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As one sequence — a cut and edit — gives way to another, so too are time and space able to elide their own eye-witness. It is not just the filmmaker who sees how perfectly these events are staged and incomplete. At the event, a man dressed in a green lycra bodysuit, that allows him to act as a human green screen, wanders around the perimeter of an academic conversation, allowing the appearance of extraneous images and sounds that then act as an interrogation of the terms in discussion, brilliantly illustrating the apex of communication and interruption.

This is realised in the form of a strange yet familiar digital landscape from artist Peter Burr. Everything, it is posited, like the Big Bang, comes from the dark: a sparkle, a light or perhaps a poem emerges, we are told. There is no better descriptor for both the magic of science and the magic of the movies. Film as a poetic object. As a unique, magical experience. Perfect Film Ken Jacobs, Copyright Desistfilm Table 1 contains the reported frequencies between and The motorway data considered here contains road traffic collisions registered on motorways in Mexico.

The data is divided for each motorway and considers, for each accident registered by the police, the distance from the starting point of the highway. Unfortunately, the data does not include in which direction of the road the accident occurred. In total, 9 motorways are considered for the study. Schematic representation of the nine roads which connect Mexico City and the five main cities in its peripheral region.

The length of the motorway and the vehicle flow rate is different for each of the 9 motorways considered.

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Both of these factors become relevant when it comes to studying road accidents. Longer roads or those with a higher number of vehicles are expected to have more accidents even if the risk for a driver is the same as compared to a shorter or less used road. Therefore, the flow, measured in vehicle kilometre per year units, makes the risk on each road comparable. Taking into account the length of the road and the number of cars using it, allows a comparison of different roads to be made. The Federal Road between Mexico City and Cuernavaca, on the other hand, has a higher accident risk and is more lethal meaning that a driver is more likely to suffer an accident and it is more likely that the accident will result in a fatality than in any other of the roads considered, but it is a short road with a reduced traffic flow and so it does not have as many accidents as the other roads.

Thus, comparing the accident risk between different roads has to be based on the length of the road and the number of vehicles that use it or its flow. The accident risk number of accidents per vehicle kilometres of travel and how lethal the accidents are, varies considerably between different roads. The road with the highest accident risk the Federal Road between Mexico City and Cuernavaca is actually It is important to determine when two accidents have occurred at the same location.

Different levels of data aggregation have been used in previous studies, from countries, provinces, counties, road segments, a point pattern process, road junctions and segments of a road with various lengths [ 26 ]. The hypothesis that road accidents are homogeneously distributed known as Complete Spatial Randomness or CSR is easily rejected [ 27 ] by measuring the nearest neighbour distance for every road accident [ 11 ]. A map of where the accidents occurred during the past ten years, in the case of the London data Fig 3 , shows a very specific pattern, highlighting main roads and congested junctions.

In the case of the urban space, we tessellate the region of analysis, that is, the city is divided into nearly 30, non-overlapping, regular hexagons, and the number of accidents within each hexagon is counted. A hexagonal tessellation is frequently used in cartography since it offers advantages in terms of the visualisation [ 28 ] and it offers equal-area units and minimal correlation with regularly spaced features, as opposed to a square grid [ 29 ]. Under this partition, Waterloo Bridge, for example, sits within four hexagons from its extremes on either side of the River Thames.

Hexagon tiles are small enough that the region they represent are clearly identifiable and, although they do not match exactly with road junctions, they clearly represent parts of streets. Smaller tiles do not capture the patterns of road accidents and larger tiles tend to blend different regions into the same tile. Also, a similar measure of 40 metres is used for urban data in other studies [ 9 , 30 ], and so this choice is likely to be close to optimal.

Rare Event Rule 1

In the case of the motorway data in Mexico, we divide the highway into non-overlapping segments of metres and count the number of accidents within each segment. Due to the precision of the data, smaller segments do not group accidents correctly and larger segments are not refined enough to identify a specific location of a highway. Also, metres has been frequently used in other studies when a highway is partitioned [ 7 , 31 ], so we use this level of partition for consistency.

In addition, although there are some vehicular entrances and exits to the motorways between their origin in Mexico City and their outer destinations, these junctions have a reduced number of vehicles compared to the main roads, we thus consider that through each segment of each motorway, the flow of vehicles is approximately the same. Although using either a tessellation in the case of the urban data or a segmentation of the road in the motorway data has its disadvantages such as a potential autocorrelation of the number of accidents it does allow a region to be clearly identified, to cluster the accidents that are nearby and to consider different levels of refinement.

Using this partition of the space transforms the data into a non-negative discrete variable, rather than a continuous measurement of the location of road accident, which is easier to analyse. Fig 4 shows the count of the number of road accidents recorded within each tile and the numbers show that there are many tiles with zero, or close to zero, accidents for the ten year period, but there are also a few tiles with more than accidents. The tiling procedure gives comparable observations in terms of the number of accidents that occur, but not in terms of the risk that a driver experience by travelling across each tile since the number of drivers that go across each tile is significantly different.

In fact, Fig 4 highlights roads in central London where most casualties occur. If our interest is to explain the reasons why a region has more accidents, a common technique is to divide the number of accidents by the traffic volume, so as to consider the Vehicle Miles of Travel VMT [ 32 ]. However, our interest here is to determine a measure of the concentration of such events. Partitioning of Central London into 29, hexagonal tiles, with sides of 40 metres, and the count of accidents between and The number of accidents within each motorway segment or within each hexagonal pixel, during a certain period of time two years in Mexico and 10 years for the London data , might be equal to zero for obvious reasons for example, for tiles which overlay a river or a park or might be much higher in regions with a higher volume of traffic [ 32 ].

An alternative approach is to use a Negative Binomial distribution [ 33 ], by using Survival Theory [ 34 ], or other statistical models [ 25 ], but here, instead of trying to explain why a region has more accidents perhaps through a regression technique we want to measure their spatial degree of concentration, so we simply assume that regions have a different accident rate, without going any further. Using a Poisson distribution for the number of road accidents observed on each segment has conceptual advantages.

In the case of the urban setting, two neighbouring tiles might have similar rates, especially if the same road goes through both of them. In the case of the analysis of motorways, two neighbouring segments might also have similar rates if they experience accidents due to similar causes. Although in our context there is a clear spatial structure that is highly relevant to the problem, we focus on the rates in each of the tiles, and we simply assume that each tile has a fixed accident rate.

Transforming the observed data of road accidents into probabilities gives a different perspective on its distribution. This procedure is known as a mixture model [ 35 ] and the non-parametric maximum likelihood estimator mle helps us compute the optimal number of groups in which the units are grouped, denoted by [ 36 ], the corresponding accident rate for each group and the relative size of each of the groups,. The results of the mixture model the number of groups, the accident rate and relative size can be computed using the statistical package CAMAN Computer Assisted Analysis of Mixtures by considering the observed number of road accidents suffered in each of the tiles or segments and a test can help us accept or reject the distribution obtained [ 37 ].

The distribution of the rates obtained from the data is useful since we could, for example, simulate accidents within each unit to understand the expected departures that simply a natural variability of the number of accidents would yield. In the case of highways, for instance, being aware of the rate of accidents from its origin to its destination gives a full description of the occurred accidents.

The RECC is defined in terms of the distribution of the rates and its expression is given by 1 which is the Gini coefficient [ 22 ] of the distribution of the rates. A value of the RECC closer to zero is interpreted as road accidents being more homogeneously distributed across the city, and a value closer to one means that road accidents are more concentrated in some regions of the city. The RECC is a coefficient comparable over different time periods, between different regions and even for different cities or type of accidents. The procedure of considering a discrete set of observations, assuming they suffer different rates and then measuring the concentration using the RECC , has been used in other contexts, for instance, for the concentration of volcanic eruptions [ 23 ] or the concentration of crime suffered by individuals [ 24 ].

A procedure to obtain a confidence interval for the observed RECC has been developed [ 24 ] based on a Monte Carlo method. It assumes that the observed distribution is the true distribution and, by simulating road accidents in the road segments, departures from the RECC are obtained which could be observed under the same true distribution of accidents. These accidents are not considered to be related to the environment, due to the small rate, and so they could have happened anywhere.

On the other hand, there are tiles with rates higher than 30 accidents over the ten year period, so they expect to have at least one accident every four months and so on. There is, however, a group of tiles with an estimated rate of , meaning that these tiles expected to have one accident every six weeks.

A: The accident rate and group sizes. B: Cumulative accident rates blue and the Lorenz curve in yellow. The RECC is represented by twice the area between the two curves. The level in which road accidents are spatially concentrated is surprisingly high. Table 3 shows the RECC for the road accidents and results are that fatal and serious accidents tend to be much more concentrated in only a few regions.

There are, on the other hand, a few tiles roughly 0. This means that in the small region represented by the tiles, we expect someone to suffer either a serious or a fatal accident every year. Table 4 shows the distribution of the accident rates. Serious and Fatal road accidents have a surprisingly high degree of concentration.

Perhaps accidents which occur at road junctions which have such a small rate cannot be attributed to the road itself and the chances are that they occurred due to causes related to the driver such as alcohol consumption, driving when fatigued or more. The RECC between and for the road accidents in London does not show a drastic change in the way accidents are distributed across the city and so a certain stability is observed, despite the decrease in the number of accidents.

Results are displayed in Table 5. Tiles with the highest rates in London have specific environmental factors which contribute to creating more dangerous roads. For instance, certain Underground stations which are transportation hubs, with a large number of pedestrians, are among the tiles with the highest rate in the city: such as Elephant and Castle, Hyde Park Corner and Camden Town. Also, some roads with a high flow have a consistent high accident rate, such as Euston Road and Kingsland Road the A10 which is a main arterial road and finally, relevant commercial streets are also among the locations with the highest accident rate, such as Oxford Street.

For the Mexican motorway data, comparing the distribution of the accident rates in the nine highways separately reveals that each road has a different pattern. This, however, does not mean that the road expects fewer accidents, but it means that from the origin to the destination, the accident rate remains practically the same at. One way to interpret this, since the units of observation are segments of a road with metres length and we are using two years of data, is that every 10 years a segment expects to observe one accident.

Alternatively, for every 1, metres one road accident is expected every year, irrespective of where on the road we start this measure from. Road accidents are rare events and there is a need to use adequate tools to deal with them. The Federal Road between Mexico City and Puebla has the lowest possible degree of concentration, but it is only when we look at the RECC that we are capable of detecting a uniform pattern. A frequently used metric to determine the concentration is the Gini coefficient.

Unfortunately, computing the Gini coefficient directly from the number of accidents observed on each road segment is not adequate due to the abundance of observations with zero accidents, since there is a correlation of Misleading interpretations also might be obtained from the Gini coefficient, directly from the number of road accidents. Furthermore, the Gini coefficient evaluated for the Federal Road between Mexico City and Puebla turns out to be the highest among the nine roads considered here, and hence it can be wrongly concluded that on this road the accidents are more concentrated than on any other road although looking at the rates, we noticed a uniform pattern.

Accidents have a low frequency and so, in the case of the Federal Road between Mexico City and Puebla we are considering only 49 accidents distributed along units of metres kilometres of road meaning that, due to the low frequency of road accidents, at least In general, the low frequency of events high count of observations with zero events increases the Gini coefficient: the share of events for a great part of the population is zero, thus meaning more inequality in their distribution.

However, by taking into account the distribution of the rates of road accidents in the Federal Road between Mexico City and Puebla and not just the number of road accidents, the results show that almost every segment of that road has the same accident rate and there is practically no concentration of accidents along that road. Another consequence of the low frequency of accidents is that the Gini coefficient computed directly from the number of accidents tends to give similar results between different roads, with small or negligible differences between them and, in the worst case scenario, with the wrong results and interpretation [ 23 ].

Other roads also have a certain degree of uniformity with regards to their accidents. The nine roads in Mexico have a different rate distribution of their accidents Fig 6.

The Federal Road between Mexico City and Pachuca has a set of road segments of five kilometres not necessarily contiguous with a rate of , meaning that there is a small number of segments which consists of 5 kilometres of the road in which we expect to observe 17 road accidents each year. Also, on the Toll Road between Mexico City and Cuernavaca, there are two segments so, one kilometre in length which have an accident rate of , much higher than in the rest of the nine roads.

On that specific kilometre again, not made of contiguous metres segments the expected number of accidents each year is more than ten. A: The accident rates and group sizes. Environmental factors that contribute to the chance of having an accident can be identified in the road segments with high accident rates. For instance, among the highest rate segments of the Federal Road between Mexico City and Pachuca, we find the segments situated at kilometres 35, 72 and 75, where we find the first segment located within an urban area, and the other two segments have junctions.

The accident rate along these two higher rate sections have been identified using the CAMAN procedure and the RECC and this high concentration is attributed to the environment and an intervention to determine whether it is related to the road conditions, its visibility, its design or speed limit should take place.

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In total, in these two segments one on the Federal Road to Pachuca and the other one, on the Toll Road to Cuernavaca which are less than 0. In addition, the different values of the RECC observed on the nine roads which originate in Mexico City are not the result of longer roads so a higher number of observations or a higher flow, nor as a result of a higher number of accidents, but due to other environmental reasons. For instance, the Toll Road to Toluca and the Federal Road to Puebla both have had less than 50 accidents in two years in fact, they are the two roads with the lowest number of accidents but the RECC in the first case is 0.

The methodology presented here, considering the distribution of the rates and the RECC , allows us firstly to overcome the low frequency of events, taking into consideration their random component and to obtain a distribution from which simulations can be easily computed. From the simulations, expected departures from the observed number of accidents can be detected, including outliers.

Results for the urban environment show that road accidents are highly concentrated, especially those that fall into the Serious and Fatal category. Results for the motorway environment show a much smaller concentration degree. In the case of the Federal Road between Mexico City and Puebla, road accidents are considered to be distributed almost uniformly along the road, meaning that statistically speaking, they have the smallest possible concentration.

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Also, the procedure introduced here, including the use of the RECC , allowed a comparison between different roads and a higher accident rate in two specific segments of the highways was observed one on the Federal Road to Pachuca and the other on the Toll Road to Cuernavaca. On these specific sections, accidents might be closely related to environmental factors and so perhaps, some of these accidents could have been avoided by a road intervention scheme, such as a reduction in the speed limit. For a city planner, a quantitative tool such as the RECC derived from the mixture model, provides the ability to compare between different severities or over different time periods to determine the effectiveness and impact of a safety program.

For events, such as road accidents, which are rare and have a high degree of concentration, a tool which allows valid comparisons between different cities becomes a valuable asset enabling us to learn from past experiences. The ability to identify regions of a road or of a city which have environmental factors that increase the risk of an accident enables infrastructures to be re-designed accordingly. Having identified a segment of a road which puts its users at a higher risk due to its environmental factors, means that something can and should be done to reduce that risk.

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The methodology presented here could be easily applied to other types of accidents by adjusting the parameters. For example, the tiling procedure could help a risk manager to identify whether there are regions in some industrial complex with an increased rate of an accident, and the RECC can be used for purposes other than the analysis of accidents, for example, by monitoring the number of people who required the assistance of the coastguard along different parts of the shoreline or it could be used by an insurance company to determine any changes observed in the distribution of accidents. Browse Subject Areas?

Click through the PLOS taxonomy to find articles in your field. Abstract Background Road accidents are one of the main causes of death around the world and yet, from a time-space perspective, they are a rare event. Methods Here, we apply a new metric, the Rare Event Concentration Coefficient RECC , to measure the concentration of road accidents based on a mixture model applied to the counts of road accidents over a discretised space. Heat maps and the random location of accidents Numerous studies have been conducted to identify the spatial patterns of road traffic accidents and develop techniques to identify crash-prone locations using, for instance, Bayesian inference [ 3 ], or data mining techniques [ 4 , 5 ].

Download: PPT. Fig 1. Heat map of a simulated point process that follows a uniform distribution. Concentration of road accidents Road accidents might happen due to a mixture of environmental elements, for example, an obstructed visibility, excessive speed of road users, the curvature or quality of the roads, the street lighting and more.

Spatial counts of the road accidents Two sources of information and two types of analysis are used here to compare the concentration of road accidents. Urban data—London The data from the Transport for London contains information on road traffic collisions that involve personal injury occurring on public highways which have been reported to the police. Table 1. Observed frequencies of collisions in Greater London between — Motorway data—Mexico The motorway data considered here contains road traffic collisions registered on motorways in Mexico. Table 2. Observed frequencies of collisions on the nine motorways which have Mexico City as origin between and Methodology It is important to determine when two accidents have occurred at the same location.

Fig 3. Registered road accidents in Central London between and Discretisation of the data Urban environment. Motorway environment. Distribution of road accidents The number of accidents within each motorway segment or within each hexagonal pixel, during a certain period of time two years in Mexico and 10 years for the London data , might be equal to zero for obvious reasons for example, for tiles which overlay a river or a park or might be much higher in regions with a higher volume of traffic [ 32 ]. Inhomogeneous distribution of road accidents. Fig 5. Table 3. RECC metric of the road accidents in London between — Table 4.

Estimated group size and accident rate for the Serious and Fatal accidents in London between and