# Technical implementation of the method of thermal potentials for the analysis of territories

In the first publication ( Using Thermal Potentials to Analyze Territories ), we described how thermal potentials can be used to analyze territories in general. In the following publications it was planned to describe how information about spatial objects is stored in databases, how models from the main components are built, and in general, what tasks of analyzing territories can be. But first things first.

Using the method of thermal potentials in the first place makes it possible to formulate a general idea of the territory of interest to us. For example, taking the initial information from the OSM in Barcelona (Catalonia), and conducting an integral analysis without selecting parameters, we can get “thermal” images of the first main components. We also spoke about “heat” maps in the first article, but it will not be out of place to recall that the term “heat” map arose due to the physical meaning of the potentials used for integral analysis. Those. in problems of physics, potential is temperature, and in problems of analysis of territories, potential is the total effect of all factors of influence on a specific point in the territory.

The following is an example of a “heat” map of Barcelona obtained as a result of an integrated analysis.

*“Heat” map of the first main component, without selecting parameters, Barcelona*

A, by setting any specific parameter (in this case, we chose industry), you can get a “heat” map directly from it.

*“Heat” map of the first main component, industry, Barcelona.*

Of course, the analysis tasks are much wider and more diverse than obtaining a general assessment of the selected territory, therefore, as an example in this article, we consider the task of finding the best place when placing a new object and the technical implementation of the thermal potentials to solve it, and in the following publications we will see others.

Solving the problem of finding the best place when placing a new object will help determine how much the territory is “ready to accept” this new object, how it will correlate with other objects already available on the territory, how much this new object will be valuable for the territory and what value it will add.

### Stages of technical implementation

Technical implementation can be represented by the sequence of procedures listed below:

- Preparation of the information environment.
- Search, collection and processing of source information.
- Building a grid of nodes in the analyzed territory.
- Dividing territory factors into fragments.
- Calculation of potentials from factors.
- Selection of factors for creating thematic integrated characteristics of the territory.
- Application of the method of principal components to obtain integral indicators of the territory.
- Creating models for choosing a place for the construction of a new facility.

#### Stage 1. Information Environment Preparation

At this stage, it is necessary to choose a database management system (DBMS), determine the sources of information, methods of collecting information, the amount of information collected.

For work, we used the PostgeSql database (DB), but it is worth noting that any other database working with SQL queries is suitable.

Initial information will be stored in the database - spatial data about objects: data types (points, lines, polygons), their coordinates and other characteristics (length, area, quantity), as well as all calculated values obtained as a result of the work and the results of the work itself .

Statistical information is also presented as spatial data (for example, districts of a region with statistics assigned to these regions).

As a result of the conversion and processing of the collected source information, tables are formed containing information on linear, point and area factors, their identifiers and coordinates.

#### Stage 2. Search, collection and processing of source information

As initial information for solving this problem, we use information from open cartographic sources containing information about the territory. The leader, in our opinion, is OSM information, updated daily around the world. However, if you manage to collect information from other sources - it will not be worse.

Information processing consists in bringing it to uniformity, eliminating inaccurate information and preparing for uploading to the database.

#### Stage 3. Building a grid of nodes in the analyzed area

To ensure the continuity of the analyzed territory, it is necessary to build a grid on it, the nodes of which have coordinates in a given coordinate system. At each node of the mesh, subsequently, the potential value will be determined. This will allow you to visualize homogeneous regions, clusters, and final analysis results.

Depending on the tasks to be solved, two options for constructing a grid are possible:

*- A grid with a regular step (S1)*is an overview throughout the territory. Using it, the potentials of factors are calculated, the integral characteristics of the territory (main components and clusters) are determined, and simulation results are displayed.

When choosing this grid, you must specify:

- grid step - the interval through which the grid nodes will be located;
- the border of the analyzed territory, which may correspond to the administrative-territorial division, or it may be the area on the map that limits the calculation area in the form of a polygon.

*- An irregular grid (S2)*describes individual points in the territory (e.g. centroids). It also calculates potentials from factors, determines the integral characteristics of the territory (main components and clusters). Modeling with the calculated main components is carried out precisely on the grid with an irregular step, and to visualize the simulation results, cluster numbers from grid nodes with an irregular step are transferred to grid nodes with a regular step according to the principle of proximity of coordinates.

In the database, information on the coordinates of grid nodes is stored in the form of a table containing the following information for each node:

- Node ID
- the coordinates of the node (x, y).

Examples of grids with regular steps to different territories with different steps are shown in the figures below.

*Grid coverage of the city of N. Novgorod (red dots). Grid coverage of the Nizhny Novgorod region (blue dots).*

#### Stage 4. Fragmentation of territory factors

For further analysis, extended territory factors must be transformed into an array of discrete factors so that each grid node contains information about each factor present in it. Linear factors are divided into segments, areal - into fragments.

The partitioning step is selected based on the area of the territory and the specifics of the factor; for large areas (oblast), the partitioning step can be 100-150 m, for smaller territories (cities) the partitioning step can be 25-50 m.

The database contains information about the results of the partitioning. a table containing for each fragment the following information:

- factor identifier;
- coordinates of the centroids of the obtained fragments of the partition (x, y);
- length / area of the fragments of the partition.

#### 5 stage. Calculation of potentials from factors

One of the possible and understandable approaches to the analysis of initial information is the consideration of factors as potentials from objects of influence.

We use the fundamental solution of the Laplace equation for the two-dimensional case - the logarithm of the distance from the point.

Taking into account the requirements of the final potential value at zero and the limitation of the potential value over large distances, the potential is determined as follows:

$$ at r

$$ for r2> r> = r1

$$for r> = r2

*Type of influence potential from a point object.*

The logarithmic function should be bounded at zero and reasonably bounded at a certain distance from the factors. If potential limitations were not made at large distances from the factor, then a huge amount of information would have to be taken into account far from the analyzed point, which practically does not affect the analysis. Therefore, we introduce the value of the radius of action of the factor, beyond which the contribution to the potential of the factor is equal to zero.

For the city, the magnitude of the radius of the factor is taken equal to half an hour

*pedestrian*access - 2,000 meters. For the region, we should talk about a half-hour

*transport*accessibility - 20,000 meters.

Thus, as a result of calculating the potential values, we have the total potential of each factor at each node of the regular grid.

#### 6 stage. Selection of factors for creating thematic integrated characteristics of the territory

At this stage, the most significant and informative factors are selected to create the thematic integrated characteristics of the territory.

The selection of factors can be carried out automatically, setting certain boundaries to the parameters (correlation, percentage of influence, etc.), or it can be done by experts, knowing the subject of the task and having some idea of the territory.

After the most significant and informative factors are selected, you can proceed to the next steps - interpretation of the main components.

#### 7th stage. Application of the method of principal components to obtain integral indicators of the territory. Clustering

The initial information about the factors of the territory, transformed at the previous stage into the potentials calculated for each grid node, is combined into new integral indicators - the main components.

The method of principal components analyzes the variability of factors in the study area and finds from the results of this analysis their most variable linear combination, which allows calculating the measure of their change - variance over the territory.

We take the general problem for compiling a model of approximation of a linear model function to given values

$$

**(2)**

Where i is the component number,

n is the number of components involved in the calculation,

j is the index of the site of the point of the territory, j = 1..k

k is the number of all nodes of the grid of the territory, according to which the main components

$$ - coefficient at the i-th main component of the model

$$- value of the i-th main component at the j-th point

B - free member of the model

$$Is the potential at the jth point of the factor for which we are building the model.

We determine the unknowns in equation

**(2)**by the least squares method, using the properties of the main components:

$$

**(3)**

Where i and i2 are component numbers, i <> i2

j is the territory node index

k is the number of all territory nodes

$$

**(4)**

(3) means the absence of correlation between components

(4) - the total value of any component is zero.

We get:

$$

$$

**(5)**

Here, the notation as in equation

**(2)**,$$means the average value of the potential.

This result can be interpreted as follows:

The model is a simple expression consisting of the average value of the modeled value and simple corrections to it of each of the components. At a minimum, the result should include the free term B and the first principal component. Below are examples of heat maps of the first major components in the Nizhny Novgorod region.

Based on the calculated main components, homogeneous regions can be constructed. this can be done both in all respects and, for example, only in pricing - i.e. perform clustering. To do this, you can use the K-means method . For each homogeneous region, the average value of the 1st main component characterizing the level of development of the territory is calculated.

An example of clustering by pricing parameters for the Nizhny Novgorod region is given below.

Also, using the obtained main components as parameters of the cost model, we can get the price surface of the territory.

*Price surface of the city of N. Novgorod*

#### 8 stage. Creating models for choosing a place for the construction of a new facility

To select the location that is most attractive for the location of a new facility (hereinafter referred to as the “facility”), it is necessary to compare the location of the “facility” with the surrounding infrastructure. For the functioning of the “object” there must be enough resources to ensure its functioning, a large number of factors must be taken into account, both positive and negative impact on the “object”. The totality of these factors can be defined as a "nutrient" environment for the functioning of the "object". Correspondence of the number of objects to the amount of resources of the territory is the basis for the stable functioning of the “object”.

The result of this comparison is the potential calculated for each point of the territory and allowing visually and analytically to analyze the choice of location for the placement of a new "object".

For trade, for example, among other things, a constant flow of buyers is important, which means that the list of factors that must be taken into account for objects of trade should include those that provide this flow (for example, social infrastructure, places of work, places of residence, highways, etc. )

On the other hand, when all conditions are met to ensure the functioning of trade objects, one must take into account the density of trade objects, since the “consumption” of the environment leads to a decrease in the possibility of purchases. The flow of people is not unlimited, the same applies to their financial resources and physical capabilities.

The algorithm for solving the problem of choosing the best location for an object is that the potential obtained as a function of the main components is as close as possible to the potential of a collection of objects of the "object" type; then the difference between the potential of the model and the potential of objects of the type "object" is calculated; the value of the contribution potential of one “object” is subtracted from the resulting difference; the negative values obtained in this case are replaced by zero, that is, those places in which there are not enough resources for the functioning of the new “object” are eliminated.

As a result of the actions, we obtain points of the territory with a positive potential value, that is, the location of the favorable location of our “object”.

In other words, we have the calculated potentials of all the factors at our disposal and the factor by which we want to build a model and analyze the selected thematic area (trade, industry, culture, social sphere, etc.).

For this, it is necessary to select factors for build environment variables - the main components - and continue to calculate models from them.

We propose to select factors by analyzing the correlations of all factors with the reference factor of the thematic area. For example, for culture, it can be theaters, for the educational system of the school, etc.

We calculate the correlation of the potential of the standard with the potentials of all factors. We select those factors whose correlation coefficients are in absolute value greater than a certain value (often the value of the minimum correlation coefficient = 0. 3).

$$

**(6)**

where$$- the absolute value of the correlation coefficient of the i-th factor with the standard.

Correlation is considered for all nodes of the grid covering the territory.

The difference between the potential of the model and the potential of objects of the same type as the new object in equation

**(2)**shows the potential of the territory that can be used to locate new objects.

As a result, we obtain the value of the potential, which characterizes the degree of benefit of the location of the "object" in the study area.

An example of how to graphically display the recommended areas for the location of a new “object” is given below.

Thus, the result of solving the problem of choosing the best location for a new object can be represented as an estimate of the territory in points at each point, giving an idea of the potential for placing an investment object, i.e. the higher the score, the more profitable it is to position the object.

In conclusion, it is worth mentioning that in this article we examined only one problem that can be solved using the analysis of territories, having data from public sources on hand. In fact, there are a lot of problems that can be solved with its help; their number is limited only by your imagination.