Statistical Physics of Crime

Part of Statistical Methods — MORIE’s statistical-methods reference.

The morie.tps_statphysics module collects four statistical-physics models of urban crime that are routinely fit on Toronto Police Service open-data feeds. The framing follows the D’Orsogna-Perc (2015) review.

Methods

Short-Brantingham reaction-diffusion PDE

Short, D’Orsogna, and Brantingham (2008) derive a reaction-diffusion PDE for the attractiveness field \(A(\mathbf{x}, t)\) — a scalar that tracks how attractive a location is to offenders. The discretised lattice form admits a closed-form steady-state and reproduces the empirically-observed hot-spot patterns in burglary and theft.

  • morie.tps_statphysics.sdb_reaction_diffusion — fits the diffusion coefficient \(\eta\), the activity rate \(\theta\), and the attractiveness decay \(\omega\) from observed incident-density grids.

  • Reference: Short, D’Orsogna, Brantingham, Schoenberg, Tita (2008). A statistical model of criminal behavior. MMMAS 18(suppl):1249–1267.

Lévy-flight tail index

Brockmann-Hufnagel-Geisel (2006) showed that human displacement distributions follow a power-law tail \(P(\Delta r) \sim \Delta r^{-(1 + \alpha)}\) — a Lévy-flight signature. The same estimator on inter-event spatial displacements between consecutive crime incidents diagnoses whether offenders’ movement is Brownian (\(\alpha \ge 2\)) or Lévy (\(\alpha < 2\)).

  • morie.tps_statphysics.levy_flight_alpha — Hill estimator on inter-event displacements with bootstrap CI.

  • Reference: Brockmann, Hufnagel, Geisel (2006). The scaling laws of human travel. Nature 439:462–465.

Bettencourt urban scaling

Bettencourt, Lobo, West (2007) showed that many city-level metrics \(Y\) scale as a power of population \(N\):

\[Y(N) \;=\; Y_0\, N^{\beta}.\]

For socio-economic metrics (including crime totals), \(\beta > 1\) (super-linear). For infrastructure metrics \(\beta < 1\) (sub-linear). The estimator is HC3-robust OLS on \(\log Y\) vs. \(\log N\).

  • morie.tps_statphysics.urban_scaling_beta — HC3-OLS fit of \(\beta\) with sandwich SE.

  • Reference: Bettencourt, Lobo, Helbing, Kühnert, West (2007). Growth, innovation, scaling, and the pace of life in cities. PNAS 104(17):7301–7306.

Lotka-Volterra predator-prey

D’Orsogna-Perc (2015, §3.4) frame the crime-and-policing dynamic as a Lotka-Volterra system: crime totals as prey, police-officer counts as predators. The fit recovers the four classical parameters (\(\alpha, \beta, \gamma, \delta\)) plus their stability classification (stable focus / spiral / saddle).

  • morie.tps_statphysics.lotka_volterra_police_crime — nonlinear least-squares on annual paired police / crime time series.

  • Reference: D’Orsogna, Perc (2015). Statistical physics of crime: A review. Physics of Life Reviews 12:1–21, §3.4.

Companion methods (under Spatial Statistics)

The following are spatial-statistics tools that the above statistical-physics methods often pair with:

  • Moran’s \(I\) global / local indicators of spatial autocorrelation

  • Ripley’s \(K\) and \(L\) for point-pattern second-order intensity

  • Getis-Ord \(G^{*}\) hot-spot tests

  • DBSCAN density-based clustering

  • Kulldorff space-time scan for significant spatio-temporal clusters