Residential burglary is a social problem in every major urban area. As such, progress has been to develop quantitative, informative and applicable models for this type of crime: (1) the Deterministic-time-step (DTS) model [Short, D’Orsogna, Pasour, Tita, Brantingham, Bertozzi & Chayes (2008) Math. Models Methods Appl. Sci.18, 1249–1267], a pioneering agent-based statistical model of residential burglary criminal behaviour, with deterministic time steps assumed for arrivals of events in which the residential burglary aggregate pattern formation is quantitatively studied for the first time; (2) the SSRB model (agent-based stochastic-statistical model of residential burglary crime) [Wang, Zhang, Bertozzi & Short (2019) Active Particles, Vol. 2, Springer Nature Switzerland AG, in press], in which the stochastic component of the model is theoretically analysed by introduction of a Poisson clock with time steps turned into exponentially distributed random variables. To incorporate independence of agents, in this work, five types of Poisson clocks are taken into consideration. Poisson clocks (I), (II) and (III) govern independent agent actions of burglary behaviour, and Poisson clocks (IV) and (V) govern interactions of agents with the environment. All the Poisson clocks are independent. The time increments are independently exponentially distributed, which are more suitable to model individual actions of agents. Applying the method of merging and splitting of Poisson processes, the independent Poisson clocks can be treated as one, making the analysis and simulation similar to the SSRB model. A Martingale formula is derived, which consists of a deterministic and a stochastic component. A scaling property of the Martingale formulation with varying burglar population is found, which provides a theory to the finite size effects. The theory is supported by quantitative numerical simulations using the pattern-formation quantifying statistics. Results presented here will be transformative for both elements of application and analysis of agent-based models for residential burglary or in other domains.

中文翻译：

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具有独立泊松钟的随机统计住宅盗窃模型

入室盗窃是每个主要城市地区的社会问题。因此，在为此类犯罪开发定量、信息丰富和适用的模型方面取得了进展：(1) 确定性时间步长 (DTS) 模型 [Short, D'Orsogna, Pasour, Tita, Brantingham, Bertozzi & Chayes ( 2008)数学。模型方法应用程序。科学。18, 1249–1267]，一个开创性的基于代理的住宅盗窃犯罪行为统计模型，首次对住宅盗窃聚集模式的形成进行了定量研究，并假设事件到达的确定性时间步长；(2) SSRB 模型（住宅入室盗窃犯罪的基于代理的随机统计模型）[Wang, Zhang, Bertozzi & Short (2019)活性粒子，卷。2, Springer Nature Switzerland AG, in press]，其中模型的随机分量通过引入泊松时钟进行了理论上的分析，时间步长变成指数分布的随机变量。为了结合代理的独立性，在这项工作中，考虑了五种类型的泊松时钟。泊松钟 (I)、(II) 和 (III) 管理盗窃行为的独立代理行为，而泊松钟 (IV) 和 (V) 管理代理与环境的交互。所有的泊松钟都是独立的。时间增量是独立指数分布的，更适合于对代理的单个动作进行建模。应用泊松过程的合并和分裂的方法，独立的泊松时钟可以被视为一个，使分析和模拟类似于SSRB模型。推导出一个鞅公式，它由一个确定性和一个随机分量组成。发现了具有不同窃贼种群的 Martingale 公式的缩放特性，这为有限尺寸效应. 该理论得到了使用模式形成量化统计的定量数值模拟的支持。这里展示的结果将对住宅入室盗窃或其他领域的基于代理的模型的应用和分析要素具有变革性。