Parametric model for the analysis of urban planning activities

. Various mathematical approaches can be used to analyze the performance of decision-makers. Our parametric model is based on comparing the distribution functions of the public green spaces areas (PGSA) of urban signiﬁcance with a lognormal distribution. The data from the St. Petersburg Law No. 430-85 of October 08, 2007, were taken as the initial data. "On public green spaces." The values of the logarithms of the urban PGSA were taken to construct the empirical and theoretical distribution functions. The maximum moduli of diﬀerence are calculated to compare empirical functions with each other, theoretical functions with each other, and theoretical and empirical functions. A comparison of areas using our model showed some correlations between trends in the formation of PGSA of urban signiﬁcance and morphotypes of housing development.


Introduction
In the conditions of sustainable development of urban areas, a special place is occupied by the problem of ensuring their environmental safety [1]. Preserving the sustainable functioning of natural and urban environments allows the natural and ecological framework of the territory to be formed as part of the master plan of the city [2]. Urban landscapes' most important ecosystem function is to create a comfortable living environment for the population [3,4]. Since both the size of the urban population and the built-up area continue to grow over time, it is imperative to control the number of green spaces available to residents and the evenness of their distribution within the city limits [5,6]. The peculiarities of the urban geosystems development are that their behavior is subordinate to achieving certain goals [7]. Comprehensive assessment systems are developed, and sociological surveys of the population are conducted to analyze the current urban development situation [8,9]. However, existing ways and tools for understanding expanding urban systems need continuous improvement. Decision-makers use the more useful information and accumulated experience, the more effective the processes of self-organization and self-development of urban geosystems. A study of decision-makers performance in forming public green spaces areas (PGSA) of urban significance is based on data from St. Petersburg Law No. 430-85 of October 08, 2007. "On public green spaces" -the number and area of PGSAs of urban significance [10].
The study aims to identify common dependencies characteristic of the districts of St. Petersburg in the formation of PGSAs of urban significance.

Materials and methods
The study of the functions of the PGSA distribution for the districts of St. Petersburg is carried out sequentially. The first step is determining the empirical functions of the PGSA of urban significance distribution. The empirical distribution functions are calculated for each district of St. Petersburg. The obtained functions for the districts are compared with each other. The maximum modulus of the difference is taken as a measure of function comparison.
The empirical distribution function is determined by the formula: , where n is the total number of PGSA territories of urban significance located in the district, S i is the area of the i-th territory of PGSA of urban significance, and are the variable values of the areas of PGSA of urban significance, I is the indicator function of the set, which has the form: . The maximum modulus of the difference is calculated by the formula: , where a and b denote a particular city area.
In the second stage, the empirical functions obtained for the city districts for the distribution of PGSA of urban significance are compared with their corresponding functions for the log-normal distribution: (LN(μ, σ 2 )), whose function has a probability distribution density , where μ is the shift factor, σ is the scale factor. First, the parameters for the log-normal distribution are determined (the arithmetic mean, variance, and median, respectively, are calculated for logarithms of areas). Then a theoretical distribution function is constructed using the calculated parameters. The arithmetic mean of the logarithms of the areas is chosen as an estimate of the parameter μ, and the square root of the sampling variance is chosen as an estimate of the parameter σ. Next, the maximum modulus of the difference is calculated to compare the theoretical functions of the districts with each other.
The third stage compares the obtained theoretical functions of the distribution of the PGSA of urban significance with the empirical functions of the distribution for the various city districts. The maximum modulus of the difference between the empirical and theoretical distribution functions is calculated for comparison.
The steps of the algorithm are performed using the Microsoft Excel tools.
The mathematical results of the study are interpreted to assess the current urban situation in the areas.
Let us perform a comparative analysis of the obtained empirical and theoretical functions of PGSA of urban significance by groups of districts, transforming the graphs of functions for clarity. In Figure 2, on the abscissa axis, we plot the values of the logarithms of the areas of PGSA of urban significance on the axis of ordinates -the relative number of areas of PGSA of urban significance. According to the calculation results, the maximum modulus of difference between the empirical and theoretical functions for the districts is ( Figure 2

Discussion
To assess the processes of self-organization and self-development of urban systems, data from various sources are used, including regional statistical databases, remote sensing data, and the results of field surveys [11,12]. Our study examines the distribution of areas of urban PGSA in districts of the city of St. Petersburg. Various mathematical approaches, including probability theory and the law of probability distribution, are used in modeling the size distribution of phenomena occurring in urban geosystems [13,14]. Models involving mixtures of log-normal distributions are discussed, for example, in [15]. However, in [15], mixtures of log-normal distributions are used to describe the size of sand particles, oil fields, Internet sites, and the size of populated places. When we turn to the study of the distribution of the areas of PGSA of urban significance within the city limits, mixtures of log-normal distributions will also arise.
Our study, in contrast to [5,8], examines only a portion of the green areas within the city limits. We did not consider such types of urban green spaces as areas of public green spaces of local importance and landscaping reserve; areas of green spaces performing special functions; areas of green spaces of limited use; areas of protective forests; areas of green spaces of specially protected natural areas.
For the majority of districts of the city, the number of PGSAs of urban significance is more than 100 units, except for Kolpinsky (49 PGSA of urban significance), Krasnoselsky (49 territories), and Kronstadtsky (35 territories) districts. Analyzing the data for the logarithm of PGSA of urban significance, we determined the following: • the arithmetic mean values for most districts are below zero, except for Kolpinsky (0.835), Petrodvortsovsky (0.393), and Krasnoselsky (0.182) districts; • the arithmetic mean values for the districts of the central part of the city (historical buildings) are close to each other -Admiralteisky (-1,676), Vasileostrovsky (-1,549), Petrogradsky (-1,811) and Tsentralny (-1,888) districts, and for non-central (later buildings) districts the values increase by almost one.
• the standard deviation for central and non-central areas differs insignificantly.

Conclusion
Developed by the authors, the parametric model allows us to describe the urban development situation in St. Petersburg, which has developed as a result of the activities of decision-makers in the formation within the boundaries of the city PGSA of urban importance. Analyzing the list of the PGSA presented in the law, we concluded that the formation of small areas of PGSA in the city districts implements the process of their self-organization, which is expressed in close distributions for the territories of areas with similar characteristics of housing development morphotypes. For example, when comparing with each other the empirical distributions of PGSA of urban significance between the empirical functions of distributions of Vyborgsky and Krasnogvardeisky districts, the value of the maximum modulus of difference 0.07; of Kurortny and Primorsky districts -0.13; of Vyborgsky and Primorsky -0.21 was obtained. When comparing the degree of correspondence between the empirical distribution functions and the log-normal distribution functions, the maximum modulus of difference for the Frunzensky district is 0.06, and for the Krasnoselsky district is 0.19. Gradation of an estimation accepted by authors is expressed in relative values and looks like less than 0.1 -good compliance, 0.1-0.15 -satisfactory compliance, and more than 0.15 -bad compliance (gradation of estimation is accepted from the condition that at the research of 100 objects and more the deviation within 10 % is admissible). The parametric model allows identifying some of the dependencies characteristic of St. Petersburg districts in forming PGSAs of urban significance. The hypothesis that requires further research is the assumption that the proximity of the empirical to the log-normal distribution may be due to the degree of area development at the current stage of the city development.