Sequential data assimilation for 1D self-exciting processes with application to urban crime data
Sequential data assimilation for 1D self-exciting processes with application to urban crime data
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DOI：
10.1016/j.csda.2018.06.014
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发表时间：
2018-12
期刊：
影响因子：
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通讯作者：
N. Santitissadeekorn;M. Short;D. Lloyd
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文献类型：
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作者：
N. Santitissadeekorn;M. Short;D. Lloyd
许多模型——如霍克斯过程和对数高斯考克斯过程——已经被用来理解犯罪率是如何随时间和/或空间演变的。在这些模型和实际犯罪数据的背景下，通常使用最大似然估计（MLE）对批量数据进行参数估计，但这种方法存在一些局限性，例如实时跟踪受限和不确定性量化。出于实际目的，最好从批处理数据估计转向顺序数据同化。通过推导允许不确定性量化的近似泊松-伽马“卡尔曼”滤波器，开发了一种新的通用贝叶斯序列数据同化算法，用于非齐次泊松过程的联合状态参数估计。该滤波器的基于集成的实现采用与集成卡尔曼滤波器类似的方法开发，使该滤波器适用于大规模的现实世界应用，而不像非线性滤波器（如粒子滤波器）。过滤器的优点是它独立于处理强度的底层模型，因此可以用于许多不同的犯罪模型以及其他应用程序领域。在合成数据和真实的洛杉矶帮派犯罪数据上验证了该滤波器的性能，并与一个非常大样本的粒子滤波器进行了比较，证明了该滤波器在实践中的有效性。此外，霍克斯模型的预测技能被用于使用接受者工作特征（ROC）的预测系统，以提供一个有用的指标，用于预测犯罪类型的警务软件何时可能有用。采用ROC和Brier评分比较分析序列同化和MLE的预测能力。结果表明，序贯数据同化能提高MLE的概率预测。
A number of models – such as the Hawkes process and log Gaussian Cox process – have been used to understand how crime rates evolve in time and/or space. Within the context of these models and actual crime data, parameters are often estimated using maximum likelihood estimation (MLE) on batch data, but this approach has several limitations such as limited tracking in real-time and uncertainty quantification. For practical purposes, it would be desirable to move beyond batch data estimation to sequential data assimilation. A novel and general Bayesian sequential data assimilation algorithm is developed for joint state-parameter estimation for an inhomogeneous Poisson process by deriving an approximating Poisson-Gamma ‘Kalman’ filter that allows for uncertainty quantification. The ensemble-based implementation of the filter is developed in a similar approach to the ensemble Kalman filter, making the filter applicable to large-scale real world applications unlike nonlinear filters such as the particle filter. The filter has the advantage that it is independent of the underlying model for the process intensity, and can therefore be used for many different crime models, as well as other application domains. The performance of the filter is demonstrated on synthetic data and real Los Angeles gang crime data and compared against a very large sample-size particle filter, showing its effectiveness in practice. In addition the forecast skill of the Hawkes model is investigated for a forecast system using the Receiver Operating Characteristic (ROC) to provide a useful indicator for when predictive policing software for a crime type is likely to be useful. The ROC and Brier scores are used to compare and analyze the forecast skill of sequential data assimilation and MLE. It is found that sequential data assimilation produces improved probabilistic forecasts over the MLE.