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3. Method for queue size assessment at crossings 
This paper proposes to apply the method of correlation and regression analysis to assess traffic and determine the 
level of dependence on the relevant affecting factors. The mathematical tools of the proposed method make it 
possible to determine the dependence of the feature on several factors, including those that require expert 
assessment procedures. In this case, many factors influence the analyzed feature (Fig. 1). The above dependence can 
be represented by the following expression: 
y = f (x
1
, x
2
, …., x
k
, u) 
(1) 
where y – analyzed feature; 
x
1
, x
2
, …., x
k
– factors affecting the feature; 
k – a number of factors; 
u – unsuspected error; 
In this case, the feature (y) and error (u) are random variables, the factors affecting the feature (x
1
, x
2
, …., x
k
) are 
non-random. 
From a mathematical point of view, the task is to find the function of dependence of the analyzed feature on the 
relevant affecting factors by processing an array data on the parameters of analog objects. The algorithm for 
building a regression type mathematical model is presented in (Fig. 3). 

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Denis Lomakin et al. / Transportation Research Procedia 36 (2018) 446–452 
449
4 
Denis Lomakin, Evgenii Fabrichnyi, Alexander Novikov / Transportation Research Procedia 00 (2018) 000–000 
Fig. 3. Building of a regression type mathematical model. 
The data array was processed using Microsoft Excel spreadsheets (data analysis package). The calculation is 
made by the least-squares method to calculate the curve that approximates the available data to the maximum. The 
function returns the array describing a straight line which equation takes the form: 
у = m
1
 

 х
1
 + m
2
 

 х
2
 + b 
(2) 
where у – traffic quality level; 
mi – variable of the i-th factor; 
хi – i-th factor value; 
b – free term of the regression equation. 
The factors х
1, 
х
2, 
…., х
k
simultaneously have a mutual effect on the dependent variable у. Due to the impossibility 
to cover the whole complex of affecting factors, as well as the difficulties in accounting the randomness of the 
action and the following consequence. In connection with the above, it is necessary to fix on the most important 
factors for which there is the array data, and in the regression function contains a random component b, which 
allows taking into account the effects of unaccounted factors.
Isaeva (2017) has revealed the most significant factors affecting the efficiency of using vehicles, which include 
speed, time of goods and passengers’ delivery, as well as idle vehicles at the traffic route. These factors depend on 
traffic flow parameters, as well as on road parameters, SRN characteristics and transport delays.
When performing the regression analysis procedure, this paper proposes to use the array data of statistical 
observations obtained using geoinformation databases (Kotikov, 2017). Based on the available data, the following 
factors were selected: 

transport flow speed, km/h; 

time of day. 
A random variable variation factor is a measure of the relative spread of a random variable; it shows what 
average proportion of this value is its average spread. The random variable variation factor is equal to the ratio of 
the standard deviation to the mathematical expectation. 

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Denis Lomakin et al. / Transportation Research Procedia 36 (2018) 446–452
 
Denis Lomakin, Evgenii Fabrichnyi, Alexander Novikov / Transportation Research Procedia 00 (2018) 000–000 
 5 
The standard deviation is, as per the probability theory and statistics, the most common indicator of the random 
variable variation factor dispersion with respect to its mathematical expectation. It is measured in units of the most 
random variable. The standard deviation is equal to the square root of the random variable variance. The standard 
deviation is used in calculating the standard error of the arithmetic mean, in constructing confidence intervals during 
statistical testing of hypotheses when measuring the linear relationship between random variables (Lomakin et al., 
2015). 
The mathematical expectation is the concept of the random variable mean value in the probability theory. 
A number of analogs n which are minimally necessary for the formation of adequate models of multiple linear 
regression, can be determined by a number related to the number of used factors k as n=2(k+2) or even n = 2(k+1). 
The Pearson’s coefficient (multiple R) allows establishing the tightness of the relationships between the features. 
If the relationship between the features is linear, then the Pearson’s coefficient determines the tightness of this 
relationship with high accuracy. Pearson's correlation assumes that two variables in question are measured at least 
within the interval scale. The Pearson’s coefficient takes the value from -1 to +1. The absolute value <0.5 means the 
absence of a stable relationship, 0.5 – 0.7 is the average level, > 0.7 is the presence of a close (strong) relationship. 
The coefficient of determination (R- square) shows a variation part of dependent variable, which is explained by 
the variation of the independent variable (values from 0 to 1). The coefficient of determination has a normal value 
≥ 0.61 (Table 1). 
Table 1. The relationship strength depending on the correlation coefficient. 
Correlation coefficient 
Relationship strength 
from ± 0.81 to ± 1.00 
strong 
from ± 0.61 to ± 0.80 
moderate 
from ± 0.41 to ± 0.60 
low 
from ± 0.21 to ± 0.40 
lowest 
from ± 0.00 to ± 0.19 
N/A