A Stochastic-Statistical Residential Burglary Model with Finite Size Effects

Wang, Chuntian; Zhang, Yuan; Bertozzi, Andrea L.; Short, Martin B.

doi:10.1007/978-3-030-20297-2_8

Cited by 5 publications

(27 citation statements)

References 74 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The deterministic component leads to a hydrodynamic-limit-type continuum analogue of the SSRB-IPC model. The continuum analogue of the SSRB-IPC model coincides with that of the DTS model [93] and of the SSRB model [104]. The stochastic component of the SSRB-IPC model is less complicated to compute and analyse than that of the SSRB model, as all the covariances vanish, thanks to the independence of the Poisson clocks.…”

Section: Introductionmentioning

confidence: 77%

“…Since residential burglary is a critical social issue in every major urban area, quantitative, informative and applicable crime models are called for inside and outside of the scientific community. Mathematical modelling and prediction of crime have been a burgeoning area for more than 10 years and many works have been done (e.g., [3,4,6,11,12,38,39,56,59,60,66,70,72,76,77,78,81,84,86,87,88,90,91,92,93,94,98,99,104,106,108]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…To incorporate randomness of arrivals of events into the DTS model, in [104], a Poisson clock is applied to the DTS model [93] and the time steps are turned into exponentially distributed random variables, and further stochasticity is brought into the DTS model. The model in [104] is referred to as the SSRB model (agent-based stochastic-statistical model of residential burglary crime). The introduction of a Poisson clock into the residential burglary model brings in theoretically the mathematical framework of Markov pure jump processes and interacting particle systems ( [20,62,63,64,65]).…”

Section: Introductionmentioning

confidence: 99%

“…The methodology of the theoretical analysis and numerical simulations of the SSRB-IPC model employs theory regarding the merging and splitting of Poisson processes [7], and the independent Poisson processes associated with the independent Poisson clocks can be treated as one merged Poisson process wherein the Poisson clocks compete to advance first in the merged Poisson process with probability in proportion to the rate of each Poisson clock. This method allows the analysis and computation of the SSRB-IPC model to be very similar to that of the SSRB model [104], and the simulation cost of the SSRB-IPC model is reduced.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A stochastic-statistical residential burglary model with independent Poisson clocks

et al. 2020

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A stochastic-statistical residential burglary model with independent Poisson clocks

et al. 2020

Self Cite

View full text Add to dashboard Cite

show abstract

Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness

Lindstrom

Bertozzi

2020

Math. Models Methods Appl. Sci.

Self Cite

View full text Add to dashboard Cite

In this paper, we develop a continuum model for the movement of agents on a lattice, taking into account location desirability, local and far-range migration, and localized entry and exit rates. Specifically, our motivation is to qualitatively describe the homeless population in Los Angeles. The model takes the form of a fully nonlinear, nonlocal, non-degenerate parabolic partial differential equation. We derive the model and prove useful properties of smooth solutions, including uniqueness and [Formula: see text]-stability under certain hypotheses. We also illustrate numerical solutions to the model and find that a simple model can be qualitatively similar in behavior to observed homeless encampments.

show abstract

The existence and stability of spike solutions for a chemotax is system modeling crime pattern formation

Mei

Wei

2020

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

Urban crime such as residential burglary is a social problem in every major urban area. As such, many mathematical models have been proposed to study the collective behavior of these crimes. In [V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, M. B. Short, M. R. D’Orsogna and L. B. Chayes, A statistical model of crime behavior, Math. Methods Appl. Sci 107 (2008) 1249–1267; M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst. 9 (2010) 462–483], Short et al. proposed an agent-based statistical model of residential burglary to model the crime hotspot phenomena. From the point of view of reaction–diffusion systems, the model is a chemotactic system with cross diffusion that exhibit hotspot phenomena. In this paper, we first construct a radial hotspot solution of this system, then study the linear stability of this hotspot solution by studying a nonlocal eigenvalue problem. It turns out that the stability of the hotspot is completely different depending on which spatial dimension the system is on. The main mathematical difficulty of the system involves treating the steep change of diffusion near the core of the hotspot, because of the quasilinearity induced by the cross diffusion. We believe that the techniques used in this paper can be developed to treat many other chemotactic systems.

show abstract

A Stochastic-Statistical Residential Burglary Model with Finite Size Effects

Cited by 5 publications

References 74 publications

A stochastic-statistical residential burglary model with independent Poisson clocks

A stochastic-statistical residential burglary model with independent Poisson clocks

Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness

The existence and stability of spike solutions for a chemotax is system modeling crime pattern formation

Contact Info

Product

Resources

About