Nonhomogeneus Generalisations of Poisson Process in the Modeling of Random Processes Related to Road Accidents

Abstract The stochastic processes theory provides concepts, and theorems, which allow to build the probabilistic models concerning accidents. “Counting process” can be applied for modelling the number of road, sea, and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The nonhomogeneous Poisson process, and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number, and number of people injured and killed in Polish roads. To estimate model parameters were used data coming from the annual reports of the Polish police.


INTRODUCTION
Von Bortkiewitsch (1898) calculated, using the data of the Prussian army, the number of soldiers who died, during 20 consecutive years, because of being kicked by a horse. He noticed that a random variable, say X, denoting the number of solders killed accidentally by the horse kick per year, has approximately Poisson distribution with parameter λ=0.61 [1/year]. Since then Poisson's distribution, and the corresponding stochastic Poisson process, have found use in various fields of science and technology.
A Poisson process and its extensions, are used in safety and reliability problems. They allow to construct the models denoting number of road, sea, and railway accidents in the given time intervals.
It should be mentioned, that this paper is an extension of article [3], because of the new data concerning the Polish road accidents in 2019 [11].
A stochastic process { ( ): ≥ 0} defined by the formula is called a counting process corresponding to a random sequence { : ∈ ℕ 0 }.
2. The process { ( ): ≥ 0} is the stochastic process with independent increments, the right continuous and piecewise constant trajectories; From the definition it follows, that the one dimensional distribution of NPP is given by the rule: The expectation and variance of NPP are the functions: The corresponding standard deviation is: The expected value of the increment ( + ℎ) − ( ) is: The corresponding standard deviation is : A nonhomogeneous Poisson process with ( ) = , ≥ 0 for each t ≥ 0, is a regular Poisson process. The increments of nonhomogeneous Poisson process are independent, but not necessarily stationary. A nonhomogeneous Poisson process is a Markov process. Scientific Journal of PNA

DATA ON MOTORIZATION AND ROAD ACCIDENTS IN POLAND
Quoted data from the Central Statistical Office from 2017, 2018, and 2019 were presented in reports of the Polish Police [9], [10], [11].

GENERAL DATA ON MOTORIZATION
Since the beginning of the 90's, the number of vehicles registered in Poland has been systematically growing

GENERAL DATA ON ROAD ACCIDENTS
A

MODEL OF THE ROAD ACCIDENT NUMBERS
Due to the nature of these events, pre-assumption that it is a nonhomogeneous Poisson process with some parameter ( ) > 0 , seems to be justified. The expected value of increment of this process is given by (9), while its one dimensional distribution is determined by (5). We can use practically these rules if the intensity function ( ) > 0 in known. To define this function one utilize information presented in table 2. The statistical analysis of the data shows that the intensity function ( ) can be approximated by the linear function ( ) = + .

ESTIMATION OF THE MODELS PARAMETERS
Dividing the number of accidents in each year, by 365 or 366 we get the intensity in units of [1 / day].
We approximate the empirical intensity by a linear regression function = + that satisfied condition Applying the rules (26) for the data from Table 2 and using Excel system we obtain: The linear intensity of accidents is: This function is shown in figure 1.
Therefore the one dimensional distribution of NPP is: where Λ( ) is given by (25).
Finally one can say that the model of the accident number on Polish roads is the nonhomogeneous Poisson process with the parameter Λ( ), ≥ 0, determined by (25).
Using data from the Table 1

ANTICIPATION OF ACCIDENT NUMBER
From (5) The corresponding standard deviation is:

Example 1.
We want to predict the number of accidents from June 1 st of 2020 to August 30 th of 2020. We also want to calculate the probability of a given number of accidents. . We assume that the NCPP is homogenous in this time interval, and a mean is calculateted in center of interval.
Finally we obtain = .
For the same data we obtain the expected value of injured number, denoted by , and corresponding standard deviation ( ) in the time interval [4900, 4992). We assume that the NCPP is homogenous in this time interval, and a mean is calculateted in center of that one. In this case Δ( ; ℎ) = 626.62. Using the same formulas we get the expectation , and the standard deviation = 25,03 of injured people number.

CONCLUSIONS
The nonhomogeneous Poisson process, and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number, and the number of injured and fatalities on Polish roads. To estimate model parameters one used data coming from the annual reports of the Polish police. Constructed models allowed to anticipate number of accidents at any time interval, with a length of h and the accident consequences. One obtained the expected value of fatalities or injured, and the corresponding standard deviation in the time interval [ , + ℎ).
The statistical distribution of fatalities number in a single accident, and statistical distribution of injured people number, and also statistical distribution of fatalities or injured number in a single accident, are computed. Scientific Journal of PNA Słowa kluczowe: wypadek drogowy, niejednorodny proces Poissona, niejednorodny złożony proces Poissona