Observations on the Relationship between Crash Frequency and Traffic Flow

This work analyzes the relationship between crash frequency <i>N</i> (crashes per hour) and exposure <i>Q</i> (cars per hour) on the macroscopic level of a whole city. As exposure, the traffic flow is used here. Therefore, it analyzes a large crash database of the city of Berlin, Germany, together with a novel traffic flow database. Both data display a strong weekly pattern, and, if taken together, show that the relationship <inline-formula><math display="inline"><semantics><mrow><mi>N</mi><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></semantics></math></inline-formula> is not a linear one. When <i>Q</i> is small, <i>N</i> grows like a second-order polynomial, while at large <i>Q</i> there is a tendency towards saturation, leading to an S-shaped relationship. Although visible in all data from all crashes, the data for the severe crashes display a less prominent saturation. As a by-product, the analysis performed here also demonstrates that the crash frequencies follow a negative binomial distribution, where both parameters of the distribution depend on the hour of the week, and, presumably, on the traffic state in this hour. The work presented in this paper aims at giving the reader a better understanding on how crash rates depend on exposure.