Development of a gray box system identification model to estimate the parameters affecting traffic accidents

Shahriar Afandizadeh Zargari; Hamid Bigdeli Rad

doi:10.1515/nleng-2022-0218

Article Open Access

Development of a gray box system identification model to estimate the parameters affecting traffic accidents

Shahriar Afandizadeh Zargari and Hamid Bigdeli Rad

Published/Copyright: July 25, 2023

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From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

In this study, the gray box method has been used to model traffic accidents for the first time. This work examines the problem of estimating and identifying a single-input single-output state-space system. In this way, the state-space model was used, which has both a black box section (experimental data) and the parameters have been estimated by acquiring prior knowledge (white box). First, the state-space of the desired system is formed, and the algorithm for estimating the parameters and their convergence and the state vector estimation algorithm are written. In comparison, the system changes from nonlinear to linear. The parameters and prior knowledge are entered from the system. Finally, by implementing the presented method on the data related to the factors affecting accidents in Qazvin (Iran), the accuracy of the presented materials is investigated. The error output shows that initially, the error increased slightly, but then it shows a downward trend, and with the increase in the data, the error tends to zero (0.658). The results also show good fit and optimal accuracy of the model in less processing time.

Keywords: system identification; gray box model; state-space model; human factors

1 Introduction

Since crash data are discrete data, it is more appropriate to use the distribution of discrete variables such as Poisson or negative binomial to explain them [1,2,3]. For this modeling, if the mean and variance values of the response variable data are equal to each other, the Poisson distribution is suitable, and if the variance value is greater than the mean value (over-dispersion), a negative binomial model is preferred [4,5]. Suppose the variance of the numerical response variable is less than the mean value of this variable (under-dispersion), in that case, specially modified models (Hurdle Poisson or Negative binomial models) or generalized linear counting method should be used [6]. Abdel-Aty and Radwan used Poisson regression in 2000 to model the crashes. But they did not find a suitable model because of the difference between the mean value and variance of the dependent variable, which caused the over-dispersion [7]. Therefore, the Poisson-gamma model or (negative binomials) was presented as an option to solve this problem. The negative binomial model has been repeatedly used in accident analysis topics on urban freeways, urban roads, and arteries [8,9,10,11]. The expansion of Poisson and negative binomial models have been used to solve the problem of many zeros in crash data, called zero-negative binomial models [1,12]. A new system identification method is used in this study to overcome the limitations of the proposed models mentioned above. It can better account for heterogeneity issues in traffic crash prediction and could be applied to other roadway networks, and it has more accuracy than others. But to get accurate results, a lot of data are needed, and there is a need for up-to-date and robust computer system.

The system identification is to obtain a mathematical model of a phenomenon (e.g., dynamic system) with the help of laboratory information [13,14]. The purpose of system identification is to find the connection between input and output. This interface can include different models. By increasing the accuracy of these models, system identification is performed with greater accuracy [15,16]. The gray box system identification method consists of two parts. One part deals with the system’s dynamics (based on fixed rules such as Newton’s laws), and the other part deals with empirical data [17,18,19]. To obtain the first part for stochastic systems, one can go to the state-space [20,21]. State space is related to control systems. The first step in designing control systems is to formulate mathematical models suitable for the system to be controlled. These models can be derived from physical laws or experimental data [22,23]. After preparing the data, the state-space equations are extracted and optimized (system order). The data estimate the optimal parameters to perform the gray box method, which is the space between the black box and white box method, by optimizing the system state-space equations. Modeling control systems using state variables perform higher than the transfer function [24]. The transfer function model provides only a description of the input–output behavior of the system; hence it is called an external description. If the state variables also describe the system’s internal dynamics, the state space’s modeling is called the system’s internal description. The description of the state-space system gives a complete picture of its internal structure [25]. This model shows how state variables interact, how system input affects system state variables, and how system output can be calculated with different combinations of state variables [26]. Another advantage of using state-space representation is introducing prior knowledge in system analysis [27,28,29]. Also, optimizing the performance of the closed-loop system in the formulation of state space is easily done, and therefore, optimal control systems can be designed in state space.

Scardovi et al. investigated that learning capability of neural networks is equivalent to modeling physical events that occur in the real environment. Several early works have demonstrated that neural networks belonging to some classes are universal approximators of input–output deterministic functions. Recent works extend the ability of neural networks in approximating random functions using a class of networks named stochastic neural networks (SNNs). In the language of system theory, the approximation of both deterministic and stochastic functions falls within the identification of nonlinear no-memory systems. However, all the results presented so far are restricted to the case of Gaussian stochastic processes only, or to linear transformations that guarantee this property. This method aims at investigating the ability of SNNs to approximate nonlinear input–output random transformations, thus widening the range of applicability of these networks to nonlinear systems with memory [30]. Turchetti et al. address the problem of actively estimating the state of a stochastic dynamic system. By active estimation we mean the problem of finding a feedback control law that aims at “maximizing the amount of information” on the state of the system. In particular, they formalize the problem within the framework of stochastic optimal control where a suitable uncertainty measure is added to the process cost to be minimized. They do not assume that the classical linear Gaussian (LG) hypotheses are verified. Indeed, if these hypotheses are satisfied, the well-known separation principle states that the choice of the control law does not affect the estimation process, that is, any control law is “equally informative.” Moreover, in the case of linear systems, it is well known that, under the hypothesis of Gaussian noises, a posteriori probability density function (pdf) of the state of the system is Gaussian as well [31]. Therefore, in order to complete the present study, we can use alternative neural network-based techniques, which are also valid when sufficient statistics and tractable distributions based on them for sampling are not available.

Most early human error studies were based on psychological and behavioral theories. Psychological and behavioral theories have proven to be a good basis for human error theories [32]. In general, both individual and systemic approaches are used to investigate the human error. Most of the initial human error research of the systems was based on the error made in the first stage of the accident by the operator and emphasized the error with an individual approach, and considered human error as the leading cause of accidents. In recent years, to investigate human error in the road transport system, the interaction between hidden error conditions in the system, and errors made by the person operating the system, is often considered. For a short time, human error has been considered a system defect instead of an individual defect [33]. Conducting human error research with a systems approach is a complex task. Although more attention has been paid to this approach, the main view of human error in the road transport system is still individual, and the crimes of accidents are focused only on the driver. The role of hidden circumstances is not considered. This issue is undesirable for error management and safety improvement [34,35]. Therefore, in this study, due to the gap in identifying systems with appropriate accuracy in traffic safety, an attempt has been made to address the gray box method concerning human factors. State-space modeling has been used in this study to help reduce hidden errors and noise despite the complexities of analyzing the factors affecting accidents to provide a more accurate model.

2 Methodology

The purpose of this study is to identify the system of traffic accidents concerning the influential human factor. According to Figure 1, an analysis of research gap in traffic accidents was conducted by reviewing the literature. So far, no comprehensive study has been conducted on identifying traffic accident systems by the gray box method, and they are generally in the form of black boxes. In the following, a case study is introduced, and data are collected. The state-space model has been used to identify the gray box system, which has a black box section (experimental data) and has estimated the parameters (gray box) by acquiring prior knowledge. In this system, the state-space model is used. In the proposed model, instead of using changes per unit time, parameter changes are used for each input data (person). The steps of the state-space method and its linearization are explained step by step in the following sections.

Figure 1

Methodology flowchart.

2.1 State space

The general equation of state space is as follows:

(1) x ( p + 1 ) = A x ( p ) + B u ( p ) y ( p ) = C x ( p ) + D u ( p ) + e ( p ) ,

where x(p) ∈ ℝⁿ is the state vector, u(p) ∈ ℝ is the system input, y(p) ∈ ℝ is the system output, A is the state matrix, B is the input matrix, C is the output matrix, and e(p) ∈ ℝ is the error with zero mean value. It is assumed that the order n is definite and p represents each driver.

u ( p ) = 0 , y ( p ) = 0 and e ( p ) = 0 for p ⩽ 0 ,

If the numerator of the transfer function is less than its denominator, then D = 0.

(2) A = 0 1 0 0 0 0 0 1 ⋱ 0 0 0 ⋱ ⋱ ⋮ ⋮ ⋮ ⋱ 0 1 − a n − a n − 1 − a n − 2 … − a 1 ∈ ℝ n × n B = b 1 b 2 ⋮ b n − 1 b n ∈ ℝ n C = [ 1 , 0 , 0 , … , 0 ] ∈ ℝ 1 ⁎ n ,

(3) s = c cA ⋮ cA n − 1 = I n .

The purpose of this study is to develop a new algorithm to estimate the parameters of matrices A and B (i.e., parameters a i and b i ) and the state vector of the system x(p) using input–output data {u(p), y(p)}. Only input and output data {u(p), y(p)} are available, but the state vector x(p) is not available. Therefore, the state vector x(p) must be eliminated from Eqs. (1) and (2) to obtain a new identification model that includes only inputs and outputs. A new algorithm is created to identify the parameters of the matrices A and B (i.e., parameters a i and b i ) and to estimate the system vector x(p) from the available input and output data {u(p), y(p)} using Eqs. (5) and (6):

(4) y ( p ) = c x ( p ) + e ( p ) ,

(5) y ( p + 1 ) = c x ( p + 1 ) + e ( p + 1 ) = c [ A x ( p ) + b u ( p ) ] + e ( p + 1 ) = c A x ( p ) + c b u ( p ) + e ( p + 1 ) ,

(6) y ( p + 2 ) = c A x ( p + 1 ) + c b u ( p + 1 ) + e ( p + 2 ) = c A [ A x ( p ) + b u ( p ) ] + c b u ( p + 1 ) + e ( p + 2 ) = c A 2 x ( p ) + c A b u ( p ) + c b u ( p + 1 ) + e ( p + 2 ) ,

(7) y ( p + n − 1 ) = c A ( n − 1 ) x ( p ) + c A ( n − 2 ) b u ( p ) + c A ( n − 3 ) b u ( p − 1 ) + c b u ( p + n − 2 ) + e ( p + n − 1 ) ,

(8) y ( p + n ) = c A n x ( p ) + c A n − 1 b u ( p ) + c A n − 2 b u ( p − 1 ) + c b u ( p + n − 1 ) + e ( p + n ) .

2.1.1 Defining vectors and matrices

(9) Y ( p + n ) = y ( p ) y ( p + 1 ) ⋮ y ( p + n − 1 ) ∈ R n ,

(10) U ( p + n ) = u ( p ) u ( p + 1 ) ⋮ u ( p + n − 1 ) ∈ R n ,

(11) E ( p + n ) = e ( p ) e ( p + 1 ) ⋮ e ( p + n − 1 ) ∈ R n ,

(12) H = 0 0 … 0 0 c b 0 … 0 0 c A b c b ⋱ ⋮ ⋮ ⋮ ⋮ ⋱ 0 0 c A n − 2 b c A n − 3 b … c b 0 ∈ ℝ n × n .

Eq. (8) is written in the form of Eq. (13).

(13) Y ( p + n ) = S x ( p ) + M U ( p + n ) + E ( p + n ) = x ( n ) + M U ( p + n ) + E ( p + n ) .

(14) x ( n ) = Y ( p + n ) − M U ( p + n ) − E ( p + n ) .

Determining the vector of the parameter θ using the information of the vector w(p)

(15) w ( p + n ) = Y ( p + n ) − E ( p + n ) U ( p + n ) ∈ R 2 n ,

(16) θ = θ a θ b ∈ R 2 n ,

(17) θ a = [ cA n ] T ∈ R n ,

(18) θ b = [ − cA n H + [ cA n − 1 b , cA n − 2 b , … , cb ] ] T ∈ R n .

By substituting Eq. (14) in Eq. (8)

(19) y ( p + n ) = cA n [ Y ( p + n ) − MU ( p + n ) − E ( p + n ) ] + cA n − 1 bu ( p ) + cA n − 2 bu ( p + 1 ) + ⋯ + cbu ( p + n − 1 ) + e ( p + n ) = cA n [ Y ( p + n ) − E ( p + n ) ] − cA n MU ( p + n ) + [ cA n − 1 b , c A n − 2 b , … , cb ] U ( p + n ) + e ( p + n ) = [ Y T ( p + n ) − V T ( p + n ) , U T ( p + n ) θ a θ b + e ( p + n ) = w T ( p + n ) θ + e ( p + n ) .

By replacing p in Eq. (19) to p–n, we obtain

(20) y ( p ) = w T ( p ) θ + e ( p ) .

2.1.2 Algorithm for estimating parameters and its convergence

The criterion function is determined according to Eq. (21).

(21) L ( θ ) = ∑ i = 1 n e 2 ( i ) = ∑ i = 1 n [ y ( i ) − w T ( i ) θ ] 2 .

The recursive least squares algorithm uses the minimization of the criterion function L(θ) to estimate the parameter θ.

(22) θ ˆ ( p ) = θ ˆ ( p − 1 ) + G ( p ) [ y ( t ) − w T ( p ) θ ˆ ( p − 1 ) ] ,

(23) G ( p ) = Z ( p ) w ( p ) = ( Z ( p − 1 ) w ( p ) ) ( 1 + w T ( p ) Z ( p − 1 ) w ( p ) ) ,

(24) Z ( p ) = [ I − G ( p ) w T ( p ) ] Z ( p − 1 ) , Z ( 0 ) = z 0 I .

where G(p) ∈ ℝ²ⁿ is the gain vector and Z ( p ) ∈ R 2 n × 2 n is the covariance matrix.

Since the information vector w(p) in Eq. (24) contains unknown cases of noise e(p − i), the least-squares return algorithm does not apply in Eqs. (26)–(28). For this purpose, their estimates are used to define vectors,

(25) w ˆ ( p ) = Y ( p ) − E ˆ ( p ) U ( p ) ∈ R 2 n ,

(26) E ˆ ( p ) = e ˆ ( p − n ) e ˆ ( p − n + 1 ) ⋮ e ˆ ( p − 1 ) ∈ R n .

If θ ˆ ( p ) = θ ˆ a ( p ) θ ˆ b ( p ) is an estimate of θ = θ a θ b for the person p, according to Eq. (20), the estimate e(p) is calculated by Eq. (27):

(27) e ˆ ( p ) = y ( p ) − w T ( p ) θ ˆ ( p ) .

Replacing w(p) in Eqs. (22)–(24) with w ˆ ( p ) provides the following least squares parameter estimation algorithm for estimating θ:

(28) θ ˆ ( p ) = θ ˆ ( p − 1 ) + G ( p ) [ y ( t ) − w T ( p ) θ ˆ ( p − 1 ) ] ,

(29) G ( p ) = Z ( p ) w ( p ) = Z ( p − 1 ) w ( p ) 1 + w T ( p ) Z ( p − 1 ) w ( p ) ,

(30) Z ( p ) = [ I − G ( p ) w T ( p ) ] Z ( p − 1 ) , Z ( 0 ) = z 0 I ,

(31) e ˆ ( p ) = y ( p ) − w T ( p ) θ ˆ ( p ) ,

(32) W ˆ ( p ) = [ y ( p − n ) − e ˆ ( p − n ) , y ( p − n + 1 ) − e ˆ ( p − n + 1 ) , … , y ( p − 1 ) − e ˆ ( p − 1 ) , u ( p − n ) , u ( p − n + 1 ) , ⋯ , u ( p − 1 ) ] T .

2.2 State vector estimation algorithm

The relationship between the parameter θ vector elements, and the coefficients of the matrix or vectors a, b, and c is established. Multiplying Eq. (3) by b, we obtain

(33) cb cAb ⋮ cA n − 1 b = b ,

(34) cA k − 1 b = b k , k = 1 , 2 … n .

Using the above equations, the H matrix can be expressed as follows:

(35) H = 0 0 … 0 0 b 1 0 … 0 0 b 2 b 1 ⋱ ⋮ ⋮ ⋮ ⋮ ⋱ 0 0 b n − 1 b n − 2 … b 1 0 .

Multiplying Eq. (3) by A, we obtain

(36) cA cA 2 ⋮ cA n = A .

According to definition A, the last row of Eq. (36) creates Eq. (39):

(37) cA n = [ − a n , − a n − 1 , … , − a 1 ] .

After multiplying both sides of Eq. (39) with H:

(38) cA n H = − ( b 1 a n − 1 + b 2 a n − 2 + … + b n − 1 a 1 ) − ( b 1 a n − 2 + b 2 a n − 3 + … + b n − 2 a 1 ) ⋮ − b 1 a 1 0 T .

Using the above equations, the parameter vectors θ _a and θ _b in Eqs. (17) and (18) can be expressed as follows:

(39) θ a = [ cA n ] T ∈ = [ − a n , − a n − 1 , … , − a 1 ] T ,

(40) θ b = [ − cA n H + [ cA n − 1 b , cA n − 2 b , … , cb ] ] T ,

(41) = b 1 a n − 1 + b 2 a n − 2 + … + b n − 1 a 1 + b n b 1 a n − 2 + b 2 a n − 3 + … + b n − 2 a 1 + b n − 1 ⋮ b 1 a 1 + b 2 b 1 = a n − 1 a n − 2 … a 1 1 a n − 2 a n − 3 … 1 0 ⋮ ⋮ ⋮ ⋮ a 1 1 … 0 0 1 0 … 0 0 b 1 b 2 ⋮ b n − 1 b n .

If θ ˆ b ( p ) is the estimated θ b , the estimates of b and θ a are as follows:

(42) b ˆ ( p ) = b ˆ 1 ( p ) b ˆ 2 ( p ) ⋮ b ˆ n ( p ) ∈ R n ,

(43) θ ˆ a ( p ) = a ˆ n ( p ) − a ˆ n − 1 ( p ) ⋮ − a ˆ 1 ( p ) ∈ R n .

When θ ˆ ( p ) = θ ˆ a ( p ) θ ˆ b ( p ) is the vector estimation of the parameter θ (relations (28)–(32)), the estimate of a ˆ i ( p ) in relation to Eq. (43) is obtained. By replacing a ˆ i ( p ) and b ˆ i ( p ) in Eqs. (39)–(41), the following results are obtained:

(44) a ˆ n − 1 ( p ) a ˆ n − 2 ( p ) … a ˆ 1 ( p ) 1 a ˆ n − 2 ( p ) a ˆ n − 3 ( p ) … 1 0 ⋮ ⋮ ⋮ ⋮ a ˆ 1 ( p ) 1 … 0 0 1 0 0 0 0 b ˆ 1 ( p ) b ˆ 2 ( p ) ⋮ b ˆ n − 1 ( p ) b ˆ n ( p ) = θ ˆ b ( p ) ,

(45) b ˆ ( p ) = b ˆ 1 ( p ) b ˆ 2 ( p ) ⋮ b ˆ n − 1 ( p ) b ˆ n ( p ) = a ˆ n − 1 ( p ) a ˆ n − 2 ( p ) … a ˆ 1 ( p ) 1 a ˆ n − 2 ( p ) a ˆ n − 3 ( p ) … 1 0 ⋮ ⋮ ⋮ ⋮ a ˆ 1 ( p ) 1 … 0 0 1 0 … 0 0 − 1 θ ˆ b ( p ) .

By substituting p–n for p in Eq. (18), we obtain

(46) x ( p − n ) = Y ( p ) − H U ( p ) − E ( p ) .

By substituting the values of H and E (p) with their estimated values H ˆ ( p ) and E ˆ ( p ) , the state estimation vector is obtained.

(47) x ˆ ( p − n ) = Y ( p ) − H ˆ U ( p ) − E ˆ ( p ) .

Using Eqs. (35), (42), (43), (45), and (47), the state estimation algorithm can be summarized as follows:

(48) x ˆ ( p − n ) = Y ( p ) − H ˆ U ( p ) − E ˆ ( p ) ,

(49) Y ( p ) = [ y ( p − n ) , y ( p − n + 1 ) , … , y ( p − 1 ) ] T ,

(50) U ( p ) = [ u ( p − n ) , u ( p − n + 1 ) , … , u ( p − 1 ) ] T ,

(51) E ˆ ( p ) = [ e ˆ ( p − n ) , e ˆ ( p − n + 1 ) , … , e ˆ ( p − 1 ) ] T ,

(52) H ˆ ( p ) = 0 0 … 0 0 b ˆ 1 ( p ) 0 … 0 0 b ˆ 2 ( p ) b ˆ 1 ( p ) ⋱ ⋮ ⋮ ⋮ ⋮ ⋱ 0 0 b ˆ n − 1 ( p ) b ˆ n − 2 ( p ) … b ˆ 1 ( p ) 0 ,

(53) b ˆ 1 ( p ) b ˆ 2 ( p ) ⋮ b ˆ n − 1 ( p ) b ˆ n ( p ) = a ˆ n − 1 ( p ) a ˆ n − 2 ( p ) … a ˆ 1 ( p ) 1 a ˆ n − 2 ( p ) a ˆ n − 3 ( p ) … 1 0 ⋮ ⋮ ⋮ ⋮ a ˆ 1 ( p ) 1 … 0 0 1 0 … 0 0 − 1 × θ ˆ b ( p ) ,

(54) θ ˆ ( p ) = θ ˆ a ( p ) θ ˆ b ( p ) ,

(55) θ ˆ a ( p ) = [ − a ˆ n ( p ) , − a ˆ n − 1 ( p ) , … , − a ˆ 1 ( p ) ] T .

2.3 Linearization

Although the dynamics of most industrial processes and real systems are nonlinear, the analysis and design of control systems for the nonlinear model is complicated, and the implementation of nonlinear controllers in many practical cases is unnecessary. Linear control systems have been shown in practice to control an extensive range of real systems and complex industrial processes. Therefore, obtaining accurate linear models of nonlinear systems is very important and inevitable. In control systems, the linear approximation of a nonlinear mathematical model works well when system output changes around the system’s operating point. Therefore, linear approximation has a reasonable validity as long as the system response is always close to the system operating point. Linear control systems successfully control complex nonlinear processes at the operating point. A practical and successful method in linearizing the nonlinear equations of the system is based on the Taylor series. Consider the general function f(t). Assuming the Taylor series function is analytical, it describes the following infinite series [36]:

(56) f ( x ) = ∑ n = 0 ∞ 1 n ! d n f ( x ) d x n x = x 0 .

The point x 0 is called the operating point.

(57) f ( x ) = f ( x 0 ) + d f ( x 0 ) d x x = x 0 ( x − x 0 ) + 1 2 ! d 2 f ( x 0 ) d x 2 x = x 0 ( x − x 0 ) 2 + ⋯ .

Ignoring higher order value

(58) f ( x ) = f ( x 0 ) + d f ( x 0 ) d x x = x 0 ( x − x 0 ) .

More generally, the vector function f depends on several variables such as x1, x2,…, xn

(59) f ( x ) ≈ f ( x 0 ) + ∂ f ( x ) ∂ x x = x 0 ( x − x 0 ) ,

(60) f ( x ) = f 1 ( x ) f 2 ( x ) ⋮ f n ( x ) ,

(61) x = | x 1 x 2 … x n | T .

The ∂ f ∂ x matrix is called the Jacobian matrix, and is represented as J _x. The object (i, j) of the matrix J _x is equal to

(62) ( J x ) ij = ∂ f i ∂ x j .

(63)By applying the above linearization process to nonlinear equations of the system x ̇ ( p ) = f [ x ( p ) , u ( p ) , p ] , y ( p ) = g [ x ( p ) , u ( p ) , p ] .

Around the operating point ( u 0 , x 0 )

(64) x ( p ) = x 0 ( p ) + ∆ x ( p ) , u ( p ) = u 0 ( p ) + ∆ u ( p ) ,

where Δu and Δx give small deviations around the operating point, respectively.

(65) x ̇ ( p ) + ∆ x ̇ ( p ) = f ( u 0 , x 0 , p ) + J x ( u 0 , x 0 , p ) ∆ x ( p ) − J u ( u 0 , x 0 , p ) ∆ u ( p ) + higher order value .

Ignoring higher order value

(66) ∆ x ̇ ( p ) = A ( p ) ∆ x ( p ) + B ( p ) ∆ u ( p ) , A ( p ) = J x ( u 0 , x 0 , p ) , B ( p ) = J u ( u 0 , x 0 , p ) .

And the output nonlinear equation is obtained similarly as follows:

(67) ∆ y ( p ) = C ( p ) ∆ x ( p ) + D ( p ) ∆ u ( p ) , C ( p ) = g x ( u 0 , x 0 , p ) , D ( p ) = g u ( u 0 , x 0 , p ) ,

g x and g u are Jacobian matrices g(.) in terms of x(p) and u(p), respectively.

If the system is time invariable, the state equations and the linearized output of the system around the operating point are obtained by rewriting the above equations:

(68) x ̇ ( p ) = Ax ( p ) + Bu ( p ) , y ( p ) = Cx ( p ) + Du ( p ) .

2.4 Summary of identification algorithm

After linearization, set the value p = 1; initial values are θ ˆ ( 0 ) = 1 n / z 0 , Z(0) = z ₀ I, z 0 = 10 6 u(i) = 0 and y(i) = 0, and e ˆ ( i ) = 1 z 0 with i ≤ 0. A small positive number is given to ε.

Input and output data u(t) and y(t) are derived from w ˆ ( p ) using Eqs. (32), (49), and (50).

The gain vector G(p) is calculated using Eq. (23), and the covariance matrix Z(p) is calculated using Eq. (24).

The vector for estimating the parameter θ ˆ ( p ) is updated using Eq. (22).

e ˆ ( p ) is calculated using Eq. (31) and E ˆ ( p ) is obtained using Eq. (51).

a ˆ i ( p ) is calculated using (55), and b ˆ i ( p ) is calculated using Eq. (53). H ˆ ( p ) is obtained using Eq. (52).

Using Eq. (48), an estimate of the state x ˆ ( p − n ) is obtained.

The values of θ ˆ ( p ) are compared with θ ˆ ( p − 1 ) . If they are close enough, if ∥ θ ˆ ( p ) − θ ˆ ( p − 1 ) ∥ ≤ ε , the algorithm is finished, and the estimation of θ ˆ ( p ) is obtained, otherwise p is increased by 1 unit and transferred to step 2.

3 Case study

In this study, several accidents on the roads leading to the city of Qazvin (Figure 2), which is the capital of Qazvin province in Iran, were investigated. In cooperation with the Traffic Department and Shahid Rajaei Hospital (Accidents) in Qazvin, data are collected from medical records and the driving history of passengers and interviews. Part of the data was obtained by examining the medical records of several injured passengers. By telephone, it was established that their information was used solely to conduct scientific research. The other part of the data was obtained using the Instagram social network. Considering promotions, people participated in this virtual interview and answered questions. By explaining the reason for using the data, they were given the insight to provide accurate information and cooperate the most.

Figure 2

Case study map.

4 Method implementation and results

4.1 Introducing the parameters

Factors affecting accidents were reviewed in the literature section and the parameters such as “driving hour,” “number of occupants,” “driver age,” “accident history,” “number of family members,” “vehicle type,” and “age difference with father” were considered as human factors. Figure 3(a)–(g) show the distribution of parameters in the collected data. As shown from Figure 3(a), a dense range of driving is observed at 15–22 h. Therefore, more attention should be paid to drivers and road conditions during these hours. According to Figure 3(b), the number of single-passenger vehicles has a high share, and of course, cars with two and three passengers also have a high number. According to Figure 3(c), the age range of 26–39 years has the highest frequency, and according to Figure 3(d), most users either have no history of accidents or have had only one history of accident. Also, according to Figure 3(e), the majority of people live in families of 3 or 4 people. As seen in Figure 3(f), the quality of their cars in all four groups (low, medium, appropriate quality, and high quality) is distributed. Finally, the last parameter is seen in Figure 3(g), related to the age difference between the driver and his father. Thus, 23–32 years has the highest share.

Figure 3

Changes in the factors affecting accidents. Driving hours (a), number of occupants (b), age of the driver (c), history of accidents (d), number of family members (e), type of vehicle (f), and age difference with father (g).

Figure 4 also shows a graph of the output values of the collected data. The accident probability of the examined persons (R) is given in a frame.

Figure 4

Graph of changes in output values.

4.2 Prior knowledge

In the first step of implementing the gray box method, prior knowledge of the existing system is required [37,38]. Thus, the relationship between the influential factors and the response variable is shown in Figures 5(a)–7(e). As shown in Figure 5(a), the probability of an accident increases from 5 to 10 p.m. Figure 5(b) shows that single-passenger and four-passengers vehicles are more likely to crash than two- and three-passengers vehicles. As shown in Figure 5(c), drivers in ages between 18 and 19 are more prone to crash, and as their age increases, this trend decreases and rises again. Figure 5(d) also shows that people with a history of two accidents are more likely to have an accident again. It is interesting to note that people without a history of accidents are more prone to have an accident than those with a history of accident. According to Figure 5(e), the probability of an accident increases with the number of family members up to three persons, and then this trend decreases. Figure 5(f) shows that low-quality, high-quality vehicles are less likely to crash, with level 2 vehicles more likely to be involved in accidents. As shown in Figure 5(g), people with a minor age difference (with the father) are more prone to have an accident, and then the trend decreases to a 30 year difference, and then the trend rises again.

Figure 5

Basic knowledge of the relationship (a) of the driving hour variable (h) with the response variable R (accident probability); (b) the number of occupants variable (T) with response variable R; (c) driver age variable (s) with response variable R; (d) crash history variable (a) with response variable R; (e) the number of family members variable (f) with the response variable R; (f) vehicle type variable (v) with response variable R; (g) the age difference variable with the father (e) with the response variable R.

In Figures 6 and 7, the interaction of input variables and response variables are given. For example, Figure 6(d) shows that in the average values of the number of occupants and the age of the driver, the value of the response variable is at its lowest value, and they report a low probability of an accident. Increasing or decreasing these two variables increases the R values.

Figure 6

Prior knowledge of the three-dimensional relationship (a) of the driving hour variable (h) and the driver age variable (s) with the response variable R (accident probability); (b) driving variable (h) and number of family members (f) with response variable R; (c) driving variable (h) and age difference variable with father (e) with response variable R; (d) driver age variable (s) and occupant number variable (T) with response variable R; (e) occupant number variable (T) and crash history variable (a) with response variable R.

Figure 7

Prior knowledge of the three-dimensional relationship (a) of the driver age variable (s) and the crash history variable (a) with the response variable R (crash probability); (b) the driver age variable (s) and the family member variable (f) with the response variable R; (c) driver age variable (s) and vehicle type variable (v) with response variable R; (d) driver age variable (s) and age difference variable with father (e) with response variable R; (e) crash history variable (a) and vehicle type variable (v) with response variable R.

Through trial and error of different combinations by experimental design software (RSM-Design Expert), the best combination according to Eq. (69) was obtained as prior knowledge:

(69) R = θ 1 h + θ 2 t + θ 3 s + θ 4 a + θ 5 f + θ 6 v + θ 7 e + θ 8 hs + θ 9 hf + θ 10 he + θ 11 ts + θ 12 ta + θ 13 sa + θ 14 sf + θ 15 sv + θ 16 se + θ 17 av + θ 18 h 2 + θ 19 t 2 + θ 20 s 2 + θ 21 f 2 + θ 22 e 2 + θ 23 h f 2 + θ 24 he 2 + θ 25 t 2 s + θ 26 t 2 a + θ 27 s 2 a + θ 28 se 2 + θ 29 h 3 ,

R: accident probability, θ i : parameters value, h: driving hours, t: number of occupants, a: driver age, f: number of family members, v: vehicle type, and e: accident history.

4.3 Model implementation

After developing and executing the gray box state space model, the value of the parameter estimation error is illustrated in Figure 8. Initially, the error increased slightly, but then it is in a downward trend, and as the data increases, the error tends to zero.

Figure 8

Error values in parameter estimation.

Figure 9

Comparison of the observed values (black R) of the response variable and the values predicted by the gray box state space model (blue Y).

The parameter values, explained in Section 4.1 and Eq. (69), and the amount of error are shown in Table 1. Values of θ i with decreasing value δ are converged, and the final values are reported in the True Value row.

Table 1

Parameter values

δ	θ ₁₀	θ ₉	θ ₈	θ ₇	θ ₆	θ ₅	θ ₄	θ ₃	θ ₂	θ ₁	P
6.2457	0.0218	0.0168	0.0005	0.0675	−0.0405	0.0448	0.5869	0.2135	0.7541	−0.4253	200
2.1143	0.0223	0.0173	0.00052	0.0672	−0.0426	0.046	0.5852	0.2136	0.7491	−0.4211	500
1.8563	0.0229	0.0179	0.00053	0.0664	−0.0432	0.0455	0.5845	0.2128	0.7499	−0.4214	1,000
1.2814	0.0234	0.0184	0.00056	0.0671	−0.0449	0.0479	0.5856	0.2132	0.7402	−0.4181	2,000
0.7921	0.0238	0.0189	0.00055	0.0674	−0.0441	0.0488	0.5819	0.2111	0.7388	−0.4183	3,000
0.7138	0.024	0.0196	0.00059	0.0699	−0.0445	0.0493	0.5832	0.2116	0.7364	−0.4161	4,000
0.6587	0.0241	0.0199	0.00058	0.0689	−0.0442	0.0498	0.5791	0.2071	0.7345	−0.4151	5,000
	0.0239	0.0197	0.00059	0.0697	−0.047	0.0502	0.577	0.2045	0.7324	−0.4152	True value

δ	θ ₂₀	θ ₁₉	θ ₁₈	θ ₁₇	θ ₁₆	θ ₁₅	θ ₁₄	θ ₁₃	θ ₁₂	θ ₁₁	P
6.2457	0.00055	−0.1659	0.0073	−0.0184	−0.0151	0.00317	−0.0008	−0.0187	−0.1632	−0.0201	200
2.1143	0.00059	−0.1668	0.0074	−0.0191	−0.0158	0.00301	−0.001	−0.0201	−0.1638	−0.0209	500
1.8563	0.00063	−0.1674	0.0071	−0.0201	−0.0165	0.00311	−0.0018	−0.0211	−0.1648	−0.0218	1,000
1.2814	0.00065	−0.1679	0.0059	−0.0228	−0.0178	0.00291	−0.0016	−0.0219	−0.1622	−0.0225	2,000
0.7921	0.00068	−0.1684	0.0061	−0.0208	−0.0162	0.00297	−0.0015	−0.0231	−0.1669	−0.0251	3,000
0.7138	0.00069	−0.1689	0.0069	−0.0218	−0.0169	0.00301	−0.0017	−0.0229	−0.1651	−0.0234	4,000
0.6587	0.00064	−0.1691	0.0067	−0.0225	−0.0174	0.00309	−0.0021	−0.0225	−0.1659	−0.0243	5,000
	0.00066	−0.1686	0.0063	−0.022	−0.017	0.00307	−0.0019	−0.0227	−0.1666	−0.0245	True value

δ	θ ₂₉	θ ₂₈	θ ₂₇	θ ₂₆	θ ₂₅	θ ₂₄	θ ₂₃	θ ₂₂	θ ₂₁	P
6.2457	−0.00014	0.00025	0.00026	0.0358	0.0042	−0.00028	−0.0033	−0.0013	0.0049	200
2.1143	−0.00022	0.00028	0.00021	0.0341	0.0047	−0.00032	−0.0035	−0.0009	0.0053	500
1.8563	−0.00013	0.00033	0.00025	0.0347	0.0049	−0.00037	−0.002	−0.001	0.0057	1,000
1.2814	−0.00015	0.00038	0.00028	0.0332	0.0058	−0.0004	−0.0021	−0.0015	0.0061	2,000
0.7921	−0.00016	0.00031	0.00039	0.0349	0.0054	−0.00042	−0.0025	−0.0011	0.0058	3,000
0.7138	−0.00019	0.00032	0.0003	0.0342	0.0061	−0.00044	−0.0027	−0.0012	0.0052	4,000
0.6587	−0.0002	0.00036	0.00036	0.0339	0.0051	−0.00049	−0.0032	−0.0014	0.0054	5,000
	−0.00017	0.0003	0.00032	0.0337	0.0054	−0.00045	−0.0029	−0.0013	0.0051	True Value

4.4 Validation

To test and validate the model, another set of data (except the data used in model development) was entered into modeling, and the results are reported in Figures 9 and 10. Figure 9 shows the observed values of the response variable and the values predicted by the gray box state-space model. As can be seen, the black and blue values, which represent the observed and modeled values, respectively, have good overlap and report a limited error.

Figure 10

Fitting the observed values of the response variable and the values predicted by the gray box state-space model.

According to Figure 10, the observed values of the response variable and the predicted values are plotted by the gray box state-space model. The drawn points indicate a good fit of these two types of data and indicate the desired accuracy of the model.

5 Conclusion

The purpose of this study is to identify the system of traffic accidents according to the human factor. The study area is introduced, and data are collected. The state-space model has been used to identify the gray box system, which has a black box section (experimental data), and the parameters have been estimated through prior knowledge. As a case study, several accidents on the roads leading to Qazvin (in Iran) were studied. After developing and executing the gray box state-space model, the error value of estimating the parameters was calculated and plotted. The error output indicates that the error increased slightly at first but then is downward and tends to zero as the data increase. The results also indicate a good fit of these two types of data and display the desired accuracy of the model. State-of-the-art modeling was used to meet more stringent constraints on control system performance, increase complexity, and facilitate access to computers. Simple modeling of a complex nonlinear system of traffic accidents was one of the achievements of this method and the current study. One of the advantages of this method is that prior knowledge of the system can be introduced to the model to increase the speed of convergence and access to the desired results. This feature is the distinguishing and superior point of the gray box method. In addition to experimental and historical data, prior knowledge can be given to the algorithm and system. Finally, it was possible to create a favorable model with a minimum error (0.658) with the help of Eq. (69) and the values of the parameters mentioned in Table 1. In this study, an attempt was made to improve the models of traffic accidents and increase their accuracy. Previous models by various researchers have generally dealt with the methods of identifying the black box. Due to the lack of prior knowledge in these cases, the data processing time is long, and reasonable accuracy is not achieved. The method presented in this study can predict the probability of traffic accidents in less time, with higher accuracy, so that the application of the model is more attractive for decision-makers in the field of transportation safety.

tel: Ex: 7138, +98-09122059097, fax: +98-21-77240398

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Appendix

A.1 Pseudocode of proposed methodology

Assuming the Taylor series function is analytical:

In order to linearization:

f ( x ) = ∑ n = 0 ∞ 1 n ! d n f ( x ) d x n | x = x 0

x 0 ← Operating point

So,

f ( x ) = f ( x 0 ) + df ( x 0 ) dx x = x 0 ( x − x 0 )

Vector function f → x 1 , x 2 , … . , x n :

f ( x ) ≈ f ( x 0 ) + ∂ f ( x ) ∂ x x = x 0 ( x − x 0 )

∂ f ∂ x → Jacobian matrix , → J x 0

( J x ) ij = ∂ f i ∂ x j ← J x

Applying the process → nonlinear equations of the system

x ( p ) = x 0 ( p ) + ∆ x ( p )

u ( p ) = u 0 ( p ) + ∆ u ( p )

∆ u , ∆ x ← Give small deviations around the operating point

x ( p ) + ∆ x ̇ ( p ) = f ( u 0 , x 0 , p ) + J x ( u 0 , x 0 , p ) ∆ x ( p ) + J u ( u 0 , x 0 , p ) ∆ u ( p ) + higher order value

Output nonlinear equation is obtained similarly:

∆ y ( p ) = C ( p ) ∆ x ( p ) + D ( p ) ∆ u ( p )

C ( p ) = g x ( u 0 , x 0 , p )

D ( p ) = g u ( u 0 , x 0 , p )

If the system is time invariable, the state equations and the linearized output of the system around the operating point are obtained by rewriting the above equations:

x ( p ) = Ax ( p ) + Bu ( p )

y ( p ) = Cx ( p ) + Du ( p )

p = 1 ; θ ˆ ( 0 ) = 1 n Z 0 , Z ( 0 ) = Z 0 I , Z 0 = 10 6 u ( i ) = 0 and y ( i ) = 0 and e ˆ ( i ) = 1 Z 0 Set to i ≤ 0 : A small positive number is given to ε .

Input and output data u ( t ) and y ( t ) ← w ˆ ( p ) Equation .

The gain vector G ( p ) ← Z ( p )

The vector for estimating the parameter θ ˆ ( p ) is updated.

e ˆ ( p ) is calculated and E ˆ ( p ) is obtained .

Table A1

Comparison of Gray box system identification with other service-oriented architecture techniques (Turchetti et al., [31])

Techniques	Advantage	Limitation
Polynomial approximation-based identifier – FRMply	Is defined by means of an approximating mapping based on a traditional approximation technique – the polynomial interpolation. They might represent a suitable choice as basis functions to model the nonlinearity in the identification problem	It is useful if the number of oscillations of the function, and therefore the polynomial degree, is not too large
Spline approximation-based identifier – FRMspl	Is defined by means of an approximating mapping based on another traditional approximation technique – the splines. This linear-in-the-parameter identifier is an extension of FRMply. Is able to capture more complicated dynamics without incurring the numerical instabilities of high-order polynomials.	Spline approximations of functions are a logical extension of using simple polynomials
Wavelet approximation-based identifier – FRMwlt	Is defined by means of an approximating mapping based on multiresolution decompositions. Wavelets are a set of self-similar mathematical functions that are used to approximate more complex functions via superpositioning principles. Wavelets also allow a signal to be divided into different frequency and time components.	This linear-in-the-parameter identifier is based on wavelets
RBF approximation-based identifier – FRMrbf	Is defined by means of an approximating mapping based on neural networks. RBFs are similar to MLPs with three layers (input, middle or “hidden” layer, and output). Also, like MLPs, RBFs can model any nonlinear function easily.	This nonlinear-in-the-parameter identifier is based on radial basis function networks. RBF does not input raw input data but rather it passes a distance measure from the inputs to the hidden layer. This distance is measured from some center value in the range of the variable (sometimes the mean) to a given input value in terms of a Gaussian function
Gray box system identification	This system has no restrictions on the type of data and has the following advantages: In gray box penetration testing, the tester does not have access to a system’s internal code; this means the tester will remain unbiased and unintrusive. Since the tester has a basic idea of how the system operates but not a thorough knowledge of its code, the testing reflects users and potential attackers accurately. Gray box penetration testing takes the relatively straightforward black-box testing technique and combines it with the code-targeted systems in white-box testing. Using the correct type of testing is essential for accurate results. Web apps have distributed systems. The absence of a source code means white box testing is impossible (while gray box testing relies on the definition and functional specifications, not code), as is black box testing. Gray box penetration testing conducts detailed studies on the internal system structure with a list of parameters that makes building technique easy and allows for good quality results that arrive relatively quickly.	−

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Received: 2022-04-28

Revised: 2022-06-07

Accepted: 2022-07-01

Published Online: 2023-07-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0218

Keywords for this article

system identification; gray box model; state-space model; human factors

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