  A self-exciting spatio-temporal point process is fitted to incident data from
the UK National Traffic Information Service to model the rates of primary and
secondary accidents on the M25 motorway in a 12-month period during 2017-18.
This process uses a background component to represent primary accidents, and a
self-exciting component to represent secondary accidents. The background
consists of periodic daily and weekly components, a spatial component and a
long-term trend. The self-exciting components are decaying, unidirectional
functions of space and time. These components are determined via kernel
smoothing and likelihood estimation. Temporally, the background is stable
across seasons with a daily double peak structure reflecting commuting
patterns. Spatially, there are two peaks in intensity, one of which becomes
more pronounced during the study period. Self-excitation accounts for 6-7% of
the data with associated time and length scales around 100 minutes and 1
kilometre respectively. In-sample and out-of-sample validation are performed to
assess the model fit. When we restrict the data to incidents that resulted in
large speed drops on the network, the results remain coherent.
