ACEAdvances in Civil Engineering1687-80941687-8086Hindawi Publishing Corporation56750910.1155/2012/567509567509Research ArticleObstacle Emergence Risk and Road Patrol PolicyKobayashiKiyoshi1KaitoKiyoyuki2PendyalaRam M.1Graduate School of ManagementKyoto UniversityYoshida-HonmachiSakyo-kuKyoto 606-8501Japankyoto-u.ac.jp2Department of Civil EngineeringOsaka University2-1 Yamada-okaSuitaOsaka 565-0871Japanosaka-u.ac.jp20121212012201201082011170920112012Copyright © 2012 Kiyoshi Kobayashi and Kiyoyuki Kaito.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

2. Basic Ideas of This Study2.1. Overview of Conventional Study

The research into mixture Poisson processes were pioneered by Fisher , and then various modifications were attempted [4, 5]. Poisson processes have been applied to the evaluation of operational risk and accident risk. In general, the model structure for a mixture Poisson process becomes extremely complicated, because an event probability distribution and an event interval probability distribution are combined. However, the Poisson-Gamma model used in this study has the simplest model structure among mixture Poisson process models and has the advantage that it can express a model theoretically. Another advantage is that it is possible to readily obtain indicators for controlling the road obstacle risk. For these reasons, the Poisson-Gamma model is employed in this study, to describe the emergence processes of road obstacles.

Here, suppose that a road obstacle follows the Poisson arrival with a certain arrival rate λ. In Figure 1, the number of road obstacles that emerged on a certain road section during the period from time τA to time t is expressed by a counting process n(t). Obviously, n(τA)=0 at the initial time τA. There emerged n road obstacles by time τB, and the n+1th obstacle did not emerge. From time τB, the second cycle starts with the initial time being τB. In the case where a road obstacle follows the Poisson arrival, the time of the emergence of the first road obstacle in the second cycle does not depend on the time of the emergence of the last road obstacle in the first cycle. Namely, the emergence of a road obstacle is not dependent on the history of road obstacle emergence.

3. Poisson Gamma Model3.1. Mixture Poisson Process

In the Poisson-Gamma model, the number of road obstacles observed in a certain unit period is expressed by a probability distribution. However, in the data obtained after road patrol, observation period varies according to road sections. Moreover, in order to discuss road patrol frequency, it is necessary to model the effects of patrol cycle on the obstacle emergence risk. In this circumstance, the authors propose a Poisson-Gamma model that explicitly considers patrol cycle (observation period). In addition, in order to secure the operability of the risk control indicators, the arrival rate distribution is expressed by a Gamma distribution with mean 1.

3.2. Formulation of the Model

3.3. Estimation Method of the Model

By conducting road patrol, it is possible to obtain the information on road obstacles. Suppose that a total of K patrol data samples have been obtained after road patrol. The information ej of the patrol sample j  (j=1,,K) is represented by el=(nj,zj,xi(j)), where i(j) denotes the code number of the target road section i of the data of the patrol sample j. In addition, nj,  zj, and xi(j) represent the number of observed road obstacles, the cycle of road patrol, and the characteristic vector of the road section i(j), respectively.

In the Poisson-Gamma model (9), unknown parameters are β and the dispersion parameter is ϕ. At this time, assume that the actual measurement information e̅={e̅j  (j=1,,K)} of the patrol sample j  (j=1,,K) has been obtained. The log likelihood function of the Poisson-Gamma model can be expressed by the following equation: ln{L(β,ϕ):e̅}=j=1K{ln[Γ(ϕ+n̅j)Γ(ϕ)]+n̅jln(μi(j)z̅j)-(n̅j+ϕ)ln(μi(j)z̅j+ϕ)+ϕlnϕ-lnn̅j![Γ(ϕ+n̅j)Γ(ϕ)]}. Here, there is the following relation regarding the gamma function: ln{Γ(ϕ+n̅j)Γ(ϕ)}=k=0n̅j-1ln(ϕ+k). Therefore, the log likelihood function can be converted as follows: ln{L(β,ϕ):e̅}=j=1K{k=0n̅j-1ln(ϕ+k)+n̅jln(μi(j)z̅j)-(n̅j+ϕ)ln(μi(j)z̅j+ϕ)+ϕlnϕ-lnn̅j!k=0n̅j-1ln(ϕ+k)}, where μi(j)=exp(x̅i(j)β). By using the log likelihood function (16), it is possible to obtain the maximum likelihood estimates of the parameters β and ϕ of the Poisson-Gamma model by means of the maximum-likelihood method. That is, the maximum likelihood estimate of the parameter β that maximizes the log likelihood function (16) is calculated as the parameter β̂=(β̂1,,β̂M) that satisfies the following condition: ln{L(β̂:e̅)}βm=0(m=1,,M). The optimization condition is Mth degree simultaneous nonlinear equations, which can be solved by using the sequential iteration method based on the Newton method. Furthermore, the estimator Σ̂(β̂) of the asymptotic covariance matrix of the parameter can be expressed by the following equation: Σ̂(β̂)=[2ln{L(β̂:e̅)}ββ]-1, where the right-hand side of (19) is the inverse matrix of the M×M Fisher information matrix whose (l,m)th element is 2ln{L(β̂:e̅)}/βlβm.

4. Model for Controlling the Risk of Obstacle Emergence4.1. Purpose in Risk Management

4.3. Extraction of Intensive Management Sections

As mentioned above, in the road section having the characteristic xi with the patrol cycle zi, the expected number of road obstacles E[nizi] can be expressed by the following equation: E[nizi]=exp(xiβ)zi. On the other hand, the sample mean X̅i* is defined as follows, using the number of road obstacles ni*li*(li*=1,,Li*) measured in the Li-time patrols in the target road section i*: X̅i*=li*=1Li*ni*li*Li*. At this time, there is the following relation with regard to the sample mean X̅i*: E[X̅i*]=Li*-1li*=1Li*E[ni*li*zi*]=E[ni*zi*]. Therefore, it can be understood that the sample mean X̅i* is identical to the unbiased estimator of E[ni*zi*]. The sample variance also can be expressed as follows, using the variance V[nizi] of the negative binomial distribution in the same way: V[X̅i*]=Li*-2li*=1Li*V[ni*li*zi*]=Li*-1V[ni*zi*]. These two equations indicate that as the number of times of patrol Li* becomes somewhat large, the variance of the sample mean X̅i* approaches 0 and X̅i* converges to E[ni*zi*], making stochastic convergence. Therefore, it is possible to statistically judge whether or not road obstacles emerge while following the Poisson-Gamma model, that is, whether or not road obstacles emerge in a normal manner, by specifying a probability distribution of the sample mean and evaluating whether the predetermined risk level p (e.g., p=0.9) satisfies the following relation: Pr(|X̅i*-E[ni*zi*]|y(i*))p. Here, the central limit theorem guarantees that the probability distribution of the sample mean X̅i* becomes a normal distribution as a whole when the number of times of patrol is large. Furthermore, when the normalization constant Z is defined as follows: Z=X̅i*-E[ni*zi*](Li*)-1V[ni*zi*], the normalization constant Z is subject to the standard normal distribution N(0,1). Therefore, (27) can be expressed as follows: Pr(|Z|y(i*)(Li*)-1V[ni*zi*]  )=Pr(|Z|yp(i*))p. The cumulative distribution function of the standard normal distribution is available in the form of a table of figures in a lot of literatures, and so it is possible to uniquely specify yp(i*) in the above equation. When the following condition is not satisfied with a given risk level p, |Z|yp(i*), it can be concluded that the road section concerned has outstanding characteristic regarding the emergence of road obstacles compared with other road sections.

5. Empirical Study5.1. Outline of the Application Cases

In this study, the authors analyze the database of road patrol conducted on general national road, Route A (target section length: 82.6 km), Route B (88.3 km), and Route C (93.6 km). This database has accumulated the data of daily and night-time patrols from April 1, 2009 to March 31, 2010. The daily logs of road patrol record not only the type and number of road obstacles but also the routes and road sections where road obstacles were discovered and the patrol times. As tabulated in Table 1, the three routes, which are the targets of the empirical analysis, are composed of 413, 442, and 468 unit sections, respectively. Here, the length of one road section is 200 m. In addition, as concrete road obstacles, fallen objects and pavement abnormalities are focused on. The road obstacles that emerged on the three national routes during the analysis-target period include 3,501, 3,079, and 3,817 samples on each route.

Route data.

RouteA1 and A2B1 and B2C1 and C2
Route length [km]82.688.393.6
Number of sample3,8203,2964,140
Number of fallen objects3,5013,0793,817
Number of average fallen objects8.486.978.16
5.2. Estimation of the Poisson-Gamma Model

In order to model the emergence processes of road obstacles, the Poisson-Gamma model was estimated. For this estimation, 11,256 sample data of normal patrol were available. As explanatory variables, the following parameters were adopted. Namely, the estimation equation can be expressed as follows: λi=exp(β1+β2xi2+β3xi3+β4xi4+βixi5)εi, where β1,  β2,  β3,  β4,  β5 represent parameters, xi2 represents large-size automobile traffic volume in the day time, xi3 denotes travel speed, xi4 represents the average amount of rutting depth, and xi5 represents roadside classification. β1 is a constant term. In this case, the arrival rate λi of the emergence of road obstacles is composed of the common characteristic β1 in all road sections, the common characteristic β2xi2+β3xi3+β4xi4+βixi5 in the same environmental condition, and the heterogeneous characteristic εj of each road section. The first-order optimization condition of (18) regarding the Poisson-Gamma model is given as simultaneous nonlinear equations, and the maximum likelihood estimator of unknown parameters was calculated with the Newton-Raphson method. The estimation results and t-value are shown in Table 2. This table tabulates the estimation results.

Estimation results of the Poisson-Gamma model.

 Poisson-Gamma model β1 β2 β3 β4 β5 ϕ Maximum likelihood estimate −4.82 8.51×10-5 1.57×10-2 9.49×10-3 0.11 3.20 (t-value) (−15.51) (14.56) (17.25) (4.22) (3.99) (20.22) Log likelihood −18,325 AIC 36,663

Cumulative discovery probability in the Poisson-Gamma model.

5.3. Analytical Results

In order to determine the optimum patrol cycle of the target route for road patrol, the risk management indicator for the patrol cycle mentioned and the optimum patrol cycle for the risk management limit are calculated for all road sections.

Firstly, the VaR index regarding the number of road obstacles VaRωα(zi) (refer to (21)) was obtained, based on the Poisson-Gamma model. The VaR index can be calculated for all road sections, but the volume of the results becomes enormous, and so the authors target the case of a certain road section, where the number of road obstacles is relatively large in each section.

Figures 4 and 5 show the relation between the risk management limit U̅i° regarding the number of road obstacles and the minimum patrol cycle defined as the minimum value of the set Ωω(U̅i°) (refer to (22)). Figure 4 is focused on that. In route B2, if the confidence level is set as ω=0.01 and the risk management limit is set to be 1, though overlap lines in the graph, the minimum patrol cycle becomes 2 days. Next, Figure 5 is focused on that. In route B2, if the confidence level is set as ω=0.05 and the risk management limit is set to be 1, though overlap lines in the graph, the minimum patrol cycle becomes 5 days.

Risk management limit (number of road obstacles) and the minimum patrol cycle (confidence level ω=0.01).

Risk management limit (number of road obstacles) and the minimum patrol cycle (confidence level ω=0.05).

The above passages discussed the minimum patrol cycle that satisfies the risk management limit for each road section, for the proposed risk management indicator. It can be inferred that which is adopted as a risk management indicator depends on the situation of the roads managed by the road administrator concerned, but it can be considered that the proposed indicator is practical for gauging the risk management limit for road obstacles. In addition, even if another indicator is adopted, the risk management limit can be defined by adopting the same idea.

6. Conclusion