modern conditions of transitioning to the sustainable development of territories, the issues of land tenure regulation, especially in historical cities, are becoming more and more relevant. Efficient and rational use of land as a limited resource is becoming a priority in the process of managing land and resource potential of a territory, while the most important problems of preserving immovable monuments as part of a system for sustainable development of territories are addressed with new solutions. One of these solutions is the use of mathematical and statistical methods for modeling the basic sizes of protection zones of cultural heritage sites, obtained from the data of real border projects using the city of Orenburg as an example.
Thus, according to the results of the study, we can conclude: First, by means of mathematical modeling, the differentiation of historical and cultural evaluative factors for each type of protection zone of cultural heritage sites has been proved. Second, the model for determining the area of DCZ&BACZ is characterized by a linear dependence, while the area of the protection zone is determined by a nonlinear multiplicative model.
Third, the obtained models of the dependence of historical and cultural evaluative factors and areas of the buffer zone, as well as DCZ&BACZ, are objectively interpreted and have satisfactory quality criteria.
The obtained modeling results can be used as a mathematical justification for determining the recommended (necessary and sufficient) sizes of zones for the protection of cultural heritage sites in the process of developing projects for the boundaries of such zones. The results of the research can be used to study the current state in the field of approving boundaries for protection zones, as well as in identifying the trends in developed zone designs, which improves the management of measured lands that contain cultural heritage sites. The obtained models can be useful in the process of carrying out the activities of scientific and design organizations, as well as bodies for the protection of cultural heritage sites in the city of Orenburg.