A continuous binning for discrete, sparse and concentrated observations.

Discrete observations from data which are obtained from sparse, and yet concentrated events are often observed (e.g. road accidents or murders). Traditional methods to compute summary statistics often include placing the data in discrete bins but for this type of data this approach often results in large numbers of empty bins for which no function or summary statistic can be computed. Here, a method for dealing with sparse and concentrated observations is constructed, based on a sequence of non-overlapping bins of varying size, which gives a continuous interpolation of data for computing summary statistics of the values for the data, such as the mean. The method presented here overcomes the problem which sparsity and concentration present when computing functions to represent the data. Implementation of the method presented here is facilitated via open access to the code. •A new method for computing functions over sparse and concentrated data is constructed.•The method allows straightforward functions to be computed over partitions of the data, such as the mean, but also more complicated functions, such as coefficients, ratios, correlations, regressions and others.

A novel rare event approach to measure the randomness and concentration of road accidents.

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. To help us prevent accidents, a metric to determine the level of concentration of road accidents in a city could aid us to determine whether most of the accidents are constrained in a small number of places (hence, the environment plays a leading role) or whether accidents are dispersed over a city as a whole (hence, the driver has the biggest influence). 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. A test application of a tessellation of the space and mixture model is shown using two types of road accident data: an urban environment recorded in London between 2005 and 2014 and a motorway environment recorded in Mexico between 2015 and 2016. FINDINGS: In terms of their concentration, about 5% of the road junctions are the site of 50% of the accidents while around 80% of the road junctions expect close to zero accidents. Accidents which occur in regions with a high accident rate can be considered to have a strong component related to the environment and therefore changes, such as a road intervention or a change in the speed limit, might be introduced and their impact measured by changes to the RECC metric. This new procedure helps us identify regions with a high accident rate and determine whether the observed number of road accidents at a road junction has decreased over time and hence track structural changes in the road accident settings.

Gravity and scaling laws of city to city migration.

Models of human migration provide powerful tools to forecast the flow of migrants, measure the impact of a policy, determine the cost of physical and political frictions and more. Here, we analyse the migration of individuals from and to cities in the US, finding that city to city migration follows scaling laws, so that the city size is a significant factor in determining whether, or not, an individual decides to migrate and the city size of both the origin and destination play key roles in the selection of the destination. We observe that individuals from small cities tend to migrate more frequently, tending to move to similar-sized cities, whereas individuals from large cities do not migrate so often, but when they do, they tend to move to other large cities. Building upon these findings we develop a scaling model which describes internal migration as a two-step decision process, demonstrating that it can partially explain migration fluxes based solely on city size. We then consider the impact of distance and construct a gravity-scaling model by combining the observed scaling patterns with the gravity law of migration. Results show that the scaling laws are a significant feature of human migration and that the inclusion of scaling can overcome the limits of the gravity and the radiation models of human migration.

A metric of the difference between perception of security and victimisation rates.

A person's perception of the level of security at a specific location depends on many factors, including past experiences in that location, the actual crime suffered by the population and more. Thus, when the individual perception that a location is insecure becomes the general rule is when the perception of security becomes an attribute of the region rather than the fears of some of its individuals, hence the relevance of aggregating individual perceptions of security into a single regional perception of security. Residents of two different regions, which have the same levels of crime, of a similar nature, may have different perceptions of the level of security. The perception of security associated with a particular place is relevant by itself but is much more useful when compared to the perception of other regions or when the perception changes over time and hence a ranking of the perception levels from different places would be a useful tool. A metric is suggested here to determine first the regional perception of security from a location and then to quantify its relationship with different victimisation rates. We quantify the relationship between the perception of security and different victimisation rates, based on data obtained from Mexico through victimisation surveys.