---
category: literaturenote
citekey: delucaroadsafetymanagementusing2012
title: Road Safety Management Using Bayesian and Cluster analysis
authors: "De Luca, Mario; Mauro, Raffaele; Lamberti, Renato; Dell’Acqua, Gianluca"
year: 2012
date: "2012-10-04 October 4, 2012"
doi: 10.1016/j.sbspro.2012.09.840
publication: Procedia - Social and Behavioral Sciences
url: "http://www.sciencedirect.com/science/article/pii/S1877042812043029"
zotero_key: 5QDWK9RA
zotero_storage: N6ASGI4Q
collections: magistritöö
folder: 001_artiklid
firstAuthor: "De Luca, Mario"
status: converted
---
![](_page_0_Picture_0.jpeg)

Available online at www.sciencedirect.com

Procedia - Social and Behavioral Sciences 54 ( 2012 ) 1260 – 1269

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# EWGT 2012

15th meeting of the EURO Working Group on Transportation

#### Road safety management using Bayesian and cluster analysis. Mario De Lucaa,\* , Raffaele Mauro<sup>b</sup> ,Renato Lamberti<sup>a</sup> , Gianluca Dell'Acqua<sup>a</sup>

*University of Napoli Federico II, Via Claudio 21; 80125 Napoli, Italy b University of Trento, Via Mesiano 77; 38123 Trento, Italy* 

#### **Abstract**

The paper reports the results of an application of the Bayesian approach-based cluster analysis applied to a problem of road safety. 1000 accidents were recorded (from 1 January 2003 to December 31, 2006) on a stretch of about 100 km. The incidents belonging to the years 2003-2004-2005 were used to construct the Bayesian model (EB) and accidents belonging to the year 2006, were used to check for the reliability of the EB model. The Bayesian model was constructed with the help of cluster analysis. In particular, Cluster Analysis was used to identify the entity on which the Empirical Bayesian was subsequently applied. From the model, obtained by combining the two techniques, the accident waiting in the different entities for the year 2006 was estimated. The reliability of this model was very good. In fact, by comparing accident rates estimated by the EB model(for the year 2006) with the observed accident, a very low error was found. With the help of this procedure (EB technique combined with Cluster Analysis) it was also possible to identify the more dangerous "Black Spot"; so as to have the necessary support to plan infrastructure projects designed to reduce danger.

© 2012 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Program Committee. © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Program Committee Open access under [CC BY-NC-ND license.](http://creativecommons.org/licenses/by-nc-nd/3.0/)

*Keywords:* Safety, Empirical Bayesiana, Cluster Analysis.

# **1. Introduction And Literature Reviews**

The planning of safety works is always a difficult choice. One of the most common problems is represented by the identification of models which are capable of correctly representing the phenomenon of the accident. Much research has shown that crashes are often due to bad decisions by drivers made in environments created by engineers (Dell'Acqua, 2011). International research (Esposito et al., 2011) has thus suggested a variety of approaches to analyze the road traffic safety level on the basis of an assessment of accident rates and frequency (Discetti et al., 2011).

<sup>\*</sup>Corresponding author. Tel.: +00390817683936; fax: +39 081 7683946.  *E-mail address*: mario.deluca@unina.it.

Over time different formulas have been proposed: initially many researches were oriented towards linear type models; these models have now been abandoned due to a series of limits, even of a conceptual nature. Poissonian models were successively taken into consideration; however this models has limits in that it does not interpret well the experimental data which, as is known (in the case of accidents) are affected by regression to the mean phenomenon. For such a reason it was thought to assume the *Binomial Negative* as the probability function for the incident count. In this way an error term is added to the *Poissonian model* which identifies and takes into account, by means of a dispersion coefficient, the regression towards the mean phenomenon. Nevertheless, it was observed that such a formulation improve the estimation of accidents yet it does not efficiently resolve the problem. For this reason, some researchers *(Hauer et al. 1997)* have recently applied the EB technique (*Empirical Bayesian*) to the study of accident rates, which resolves the regression to the mean problem in a definitive manner. In particular, in order to estimate the accident rate, the EB method is based on two items of information: the first regards the history of accidents on the entity in question (cross road, curves stretch, etc); and the second regards the history of accidents relative to a group of entities similar to those being examined. The estimation is much better when the reference group is numerous. It is evident that in order not to banally consider the entities on which the Bayesian technique should be applied, it is useful to couple the EB technique to a technique which allows the aggregation of data in clusters (entities) which have significance in terms of accident rates. A very useful technique that we believe (for this purpose) may be the Cluster Analysis. *De Luca et al. (2011)* illustrates an application of Cluster Analysis to a "Road Safety "problem". Experimental analysis, by using Cluster algorithms, was carried out on segments situated in the Southern Italy freeway. Through these algorithms, it was possible to build a partition (hazardous zone) and estimate the relative hazard. These groupings were used, after introducing the "hazardous zone index", to build a predictive model of accidents (obtained through a multiple regression). The reliability of this model, used to simulate the Before After situation resulted as being very interesting; in fact the results were very hopeful because the maximum error returned by model is about 10%. *Depaire et al. (2008)*, always in the field of road safety, demonstrated cluster analysis in order to identify homogeneous classes of accidents that allowed for a very effective analysis. Similarly, Kwok-Suen et al. (2002) used cluster analysis to group homogeneous data in an experimental analysis to develop an algorithm to estimate the number of road accidents and to assess the risk of accidents. In the same area (road safety), some Greek researchers *(Yannis G. et al., 2007)* used this technique to build clusters to conduct a series of evaluations of alcohol-accident reports. Within the transport sector *Schweitzer (2006)* explores whether the risk of a toxic release during transport is greater in poor and minority neighborhoods using a combination of mapping and statistical methods. Cluster analysis is used to examine the density of facilities and transport spill events, as well as test for the spatial covariance between facilities and spills. Strong clustering of transport spills is evident, as well as clustering between factory sites and transport spills. A spatial model demonstrates raised rates of transport spills surrounding clusters of toxic firms. *Hesham Rakha et alt. (2011)* in a study conducted in the USA, proposes a procedure in which the Bayesian technique is applied together with the bootstrap technique. In particular a Bayesian and Bootstrap logistic left-turn gap acceptance model is developed using 2,730 field observations. The variables that are considered in the model include the gap duration; the driver's wait time in search of an appropriate acceptable gap; the time traveled by a driver to clear the conflict point; and the rain intensity. The model demonstrates that the acceptable time gap decreases as a function of the driver's wait time and increases with the rain intensity increases. The Bayesian and Bootstrap approaches are demonstrated to estimate consistent model parameters. In this paper a procedure for the identification of "Black Spots", by means of Cluster Analysis and Empirical Bayesian, is proposed. Cluster Analysis is used in order to define the *"Entity"* on which the empirical Bayesian should be applied.

## **2. Empirical Bayesian**

In order to correctly evaluate the accident phenomenon, with the Bayesian approach, it is necessary to have information both on the element that is being analyzed and on similar elements. For example, if the safety/accident rate of a intersection is being evaluated it is necessary to have information both on the intersection in question (number of accidents, geometry, traffic, etc.) and on a series of intersections similar to that being analyzed. The intersection element is therefore defined as an "entity" to be studied; similar entities become the reference population. It is possible to state, in a more rigorous manner, that the reference population for an entity is formed by the entity group that has the same set of characteristics of the entity being examined. Therefore, the Bayesian approach has two types of clues as its starting point:

*Information that come from characteristics of the entity, which render it similar to other entities for which data regarding safety is available;* 

*Information from the recording of accidents for the entity which is object of the study.*

It is necessary to understand how to make these two items of information interact in order to obtain a univocal estimation. It is necessary to estimate k (number of accidents) for a date entity, the number of accidents expected for a certain accident typology and referring to a certain period. For the entity being examine two things are known: first of all, through knowledge of its characteristics, we know that the entity belongs to reference population data, formed by various entities each with its own k, with mean E ^K`, and variance VAR ^K`; moreover we know that the considered entity has had K accidents during the reference period. Both items of information must be joined. Therefore let us consider the reference population entity: some will present zero accidents, some only one accident, others two accidents, and so on. From these, only those which present precisely K incidents in the period under examination are considered; we call E^kലK` and VAR^kലK` respectively the mean and the variance of *k* (number of accidents) in this in this subpopulations. On the basis of the previous considerations it is possible to affirm that if the entity being examine has the same characteristics and the same number of accidents as the entity which form this subpopulation, then its k value can be, with equal probability, any one of the k of this subpopulation of entities which registered all K incidents. Therefore, the best estimation of k for our entity is equal to E^kലK`,and the variance o this estimation is *VAR*^*k*ല*K*`.<sup>E</sup>^kലK` is a mix of two elements, or rather of E^k` relative to the reference population, and of the value K relative to the entity being examine. Considering a value that is between the two, we have:

$$E\{k|K\} = \alpha \cdot E\{k\} + (1-\alpha) \cdot K \tag{1}$$

The term D is a value between 0 and 1, furthermore:

- x if D # 1 then the value of k, estimated by E^k|K` is in proximity to the mean E^k` of the reference population;
- x if D # 0 then k will reflect the influence of the number of accidents K of the entity in consideration.

The problem consists in how to choose the value for the "weight" D. In (12) how to obtain an estimation of k that is as precise as possible is shown. The value to be assigned to D is obtained as:

$$\alpha = \frac{1}{1 + \frac{VAR\left\{K\right\}}{E\left\{K\right\}}}\tag{2}$$

The term D is a function only of the mean and the variance of the *k* and it is always a value of between 0 and 1. It is necessary to state that K and k must refer to the same temporal period. If, when working, a case is encountered where, for example, the information relative to the reference population refer to an annual accident frequency, while the number of *K* incidents refers to a period of 2 or 3 years, then it is necessary to use the following relationship:

$$\alpha = \frac{1}{1+r \cdot \frac{VAR\{K\}}{E\{K\}}} \tag{3}$$

where the term r represents the relation between the number of years that K refers to and the number of years that k refers to. In this way, taking the information on E^k`, VAR^k`, and K as a starting point, it is possible to obtain an estimate relative to the subgroup E^kലK` which will be used to evaluate k for our entity.

*Estimation of E*^*k*` *and VAR*^*k*`

These two parameters can be calculated by different methods. In the study in question the sample *moments method* was used. By using this method it is possible to define the mean and the sample variance respectively as:

$$\bar{K} = \frac{\sum K \cdot n(K)}{n} \tag{4}$$

$$s^2 = \frac{\sum (K - \bar{K})^2 \cdot n(k)}{n} \tag{5}$$

The summation is extended to all the values of K = ( 0,1,2,3,….). When n increases, ܭഥ approaches E^k`, and s VAR^k`. Therefore, by substituting, E^k` with ܭഥ , and VAR^k` with s2 , the starting equations become:

$$\hat{E}\{k\} = \bar{K} \tag{6}$$

$$\widehat{VAR}\{k\} = s^2 - \overline{K} \tag{7}$$

The precision and accuracy of the estimate is linked to the size of the reference population. As the estimates of E^k` and VAR^k` are based on ܭഥ and s2 , this estimation approach is called "*Moments Method*".

## **3. Cluster Analysis**

The term cluster analysis was initially used by Tryon (Tryon,1939) meaning a number of different algorithms and methods to assemble objects into their respective categories. A general question is how to organize observed data into meaningful structures that will be taxonomic. In other words, cluster analysis is an exploratory data analysis tool which aims to assemble different objects into groups in such a way that the degree of association between two objects is maximal if belonging to the same group and minimal otherwise. Thus, cluster analysis can be used to discover structures in data without providing an explanation/interpretation. In other words, cluster analysis simply discovers structures in data without explaining why they exist. Two following types of data partition exist in cluster analysis: a) strong or precise (crisp), a bivalent approach; b) weak or blurred (fuzzy), a polyvalent approach. The analysis shown in this paper concerns the first type of partition (logical type of hard c means).

## *Hard c-Means Method*

The principles of this technique are as follows. The aim of the group analysis consists in identifying a specific U partition, in c groups (2İ c İ n) of the U collection space constituted by n-elements. The hypothesis upon which this method is based is the following: the elements of the X space, that belong to a group, are characterized by a mathematical affinity and this affinity is greater than the elements of the different groups. Each element in the sample can be schematized as a point identified by m-coordinates, and each coordinate constitutes an attribute of the same element. One of the simpler measures of affinity is represented by the distance measured between two points and these belong to the data-space. We define an appropriate measurement for distance and we measure this between each unit of observation and all the units as a whole. Of course the distance between points belonging to the same group is smaller than the distance between points contained in different groups. Let X = {x1, x2, x3, … xn}, the set of n data to be divided into c groups. Each element xi, is defined by m characteristics (xi = {x1, x2, x3, … xim}). For this reason xi (where xi represents the accident "i") can be represented by a point on the Rm space. This method is based on the use of a J objective function that tends to create "spherical" groups for successive approximations. The objective function follows two results simultaneously: firstly it minimizes the Euclidean distance between the points of each group and the center of the same group (which generally doesn't coincide with any of the collection points) and in the second place, it maximizes the Euclidean distance between the centers of all the groups, U indicates the generic partition and U\* is the optimum that belongs to the Mc space of the possible partition of X. The J = J(U) value, assumed by the objective Function for each U partition, constitutes a relative measure of how close it is to the optimum. The objective function is to minimize the square addition of the Euclidean distances measured between all points and the center of each group. It is difficult to find the U partition because the cardinality of the Mc space of X's possible partitions tends rapidly to infinity. The search for the global optimum in problems of significant dimensions is not possible without laborious computation so the problem is resolved using an iterative optimization algorithm. Hypothesizing a first attempt with a U (r = 0) partition, number "c" groups and an iteration tolerance value ε (accuracy required for the solution) the position of the group center can be determined. Starting from these, we calculate again the attribution of each point to the different groups, and we obtain a new calculation for the matrix U (r = 1). Then we compare the two successive determinations of the U matrix and we repeat the process until the difference between the partitions, obtained over two successive cycles exceeds the predefined level of tolerance. This technique presupposes that the number of clusters is known beforehand, but as the optimum number of clusters with which to make the definitive classification is not known (this is due to the substantial lack of initial information on the structure of the clusters within which the units of observation are to be placed), we proceeded at random. We hypothesized different divisions of the database and then chose a value for an "S" index, defined as the best grouping index.

# **4. Data collections**

Many researchers have verified that one of the parameters that most influence safe driving is the speed variable and in the scientific literature some research works have dealt with speed prediction models to analyze real driver behavior (*Dell'Acqua and Russo*, 2011 a). The experimental analysis presented here is only one component of a larger study which has been under way on a number of roads for several years now with a view to improving performance, road management and safety (*Dell'Acqua and Russo* 2011 b*, Dell'Acqua et al.*, 2011a-b-c). The segment analyzed (see fig.1) belongs to the A3 (Salerno-Reggio Calabria freeway ) situated in the south of Italy. The stretch analyzed is situated between distance "226.000 km", near the "Tarsia interchange", and distance "285.000 km" near the "Grimaldi interchange)". Each accident (occurring between 1/1/2004 and 12/31/2006) is marked by date and coordinates (distance) on the road, with the environmental situations and the geometrical characteristics of the stretch where the accident happened. In this way we can obtain a structured matrix as shown in table 1.

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Fig.1. Segment analyzed

The variables shown in Table 1 were determined by the following procedure: The variables were initially introduced and subsequently removed and reintroduced until the analysis was significant following a feedback process.

Table 1. Abstract of data matrix

| Number<br>Record | Datum        | hours | Distance | Dir | N. vehicol | Injured        | deaths<br>N. of | Curving  | CCR   | Slope | Freeway<br>exit | Section<br>Width | Paving | Heifgt of<br>rain | Tunnel |
|------------------|--------------|-------|----------|-----|------------|----------------|-----------------|----------|-------|-------|-----------------|------------------|--------|-------------------|--------|
| 1                | 25/10/0      | 9.30  | 226.00   | N   |            | 1.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 2                | 31/10/0      | 5.55  | 226.00   | N   |            | 2.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 3                | 4<br>16/11/0 | 16.00 | 226.00   | S   |            | 1.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 4                | 4<br>14/12/0 | 20.10 | 226.00   | S   |            | 1.00 1.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 5                | 4<br>12/03/0 | 18.10 | 226.00   | S   |            | 1.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 6                | 5<br>20/03/0 | 9.06  | 226.00   | N   |            | 1.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 7                | 5<br>28/04/0 | 19.50 | 226.00   | S   |            | 1.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 8                | 5<br>29/04/0 | 18.50 | 226.00   | S   |            | 1.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 9                | 5<br>14/07/0 | 3.35  | 226.00   | N   |            | 2.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 10               | 5<br>30/07/0 | 18.05 | 226.00   | S   |            | 1.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Porous | 0.0               | no     |
| 11               | 5<br>05/09/0 | 13.45 | 226.00   | N   |            | 3.00 0.00 0.00 |                 | 0.000667 | 0.042 | -0.40 | Yes             | 11.50            | Dense  | 0.0               | no     |

## **5. Results and discussion: Cluster Analysis and Empirical Bayesian Applications**

The "Cluster Analysis" has been applied to the data matrix indicated in table 1. Nineteen clusters were obtained; each of these clusters represents an "entity " on which to apply the Bayesian technique. Table 2 shows an extract of a "Cluster" obtained with the "Cluster Analysis". As can be seen (in the cluster) there are accidents occurring in the same geometric, environmental and traffic conditions. In particular, in table 2 the rows represent an "entity" with similar characteristics. So if, for example, we want to know the accident in entity 1, shown in row 1, 2 and 3, other entities (reported in other rows) can be used as the reference group. For each incident an influence area of 1 km was considered.

The Bayesian technique, illustrated in paragraph 2, has been applied to these entities. Referring to incidents for the years 2003-2004 2005, through Eq. (1) a Bayesian predictive model was constructed. In Table 3 the Variables used in the construction of the model for each group (cluster) are shown. The meaning of these variables is reported in Chapter 3.

In Table 4 the estimation of accidents for the year 2006 is shown, obtained with the model (1), for the group 1 and 2 (cluster n.1and 2). The same was carried out for all other groups (clusters). Accidents estimated with the Bayesian model (which in the case of the example of Table 5 relate to group 1) were compared, for each group (entity), with accidents which actually occurred in 2006 in the same entity. A synthetic index (indicated as eq .1) was used to compare the safety conditions of the different groups where the crashes happened:

$$I_{i} = \frac{Nv \cdot Sev \cdot 10^{8}}{(AADT \cdot 365 \cdot L \cdot K1 \cdot K2)} \tag{8}$$

where:

**Nv**, is the number of accidents. **L**, is the length of the "Entity" and was considered equal to 1 km. **K1**, is a coefficient that takes into account the road surface conditions and has a value of 0.75 for a dry road surface, and 0.25 for a wet road surface;

| Table 2. Extract of a Cluster Obtained with | "Cluster Analysis" |
|---------------------------------------------|--------------------|
|                                             |                    |

| Accident<br>ID | Cluster<br>Label | Date     | Hours | Distance | Dir. | Vheicols<br>N. | Injured | death<br>N. of | Curving | CCR   | Slope | Freeway<br>Exit | State of<br>Paving | Height of<br>rain | Tunnel |
|----------------|------------------|----------|-------|----------|------|----------------|---------|----------------|---------|-------|-------|-----------------|--------------------|-------------------|--------|
| 770            | 19c              | 10/16/03 | 16:44 | 277.30   | S    | 1.00           | 1.00    | 0.00           | 0.3125  | 0.199 | -4.0  | No              | Dense              | 0.6               | No     |
| 772            | 19c              | 01/31/04 | 06:12 | 277.00   | S    | 1.00           | 1.00    | 0.00           | 0.3125  | 0.199 | -4.0  | No              | Dense              | 0.0               | No     |
| 773            | 19c              | 02/03/04 | 20:35 | 277.20   | S    | 1.00           | 0.00    | 0.00           | 0.3125  | 0.199 | -4.0  | No              | Dense              | 0.0               | No     |
| 991            | 19c              | 06/5/04  | 07.05 | 280.00   | S    | 3.0            | 0.0     | 0.0            | 0.385   | 0.098 | -3.5  | No              | Dense              | 0.2               | No     |
| 993            | 19c              | 8/23/04  | 17.25 | 280.00   | S    | 1.0            | 0.0     | 0.0            | 0.385   | 0.098 | -3.5  | No              | Dense.             | 0.0               | No     |
| 1060           | 19c              | 12/20/04 | 12.30 | 282.30   | N    | 4.0            | 1.0     | 0.0            | 0.397   | 0.101 | -3.0  | No              | Dense.             | 0.0               | No     |
| 1061           | 19c              | 1/26/05  | 14.00 | 282.00   | N    | 1.0            | 0.0     | 0.0            | 0.397   | 0.101 | -3.0  | No              | Dense.             | 0.0               | No     |
| 1062           | 19c              | 4/18/05  | 13.10 | 282.10   | N    | 1.0            | 4.0     | 0.0            | 0.397   | 0.101 | -3.0  | No              | Dense              | 0.2               | No     |
| 1128           | 19c              | 8/28/04  | 13.00 | 285.00   | S    | 2.0            | 0.0     | 0.0            | 0.500   | 0.127 | -3.3  | No              | Dense              | 0.0               | No     |
| 1129           | 19c              | 8/29/04  | 8.15  | 285.20   | S    | 3.0            | 0.0     | 0.0            | 0.500   | 0.127 | -3.3  | No              | Dense              | 0.0               | No     |
| 1130           | 19c              | 10/2/04  | 19.50 | 285.40   | S    | 1.0            | 0.0     | 0.0            | 0.500   | 0.127 | -3.3  | No              | Dense              | 0.0               | No     |
| 1131           | 19c              | 11/20/04 | 08.45 | 285.00   | S    | 1.0            | 0.0     | 0.0            | 0.500   | 0.127 | -3.3  | No              | Dense              | 0.0               | No     |
|                |                  |          |       |          |      |                |         |                |         |       |       |                 |                    |                   |        |

**K2**, is a coefficient that takes into account light conditions and has a value of 0.67 for daylight and 0.33 for nocturnal light. **Sev**, is the severity and the following values: 1 for 0 injured; 1.5 for 2 or 3 injured; 2.5 for more than 3 injured and 3 for with dead men. **AADT**, is the average daily traffic; In this way it was possible to identify the entities at a higher risk of accident and then identify the "black spots" most dangerous. Table 5 shows the "Entities" more dangerous in terms of accident. Also in Figure 2 shows schematically the analyzed stretch with the different zones of risk accident. In particular in red was indicated most dangerous "Black spots".

Table 3. Variable used to construction the model **EB** 

| Centroid of<br>cluster [km] | Number of<br>Cluster | ݇ത    | S 2     | E[݇ത] | VAR[݇ത] | a    |
|-----------------------------|----------------------|-------|---------|-------|---------|------|
| 252.090                     | 1                    | 10.63 | 56.73   | 10.63 | 46.11   | 0.07 |
| 252.000                     | 2                    | 12.40 | 476.00  | 12.40 | 463.60  | 0.01 |
| 252,734                     | 3                    | 14.17 | 129.81  | 14.17 | 115.64  | 0.04 |
| 242.990                     | 4                    | 7.670 | 135.00  | 7.67  | 127.33  | 0.02 |
| 280.100                     | 5                    | 28.83 | 1828.00 | 28.83 | 1799.17 | 0.01 |
| 279.020                     | 6                    | 16.00 | 676.67  | 16.00 | 660.67  | 0.01 |
| 268.510                     | 7                    | 6.330 | 64.00   | 6.33  | 57.67   | 0.04 |
| 260.460                     | 8                    | 24.00 | 580.00  | 24.00 | 556.00  | 0.01 |
| 253.000                     | 9                    | 4.000 | 16.00   | 4.00  | 12.00   | 0.10 |
| 266.170                     | 10                   | 9.000 | 117.33  | 9.00  | 108.33  | 0.03 |
| 244.530                     | 11                   | 5.000 | 49.67   | 5.00  | 44.67   | 0.04 |
| 226.000                     | 12                   | 34.00 | 1156.00 | 34.00 | 1122.00 | 0.01 |
| 280.130                     | 13                   | 13.89 | 548.78  | 13.89 | 534.89  | 0.01 |
| 276.690                     | 14                   | 16.00 | 342.80  | 16.00 | 326.80  | 0.02 |
| 259.000                     | 15                   | 46.00 | 2116.00 | 46.00 | 2070.00 | 0.01 |
| 233.600                     | 16                   | 3.210 | 17.50   | 3.21  | 14.29   | 0.07 |
| 233.820                     | 17                   | 4.200 | 26.20   | 4.20  | 22.00   | 0.06 |
| 234.810                     | 18                   | 6.750 | 47.25   | 6.75  | 40.50   | 0.05 |
| 262.316                     | 19                   | 11.67 | 225.67  | 11.67 | 214.00  | 0.02 |

Table 4. Accidents estimated by Model **EB** (refer to the cluster No. 1 and No. 2)

| Cluster | Distance   | n(y)       | Y         | Y/3          | y · n(y) | 2<br>(y-my)<br>* n(y) | Average number   |
|---------|------------|------------|-----------|--------------|----------|-----------------------|------------------|
|         | Entity     | number of  | Total     | Average of   |          |                       | of accidents     |
|         | [Km]       | Entity in  | accidents | accidents in |          |                       | estimated by the |
|         |            | each group | 03/04/05  | 03/04/05     |          |                       | model EB         |
| 1       | 251,00±0.5 | 2.00       | 1.00      | 0.33         | 2.00     | 185.28                | 0.56             |
| 1       | 250,00±0.5 | 1.00       | 3.00      | 1.00         | 3.00     | 58.14                 | 1.18             |
| 1       | 252,00±0.5 | 2.00       | 12.00     | 4.00         | 24.00    | 3.78                  | 3.97             |
| 1       | 255,00±0.5 | 1.00       | 16.00     | 5.33         | 16.00    | 28.89                 | 5.21             |
| 1       | 245,00±0.5 | 1.00       | 19.00     | 6.33         | 19.00    | 70.14                 | 6.13             |
| 1       | 258,00±0.5 | 1.00       | 21.00     | 7.00         | 21.00    | 107.64                | 6.75             |
| 2       | 243.80±0.5 | 2.00       | 1.00      | 0.33         | 2.00     | 2.00                  | 0.37             |
| 2       | 258.50±0.5 | 1.00       | 5.00      | 1.67         | 5.00     | 25.00                 | 1.69             |
| 2       | 243.00±0.5 | 1.00       | 7.00      | 2.33         | 7.00     | 49.00                 | 2.35             |
| 2       | 256.00±0.5 | 1.00       | 48.00     | 16.00        | 48.00    | 2304.00               | 15.90            |
| 2       | 253.00±0.5 | 2.00       | 1.00      | 0.33         | 2.00     | 2.00                  | 0.37             |
|         |            |            |           |              |          |                       |                  |

## **Conclusions**

This work has shown the application of Empirical Bayesian through the support of Cluster Analysis. In particular, the cluster has been used to identify the entity which was subsequently studied with the EB. By the application of Empirical-Bayesian techniques predictive models that have allowed to estimate the expected accident rate in the different entities for the year 2006 were obtained . The reliability of these models, as reported in Table 3, was very good. In fact, by comparing accident rates estimated by the model (for the year 2006) with the observed accidents in the same year there was a very small error. With these models it is possible to identify the most dangerous "Black Spots" and then schedule some infrastructural projects to reduce the danger. The study is continuing with the extension of to procedure to an entire road network characterized by roads with different functional importance. From initial analyses carried out important indications on the modes of intervention on "Black Spots" were obtained

![](_page_8_Figure_3.jpeg)

Fig. 2. Analyzed stretch with the different zones of risk accident

Table 5 . List of the Entities "more dangerous" in terms of accident

| Distance<br>Km | Accidents<br>from<br>03 to 05 | Average of<br>accidents in<br>03/ 04/05 | N. accidents<br>expected in<br>2006<br>(EB models) | Incidents<br>observed<br>in 2006 | AADT<br>Average<br>from 03<br>to 05 | AADT<br>2006 | Ii<br>expected<br>in 2006 | Ii<br>Observe<br>d in<br>2006 | Risk     |
|----------------|-------------------------------|-----------------------------------------|----------------------------------------------------|----------------------------------|-------------------------------------|--------------|---------------------------|-------------------------------|----------|
| 277.00±0.5     | 76                            | 25.3                                    | 25.22                                              | 19                               | 12360                               | 12720        | 543.21                    | 409.24                        | High     |
| 279.00±0.5     | 30                            | 10.0                                    | 10.33                                              | 9                                | 12200                               | 12555        | 225.41                    | 196.39                        | High     |
| 253.00±0.5     | 30                            | 10.0                                    | 9.81                                               | 11                               | 15450                               | 15900        | 169.04                    | 189.54                        | High     |
| 278.00±0.5     | 29                            | 9.7                                     | 9.78                                               | 8                                | 11900                               | 12247        | 218.79                    | 178.97                        | High     |
| 287.00±0.5     | 21                            | 7.0                                     | 6.67                                               | 6                                | 12450                               | 12813        | 142.63                    | 128.30                        | High     |
| 274.00±0.5     | 27                            | 9.0                                     | 9.00                                               | 6                                | 12350                               | 12710        | 194.01                    | 129.34                        | High     |
| 282.00±0.5     | 18                            | 6.0                                     | 6.14                                               | 5                                | 11870                               | 12216        | 137.71                    | 112.14                        | High     |
| 261.00±0.5     | 14                            | 4.7                                     | 4.79                                               | 4                                | 12200                               | 12555        | 104.52                    | 87.29                         | High     |
| 262.00±0.5     | 12                            | 4.0                                     | 3.99                                               | 3                                | 12360                               | 12720        | 85.94                     | 64.62                         | medium   |
| 281.00±0.5     | 12                            | 4.0                                     | 4.24                                               | 3                                | 12890                               | 13265        | 87.57                     | 61.96                         | medium   |
| 262.00±0.5     | 12                            | 4.0                                     | 3.99                                               | 3                                | 12900                               | 13276        | 82.34                     | 61.91                         | medium   |
| 276.00±0.5     | 11                            | 3.7                                     | 3.88                                               | 2                                | 11450                               | 11783        | 90.21                     | 46.50                         | medium   |
| <br>245.00±0.5 | <br>1                         | <br>0.33                                | <br>0.38                                           | …<br>0                           | …<br>12900                          | <br>13276    | <br>7.84                  | …<br>0.00                     | …<br>Low |

# **Acknowledgements**

The research was conducted under the Italian National Research Project "Driver speed behaviour evaluation using operating speed profile and crash predicting models".

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