The potential of clustering methods to define intersection test scenarios: Assessing real-life performance of AEB

Accident Analysis and Prevention 113 (2018) 1–11

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Accident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap

The potential of clustering methods to define intersection test scenarios: Assessing real-life performance of AEB

T

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Ulrich Sandera,b, , Nils Lubbea a b

Autoliv Research, Wallentinsvägen 22, 447 83, Vårgårda, Sweden Chalmers University of Technology, 412 96, Gothenburg, Sweden

A R T I C L E I N F O

A B S T R A C T

Keywords: Straight crossing path SCP Left turn across path LTAP/OD Intersection Junction AEBS AEB Test scenarios Clustering

Intersection accidents are frequent and harmful. The accident types ‘straight crossing path’ (SCP), ‘left turn across path – oncoming direction’ (LTAP/OD), and ‘left-turn across path – lateral direction’ (LTAP/LD) represent around 95% of all intersection accidents and one-third of all police-reported car-to-car accidents in Germany. The European New Car Assessment Program (Euro NCAP) have announced that intersection scenarios will be included in their rating from 2020; however, how these scenarios are to be tested has not been defined. This study investigates whether clustering methods can be used to identify a small number of test scenarios sufficiently representative of the accident dataset to evaluate Intersection Automated Emergency Braking (AEB). Data from the German In-Depth Accident Study (GIDAS) and the GIDAS-based Pre-Crash Matrix (PCM) from 1999 to 2016, containing 784 SCP and 453 LTAP/OD accidents, were analyzed with principal component methods to identify variables that account for the relevant total variances of the sample. Three different methods for data clustering were applied to each of the accident types, two similarity-based approaches, namely Hierarchical Clustering (HC) and Partitioning Around Medoids (PAM), and the probability-based Latent Class Clustering (LCC). The optimum number of clusters was derived for HC and PAM with the silhouette method. The PAM algorithm was both initiated with random start medoid selection and medoids from HC. For LCC, the Bayesian Information Criterion (BIC) was used to determine the optimal number of clusters. Test scenarios were defined from optimal cluster medoids weighted by their real-life representation in GIDAS. The set of variables for clustering was further varied to investigate the influence of variable type and character. We quantified how accurately each cluster variation represents real-life AEB performance using pre-crash simulations with PCM data and a generic algorithm for AEB intervention. The usage of different sets of clustering variables resulted in substantially different numbers of clusters. The stability of the resulting clusters increased with prioritization of categorical over continuous variables. For each different set of cluster variables, a strong in-cluster variance of avoided versus non-avoided accidents for the specified Intersection AEB was present. The medoids did not predict the most common Intersection AEB behavior in each cluster. Despite thorough analysis using various cluster methods and variable sets, it was impossible to reduce the diversity of intersection accidents into a set of test scenarios without compromising the ability to predict real-life performance of Intersection AEB. Although this does not imply that other methods cannot succeed, it was observed that small changes in the definition of a scenario resulted in a different avoidance outcome. Therefore, we suggest using limited physical testing to validate more extensive virtual simulations to evaluate vehicle safety.

1. Introduction Detailed understanding of the real-life performance of active safety and Advanced Driver Assistance Systems requires the evaluation of all possible scenarios. This takes effort, and not all scenarios are equally important. The more common a scenario, the more influence it will have on the safety benefit. Hence, performance evaluation is commonly ⁎

restricted to a set of frequent scenarios, depending on the required accuracy of the performance estimation and the effort needed and tolerable to add more scenarios. Computer simulations have removed some of the need for physical testing. However, simulations with complex, accurate models still take time, and the models themselves need to be created and validated. Furthermore, it is still common for performance evaluation and

Corresponding author at: Autoliv Research, Wallentinsvägen 22, 447 83, Vårgårda, Sweden. E-mail address: [email protected] (U. Sander).

https://doi.org/10.1016/j.aap.2018.01.010 Received 12 July 2017; Received in revised form 24 November 2017; Accepted 9 January 2018 0001-4575/ © 2018 Elsevier Ltd. All rights reserved.

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crossing a junction. Consequently, Euro NCAP is planning to incorporate updated car-to-car AEB test and assessment procedures reflecting real-life scenarios and addressing, besides others, junction and intersection crossing accidents, by 2020 (Euro NCAP, 2015). In this paper, we provide the much-needed answer to the question: Is it possible to reduce the diversity of intersection accidents into a set of test scenarios without compromising the ability to predict real-life performance of Intersection AEB? We do this in three steps. First, we apply three different clustering methods and choose medoids, the cluster centers, as cluster representatives. Second, we run virtual simulations of real-world intersection accidents with a generic Intersection AEB system and compare how well the cluster representatives predict the collision avoidance potential compared to the whole accident sample. Third, we investigate whether there is a demonstrable advantage in using predictions made by cluster representatives over those made through random sampling.

validation to be carried out by hardware testing, as the example of the European New Car Assessment Program (Euro NCAP, 2017) shows. Euro NCAP provides consumers with safety ratings for selected new cars and thus has become an important driver for advances in vehicle safety. The consumer organization applies more stringent tests of safety performance compared to regulations and includes safety assessments in new areas not yet regulated (van Ratingen et al., 2016). For one of these new assessments, the Automated Emergency Brake (AEB) assessment, test conditions and assessment criteria have been defined for car-to-car rear-end accidents (Schram et al., 2013) and carto-pedestrian accidents (Schram et al., 2015). Tests are carried out with the car under assessment and hardware targets on a track. Recently, Euro NCAP has introduced a “grid approach” for active safety testing, requiring the vehicle manufacturer to provide performance information for a wide range of scenarios while only verifying some (Euro NCAP, 2017). Additional information is therefore collected at no extra cost to Euro NCAP, as the effort is shifted to the manufacturer. This may be a result of the expectation that, for a manufacturer, the additional effort of testing more scenarios is negligible as simulation methods are utilized. However, it is unclear which scenarios need to be tested to give a reliable estimate of real-life performance. A common approach is to segment accident data into homogenous groups using expert knowledge and to define a limited number of tests for the most frequent groups. Decisions are often made on the basis of what can technically be tested and how much can economically be tested; in consequence, real life performance receives less attention. Some years after active safety systems have come to the market and accidents have been observed, real-life performance can be estimated and compared with test performances. Good correlation between real-life and test performances indicates that the tests are relevant and valid (Kullgren et al., 2010; Pastor, 2013; Strandroth et al., 2011). However, until sufficient accidents are observed, the test scenarios might lead to system designs that are irrelevant or even harmful to real-life safety. Therefore, the selection of test scenarios needs careful attention even at the introduction stage. Intersection accidents are frequent and have severe consequences. Approximately fifty percent of all injury accidents in the US occur at intersections or are intersection-related; further, approximately thirty percent of fatal road traffic accidents occur at these locations (NHTSA, 2016). In Europe, about twenty-four percent of road traffic fatalities are caused by junction accidents (European Commission, 2015). The most frequent types of car-to-car intersection conflicts in Germany are: straight crossing path (SCP), left turn across path – oncoming direction (LTAP/OD), and left turn across path – lateral direction (LTAP/LD), illustrated in Fig. 1 (Sander, 2017). Euro NCAP expects next-generation AEB to be able to address more complex accident scenarios, such as turning into oncoming traffic or

2. Background: clustering methods This section provides the background to the clustering methods used in our analysis. 2.1. Distance- or similarity-based clustering For many cluster analysis methods, the clustering is done by evaluating the distances or (dis-) similarities between observations, such as Euclidean or Manhattan distance for continuous variables, Jaccard similarity (coefficient) for categorical variables, and Gower’s generalized coefficient of similarity for mixed-type data. The input is usually an object-by-attribute matrix, where the rows stand for the observations and the columns stand for attributes. Intermediate output can be a distance or dissimilarity matrix, which is a symmetric matrix where d (i,i’) = d(i’,i), measuring the distance and dissimilarity, respectively, between the observations i and i’. Euclidean and Manhattan are true distances since they obey the triangle inequality with 1 J

d ii′ =

⎞ ⎛ |x −x ′ | ⎜ ∑ ij i j ⎟ ⎠ ⎝ j=1

p

(1)

where d = distance, J = number of attributes, j = attribute index, i,i’ = object indices, and p = 1 for Manhattan and p = 2 for Euclidean distance. In general, the input data is standardized and adjusted to the mean to give all attributes the same weight. Jaccard index dissimilarity is used for dichotomous data; thus for polytomous data, zero-unity dummy variables have to be created which can only take the value ‘0’ or ‘1’. The Jaccard index ignores the co-

Fig. 1. Pictograms of most frequent car-to-car intersection accident scenarios in Germany: A) Straight Crossing Path, B) Left Turn Across Path / Oncoming Direction, and C) Left Turn Across Path / Lateral Direction; whereα = collision angle, L = lateral offset, and R = turning radius.

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2.1.2. K-means, k-medoids, and partitioning around medoids The k-means algorithm is one of the most popular partitioning methods, intended for quantitative variables, and using squared Euclidean distance as the dissimilarity measure. The results depend on the a priori user determination of the number of clusters that the data is to be grouped into (Hastie et al., 2009). In each step, an observation is assigned to the cluster with the closest mean, called the centroid of the observations in the cluster, by minimizing the total squared error. This requires, in contrast to hierarchical clustering, a priori determination of cluster number. Start values for the centroids are often randomly chosen, though an alternative method proposes a combination of kmeans and HC, whereby initial centroids are taken from an HC analysis with same number of clusters (Husson et al., 2010). Where categorical variables have to be considered for clustering, the calculation of centroids is often not meaningful, as it is not possible to interpret the mean value of a categorical variable. Instead, the available data points at the center of the cluster, the medoids, can be used. Thus, the k-medoids algorithm is closely related to the k-means algorithm, but it minimizes the sum of dissimilarities of observations assigned to a cluster to the corresponding medoids instead of the sum of squared Euclidean distances. A common application of the k-medoids method is Partitioning Around Medoids (PAM). During a build phase each observation is assigned to the closest medoid. In the swap phase the algorithm identifies for each cluster, whether any other member lowers the dissimilarity coefficient. If so, this member becomes the new medoid. If at least one medoid has changed, the build phase is started again and each observation is assigned to one of the changed medoids (Kaufman and Rousseeuw, 2005).

absence and calculates the similarities in terms of co-presence. Further, the co-presence count is expressed relative to the number of attributes present in at least one of the pairs under consideration. For example, the values of a variable can take ‘red’ and ‘not-red’. Co-presence is not considered if two observations are coded as ‘not-red’ as it cannot be stated whether or not they have a similar color. Gower’s generalized coefficient of similarity allows for similarity calculations when dichotomous, qualitative, and quantitative variables are present in a dataset (Gower, 1971). In general, the similarity coefficient between i and i’ is defined as the average score taken over all possible combinations: J

Sii′ = ∑ j = 1 sii′j ∑J δ ′ j = 1 ii j

(2)

For dichotomous variables, sii’j = 1 when the values of a variable match and sii’j = 0 when there is a mismatch. Similar to the Jaccard index, possible combinations δii’j are considered in case of co-presence. For qualitative variables, sii’j = 1 in case two observations agree, otherwise sii’j = 0. For quantitative variables

sii′j = 1− |x i−x i′ | Rj

(3)

where R is the range of the variable in the sample or the population. Where the dataset does not contain missing values, the similarity matrix Sii’ is positive semi-definite, and a true Euclidean representation with distances can be derived:

d ii′ =

1−Sii′

(4)

2.2. Model-based clustering

We now introduce the two distance-based clustering methods applied in this study.

Model-based clustering (Banfield and Raftery, 1993) has been also referred to as latent profile analysis (Gibson, 1959), or latent class cluster analysis (Vermunt and Magidson, 2002), and is a based on a generalized Finite Mixture Model (FMM) (McLachlan and Peel, 2000). The method derives clusters using a probabilistic model to explain the distributions in the data instead of distance measures. It is assumed that the data are generated by a mixture of underlying probability distributions in which each component stands for a separate cluster (Fraley and Raftery, 1998; Vermunt and Magidson, 2002).

2.1.1. Hierarchical clustering Hierarchical Clustering (HC) is a popular clustering method based on distances between observations. As the name suggests, HC produces structures in which the clusters at each level of the hierarchy are built from merging groups at the next lower level. At the lowest level, each cluster contains one single observation. At the highest level, one single cluster contains all of the data. There are two basic approaches for HC: agglomerative (bottom-up) and divisive (top-down). Agglomerative clustering starts at the bottom and recursively merges a selected pair of clusters into a single cluster at each level while the divisive method starts at the top and at each level recursively splits one of the existing clusters at that level into two new clusters (Hastie et al., 2009). The distance or dissimilarity matrix is further processed with hierarchical clustering algorithms to derive the hierarchical structure by usage of a specific linkage criterion, either single, complete, average, or Ward. Single and complete linkage considers the minimum and maximum, respectively, of the pair of dissimilarities between the clusters and others. For average, as the name indicates, the average dissimilarity is chosen. With the Ward method those clusters are merged that result in the smallest increase in the overall sum of the squared within-cluster distances (Ward, 1963). The selection of the linkage criterion has a substantial effect on the design of the derived structure and clusters. It is important to note that the term cluster analysis is not associated with a specific statistical model and there is no need to make assumptions about the underlying data distribution. A dendrogram is a compact visualization of the dissimilarity matrix where the observations are joined together in a hierarchical way from the closest, which means the most similar, to the furthest, the most dissimilar. This binary tree structure can be cut at a certain dissimilarity height to create clusters, where the nodes below the cut line define the assignment of each object to a cluster.

2.2.1. Latent class clustering The model-based clustering utilized in this study is named Latent Class Clustering (LCC), as the model gives probabilities of the observations belonging to a certain cluster (latent class). Mixed-mode data, such as continuous and categorical data, can be handled in FMM. For LCC, local independence of the attributes in FFM is assumed. This means that instead of identifying a single multivariate distribution for each set of observed attributes, an appropriate univariate distribution for each attribute is formulated (Vermunt and Magidson, 2002). Hence, by knowing or assuming the underlying distributions for each cluster, the problem of finding the clusters can reduce to a parameter estimation problem. Following the maximum likelihood approach, the unknown parameter vector is often estimated by means of the Expectation–Maximization (EM) algorithm (Depaire et al., 2008). One challenge with the maximum likelihood approach is the potential identification of a local maximum instead of a global maximum, as estimation algorithms are designed to only increase likelihood (Dempster et al., 1977). A global maximum is indicated when the same solution is reached with different start parameters. The main advantage of LCC compared to similarity-based clustering is the ability to represent overlap across clusters rather than independent or nested clusters only. This overlap is denoted by cluster membership probabilities. Cluster membership is identified by the most likely assignment from the posterior state sequence. Further, the 3

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data with geo-spatial information. Anderson (2009) characterized accident hotspots in the London area through kernel density estimation and compared the results to k-means clustering. The author did not describe the variables used for clustering, though the application of kmeans suggests that only quantitative ones were used. The description of the clustering process indicates that hierarchical clustering was used to some extent. Accidents in Honolulu, Hawaii, involving pedestrians were investigated using k-means partitioning by Kim and Yamashita (2007). Input variables covered the spatial information of the accidents and the output comprised the centroids for each cluster, extension along X and Y axes, rotation, and the area covered by the standard deviational ellipse. Socio-economic attributes, including traffic volume, speed and roadway type, were mentioned as important, but not considered in the analysis. It is noted that k-means clustering provides an improvement over simplistic pin-map approaches to accident location analysis, but that hierarchical clustering may have advantages when applied to a larger area. Saunier et al. (2011) applied a combination of the two methods in the analysis of traffic observations at an intersection. The number of clusters was obtained using the k-means algorithm and a hierarchical agglomerative clustering (HAC) method sequentially, and the “best” number of cluster was derived from the biggest step in the HAC dendrogram. Nowakowska (2012) analyzed accidents on undivided two-lane twodirectional roads using a k-means algorithm and a batch-learning Kohonen neural network. Qualitative variables were re-coded by introducing dummy zero-unity variables and outlier observations were removed on the basis that k-means is sensitive to outliers. The author stated that the decision regarding the final number of clusters was not straightforward and required a series of trials. As a result, the k-means method showed a slightly worse separation than the Kohonen maps. Another study used geometrical, environmental, and accident-related characteristics to cluster motorway accidents through a k-means algorithm (Mauro et al., 2013). The input variable set contained quantitative and qualitative variables; one of the qualitative variables was polytomous and the rest dichotomous. The result of the cluster analysis was given as mean values of the qualitative variables, which are difficult to interpret. LCC has been applied to traffic safety analyses, for example to police-reported traffic accidents in Belgium (Depaire et al., 2008), cyclist–motorist accident patterns in Denmark (Kaplan and Prato, 2013), highway accidents in Spain (De Oña et al., 2013), young drivers in New Zealand (Weiss et al., 2015), and pedestrian accidents in Switzerland (Sasidharan et al., 2015). Notably, none of the studies defined continuous variables; for computational simplicity, variables were defined as nominal and ordinal. Furthermore, data elements were commonly assumed to be independent within a cluster so that the variance-covariance matrix becomes diagonal (all in-cluster co-variances are equal zero).

existence of an underlying statistical model allows for the calculation of cluster probabilities for new cases, and the provision of several goodness-of-fit criteria facilitate the decision regarding the number of clusters (Kaplan and Prato, 2013). 2.3. Assessment of clustering quality For distance- and model-based clustering methods, different approaches have to be considered to assess clustering quality. For distance-based clustering, we applied the Average Silhouette Width (ASW) method. For model-based-clustering, the Bayesian Information Criterion was used in combination with the chi-squared goodness of fit assessment. 2.3.1. Average silhouette width The silhouette width method was introduced by Rousseeuw (1987) and describes how well each object of a dataset fits into the cluster that it has been assigned to. This is done by comparing how close the object is to other observations in its own cluster with how close it is to observations in other clusters. Possible values range from -1 to 1 with negative values or values close to zero indicating that the object might belong to another cluster and values close to one indicating that the observation is assigned to the right cluster. The average value of all silhouette widths gives the ASW. The interpretation of the ASW value is given in Table 1. Here, the ASW was used to identify the optimal number of clusters, namely the number of clusters used in the iteration loop for HC and PAM which resulted in the highest ASW. 2.3.2. Bayesian information criterion The Bayesian Information Criterion (BIC), also called the Schwarz Bayesian Criterion, is based on the premise that where two models have equal loglikelihood, the model with the fewest parameters is better. It is calculated as follows:

BIC = −2∙ln(L) + p∙ln(N )

(5)

where ln(L) is loglikelihood, p is the number of estimated model parameters, and N is the total number of observations. Smaller values correspond to better models. For the best fitting model, the goodness of fit was assessed by the likelihood ratio chi-square test (Cochran, 1952, Chapter 14). 2.4. Literature review on application of clustering for accident data analysis HC has been used to develop test scenarios from accident data for car-to-car rear-end and pedestrian AEB (Hulshof et al., 2013; Lenard et al., 2016, 2011) although these are rare examples of the application of clustering to derive test scenarios for vehicle safety assessment. For example, Lenard et al. (2016), Lenard et al. (2011) used agglomerative clustering and defined a Manhattan distance between accidents using the average linkage method. Pedestrian age and gender were arbitrarily recoded into one ordinal variable with four states: 0–7years, 8–15years, adult female, and adult male with numerical values of 0, 0.33, 0.66 and 1. As a consequence, a child is defined to be more similar to an adult female than to an adult male. The k-means algorithm has been used to group and analyze accident

3. Data and methods We identified relevant clustering variables using Principal Component Analysis (PCA) for the scenarios SCP and LTAP/OD (Section 3.2). Three clustering methods with two variations on two of the methods were applied to the data in order to compare the cluster size and representation:

Table 1 Interpretation of the Average Silhouette Width (ASW). ASW range

Interpretation

< = 0.25 0.26–0.50 0.51–0.70 0.71–1.00

No substantial structure has been found A weak structure has been found that could be artificial A reasonable structure has been found A strong structure has been found

• Hierarchical Clustering (HC) • Partitioning Around Medoids with both random start values (PAMRAN) and start values taken from HC (PAM-HC) • Latent Class Clustering using both an independent mixture model (LCC-IMM) for mixed-data and a model for polytomous data only (LCC-POL).

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condition and perception were not considered, as the scenario classification is aimed towards AEB without driver interaction. Only the information about sight obstruction was based on driver interview information. The variables ‘lateral offset’ and ‘turning radius’ in LTAP/OD (see Fig. 1C) were computed from the PCM data. Simulations of a generic Intersection AEB (see Section 3.4) showed that in about 95% of the accidents, AEB would be activated before the driver started braking. This means that driver braking does not interfere substantially with the decision making of the AEB algorithm. For this reason, kinematics were simplified by eliminating variables for brake deceleration and brake distance. In consequence, the values for the variables describing each car’s collision speed were the same as for initial speed and could be dismissed. Finally, the SCP and LTAP/OD cases from the PCM database were re-simulated with driver braking eliminated, if present. New values for the variables ‘collision angle’ between vehicles and ‘impact area’ on each vehicle were computed based on the kinematics without driver braking. Due to removal of driver braking immediately before the collision, in 11 SCP and 10 LTAP/OD scenarios the conflict was entirely avoided. As a consequence, these cases have missing variables (all those which are derived from the collision) and were excluded. Additionally, a variable was introduced to indicate if the vehicle was accelerating from quasi standstill (rolling stop with a speed equal to or less than 5 km/h) rather than driving through the intersection.

For the distance-based methods, HC, PAM-RAN, and PAM-HC, the optimum number of clusters and the cluster quality were derived from the ASW while for the model-based clustering methods, LCC-IMM and LCC-POL, BIC was used (Section 3.3). To evaluate the degree to which Intersection AEB effectiveness derived from cluster medoids is representative of the effectiveness of the whole accident sample, simulations of real-life SCP and LTAP/OD scenarios with and without an Intersection AEB were conducted to assess the accident avoidance opportunities. For both SCP and LTAP/OD, the percentage of avoided medoid accidents from all medoid accidents was then compared to the percentage of all avoided accidents from all accidents. Finally, we resampled data subsets with the size of the cluster medoids to identify the benefit, if any, gained from this methodological approach compared to random test scenario selection (Section 3.4). 3.1. Data The analysis was based on the German In-Depth Accident Study (GIDAS) data from July 1999 to July 2016. The GIDAS data is collected in and around two cities in Germany. Accidents are only added to the database where at least one person is injured and where the accident happened within the sample region and time shift (Otte et al., 2003). Only car-to-car accidents were selected with the definition of a car made according to the ECE M1 category (UN-ECE WP.29, 2014). SCP accidents covered the conflict types 271, 301, 311, 321, 331, 353, and 355, while LTAP/OD covered the types 211, 281, 351, and 354 following the classification scheme of the German Insurance Association (Gesamtverband der Deutschen Versicherungswirtschaft e.V., 2016). In the selected conflict types, participants have similar trajectories although there is variation in right-of-way and traffic light presence. Pictograms of the conflict types are summarized in Appendix A. The GIDAS sample contained 1109 SCP and 849 LTAP/OD car-to-car accidents. GIDAS is a sub-sample of the police reported accidents held at the German Federal Statistics Office, Destatis. The variables accident severity, accident type, and accident year were used in a hyper-cube weighting scheme to calculate weights accounting for the outcomebased sampling bias: for each combination of weighting variable values a cluster is generated and, subsequently a weight is computed for each cluster to achieve a match between the relative frequency in the GIDAS data and that found in the German national statistics (Hautzinger et al., 2004). If the variables were not available from the police recordings, the corresponding coding from the investigation units was utilized as substitute. A subset of the GIDAS database, the GIDAS-based Pre-Crash Matrix (PCM), was utilized to reproduce the pre-crash phase of SCP and LTAP/ OD accidents over five seconds in a virtual simulation environment (Schubert et al., 2013). The February 2017 PCM dataset consists of 8813 GIDAS accidents for which a reconstruction of the pre-crash phase was feasible. After applying the above-mentioned filter criteria, SCP and LTAP/OD scenarios accounted for 784 and 453 accidents in the PCM, respectively. A different set of variables containing impact momentum and road width were used to weight the PCM data sample and thus to make it representative of GIDAS (Sander, 2017). A two-stage weighting process was chosen, because the PCM, GIDAS and national statistics contain different sets of variables which need weighting. The two weights were then multiplied and normalized in a single weight making the PCM data, which is used for all statistics presented here, representative for the German national accident statistics while maintaining the original sample size.

• Quantitative variables consist of: • road width* • directional lane width*(width of lanes in same direction as the car is heading) • initial speed* (speed before the driver initiated braking) • collision angle (angle between the vehicles at the point of the first contact) • lateral offset (lateral distance between approaching cars in LTAP/ OD) • turning radius (for turning car in LTAP/OD). • Qualitative variables consist of: • accident location (urban / rural) • light conditions (day / night with street lamps, night without street lamps) • precipitation present (yes/no) • traffic light presence (yes/no) • sight obstruction present* (for both cars; yes/no) • impact area* (for both cars; front-left, front-mid, front-right, sidefront, side-mid, side-rear) • acceleration from standstill* (for both cars; yes/no). Variables marked with * were selected for both cars. For LTAP/OD the acceleration from standstill was only considered for the turning vehicle, as acceleration from standstill for the straight heading vehicle was not present in the dataset. The PCA was conducted on this set of variables to evaluate whether a reduced number of variables, called the principal components, could express the variance of the data. Here, the R package ‘PCAmix’ was utilized (Chavent et al., 2014). The collinearity of the input variables was investigated by computation of the angle between the variable vectors in the orthogonal component coordinate system. An arbitrary definition of correlation was introduced by the angle: (< 10°) equals strong, (> = 10° and < 25°) equals moderate, and (> = 25° and < 40°) equals slight. Gower’s generalized coefficient of similarity is sensitive to outliers in quantitative variables, as the similarity between two observations is calculated based on the variable range (Eq. (3)), and the range increase due to outliers maximizes the similarity between non-outliers. Therefore, all cases with outliers, missing values or unknown values for the variables below were excluded from the datasets. We defined outliers as values with a distance to the mean of over 1.5 times the inter-quartile

3.2. Principal component analysis An initial set of variables was selected from the GIDAS database based on their relevance in describing vehicle trajectories, road network, and environmental conditions. Variables describing driver 5

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3.4. Simulation

range. The datasets utilized for clustering of SCP and LTAP/OD accidents had a remaining size of 519 and 235, respectively (see Supplementary information – descriptive statistics).

Autoliv’s pre-crash simulation framework PRAEDICO was used to estimate the real-life effectiveness of an Intersection AEB from the selected SCP and LTAP/OD accidents in the PCM (Sander and Lubbe, 2016). PRAEDICO utilizes the statistical software R, Matlab, and the time-dependent simulation in Simulink. The PCM data specifies vehicle properties and trajectories, road edges and lines, and if present, the position and dimension of observations that could have an effect on sight obstruction during the precrash phase. A driver model was utilized to let the vehicle dynamics model follow given PCM pre-crash trajectories through the inputs of gas and brake pedal position, and steering wheel angle. The vehicle dynamics model dictated the new global position and orientation, and the longitudinal, lateral, and rotational velocities and accelerations in the vehicle coordinate system. A generic intersection AEB was specified using sensor characteristics, actuation by braking, and algorithms for path prediction, threat assessment, and decision-making. The sensor field of view was set to 180° with the origin on the longitudinal vehicle axis about 1.5 m behind the front bumper. Ideal sensing was assumed. The opponent was classified as visible when all edges of the vehicle-surrounding rectangle were within the field of view. After five visible sensor frames, the opponent was tracked and an evaluation made as to whether the vehicles were on a collision course. When a potential collision was pending, the Intersection AEB algorithm predicted whether a crash could be avoided either by braking or steering of the ego or opponent vehicle. Automatic emergency braking was activated when an avoidance of the collision was no longer possible by braking or steering by either of the conflictinvolved drivers within their comfort zone (Sander, 2017). For simplicity, the comfort boundaries were defined at 5 m/s2 longitudinal or lateral acceleration. After a brake silent time of 150 ms, maximum braking was applied with a brake jerk of 22 m/s3. AEB effectiveness E was calculated from the weighted number N’ of accidents avoided from the simulation with Intersection AEB equipped vehicles and the weighted number N of all accidents from the simulations with vehicles without Intersection AEB expressed as percentage as follows:

3.3. Clustering Based on the PCA evaluation in Sections 4.1 and 4.2, results were given for the following three set of variables: Set 1: This set comprises all relevant variables: initial speeds, collision angle, impact areas, light conditions, precipitation, sight obstruction (plus lateral offset and turning radius for LTAP/OD) Set 2: This set excludes the environmental condition variables and comprises initial speeds, collision angle, impact areas (plus lateral offset and turning radius for LTAP/OD) Set 3: This set further excludes the impact angle variable and thus comprises initial speeds, impact areas (plus lateral offset and turning radius for LTAP/OD) For SCP and LTAP/OD, separate datasets were generated. Observations with variable values coded as unknown or missing were excluded. Pairwise dissimilarity between observations was calculated utilizing the Gower’s generalized coefficient of similarity with the R package ‘cluster’, function ‘daisy’ (Gower, 1971; Kaufman and Rousseeuw, 2005). In the application of Gower’s coefficient of similarity, variables were standardized by default. For HC, agglomerative nesting was applied to the dissimilarity matrix with the Ward method in the R package ‘cluster’, function ‘agnes’ (Kaufman and Rousseeuw, 2005; Struyf et al., 1996). Here, for each case, the squared Euclidean distance to the cluster means was calculated and summed up. The two clusters were merged that resulted in the smallest increase in the overall sum of the squared within-cluster distances. The optimum number of clusters was identified by the ASW with a limit for the maximum cluster number of 150. For PAM-RAN, the initial medoids were randomly selected whereas for PAM-HC the initial medoids were taken from HC when the tree was cut at the same number of clusters (Husson et al., 2010). For both methods, the ASW was calculated for sizes from two to 150 clusters (R package ‘cluster’, function’ pam’). For HC, PAM-RAN, and PAM-HC, the medoids were determined by the minimum of the row-sums of the distance matrix of the corresponding cluster observations. LCC-IMM, capable of handling quantitative and qualitative variables (R package ‘depmixS4’, functions ‘mix’ and ‘fit’), was used to compute latent classes (Hagenaars and McCutcheon, 2002) through maximum likelihood using the EM algorithm (Dempster et al., 1977). To ensure the identification of a global maximum, the simulation loops over different states, the number of latent classes, were run with 15 different seeds for the initialization values. We chose Gaussian distribution for the quantitative variables and multinomial distribution for the qualitative variables. Transfer functions were applied to the values of the initial speed variable to achieve an improved fit with Gaussian distribution. LCC-POL was conducted using the R package poLCA (polytomous LCA) with the same named function. Here, polytomous indicates the usage of qualitative variables only. The model was specified to use 15 random start values for each number of latent classes. The initial speeds were categorized in four groups: 0–29 km/h, 30–39 km/h, 40–49 km/h, and 50+ km/h. For the collision angle, three categories were chosen: < 80°, 80°–100°, and > 100°. As a default setting, the algorithm assumed multinomial distributions for the qualitative variables. For both LCC-IMM and LCC-POL, BIC was used to identify the most suitable number of latent classes. As a goodness-of-fit test for the latent classes we chose the ratio chi-squared test (Cochran, 1952). Due to high computation requirements, BIC, chi-squared, and residual degree of freedom were computed for sizes from two to 50 clusters.

(

)

E = 1− N ′ N ∙100

(6)

AEB effectiveness for the different clustering methods and variations Em was calculated as for E above using the corresponding number of cluster medoids Nm and the number of avoided medoid accidents N’m in place of N and N’:

Em = ⎛1− N ⎝ ⎜

′ m

⎞ Nm ∙100 ⎠ ⎟

(7)

Lastly, we identified whether the Intersection AEB effectiveness derived from calculated cluster medoids Em was substantially better than that derived from random sampled accidents. We calculated the ∼ effectiveness E 10 000 times based on randomly-sampled accidents equal to the number of medoids Nm from the SCP and LTAP/OD data sets. The probability of random sampling giving an equal or better prediction of Intersection AEB effectiveness than that computed from cluster medoids was calculated as follows:

∼ p (|E −E| ≤ |Em−E|)

(8)

4. Results The results for SCP and LTAP/OD are presented separately, as the LTAP/OD scenario required two more variables for a biunique description of the vehicle trajectories (lateral offset and turning radius). Further, the distribution of variables were substantially different. 6

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Fig. 2. Eigenvalues and cumulative variance over components from PCA for SCP.

4.1.2. Clustering In Section 3.2, the sets of variables for clustering were specified as follows: Set 1: Initial speeds, collision angle, impact areas, light conditions, precipitation, sight obstruction Set 2: Initial speeds, collision angle, impact areas Set 3: Initial speeds, impact areas Table 2 gives the results for the application of HC, PAM-RAN, PAMHC, LCC-IMM, and LCC-POL for the different variable sets. In general, a higher number of clusters resulted in higher ASW. With all three sets, the application of HC resulted in the highest number of clusters. For the first variable set, the maximum cluster size was reached with HC. Cluster size and ASW were similar for PAM-RAN and PAM-HC. Though optimal cluster number for PAM-RAN and PAM-HC was smaller than for HC, the maximum in the ASW was not distinctive. For Set 2 and 3, the ASW values indicated that a reasonable structure was found. With the elimination of a continuous variable from Set 2 (giving Set 3), the ASW increased while the optimal cluster number remained constant. The application of LCC-IMM through a loop from two to 50 clusters was not possible for all sets of variables, as during the application of the EM algorithm to models with a higher cluster number, the likelihood decreased (marked with n/a). In general, it is expected that the likelihood will continuously increase (Dempster et al., 1977). It is likely that as the number of clusters increased, the model parameters grew too many relative to the sample size, so that a Gaussian distribution of continuous variables was no longer ensured. Constantly growing BIC values with increasing cluster numbers until EM algorithm abortion indicates that the most likely latent class size was the start value for cluster numbers. With the LCC-POL method, the EM algorithm aborted for the first

4.1. SCP SCP car-to-car accidents account for about 13% of all car-to-car accidents. Two-thirds of the accidents were caused by violation of the right-of-way rule for the vehicle coming from the left. 4.1.1. Principal component analysis Applying the set of variables from Section 3.2 included 26 dimensions. As Fig. 2 indicates, there is a slow increase of cumulative variance over the components. When only those components having an eigenvalue greater or equal to one are considered, then 13 components cover 71% of the total variance. Thus, a reduction to a few principal components while maintaining most of the variance was not feasible. The correlation between the variables was assessed by the angle between the variable vectors in the orthogonal component coordinate system (see Supplementary information – variable correlation and cluster homogeneity). Strong correlation was found between:

• road width and directional lane width for the car coming from the left • sight obstruction from left and right approaching cars. • Moderate correlation was present between: • road width and directional lane width for the car coming from the right • impact areas of the cars. • Finally, minor correlation was identified between: • initial speed of both cars • initial speed and acceleration from standstill for both cars • road/directional lane width and sight obstruction • accident location and light conditions • collision angle and road/directional lane width of the car coming from the right • impact area of both cars and environmental conditions.

Table 2 Optimal cluster size and quality for SCP accidents.

Based on the correlation information, the contribution of variables in the principal components, and the contribution of the principal components to the overall variance, the variables for road width, directional lane width, accident location, and acceleration from standstill were removed from the set of variables for clustering. As there was a high correlation between statements from drivers of the left and right approaching vehicles regarding sight obstruction, a vehicle-independent sight obstruction variable was introduced. 7

HC

PAM-RAN

PAM-HC

LCC-IMM

LCC-POL

Set 1

Cluster number ASW / p-value

150 0.53

73 0.42

76 0.43

n/a n/a

n/a n/a

Set 2

Cluster number ASW / p-value

20 0.60

15 0.56

15 0.56

n/a n/a

2 < 0.00001

Set 3

Cluster number ASW / p-value

21 0.68

15 0.62

15 0.62

n/a n/a

2 < 0.00001

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The correlation matrix (see Supplementary information: variable correlation and cluster homogeneity) was used to analyze the correlation between selected variablesd variables. Strong correlation was not identified. Moderate correlation was present between:

Table 3 Effectiveness of Intersection AEB on accident avoidance for different variable cluster sets for SCP. All SCP accidents

HC medoid SCP accidents

PAM-RAN medoid SCP accidents

PAM-HC medoid SCP accidents

Left approaching car

55%

Set 1 Set 2 Set 3

53% 71% 75%

54% 75% 76%

48% 75% 76%

Right approaching car

47%

Set 1 Set 2 Set 3

50% 52% 66%

47% 43% 68%

51% 43% 68%

• sight obstruction from left and right approaching cars • impact areas of both cars. Minor correlation was present between:

• road widths and directional lane widths • road widths of both cars • directional lane widths of both cars • initial speed and directional lane width for straight heading vehicle • initial speed and sight restriction for turning car • initial speed and acceleration from standstill for turning car • impact areas and lateral offset • lateral offset and turning radius • impact areas and light conditions.

variable set, but succeeded in executing for the second and third variable set. However, LCC-POL converted against a single cluster. BIC indicated that the best models were those with the lowest number of latent classes, though one latent class cannot be tested. The p-value indicated significance to the 0.01 level, which means that the null hypothesis, that the model fits the observations, had to be rejected. 4.1.3. Simulation As it was not possible to derive meaningful clusters with LCC-IMM and LCC-POL, both variants were excluded for the calculation of Intersection AEB effectiveness. The effectiveness of the Intersection AEB for accident avoidance based on all cases and based on representative clusters from HC, PAM-RAN, and PAM-HC is given in Table 3. Although for HC, PAM, and PAM-HC, variable Set 1 obtained the lowest ASW value of all variable sets, the calculated effectiveness was closest to that of all SCP accidents. However, the probability of retrieving a similar or smaller deviation by random sampling was about 50% (Table 4). Though PAM and PAM-HC only differed in the selection of the start values for clustering, the small change in cluster size had a pronounced effect on the calculated effectiveness. With variable Set 2 and 3, the effectiveness for either one or both vehicles in accident avoidance was substantially different than the effectiveness from all SCP accidents. The in-cluster distribution for the variable Sets 1–3 show that many of the clusters are inhomogeneous, which means that neither avoided nor not-avoided accidents were dominant (see Supplementary information – variable correlation and cluster homogeneity). This is particularly valid for bigger clusters.

Slight differences in correlation between variables for SCP and LTAP/OD were present when all components of the PCA were considered. These differences reduced when the projection space was limited to dominant components. Therefore, the variable sets from SCP plus variables for lateral offset and turning were selected for clustering. 4.2.2. Clustering The sets of variables for clustering specified in Section 3.2 are repeated here for convenience: Set 1: Initial speeds, lateral offset, turning radius, collision angle, impact areas, light conditions, precipitation, sight obstruction Set 2: Initial speeds, lateral offset, turning radius, collision angle, impact areas Set 3: Initial speeds, lateral offset, turning radius, impact areas For all variable sets, HC resulted in cluster sizes at the defined upper limit (Table 5). Similar to SCP, the maximum for ASW to define the optimal number of clusters with PAM and PAM-HC was not distinctive. The EM algorithm showed again a decrease of likelihood for LCC-IMM and aborted. LCC-POL calculated the lowest BIC for the lower limit of cluster numbers, where the p-value again indicated that the models did not fit the observations(Tables 6 and 7). 4.2.3. Simulation As for SCP, LCC-IMM and LCC-POL were not considered in the simulation for LTAP/OD. For all variable sets, the effectiveness of the generic Intersection AEB for accident avoidance derived from the HC medoids was of the same magnitude as the effectiveness derived from all SCP accidents. The randomly-selected accidents for HC covered 150 out of 235 accidents. Correspondingly, the probability of deriving a similar effectiveness to that derived from 150 medoid accidents was high. Here, PAM and PAM-HC showed a similar performance with fewer medoids. The probability for similar or lower effectiveness deviation being obtained by random sampling was below 43% for all variable sets. As identified for SCP, clusters were more inhomogeneous with the presence of fewer clusters. Further, inhomogeneity was higher for clusters with more cases (see Supplementary information – variable correlation and cluster homogeneity).

4.2. LTAP/OD LTAP/OD car-to-car accidents account for about 10% of all car-tocar accidents. The turning vehicle caused around 90% of all accidents by not respecting the right-of-way of the straight heading vehicle. 4.2.1. Principal component analysis The dataset included 26 dimensions. Similar to SCP, a reduction of dimensionality would substantially reduce the variance expressed by the variables in the dataset (Fig. 3). Table 4 Probability that random sampling gives equal or better prediction than clustering for SCP.

Left approaching car Right approaching car

Set Set Set Set Set Set

1 2 3 1 2 3

HC medoid SCP accidents

PAM-RAN medoid SCP accidents

PAM-HC medoid SCP accidents

0.42 0.56 0.53 0.51 0.30 0.52

0.18 0.53 0.52 0.06 0.25 0.51

0.51 0.52 0.53 0.46 0.25 0.51

5. Discussion We set out to answer the question of whether it is possible to reduce the diversity of intersection accidents into a set of test scenarios without compromising the ability to predict real-life performance of an Intersection AEB. Given our analysis and findings, we can state that it is 8

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Fig. 3. Eigenvalues and cumulated variance over components from PCA for LTAP/OD.

intersection accidents are highly diverse and particularly when vehicle kinematics are considered, it is hardly possible to reduce the information content to a set of descriptors without losing relevant content. It is not contradictory that for the most comprehensive set of variables HC tends towards the maximum number of clusters, whereas LCC tends towards the minimum. HC evaluates in-cluster variance versus the inbetween cluster variance and the selection of the highest cluster number indicates that there is no substantial difference. LCC however converges on one single cluster, which is not feasible to compute. When information on latent clusters is taken away, only randomness is left. Thus, both cluster approaches indicate that the identified structures are weak and insubstantial. For PAM and PAM-HC, the maxima in the ASW were not outstanding compared to the ASW of the surrounding cluster sizes. Variable selection has a strong influence on the cluster results. Most cluster studies in the field of road traffic safety have not stated under which presuppositions the variables for clustering have been selected and of what quality the resulting clusters are (Anderson, 2009; Lenard et al., 2016; Mauro et al., 2013; Nowakowska, 2012; Saunier et al., 2011). Only a few other studies in the area of traffic safety research have used a combination of qualitative and quantitative variables. Furthermore, variables of a quantitative nature were either pre-classified in categories or recoded into such. Although this simplifies the calculation of similarity, it is inherent with information loss. The usage of Gower’s coefficient of similarity is one way to bypass this issue, though outliers in continuous variables may affect how much contribution this variable makes to the dissimilarity between observations. Methods have been proposed to add weights to the mixed type of data; however, they are computationally expensive (Chae et al., 2006; van den Hoven, 2016). When the definition of clusters from HC, PAM-RAN, and PAM-HC were applied to the simulation results for an exemplary Intersection AEB, many of the clusters showed a similar proportion of avoided and not-avoided accidents. It might be expected that those accident scenarios grouped in one cluster are so similar to each other that all of them are either avoided or not. Further, the cluster medoid is not always representative of the dominating outcome of avoided versus notavoided accidents. The utilized Intersection AEB is only activated if the drivers of both vehicles cannot avoid the pending collision by braking or steering within a defined comfort range. In intersection accidents, only a small change in speed or steering may result in vehicles passing each other. Thus, small differences in the dissimilarity matrix between observations

Table 5 Optimal cluster size and quality for LTAP/OD accidents. HC

PAM-RAN

PAM-HC

LCC-IMM

LCC-POL

Set 1

Cluster number ASW / p-value

150 0.62

22 0.33

23 0.33

n/a n/a

2 < 0.00001

Set 2

Cluster number ASW / p-value

150 0.57

12 0.47

12 0.47

n/a n/a

2 < 0.00001

Set 3

Cluster number ASW / p-value

150 0.59

12 0.48

12 0.48

n/a n/a

2 < 0.00001

Table 6 Effectiveness of Intersection AEB on accident avoidance for different variable cluster sets for LTAP/OD. All SCP accidents

HC medoid LTAP/OD accidents

PAM-RAN medoid LTAP/OD accidents

PAM-HC medoid LTAP/OD accidents

Straight heading car

18%

Set 1 Set 2 Set 3

19% 20% 19%

18% 12% 12%

18% 12% 12%

Turning car

51%

Set 1 Set 2 Set 3

52% 48% 50%

57% 44% 44%

57% 44% 44%

Table 7 Probability that random sampling gives equal or better prediction than clustering for LTAP/OD. HC medoid LTAP/OD accidents

PAM-RAN medoid LTAP/ OD accidents

PAM-HC medoid LTAP/OD accidents

Straight heading car

Set 1 Set 2 Set 3

0.50 0.58 0.51

0.05 0.33 0.43

0.04 0.33 0.43

Turning car

Set 1 Set 2 Set 3

0.38 0.62 0.39

0.37 0.43 0.33

0.38 0.43 0.32

not possible to do so using commonly used clustering methods. It may be possible through other means, as these negative findings do not rule out other solutions can possibly exist, but the PCA results indicate that 9

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structures in the data, but it was not possible to identify scenarios that are sufficiently valid to represent the whole dataset of SCP and LTAP/ OD accidents. This does not mean that it is generally impossible to derive clusters as the basis for test scenario definitions; however, we found that accidents that were avoided differed only slightly in only one variable from accidents that were not avoided. Thus, we anticipate that reducing variation for physical testing runs the risk of inaccurately representing real-life situations. On the other hand we recognize the need for limiting quantity in physical testing. To this end, comprehensive virtual simulations, covering all variations, can support the provision of more accurate predictions of the benefits of new safety technologies. However, we deem physical testing as very necessary: it should be substantial, and defined both to give basic information about Intersection AEB performance and to verify and validate virtual simulations.

can switch the outcome from collision to avoidance, and vice versa. It would be possible to identify variables that have a strong effect on the outcome of accident avoidance for a specific Intersection AEB and then use these variables for clustering. This approach, however, would be system-dependent and not robust for real-world accident avoidance evaluation. The clusters resulting from the methods applied vary strongly in how many observations they contain. One alternative approach could be to focus on those clusters which are most frequently represented. However, the sum of small clusters is not negligible. So far, Euro NCAP has focused on test scenarios where accident avoidance with an appropriately designed safety system is possible. As several of the derived clusters are dominated by not-avoidable accidents and as the Intersection AEB investigated may be considered as one of the more advanced systems with ideal sensing capacities, those clusters may not be of interest for Euro NCAP testing.

Acknowledgement 5.1. Study limitations This research has been supported by the Swedish government agency for innovation systems (VINNOVA) in the QUADRAE project (ref 2015-04863).

This study focusses on cluster analysis as an unsupervised learning technique; the methods applied are popular in road safety analysis. Other unsupervised learning approaches can in principle also be applied: Self-organizing maps (a specific type of neural network) or association rule learning were not investigated and may deliver more stringent data separation and dimensionality reduction. Methods of supervised learning were not employed at all as they require knowledge of classification groups a priori, knowledge which was not available. By using SCP and LTAP/OD scenarios, the accident data is already pre-filtered for the clustering process. Thus, the opportunity to identify intersection clusters out of, for example, all car-to-car accidents, is neglected. However, the definitions for SCP and LTAP/OD have evolved over many years and are part of common terminology. We define test scenarios for police-recorded accidents irrespective of injury severity. This is a consequence of using PCM data and its associated sampling criteria weighted to national data without any further filtering for injury severity. Consequently, the test scenarios are designed to be most effective in assessing the ability of AEB to reduce injury accidents. This can be an objective for AEB and road safety in general; however, safety targets can also be set at different levels. The European Commission, for example, targets fatality and serious injury accidents (EC, 2010). The target of reducing fatalities by 50% from 2010 to 2020 followed a similar target for the preceding 10 year period. A target value for serious injury, now defined consistently across EU member states as MAIS3+ (EC, 2013; EC, 2015), remains to be set by the EC; a proposal from ETSC (2016) is to achieve a 35% reduction between 2014 and 2020. Tingvall et al., 2013 recommend targeting injuries at lower severity, MAIS2+, as otherwise injuries which carry long‐term consequences are otherwise neglected or given less attention. In this analysis, all SCP and LTAP/OD accidents in the PCM database have been considered to be independent of the year of occurrence. We assume that today accidents will happen with kinematics similar to those of 15 years ago. However, passive safety has evolved and older vehicles are not comparable to a modern car fleet. Selecting only MAIS2+ or MAIS3+ accidents in newer vehicles would have reduced the data sample such that a clustering analysis becomes impossible.

Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.aap.2018.01.010. References Anderson, T.K., 2009. Kernel density estimation and K-means clustering to profile road accident hotspots. Accid. Anal. Prev. 41 (3), 359–364. http://dx.doi.org/10.1016/j. aap.2008.12.014. Banfield, J.D., Raftery, A.E., 1993. Model-Based Gaussian and Non-Gaussian Clustering Author (s): Jeffrey D. Banfield and Adrian E. Raftery Published by : International Biometric Society Stable URL: http://www.jstor.org/stable/2532201 Model-Based Gaussian and Non-Gaussian Clustering. Biometrics 49 3, 803–821. Chae, S.S., Kim, J.-M., Yang, W.Y., 2006. Cluster analysis with balancing weight on mixed-type data. Korean Commun. Stat. 13 (3), 719–732. Chavent, M., Kuentz-Simonet, V., Labenne, A., Saracco, J., 2014. Multivariate analysis of mixed data : the pcamixdata R package. arXiv E-Prints, pp. 1–31. Cochran, W.G., 1952. The χ2 test of goodness of fit. Ann. Math. Stat. 23 (3), 315–345. De Oña, J., López, G., Mujalli, R., Calvo, F.J., 2013. Analysis of traffic accidents on rural highways using latent class clustering and bayesian networks. Accid. Anal. Prev. 51, 1–10. http://dx.doi.org/10.1016/j.aap.2012.10.016. Dempster, A.P., Laird, N.M., Rubin, D.B., 1977. Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B Methodol. 39 (1), 1–38. http://dx. doi.org/10.2307/2984875. Depaire, B., Wets, G., Vanhoof, K., 2008. Traffic accident segmentation by means of latent class clustering. Accid. Anal. Prev. 40 (4), 1257–1266. http://dx.doi.org/10.1016/j. aap.2008.01.007. Euro NCAP, 2015. Strategy working group. 2020 Roadmap. Euro NCAP, 2017. European New Car Assessment Programme (Euro NCAP) - Test Protocol AEB VRU Systems. European Commission, 2015. Traffic Safety Basic Facts on Junctions. Fraley, C., Raftery, A.E., 1998. How many clusters? which clustering method? answers via model-based cluster analysis. Comput. J. 41 (8), 578–588. http://dx.doi.org/10. 1093/comjnl/41.8.578. Gesamtverband der Deutschen Versicherungswirtschaft e.V, 2016. Unfalltypen-Katalog. Berlin, Germany. . Gibson, W.A., 1959. Three multivariate models: factor analysis, latent structure analysis, and latent profile analysis. Psychometrika 24 (3), 229–252. http://dx.doi.org/10. 1007/BF02289845. Gower, J.C., 1971. A general coefficient of similarity and some of its properties. Biometrics 27 (4), 857–871. http://dx.doi.org/10.1017/CBO9781107415324.004. Hagenaars, J.A., McCutcheon, A.L., 2002. Applied Latent Class Analysis. Cambridge University Press. Hastie, T., Tibshirani, R., Friedman, J., 2009. The elements of statistical learning. Elements 1, 337–387. http://dx.doi.org/10.1007/b94608. Hautzinger, H., Pfeiffer, M., Schmidt, J., 2004. Expansion of GIDAS Sample Data to the Regional Level : Statistical Methodology and Practical Experiences Hanover, Germany. 1st International Conference on ESAR “Expert Symposium on Accident Research 38–43. Hulshof, W., Knight, I., Edwards, A., Avery, M., Grover, C., 2013. Autonomous emergency braking test results. Proc. 23rd Int. Tech. Conf. Enhanc. Saf. Veh. 1–13. Husson, F., Josse, J., Pagès, J., 2010. Principal Component Methods - Hierarchical Clustering - Partitional Clustering: Why Would We Need to Choose for Visualizing Data? Technical Report Rennes Cedex, France. http://factominer.free.fr/docs/HCPC_

6. Conclusion Intersection accidents are highly variable. Employing popular and diverse methods of unsupervised learning, we found no set of clusters that could group accidents into homogenous groups. Furthermore, we found no set of clusters which accurately discriminates between accidents being avoidable or unavoidable through AEB intervention. This is an important finding for the consideration of future test scenarios for intersection accident mitigation systems. We used various variable sets and state-of-the-art clustering techniques to identify underlying 10

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U. Sander, N. Lubbe

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