Extending finite mixture models to multiple prospective and retrospective smooth trajectories

本研究将涉及开发一种新的统计模型，以允许将回顾性和前瞻性计数数据的多个轨迹估计为扩展的有限混合模型。给定一个目标事件，目的是开发一个模型，该模型将允许通过检查计数数据来识别不同的个人群体，这些数据既可以在时间上回顾（回顾性轨迹），也可以在多个计数时间序列上展望（前瞻性轨迹）。这些模型还将考虑到零通货膨胀以及数据中可能的过度分散。该模型提供的前瞻性和回顾性轨迹的链接将在应用中提供有价值的预测信息。形式上，我们可以假设对于个体i，存在K1个回顾性计数时间序列，其值Y（k1）由事件前测量的k1索引，并且存在K2个前瞻性计数时间序列，其值Y（k2）由事件后测量的k2索引。每个回顾性序列将从-1向后计数到-T（k1）;每个K2前瞻性时间序列将从1向前计数到T（k2）。因此，该模型允许每个系列的不同时间增量和不同长度的系列。然后我们可以假设有J个不同的组，每个组由一组不同的K1+K2估计轨迹组成。该模型可以被认为是有限混合模型的复杂形式。EM算法将提供用于最大化可能性的计算引擎。与所有混合模型一样，需要多个起始值，以确保找到一个全面的解决方案，该项目利用司法部的“数据第一”倡议创造的机会，将有关罪犯的定罪、缓刑和其他刑事司法行政数据库联系起来，以便能够建立犯罪历史。所开发的模型将说明在英格兰和威尔士的青少年性犯罪者的犯罪历史的检查，采取的目标事件是个人的第一次性定罪。犯罪历史包括定罪事件的记录，其中包括定罪日期、犯罪性质、抗辩和处置等信息。关于性犯罪，关于受害者年龄和性别的信息也很有限。从性犯罪的定罪来看，既可以看到先前的犯罪历史，也可以看到随后的犯罪历史，并可以估计两者的关联轨迹，将事件发生前的情况与事件发生后的情况联系在一起。应当指出的是，定罪的发生是不规则的，随着时间的推移有多个纵向系列-从目标事件向前和向后看，包括严重程度、犯罪类型、频率和处置（判刑）历史。目的是建立一个模型，适用于估计多计数数据时间序列的相关前瞻性和回顾性轨迹，目标是a）开发估计上述统计模型的软件和方法b）使用司法部数据，将该模型应用于英格兰和威尔士的首次性犯罪者的先前和随后的犯罪历史）通过将估计轨迹与缓刑数据相结合，调查建立后续犯罪预测模型的可能性。（目标期刊的皇家统计学会系列A）和犯罪学期刊（目标性虐待）这个项目是兼容的一些EPSRC的主题。它通过其创新的统计内容与数学科学主题保持一致。它还与数字经济主题一的数据、信息和知识分主题有关，涉及对大量数据的理解和解释。

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 bea) to develop software and methods for estimating the above statistical modelb) Using Ministry of Justice data, to apply the model to the prior and subsequent criminal histories of first-time sexual offenders in England and Walesc) 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.