A case-oriented approach for analyzing the uncertainty of a reconstructed result based on the evidence theory

Abstract

Uncertainty analysis is an effective methodology to improve the reliability of an accident reconstruction result. Many existing methods can be employed in this field, which can confuse a practicing engineer who does not know these methods well. To make the selection easier, a case-oriented approach was proposed based on the evidence theory. Users only need to input uncertain traces and a selected accident reconstruction model to calculate the uncertainty of reconstructed results using the proposed approach. Three basic steps of the case-oriented approach are as follows: first, all types of input traces should be transformed into their evidence form; then, focal elements of the reconstructed result and their corresponding basic probability assignment (BPA) need to be calculated; finally, the belief function (Bel) and plausibility function (Pl) of the reconstructed results are calculated. Three common conditions, which are accidents with all interval traces, accidents with all probabilistic traces, and accidents with interval and probabilistic traces, were discussed based on the basic steps of the case-oriented approach. Furthermore, methods for how to transform different traces to their evidence form, how to calculate the interval of the response efficiently, and how to fuse high conflict evidence were presented. Numerical cases showed that the approach worked well in all conditions. Finally, a vehicle collisions accident case was presented to demonstrate the application of the proposed approach in practice.

Appendices

Appendix A - Numerical methodology

Typical numerical methodology includes the upper and lower bound method (ULM) [18], the difference method (DM) [19], and the Monte Carlo method (MCM) [20, 21]. As discussed in reference [17], all methods have advantages and disadvantages. Here, some simple numerical cases will be given to show some properties of these methods. If the ULM is employed to calculate the interval of

$$ f\left(x,y\right)={x}^2+y,x\in \left[1,2\right],y\in \left[1,3\right] $$

(A1)

then the true interval of f can be calculated, which is [2, 7]; if the ULM is employed to calculate the interval of

$$ f\left(x,y\right)={x}^2+{y}^2,x\in \left[-1,1\right],y\in \left[-1,1\right] $$

(A2)

then the interval of f can be calculated, which is [2]; this result is far from the true interval [0, 2]. If the DM is employed here, then the interval of Eq. (A1) is [3, 6], while that of Eq. (A2) is [0, 1]; this means that the DM is an ordinary method. If the MCM is employed here with 10⁷ sample points, then the interval of Eq. (A1) is [2.0, 7.0], while that of Eq. (A2) is [0.0, 2.0]; this means that the MCM can work well. Back to the essence of the uncertainty analysis here, the main assignment is to determine the upper and lower bounds of the response in the definition domain. It is easy to understand that more sample points provide a more precise result. However, the other aspect is that more sample points require more calculation time. A simple and reasonable idea is to reduce sample points, so some new technology, such as the genetic algorithm (GA), can be introduced. Another numerical case is

$$ f(x)=\sin (x),x\in \left[0,7\right] $$

(A3)

By combining this case with Eqs. (A1) and (A2), the calculation results and its calculation time are listed in Appendix Table 6. From Appendix Table 6, it can be easily seen that the calculation time of the GA is shorter than the MC, but the result is the same as the MC, and it is the true interval.

Table 6 Results of the MC and GA

Conclusions based on the above discussion are to obtain a more precise result; more sample points should be generated in the definition domain and to speed up the calculation, i.e., to reduce the calculation time, an optimization algorithm, such as the GA, should be introduced in the field.

Appendix B - Theoretical methodology

Typical theoretical methodologies include the interval theory (IT) [28, 29], the grey theory (GT) [24, 30], and the affine theory (AT) [31, 32]. The reason why we named these theoretical methodologies is all these methods are based on strict and exact mathematics and all have four operations. For example, the IT is based on the interval theory, and the interval arithmetic is defined as:

(B1)

where $\underline{a},\overline{a}$ represents the lower and upper bounds of the interval a^I, and $\underline{b},\overline{b}$ represents the lower and upper bounds of the interval b^I.

Do all theoretical methods work well in the uncertainty analysis field? Some numerical cases will be given here.

If the IT is employed to analyze the interval of

$$ f\left(x,y\right)=\frac{x}{x+y},x\in \left[1,2\right],y\in \left[3,5\right] $$

(B2)

then the interval of f can be calculated, which is [0.14, 0.5], but it is different from the true value [0.17, 0.4]. If the model in Eq. (B2) is changed to

$$ f\left(x,y\right)=\frac{1}{1+y/x},x\in \left[1,2\right],y\in \left[3,5\right] $$

(B3)

then the truth interval of f can be calculated by the IT. This is the widely known interval expansion problem of the interval theory.

If the GT is employed to calculate the interval of Eqs. (B2) and (B3), the same interval [0.25, 0.29] can be obtained. This is an excellent property of the GT, and it means that results obtained from the GT will not change according to the expression of the model. Nonetheless, the results of Eqs. (B2) and (B3) are too risky. If the GT is employed to calculate the interval of

$$ f\left(x,y\right)={xy}^2,x\in \left[1,2\right],y\in \left[2,4\right] $$

(B4)

then the truth interval [4, 32] can be obtained.

If the AT is employed to calculate the interval of model (A2) in Appendix A, then the truth interval can be obtained. If the AT is employed to calculate the interval of models (B2) and (B4), their intervals [0.14, 0.41] and [− 3.5, 32], respectively, can be calculated.

Obviously, it is hard to conclude which method is the best one. Are there any other techniques that can be employed to improve the degree of accuracy of these methods? Yes, the sub-interval technique can be employed. Theoretically, if there is a sufficient sub-space domain that consists of these sub-intervals in the definition domain, the accurate interval of the response can be obtained. The model (B2) can be used as an example. The results are shown in Appendix Tables 7, 8, and 9, where

$$ \mathrm{Error}=\sqrt{{\left(a-{a}_0\right)}^2/2+{\left(b-{b}_0\right)}^2/2} $$

(B5)

And [a, b] is an arbitrary interval with the truth value [a₀, b₀].

Based on Appendix Tables 7, 8, and 9, the following conclusions are drawn. The primary conclusion is that the accurate interval of the response can be obtained if there are sufficient sub-intervals. The other conclusion is that the error of the calculation interval decreases as the number of the sub-intervals increases.

Table 7 Results of the IT

Table 8 Results of the GT

Table 9 Results of the AT

Appendix C - Solutions for fusing two arbitrary intervals

For two arbitrary intervals d₁ = [a₁, b₁] and d₂ = [a₂, b₂], solutions were given below.

• Condition 1.a₁ ≤ b₁ ≤ a₂ ≤ b₂. Then, the frame of discernment of the first organization is {d₁₁, d₁₂}, where d₁₁ = [a₁, b₁], d₁₂ = [a₂, b₂]; m(d₁₁) = 1, m(d₁₂) = 0. Similarly, the frame of discernment of the second organization is {d₂₁, d₂₂}, where d₂₁ = [a₁, b₁], d₂₂ = [a₂, b₂]; m(d₂₁) = 0, m(d₂₂) = 1. Under such conditions, the K in Eq. (3) can be calculated, which is 1 and it means that Eq. (3) cannot be used here to fuse these types of evidence. By employing Eq. (3), a solution can be given.

For a problem with m focal elements, the BPA of each focal element should be revised using Eq. (C1)

$$ m\left({A}_i\right)=\Big\{{}_{10^{m\left({A}_i\right)+1/m}\kern0.5em m\left({A}_i\right)\ge \frac{1}{m}}^{10^{m\left({A}_i\right)-1/m}\kern0.5em m\left({A}_i\right)<\frac{1}{m}} $$

(C1)

Then, the BPA obtained from Eq. (C1) should be normalized using Eq. (C2)

$$ m\left({A}_i\right)=\frac{m\left({A}_i\right)}{\sum \limits_{i=1}^mm\left({A}_i\right)} $$

(C2)

After that, Eq. (3) can be employed to fuse the evidence.

The situation given in condition 1 can be used as an example. According to Eq. (C1), m(d₁₁) = 31.6228, m(d₁₂) = 0.3162; m(d₂₁) = 0.3162, m(d₂₂) = 31.6228. According to Eq. (C2), m(d₁₁) = 0.9901, m(d₁₂) = 0.0099; m(d₂₁) = 0.0099, m(d₂₂) = 0.9901. According to Eq. (3), K = 0.9804, m(d₁₁) = m(d₂₁) = 0.5, m(d₁₂) = m(d₂₂) = 0.5. This result is reasonable. If we cannot deny the evidence obtained from different organizations, one reasonable way is to consider different BAPs of the same evidence is they are equal to each other.

• Condition 2.a₂ ≤ b₂ ≤ a₁ ≤ b₁. The solution is the same as condition 1.

• Condition 3.a₁ ≤ a₂ ≤ b₂ ≤ b₁. Then, the frame of discernment of the first organization is {d₁₁, d₁₂, d₁₃}, where d₁₁ = [a₁, a₂], d₁₂ = [a₂, b₂], d₁₃ = [b₂, b₁]. Similarly, the frame of discernment of the second organization is {d₂₁, d₂₂, d₂₃}, where d₂₁ = [a₁, a₂], d₂₂ = [a₂, b₂], d₂₃ = [b₂, b₁]; m(d₂₁) = 0, m(d₂₂) = 1, m(d₂₃) = 0. The value of m(d₁₁), m(d₁₂) and m(d₁₃) can be calculated using Eq. (C3). For an arbitrary interval [a, b] with a BPA = 1, the BPA of another arbitrary interval C = [c, d] can be calculated by

$$ m(C)=\frac{d-c}{b-a} $$

(C3)

where b ≤ c ≤ d ≤ a.

A simple case. In one traffic accident, the braking distance of the vehicle is [16, 18] m according to the police report, while it is [16.5, 17.5] m according to another report given by a traffic accident research institution. The frame of discernment of the braking distance is {[16, 16.5], [16.5, 17.5], [17.5, 18]}. According to the report from the accident research institution, m([16, 16.5]) = 0, m([16.5, 17.5]) = 1, m([17.5, 18]) = 0. According to the police report and Eq. (C3), m([16, 16.5]) = 0.25, m([16.5, 17.5]) = 0.5, m([17.5, 18]) = 0.25. Then, the K in Eq. (3) can be calculated, which is 0.5. Finally, the fused result can be given according to Eq. (3), which is m([16, 16.5]) = 0, m([16.5, 17.5]) = 1, m([17.5, 18]) = 0. It is unreasonable and intervals [16, 16.5] and [17.5, 18] cannot be deleted automatically.

The reason that some focal elements will be deleted automatically is that the BPA of these elements are equal to 0. To obtain a reasonable result, the BPA of these focal elements should be revised using Eqs. (C1) and (C2). According to Eqs. (C1) and (C2), the BPA obtained from the accident research institution can be revised, which is m([16, 16.5]) = 0.0207, m([16.5, 17.5]) = 0.9587, m([17.5, 18]) = 0.0207. Then, K = 0.5104, the fused result is m([16, 16.5]) = 0.01, m([16.5, 17.5]) = 0.98, m([17.5, 18]) = 0.01. This result is much more reasonable.

• Condition 4.a₂ ≤ a₁ ≤ b₁ ≤ b₂. The solution is the same as condition 3.

• Condition 5.a₁ ≤ a₂ ≤ b₁ ≤ b₂. Then, the frame of discernment of the first organization is {d₁₁, d₁₂, d₁₃}, where d₁₁ = [a₁, a₂], d₁₂ = [a₂, b₁], d₁₃ = [b₁, b₂]; m(d₁₃) = 0. Similarly, the frame of discernment of the second organization is {d₂₁, d₂₂, d₂₃}, where d₂₁ = [a₁, a₂], d₂₂ = [a₂, b₁], d₂₃ = [b₁, b₂]; m(d₂₁) = 0. According to the discussion above, the BPA of other focal elements can be calculated using Eq. (C3), and then the BPA should be revised using Eqs. (C1) and (C2) because the values of some BPAs are equal to 0; finally, all evidence from different organizations can be fused using Eq. (3).

A simple case. The interval of the braking distance from the police report is [16, 18] m, while from another accident research institution is [17, 19] m. The frame of discernment of the braking distance can be determined, which is {[16, 17], [17, 18], [18, 19]}. According to the police report and Eq. (C3), m([16, 17]) = 0.5, m([17, 18]) = 0.5, m([18, 19]) = 0. According to the report from the accident research institution and Eq. (C3), m([16, 17]) = 0, m([17, 18]) = 0.5, m([18, 19]) = 0.5. If Eq. (3) is employed here directly, K = 0.75, and m([16, 17]) = 0, m([17, 18]) = 1, m([18, 19]) = 0. The result is unreasonable. Hence, Eqs. (C1) and (C2) should be employed first, and then the BPA obtained from the police report can be revised, which is m([16, 17]) = 0.4835, m([17, 18]) = 0.4835, m([18, 19]) = 0.0329, while the BPA obtained from the report of the accident research institution is m([16, 17]) = 0.0329, m([17, 18]) = 0.4835, m([18, 19]) = 0.4835. Finally, Eq. (3) can be employed, and then K = 0.7342, and m([16, 17]) = 0.06, m([17, 18]) = 0.88, m([18, 19]) = 0.06.

• Condition 6.a₂ ≤ a₁ ≤ b₂ ≤ b₁. The solution is the same as condition 5.

Appendix D - Case 3

The model of the case is

$$ d= vt+\frac{v^2}{2a} $$

(D1)

It is a model used to calculate the braking distance of a vehicle. In the model, t represents the reaction time of the driver, v is the initial velocity of the vehicle, a is the deceleration, and d is the braking distance. In one case, probabilistic information of the input parameters can be obtained and all obey the normal distribution. The mean values of v, t, and a are 16 m/s, 0.8 s, and 0.749g (g is the gravitational acceleration, 9.81 m/s²), respectively; the standard deviations of v, t, and a are 1 m/s, 0.1 s, and 0.052 g, respectively. According to steps presented in the “Steps of the approach” section, set k = 50 and n₀ = 10, 50, and 500 to get the results presented in Figs. 19, 20, and 21, respectively. From Figs. 19, 20, and 21, the same conclusions as case 2 are drawn: first, the truth CDF is between the Bel and Pl; second, the Bel and Pl are close to each other and all converge to the true CDF as n₀ increases.

Fig. 19

CDF with n₀ = 10 in case 3

Fig. 20

CDF with n₀ = 50 in case 3

Fig. 21

CDF with n₀ = 500 in case 3