
Citation: Gideon Simpson, Daniel Watkins. Relative entropy minimization over Hilbert spaces via Robbins-Monro[J]. AIMS Mathematics, 2019, 4(3): 359-383. doi: 10.3934/math.2019.3.359
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Quantitative, informative and applicable models have long been proven beneficial to assist law enforcement to curb urban crimes. Following the seminal work (see reference [18]) on the mathematics of agent-based models for residential burglary, many works have been done on mathematical crime modeling and prediction, see e.g. references [1], [3], [7], [9], [12], [13], [14], [15], [16], [18], [19] and [20], and the references cited therein, and in [18].
There are roughly two classes of crime models. One is agent-based aiming to describe the individual activities, and the other is event-based without specifying the individual agents, aiming to predict the patterns of observed events (see e.g. references [15] and [16]). Here we address the first class of models.
In reference [18], "hotspots", clusters of residential burglary well documented in real life, are quantitatively studied for the first time. A discrete lattice grid is imposed, upon which residents are distributed each labeled with its level of attractiveness, and burglary agents walk over sites seeking for targets. Burglary dynamics is coupled with the environment variable based on the repeat and near-repeat victimization and the broken-windows effects. These notions maintain that a previously burgled house and its neighbors are more attractive to potential offenders, as an environment encouraging further illegal activities is more likely to be created by past crimes with visible signs (see the relevant references in references [18], [19] and [20]).
In reference [18] the time steps are set as deterministic and all types of events occur at fixed regular intervals. Random arrivals are incorporated later in references [19] and [20], governed by stochastic clocks generated from Poisson processes. In reference [19], there is one uniform Poisson-clock, and in reference [20] independent Poisson clocks are assigned to each individual agent. To summarize, these agent-based models are referred to as DTS (deterministic-timestep) Model (see reference [18]), SSRB (stochastic-statistical residential burglary) Model (see reference [20]), and SSRB-IPC (independent Poisson-clock) Model (see reference [20]), respectively. Continuum analogues parallel to these models are derived as a mean-field limit in reference [18] or a potential hydrodynamic limit in references [19] and [20], both of which lead to the same coupled deterministic reaction-diffusion equation system.
The models mentioned above serve as the first-generation agent-based criminal behavior models, whose prototype will be referred to as First-Generation Model. They are stochastic models with a continuum limit that can be viewed as a statistical average. The continuum limit is simpler to analyze and simulate by computer, and when total population is sufficiently large, the discrete and continuum simulations exhibit similar hotspot dynamics. However, as agent number decreases, the discrete simulations exhibit more transient hotspot regimes, generating the finite size effects. Quantitative analysis in references [19] and [20] show that these effects arise due to stochasticity of the system, and criminals act randomly in reality. It is called for that we combine the strengths of First-Generation Model and its continuum limit to build a second-generation criminal behavior model.
In this work, we construct a stochastic multi-scale criminal behavior model with hybrid dynamics, where agent actions and environment variables are set on different spatio-temporal scales. This model will be referred to as M-IPC (Multiscale-Independent-Poisson-Clock) Model. In contrast, the clocks in First-Generation Model (deterministic and random) all have rates at the same order of magnitude. Here we assume instead that the environment variables evolve as the fast component continuously in time and in space, while criminal agents move along discrete lattice grids following independent Poisson clocks on slow time scales. The evolution of the environment variables can be sociologically interpreted as the spreading of information, which we assume to be in a continuous mode. That is, the rate of change of information at each spatial point is "fast" while the amount of change is infinitesimal during each infinitesimally short time period. And we adopt suitable temporal-spatial scaling to make the change of information at each spatial point over a time period to have the same order of magnitude. The result is a partial differential equation whose parameter depends upon agent distributions, which switches values according to the Poisson jumps governing agent activities. The simulation cost of agent-based M-IPC Model is vastly reduced compared to the SSRB-IPC model. Moreover, transient hotspot regimes as well as stationary hotspot regimes arise in the simulations. And our M-IPC Model with the multiscale setup can also be applied to types of crimes other than residential burglary, e.g. drug smugglings.
The separation of scales not only facilitates simulations, but also brings in the mathematical framework of piecewise deterministic Markov processes (PDMP), while First-Generation Model evolves according to (discrete or continuous time) Markov processes. A martingale approach similar to that developed in references [19] and [20] is applicable. The martingale formulation can be used to represent the model as the sum of a deterministic and a stochastic part, which will be useful for further study of hotspot dynamics. PDMP, also referred to as the iterated random functions (see e.g. reference [6]), hybrid stochastic processes, or jumping Markov processes (see e.g. reference [11]), has been extensively studied (see e.g. references [2], [5], [8], [10] and [17], and the references cited therein). Also PDMP has been widely applied in physics, chemistry, biology and other fields in natural science (see e.g. references [2], [8] and [17], and the references cited therein). However, as far as we know, this is the first time that PDMP is applied to crime modeling, and our methodology can be useful for quantitative study in social and behavioral science in general.
The paper is organized as follows. We first introduce the set-up of agent-based M-IPC Model (Section 2), where the slow and the fast components are described in Sections 2.2 and 2.3, respectively. The computer simulations are shown in Section 3, where both transient and stationary hotspot regimes are displayed. We review the general characterizations of PDMP and apply them to our model (Section 4.1). Finally the martingale representation of our model is established in Section 4.2.
We start with a domain
1With minor changes we can also consider e.g. the Dirichlet boundary conditions, which is more realistic.
(ⅰ) Natural districts corresponding to those in an urban setting. For example, in Los Angeles, we can consider the natural counties as what those districts correspond to.
(ⅱ) Artificial districts to facilitate the collecting of data etc.
Attached to each
ˉAK=∫DKA(x)dx, | (1) |
We also assume that
A(x,t)=B(x,t)+A0(x), | (2) |
where
(NK(t)),B(x,0))=(N0K,B0(x)). | (3) |
Following SSRB-IPC Model in reference [20], three types of independent Poisson clocks govern the time increments of arrivals of events, and all the Poisson clocks are independent with each other. Unlike the setup in First-Generation Model (see references [18], [19] and [20]), here we assume that the attractiveness evolves over
(Ⅰ) Committing crimes.
A Type (Ⅰ) Poisson clock is assigned to each agent to govern his action of committing crimes like burgling. Suppose that a Type (Ⅰ) clock of an agent in district
For agent activity Type (Ⅰ) described as above, there is an alternative description. Suppose that each criminal has a Poisson clock with rate
(Ⅱ) Moving along districts.
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
qK→J(t)=ˉAJ(t−)TK(t−), | (4) |
where
δt≅1c2L2. | (5) |
(Ⅲ) Replacement.
A Type (Ⅲ) Poisson clock is assigned to each district
Remark 1. The Poisson clock is particularly suitable to model random arrivals and has been used to build Poisson processes which are widely applied in biology, economics, and physics (for references see those cited in references [19] and [20]).
Following the setup of First-Generation Model (see references [18], [19] and [20]), we assume that the attractiveness field react to the agent activities described above according to the repeat and near-repeat victimization, and the broken-windows effects (see the related references in references [7], [18], [19] and [20]).
To model the repeat victimization and broken windows effects, we assume that the attractiveness increases continuously in time, with an increasing rate depending upon the average number of criminal events at the current location. However, this increase has a finite lifetime and decays at a certain speed. And we model the near-repeat victimization effect by letting the attractiveness to spread diffusively in space. Summing up, between the Poisson jumps of the criminal activities, the following partial different equation describes the spatio-temporal evolution of
dB(x,t)dt=c1θNK(t)A(x,t)−ωB(x,t)−ηΔB(x,t),x∈DK,K∈D. | (6) |
Here
For
N(x,t)=∑K∈DNK1K(t)(x), | (7) |
where
dA(t,x)dt=θc1N(x,t)A(x,t)−ωB(x,t)+ηΔB(x,t). | (8) |
We note that to completely specify the coupled dynamics of criminal activities and attractiveness field evolution, it is sufficient to keep track of the total number of criminals in each district. To conclude, the scales of the criminal activity and the evolution of the attractiveness field separate automatically.
We simulate M-IPC Model described above under various combinations of the parameters, in order to give insight into the behavior regime of the model, and compare it with the DTS, SSRB, and SSRB-IPC Models. The resulting attractiveness plots are displayed in Figs. 1 and 2.
Behavioral regimes similar to those in references [18], [19] and [20] for the attractiveness field
Case 1 Stationary hotspots. In this regime, the system tends towards an almost steady state in which stationary spots of high attractiveness are found, surrounded by areas of extremely low attractiveness.
Case 2 Dynamic hotspots. In this regime, loclized 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. There are two sub categories, depending on whether the hotspots interact between each other or not.
Case 2.1. Independent hotspots. Hotspots each change in size and shape but any two hotspots rarely have interactions.
Case 2.2. Interactive hotspots. Hotspots repeatedly and constantly merge with each other and split. This regime reveals the intrinsic randomness of the model.
All the simulations were run with
In Fig. 1 (a), we set
The parameter regimes are robust for all three cases of hotspot dynamics mentioned above, and the same hotspot dynamics is also observed over other random paths with the same parameters used to create the plots in Fig. 1. In Fig. 2, distinct-stochastic-basis realizations for Fig. 1(c) are displayed.
Remark 2. The numerical scheme for (6) is similar to that in reference [18]. We use a semi-implicit scheme in which
[1+ωδt−ηδtΔ]Am+1=Am+θc1δtNmAm. | (9) |
Here
The coupled dynamics of
First introduced as in reference [4], a PDMP, roughly speaking, is a dynamic system occasionally interrupted by a pure jump process whose states then switch according to certain probability distributions (see e.g. references [2], [5], [8] and [10]).
We start with a standard stochastic basis
dU(t)dt=L(U(t)). | (10) |
Alternatively, starting from
P(τ1>t|U(0)=U0)=exp(−∫t0Λ(U(s))ds). | (11) |
Then
The infinitesimal generator of the stochastic process can be derived following e.g. references [4], [5], [8], [10] and [17] as
Tf(u)=Λ(u)[Qf(u)−∫Ef(u)Q(u;dv)]+L(u)∘∇f(u), | (12) |
for every
f(U(t))=f(U(0))+∫t0Tf(U(s))ds+M(t), | (13) |
where
For every
⟨N(t),ϕ⟩=∫MN(x,t)ϕ(x)dx=L−2∑K∈DNK(t)ˉϕK, | (14) |
where
ˉϕK:=∫DKϕ(x)dx. |
We define
⟨(N(t),B(t)),ϕ⟩:=(⟨N(t),ϕ⟩,⟨B(t),ϕ⟩). | (15) |
Then
Theorem 4.1. Starting with the initial data
{⟨N(t),ϕ⟩=⟨N0,ϕ⟩+∫t0Λ(N(s),B(s))(Q1−I1)⟨(N(s),B(s)),ϕ⟩ds+M1(⟨(N(t),B(t)),ϕ⟩),⟨B(t),ϕ⟩=⟨B0,ϕ⟩+∫t0L2(⟨(N(s),B(s)),ϕ⟩)ds+M2(⟨(N(t),B(t)),ϕ⟩), | (16) |
where
Λ(N(t),B(t))=c2L2∑K∈DNK(t)+L2c1∑K∈DNK(t)ˉAK(t)+L2Γ, | (17) |
(Q1−I1)⟨(N(t),B(t)),ϕ⟩=c2∑K∈DNK(t)(∑K′K′∼KˉAK′(t)TK(t)(ˉϕK′−ˉϕK))c2L2∑K∈DNK(t)+L2c1∑K∈DNK(t)ˉAK(t)+L2Γ−c1∑K∈DˉAK(t)NK(t)ˉϕKc2L2∑K∈DNK(t)+L2c1∑K∈DNK(t)ˉAK(t)+L2Γ+ΓL−2∑K∈DˉϕKc2L2∑K∈DNK(t)+L2c1∑K∈DNK(t)ˉAK(t)+L2Γ, | (18) |
L2(⟨(N(t),B(t)),ϕ⟩)=∫M[θc1N(x,t)A(x,t)−ωB(x,t)+ηΔB(x,t)]ϕ(x)dx. | (19) |
In other words, the deterministic flow of the PDMP driven by
Proof of Theorem 4.1. Substituting
Tu=Λ(u)[Qu−u∫EQ(u;dv)]+L(u)=Λ(u)(Q−I)u+L(u). | (20) |
Thus we have for the vector-valued stochastic process
T⟨(N(t),B(t)),ϕ⟩=(L1(⟨(N(t),B(t)),ϕ⟩),L2(⟨(N(t),B(t)),ϕ⟩))+Λ(N(t),B(t))((Q1−I1)⟨(N(t),B(t)),ϕ⟩,(Q2−I2)⟨(N(t),B(t)),ϕ⟩), | (21) |
where
In between advancements of Poisson clocks, the value of
L1(⟨(N(t),B(t)),ϕ⟩)=d⟨N(t),ϕ⟩dt=⟨dN(t)dt,ϕ⟩=0. | (22) |
Similarly as
L2(⟨(N(t),B(t)),ϕ⟩)=d⟨B(t),ϕ⟩dt=⟨dB(t)dt,ϕ⟩, | (23) |
which together with (6) implies (19).
Since all the Poisson clocks in the system are independent,
Suppose that a Poisson clock advances at time
Λ(N(t),B(t))(Q1−I1)⟨(N(t),B(t)),ϕ⟩=limδt→01δtE[⟨B(δt+t−),ϕ⟩−⟨B(t−),ϕ⟩|(N(t−),B(t−))]=L2c1∑K∈DˉAK(t)NK(t)(−L−2ˉϕK)+c2L2∑K∈D∑K′K′∼KˉAK′(t)TK(t)(L−2NK(t)(ˉϕK′−ˉϕK))+Γ∑K∈DL−2ˉϕK=−c1∑K∈DˉAK(t)NK(t)ˉϕK+c2∑K∈D∑K′K′∼KˉAK′(t)TK(t)NK(t)(ˉϕK′−ˉϕK)+L−2Γ∑K∈DˉϕK. | (24) |
This together with (17) implies (18). From (6), we infer that the flow
limδt→0[⟨B(δt+t−),ϕ⟩−⟨B(t−),ϕ⟩]=0, | (25) |
which implies that
Λ(N(t−),B(t−))(Q2−I2)⟨(N(t−),B(t−)),ϕ⟩=limδt→01δtE[⟨B(δt+t−),ϕ⟩−⟨B(t−),ϕ⟩|(N(t−),B(t−))]=Λ(N(t−),B(t−))×0=0. | (26) |
With every term on the right-hand-side of (21) derived, from (12), (13), (20), and (21) we obtain (16).
The proof of Theorem 4.1 is completed.
Remark 3. Unlike First-Generation Model, the finite size effects are circumvented here. Nevertheless, our model still generates interesting pattern formation of hotspot dynamics. Pattern formation in complex system has been studied extensively recently in natural and social sciences (see e.g. the references cited in the introduction of [20]). And the mathematical framework that we develop here based on PDMP theory and martingale formulation can be useful for quantitative study in this area, which has been lacking so far.
Ever since the pioneering agent-based residential burglary model of criminal behavior reference [18] is published, many works have been done aiming to study and improve the original model e.g. by incorporating random arrivals as in references [19] and [20]. In these models, residents are assumed to be located on the grids of a lattice field. Two components, the agent activities over the lattice and the level of attractiveness of target sites, are assumed to interact with each other on the same spatio-temporal scale. The continuum limit of First-Generation Model turns out to be reaction-advection-diffusion equations. However in the continuum simulations, transient hotspot regime is missing, which is not realistic as transient hotspots are well-document in empirical observations. A new generation of agent-based models of criminal behavior are needed.
In this work, we propose a multiscale model, where the attractiveness evolves continuously based on a diffusive partial different equation, whose parameter changes values whenever a Poisson clock associated with an agent action advances. Compared with First-Generation Model with two slow components, our model has a fast component and easier to simulate and analyze. Moreover, the stochastic nature of the model is successfully maintained and transient hotspot regimes with intrinsic randomness are picked up in the simulations as desired.
Furthermore, our multi-scale hybrid model can be characterized as a PDMP, which is applied in such a circumstance for the first time to the best of our knowledge. A martingale formula is derived, which consists of the deterministic component corresponding to the diffusive partial differential equation of attractiveness evolution, and the stochastic component corresponding to the Poisson arrivals. Results presented here will be transformative for both elements of application and analysis of agent-based models for criminal behavior.
We would like to thank the helpful discussions with Professors A. Debussche, A. L. Bertozzi, M. B. Short, T. Liggett, C. Mueller, J. Quastel, Da Kuang, Yifan Chen, Fangbo Zhang, Yatin Chow, Hangjie Ji, Yuanzhen Shao, and Kenneth Van. Yuan Zhang has been partially supported by Beijing Academy of Artificial Intelligence (BAAI).
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