A stochastic-statistical residential burglary model with independent Poisson clocks

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.

One prominent feature of urban residential burglary is that the crimes are not distributed uniformly in time and space but rather exhibit spatio-temporal aggregates called 'hotspots'. In [93], a pioneering agent-based statistical model of criminal behaviour for residential burglary (referred to as the DTS model) is developed and the aggregate pattern formation is quantitatively studied for the first time. Interactions of agent actions with the environment are described based on the near-repeat victimisation and the broken-windows effects, which are notions in criminology and sociology that have been empirically observed [10,32,43]. Follow-up works of [93] show that such simple statistical agent-based model can exhibit both crime displacement and crime suppression in the presence of police activity [90,91]. In the DTS model, time steps are discretised with a fixed duration and occur at regular intervals, such that all types of events for all agents arrive according to the same schedule. However, a more realistic model should treat all events as occurring independently, according to their own stochastic clock.
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]). A Martingale formulation is derived for the SSRB model to express the model as the summation of two components: a deterministic component and a stochastic component. Together with statistics quantifying the degree of hotspot transience, analysis of the stochastic component leads to a theory for the finite size effects present in the system. These effects are observed in the simulations of the DTS model and the SSRB model: both transient and stationary hotspot dynamical regimes appear in the simulations, and as burglar number decreases, simulations exhibit more transience. Since the number of operating criminals within any city is very likely to fall within the regime of hotspot transience observed, a deeper and quantitative understanding of the finite size effects is relevant to real crime statistics [38,76]. This is the first time that stochastic analysis is applied into residential burglary models and the stochastic component is theoretically analysed.
Despite the progress made by the SSRB model, it still has some limitations. Most notably, in the SSRB model, only one Poisson clock governs all the agents, with all agents acting simultaneously whenever the clock randomly ticks. However, in reality, criminals (or very small groups of criminals acting as teams) act independently. Therefore, models are called for in which independent actions of agents are taken into account. In fact, within the context of the DTS family of models, five types of independent Poisson clocks need to be taken into consideration to describe each agent's burglary behaviour and their interactions with the environment as well.

The SSRB-IPC model
The agent-based SSRB-IPC model consists of essentially two components -the stationary burglary sites and burglar agents jumping from site to site, occasionally committing crimes. Independent Poisson clocks will govern time increments of evolution of these components.
We assume the domain as D := [0, L] × [0, L] with periodic boundary conditions. The lattice grid over the domain D has spacing = L/N, N ∈ N. The grid points are denoted as s = (s 1 , s 2 ), s 1 = , 2 , · · ·, L, s 2 = , 2 , · · ·, L. And the collection of all the grid points is denoted as S . Attached to each grid point s ∈ S is a pair (n s (t), A s (t)) representing the number of burglar agents and attractiveness at site s at time t. The attractiveness shows the burglar's beliefs about the vulnerability and value of the site. We also assume that A s (t) consists of a dynamic term and a static background term Here A0 s is not necessarily uniform over the lattice grids. And the initial data are given to be (n s (0), B s (0)) = (n0 s , B0 s ). A type (I) Poisson clock is assigned to each agent to govern his action of burgling. Suppose that a type (I) clock of an agent advances at time t − . At time t, the agent will burgle his current location and immediately be removed from the system, representing the burglar fleeing with his trophy. Type (I) Poisson clocks advance according to independent Poisson processes with rate A s (t), s being the current location of the burglar agent. Thus, the attractiveness numerically represents the stochastic rate of burglary at a location, given that there is a criminal located there.
A type (II) Poisson clock is assigned to each agent to govern his movement. Suppose that a type (II) clock of an agent advances at time t − . At time t, the agent will jump from site s to one of the neighbouring sites, say k, with a probability, q s→k (t), which is defined by the ratio of the attractiveness of k over the combined attractiveness of the neighbouring sites of s: where T s (t) := s s ∼s A s (t), s ∼ s indicating all of the neighbouring sites of s. To create consistency with the DTS model, type (II) Poisson clocks advance according to independent Poisson processes with rate D −2 , where D is an absolute constant independent of . On average, the time increment δt for type (II) clocks is the inverse of the rate of the Poisson clock: A type (III) Poisson clock is assigned to each site to govern replacement of burglars. Suppose that the type (III) clock attached to a site s advances at time t − . At time t, a new agent will be placed at s. Type (III) Poisson clocks advance according to independent Poisson processes with rate , an absolute constant indicating the growth rate of criminal population.
A stochastic-statistical residential burglary model 5 The attractiveness field gets updated according to the repeat and near-repeat victimisation and the broken-windows effect. The 'broken windows' theory argues that the visible signs of past crimes are likely to create an environment that encourages further illegal activities [106]. The so-called repeat and near-repeat events depict the phenomenon that residential burglars prefer to return to a previously burgled house and its neighbours [30,49,50,51,92].
The repeat victimisation and broken-windows effects are modelled by letting B s depend upon previous burglary events at site s. Let θ be an absolute constant measuring strength of the repeat victimisation effect, and the attractiveness increases by θ whenever a burglary event occurs on that site via a type (I) clock as described above. However, this increase has a finite lifetime governed by type (IV) Poisson clocks, one of which is assigned to each site. Suppose that the type (IV) clock associated with site s advances at time t − . At time t, the attractiveness gets updated as follows: where ω is an absolute constant setting the speed of the decay. Type (IV) Poisson clocks advance according to independent Poisson processes with rate D σ , where σ < 0 is an absolute constant that affects the speed of this type (IV) process, and whose value we will decide later. On average, the time increment δt for the type (IV) clocks is the inverse of the Poisson clock rate The near-repeat victimisation effect is modelled by allowing B s to spread in space from each house to its neighbours via a type (V) Poisson clock, which is assigned to each site. Suppose that the type (V) clock associated with a site s advances at time t − . At time t, the attractiveness gets updated as follows: where η ∈ (0, 1) is an absolute constant that measures the significance of neighbourhood effects, and is the discrete spatial Laplace operator associated with the lattice grid, namely Again for consistency with the DTS model, type (V) Poisson clocks advance according to independent Poisson processes with rate D −2 .
The spatially homogeneous equilibrium solutions to the above described SSRB-IPC model are the same as the DTS model [93] and the SSRB model [104]. From now on, we always assume that A0 s ≡ A0 for simplicity. The homogeneous equilibrium values are deduced as 6 C. Wang et al.

Numerical simulations of the SSRB-IPC model
To compare the agent-based SSRB-IPC model with the agent-based SSRB model [104], we perform simulations of the models and display the resulting attractiveness fields in Figures 1, 2, 3 and 4. The SSRB model is shown in Figures 1(a), (b), 2(a), (b), 3(a), (b), 4(a) and (b). The SSRB-IPC model is shown in the remaining figures. The parameters are mostly the equivalent of those used to create the plots for the DTS model [93] in Figure 3 of [93] and for the SSRB model in Figures 2, 3 and 4 of [104]. The same behavioural regimes are observed: (1) Spatial homogeneity. In this regime, A s (t) does not vary essentially in space or in time. Very few visible hotspots, that is, the accumulation of A s (t) in space and in time, appear in the process.
(2) Dynamic hotspots. In this regime, localised spots of increased attractiveness form and are transient. These spots remain for varying lengths of time and may appear and disappear at seemingly random locations and move about in space and deform over time. They may also merge and split. Similar examples of transitory hotspots have been observed in the simulation output in [93] and [104].
(3) Stationary hotspots. In this regime, hotspots form and stay more or less stationary over time.
The spatially homogeneous equilibrium value of the dynamic attractivenessB in (2.7) serves as a midpoint for all the simulations and is shaded in green. To document the false colour map for the attractiveness, a colour key is given in the figures. All the simulations were run with L = 128, = 1, ω = 1/15, D = 100, and A0 = 1/30. As = 1, there is no need to specify σ for type (IV) Poisson clocks. And the initial criminal number at each site n0 s is set to ben on average, and the burglar agents are assumed to be randomly uniformly distributed over the 128 × 128 grids, with s∈S n0 s = 128 2n .
For the cases with zero hotspot formation (Figures 1(a), (b), (c) and (d) and 2(a), (b), (c) and (d)), we set η = 0.2 and B0 s ≡B for every s ∈ S . For the cases with hotspot formation (the remaining figures), η = 0.03, and B0 s is set to beB on every site except for 30 randomly chosen grid points where the attractiveness is increased by 0.002.
The same finite size effects as in the SSRB model [104] and the DTS model [93] are observed. The degree of hotspot 'transience' seems to depend on the total burglar population. The regimes of transient hotspots seem to appear associated with low or vanishing burglar numbers and low numbers of events, while the regimes of stationarity, including the stationary hotspots or homogeneity regimes, occur more likely with large numbers of burglars and burglary events.     A stochastic-statistical residential burglary model 11 3 Analysis and discussion

Martingale formulation
The Martingale formulation of a Markov pure jump process characterises the process as sum of an integral part involving the infinitesimal mean and a Martingale part involving the infinitesimal variance. For every t, we define B (t), n (t) := { B s (t), n s (t) : s ∈ S }. In a similar way, we can define the stochastic processes n (t), B (t) and A (t) associated with the SSRB-IPC model. For arbitrary scalar fields f := {f s : s ∈ S } and g := {g s : s ∈ S }, we define the discrete inner product and L p norm over the lattice S : Let φ = {φ s : s ∈ S } be an arbitrary stationary scalar field, and we define As B (t), n (t) , φ is a Markov pure jump process with state space R 2 , a Martingale approach is applicable (e.g., [20,21,54,65,96]) and a Martingale formulation can be derived as follows: Theorem 3.1 For each fixed , before the possible blow-up time, B (t), n (t) , φ can be written as

2)
where M i B (t), n (t) , φ , i = 1, 2, are Martingales that start at t = 0 as zeros, and G i B (t), n (t) , φ , i = 1, 2, are the infinitesimal means for the attractiveness and the burglar distribution, respectively, and The variances of M i B (t), n (t) , φ , i = 1, 2, can be characterised in the following way: where V i B (t), n (t) , φ , i = 1, 2, are the infinitesimal variances for the attractiveness and the burglar distribution, respectively, and where ∇ s→s φ s denotes the discrete directional derivative from s pointing towards s , that is, Proof of Theorem 3. 1 We compute the infinitesimal means and variances for the Markov pure jump process B (t), n (t) , φ for a fixed , using the methods in [1,17,22,44,46,53,62,73,74,83,85]. In the computational steps, we will drop the superscript for simplicity.

Continuum analogue of the SSRB-IPC model
With a similar derivation of the hydrodynamic limit of interacting particle systems [26,40,54,55,96,97,102,103], we can find a continuum analogue of the agent-based SSRB-IPC model based on the Martingale formulation (3.2), where the infinitesimal variances have an equal or lower order of magnitude than .
We check the validity of the continuum equation of the agent-based SSRB-IPC model through simulations. We use the same algorithm as that for the continuum analogue of the DTS model [93] (see (3.11)-(3.13) in [93]). Figure 5 shows example output of the attractiveness A(x, t) in the cases of hotspot formation. The same colour key is used as in Figure 1. We infer from (3.14) that the parameters and data used to create Figures 3(a), (b), (c) and (d), and 4(a), (b), (c) and (d) give rise to the same attractiveness field for the continuum analogue of the SSRB-IPC model. Hence, we only display the output once here in Figure 5. For the cases of zero hotspot formation, the parameters and data used to create Figures 1(a), (b), (c) and (d) and 2(a), (b), (c) and (d) give rise to the same attractiveness field for the continuum analogue of the SSRB-IPC model, which is the equilibrium as the continuum system stays at the equilibrium with equilibrium initial data.
The regimes of dynamic hotspots are absent in the continuum simulations. As the total burglar population decreases, deviations of behavioural regimes of the SSRB-IPC model from its continuum equation increase. These observations are closely linked to the finite size effects exhibited in Figures 1, 2, 3, and 4. The same hotspot pattern formations are observed in the simulations of the DTS model [93], the SSRB model [104] and their respective continuum equations.

Quantification of the pattern formation
To analyse the finite size effects mathematically, we quantify the pattern formation of hotspots by measuring the degree of hotpot transience with statistics through quantitative simulations.

Statistics measuring degree of hotspot transience
We use the same statistics as in [104] to measure degree of hotspot transience: (i) Relative Fisher information relative to the uniform measure over S , logarithm mean type (see Appendix of [16], and also [27,28,29]): (3.15) (ii) Rate of change of A (t) over time in the discrete L p norm, p 1. Fixing a time increment t > 0, we define the L p -area rate of change as (iii) Rate of change of the area of regions with attractiveness higher than 2Ā (red regions). For these regions, we define the relative overlapping area O (t) and non-overlapping area N (t) as follows: 1(s), (3.17) where 1(s) is an indicator function, and A R is the renormalisation:

3.19)
For the continuum equation (3.13), in a very similar way, we can define all the analogues of the above quantities, which will be denoted as I (t), δ p (t), O(t) and N (t). As in [104], the simulation output shows that a larger degree of hotspot transience appears with a smaller burglar population. Also, the continuum simulations exhibit the lowest degree of hotspot transience. The peaks in Figures 7(b), (d), 9(a) and (b) correspond to the initial emergence of hotspots in the SSRB model, the SSRB-IPC model, and their continuum equations. The statistics increase during the emergence period as hotspots form, and decrease and stabilise (or directly stabilise) as hotspots stabilise. The same simulation output is also observed over other random paths. The output matches well with the qualitative simulations in Figures 1, 2, 3, 4, and 5, which indicates that the above statistics are suitable to use.

Mathematical analysis and simulations of the finite size effects
With quantification of the degree of hotspot transience, we analyse the finite size effects based on the Martingale formulation, and simulations are run which supports our theoretical conclusion.  7. Examples of the L 1 -area rate of change δ 1 (t) and δ 1 (t) for SSRB and SSRB-IPC models for zero hotspot formation and hotspot formation. For the SSRB model [104], (a) shows results with cases of zero hotspot formation, and the blue, magenta, black and red lines represent the statistics of the models plotted in Figures 1(a)

A theory of the finite size effects
We analyse the deterministic and stochastic component of the Martingale formulation with varying initial burglar number, repeat victimisation strength θ and replacement rate . Fixing > 0, > 0, for q ∈ (0, D/ 2 ), we consider the agent-based SSRB-IPC model with parameters and initial data scaled in the following way: (3.20) Here,B,n are the homogeneous equilibrium values as in (2.7) with θ = and = : By (3.5) and additivity of the variance in time for Martingales, we have (3.23) This together with (3.22) implies that the infinitesimal variances are the key to estimate the standard deviation of the stochastic component, and the deviation of the trajectories of the evolution of the model from its deterministic component.
To analyse the infinitesimal variance for attractiveness, by (3.12) , we have We perform estimates at the first time step. At time zero, from (3.24) and (3.21), we infer 25) which implies that the infinitesimal variance for the attractiveness is inversely proportional to q: (3.26) We have for 0 < q <q < D / 2 , t, φ , for 0 < q <q < D 2 and t > 0, (3.29) which leads to a theory of the finite size effects at an arbitrary later time.

Numerical simulations of the theory of finite size effects
To check the validity of (3.29), we perform direct simulations of V ,(q) 1 (t), the infinitesimal standard deviation for the attractiveness. We make comparisons of the agent-based SSRB-IPC model with the SSRB model [104]. Example output can be found in Figure. 10. For all the simulations, we choose the test function as φ (x) = 1 + sin(x 1 )sin(x 2 )/20, x = (x 1 , x 2 ). (3.30) This choice of test function is arbitrary. It is a perturbation of the constant function so that the test function is non-trival. The perturbation does not need to be periodic. For the agent-based SSRB model [104], Figure 10 The output of the simulations supports the validity of (3.29). The same simulation results are also observed over other random paths.
Furthermore, we check with simulations the validity of (3.25) for an arbitrary later time t > 0, which is  This is an analog of Equation (39) [104] for the SSRB model. In Figure 11, we run direct simulations for the SSRB model [104] and the SSRB-IPC model. Taking average of both sides of (3.31) over a time period [T 1 , T 2 ], we obtain (3.32) Figure 11 shows the log-log plot with error bars for (3.32). The lines show the theoretical scaling with slope as −1 and the x-intercept as 2 2 φ 22 C. Wang et al.  [104], Figure 11 (a) shows results with no hotspot, and the error bars with x-axis as 0, 1, 2 and 3 show results with the simulations of the blue, magenta, black and red lines in Figure 10 (a), respectively. Figure 11(b) shows results with hotspot formation, and the error bars with x-axis as 0, 1, 2 and 3 show results with the simulations of the blue, magenta, black and red lines in Figure 10 (b), respectively.
For the SSRB-IPC model, Figure 11(c) shows results with no hotspot, and the error bars with x-axis as 0, 1, 2 and 3 show results with the simulations of the blue, magenta, black and red lines in Figure 10 (c), respectively. Figure 11(d) shows results with hotspot formation, and the error bars with x-axis as 0, 1, 2 and 3 show results with the simulations of the blue, magenta, black and red lines in Figure 10 (d), respectively.
The output shows that for the agent-based SSRB model [104] and the agent-based SSRB-IPC model, the error bars are short and fall mostly on the straight lines representing the theory. Moreover, the larger the total number of burglars is, the shorter the error bar is. The same simulation results are also observed over other random paths. These results support the validity of (3.32) and our theory for the finite size effects based on the Martingale formulation.

Conclusion
In this article, independent Poisson clocks are applied to individual agents in the agent-based DTS model [93]. The time increments are independently exponentially distributed random variables, which are more suitable to model independent actions of agents. The agent-based SSRB-IPC model is more realistic than the agent-based SSRB model [104] where one Poisson clock governs all the agents. The SSRB and SSRB-IPC models are both models with stochastic features.
The simulations of the SSRB-IPC model produces aggregate pattern formation of residential burglary assuming independent agent actions. And the finite size effects are observed in the simulations: dynamic hotspots are observed associated with small burglar population.
The agent-based SSRB-IPC model is also an interacting particle system, and stochastic analysis is applicable. A Martingale formulation for the SSRB-IPC model is derived. The formula consists of a deterministic and a stochastic component, that is, the infinitesimal mean and the infinitesimal variance. The infinitesimal mean is very similar to that of the SSRB model (see (11) and (12) in [104]). And the infinitesimal mean yields a continuum equation, which coincides with the that of the DTS model [93] and of the SSRB model [104]. Simulations show that the continuum equation is a good approximation of the SSRB-IPC model under the circumstance of a large number of burglars. The infinitesimal variance of the SSRB-IPC model is less complicated than that of the SSRB model (see (13) and (14) in [104]). And we get more robust and rigorous results than [104].
Moreover, the pattern formation of crime hotspots is quantified by the statistics measuring the degree of transience of hotspots. A theory for the finite size effects is developed based on the Martingale formulation. As the burglar population decreases, the stochastic component of the Martingale representation increases in size, while the deterministic component remains the same, which leads to a larger stochastic fluctuation of the SSRB-IPC model from its continuum equation. The theory can be proven at time zero with equilibrium initial data, and we conjecture that it remains to be true at arbitrary later times with negligible error. Quantitative numerical simulations support our conjecture. The finite size effects are closely related to hotspot transience, which is well documented in real crime statistics. Therefore, through the observation of hotspot transience, we could estimate the number of criminals, something that is normally difficult to predict. For general human behaviour with similar aggregation pattern formation, our finding could help with the prediction of the size of agents participating in the activity.