Extending finite mixture models to multiple prospective and retrospective smooth trajectories

This study will involve the development of a new statistical model to allow multiple trajectories of retrospective and prospective count data to be estimated as an extended finite mixture model. Given a target event, the intention is to develop a model which will allow distinct groups of individuals to be identified by examining count data both looking back in time (retrospective trajectories) and looking forward (prospective trajectories), each on multiple count time series. The models will additionally allow for zero inflation and also the likely overdispersion in the data. The linking of the prospective and retrospective trajectories provided by this model will provide valuable predictive information in applications.

Formally, we can assume that for individual i, there are K1 retrospective count times series with values Y(k1) indexed by k1 measured before the event, and K2 prospective count time series with values Y(k2) indexed by k2 measured after the event. Each retrospective series will count backwards from -1 to -T(k1); each of the K2 prospective time series will count forwards from 1 to T(k2). The model thus allows for different time increments for each series and different lengths of series. We can then assume that there are J distinct groups, each consisting of a distinct set of K1+K2 estimated trajectories.

The model can be thought of as a complex form of finite mixture model. The EM algorithm will provide the computational engine for maximising the likelihood. As with all mixture models, multiple start values will be needed to ensure a global solution is found.
This project uses the opportunity created by the Ministry of Justice's Data First initiative to bring together linked conviction, probation and other criminal justice administrative databases on offenders to enable criminal histories to be constructed.

The developed model will be illustrated by the examination of the criminal histories of juvenile sex offenders in England & Wales, taking the target event to be the first sexual conviction of an individual. Criminal histories consist of records of conviction events with information on the date of conviction, the nature of the offence or offences, the plea, and the disposal. For sexual crime there is also limited information on the age and gender of the victim. From the sexual conviction, it is possible to look at both the prior offending history and the subsequent criminal history, and to estimate linked trajectories for both, tying together what happens before an event with what happens after. It should be noted that convictions occur irregularly and there are multiple longitudinal series over time- looking both forward and backward from a target event including severity, type of offending, frequency and disposal (sentence) histories.

The aim is to build a model suitable for estimating linked prospective and retrospective trajectories for multiple count data time series.

The objectives will be
a) to develop software and methods for estimating the above statistical model
b) Using Ministry of Justice data, to apply the model to the prior and subsequent criminal histories of first-time sexual offenders in England and Wales
c) To investigate the potential for a predictive model for subsequent offending by combining the estimated trajectories with probation data.
d) To generate academic impact by publishing academic papers in highly rated and relevant statistical journals (target Journal of the Royal Statistical Society Series A) and criminological journals (target Sexual Abuse)

This project is compatible with a number of EPSRC themes. It is strongly aligned with the Mathematical sciences theme through its innovative statistical content. It is also linked to the data, information and knowledge subtheme of the Digital Economy theme I, being concerned with the understanding and interpretation of large amounts of data.