Skip to contents

Hot-spot dynamics: the Short-D'Orsogna-Brantingham model

Source: R/sdb.R
sdbsim.Rd

Simulates the model of residential burglary of Short et al. (2008) on an n by n grid of houses that wraps at its edges. Each site has an attractiveness a0 + b; b rises by theta with every burglary there, spreads to the four neighbours at rate eta and decays at rate omega, so that burglaries draw further burglaries (repeat and near-repeat victimisation). In the "agent" model offenders (rho, whole numbers per site) burgle with probability \(1 - e^{-(a_0 + b) dt}\) and leave, or else move to a neighbouring site chosen in proportion to its attractiveness; new offenders arrive at gam per site per unit time. In the "pde" model the densities follow the continuum limit, \(b_t = (\eta D / z) \Delta b - \omega b + \epsilon D \rho A\) and \(\rho_t = (D / z)(\Delta \rho - 2 \nabla \cdot (\rho \nabla A / A)) - \rho A + \gamma\) with \(A = a_0 + b\), by explicit Euler steps on a grid of spacing dx; sup marks sites where policing suppresses crime (no burglaries there). The step must satisfy dt <= z dx^2 / (4 dif max(1, eta)).

Usage

sdbsim(
  n,
  steps,
  mode = "agent",
  a0 = 1/30,
  omega = 1/15,
  eta = 0.03,
  gam = 0.002,
  theta = 1,
  dt = NULL,
  b = NULL,
  rho = NULL,
  dx = 1,
  dif = 1,
  eps = 1,
  z = 4,
  sup = NULL,
  seed = NULL
)

Arguments

n

The side of the grid: one whole number from 3 (at most 2000).

steps

The number of steps: one whole number from 0.

mode

"agent" or "pde".

a0

The static attractiveness: one positive number.

omega

The decay rate of b: one number from 0 (omega dt at most 1).

eta

The spread of b to neighbours: one number from 0 to 1.

gam

The arrival rate of offenders per site: one number from 0.

theta

The rise of b per burglary ("agent"): one number from 0.

dt

The time step: one positive number; NULL for 1 ("agent") or 0.01 ("pde").

b, rho

NULL or n by n matrices of numbers from 0: the starting attractiveness (0 by default) and offenders ("agent": whole numbers, none by default; "pde": a density, gam / a0 by default).

dx, dif, eps, z

The grid spacing, the offenders' diffusion, the rise of b per unit of crime and the lattice's neighbour count of the "pde" model: positive numbers (eps from 0).

sup

NULL or an n by n logical matrix of suppressed sites ("pde").

seed

NULL to draw the seed from R's generator, or one whole number from 0 ("agent").

Value

An object of class sdbsim, a list: b and rho, the fields at the end; tot, the burglaries at each site over the run (the integrated crime rate for "pde"); burg and meanb, the burglaries and the mean of b at each step; and mode.

Details

Started at the uniform steady state sdbthr() gives, the "pde" model stays there; started near it, a mode grows or dies at the rate sdbthr() predicts for its wavenumber. Without offenders the attractiveness only decays: each step multiplies its largest value by at most 1 - omega dt (proof("sdbsim")) and its mean by exactly that.

References

Short, M. B., D'Orsogna, M. R., Pasour, V. B., Tita, G. E., Brantingham, P. J., Bertozzi, A. L. and Chayes, L. B. (2008). A statistical model of criminal behavior. Mathematical Models and Methods in Applied Sciences, 18(supp01), 1249-1267. doi:10.1142/S0218202508003029

Examples

s <- sdbsim(20, 50, a0 = 1 / 30, omega = 1 / 15, eta = 0.03,
            gam = 0.02, theta = 0.5, seed = 1)
s
#> sdbsim: agent model, 20 x 20 grid, 50 steps; burglaries 392; mean attractiveness at the end 0.1493598
p <- sdbsim(16, 200, mode = "pde", eps = 2, dt = 0.05,
            rho = matrix(0.07, 16, 16))
range(p$b)
#> [1] 0.05691214 0.05691214