A stochastic-statistical residential burglary model with independent Poisson clocks
A stochastic-statistical residential burglary model with independent Poisson clocks
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DOI：
10.1017/s0956792520000029
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发表时间：
2020-02
影响因子：
1.9
通讯作者：
Chuntian Wang;Y. Zhang;A. Bertozzi;M. Short
中科院分区：
文献类型：
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作者：
Chuntian Wang;Y. Zhang;A. Bertozzi;M. Short
住宅盗窃在每个主要城市地区都是一个社会问题。因此，已经取得的进展是为这种类型的犯罪开发定量的、信息丰富的和适用的模型：（1）确定性时间步长（Deterministic-time-step，缩写为“确定性时间步长”）模型[Short，D 'Orsogna，Pasour，Tita，Brantingham，Bertozzi & Chayes（2008）Math. Models Methods Appl.Sci. 18，1249-1267]，一个开创性的基于代理的住宅盗窃犯罪行为的统计模型，假设事件到达的确定性时间步长，首次定量研究了住宅盗窃聚集模式的形成;（2）SSRB模型（住宅盗窃犯罪的基于代理的随机统计模型）[Wang，Zhang，Bertozzi & Short（2019）Active Particles，Vol. 2，Springer Nature Switzerland AG，in press]，其中通过引入泊松时钟从理论上分析了模型的随机分量，其中时间步长变为指数分布的随机变量。为了将独立的代理，在这项工作中，五种类型的泊松时钟被考虑在内。泊松时钟（I），（II）和（III）管理独立的代理行动的盗窃行为，和泊松时钟（IV）和（V）管理代理与环境的相互作用。所有的泊松钟都是独立的。时间增量是独立的指数分布，这是更适合于模型的个体行动的代理。应用泊松过程的合并与分裂方法，将独立的泊松钟视为一个，使分析和模拟类似于SSRB模型。一个鞅公式推导，其中包括一个确定性和随机组件。发现了具有不同窃贼数量的鞅模型的标度性，为有限尺寸效应提供了理论依据。该理论是支持定量的数值模拟使用模式形成量化统计。这里提出的结果将是变革性的应用和分析基于代理的住宅盗窃或其他领域的模型的元素。
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.