Crime Modeling with Lévy Flights

Transcription

1 Crime Modeling with Lévy Flights Jonah Breslau, 1 Sorathan (Tum) Chaturapruek, 2 Daniel Yazdi 3 Mentors: Professor Theodore Kolokolnikov, 4 Professor Scott McCalla 3 1 Pomona College 2 Harvey Mudd College 3 University of California, Los Angeles 4 Dalhousie University 8/8/2012

2 Figure: Crime Hot-Spot Pattern, Long Beach, LA. Short et al [2]

3 Hotspot Modeling Our goal is to model crime, specifically the hotspot phenomenon.

4 Hotspot Modeling Our goal is to model crime, specifically the hotspot phenomenon. Crime hotspots are when several crimes occur in a short period of time and in a small area.

5 Hotspot Modeling Our goal is to model crime, specifically the hotspot phenomenon. Crime hotspots are when several crimes occur in a short period of time and in a small area. Considerable empirical evidence behind them.

6 Hotspot Modeling Our goal is to model crime, specifically the hotspot phenomenon. Crime hotspots are when several crimes occur in a short period of time and in a small area. Considerable empirical evidence behind them. We focus on burglaries for simplicity.

7 Hotspot Modeling: Theory What causes hotspots?

8 Hotspot Modeling: Theory What causes hotspots? Repeat/ Near Repeat Effects: increased knowledge of location after successful crime. Figure: Repeat Effects. (Short et al. 2009, [1])

9 Hotspot Modeling: Theory What causes hotspots? Repeat/ Near Repeat Effects: increased knowledge of location after successful crime. Figure: Repeat Effects. (Short et al. 2009, [1]) Broken Windows Theory: crime causes sense of lawlessness.

10 The Short et al Model (2008) Starts with discrete model

11 The Short et al Model (2008) Starts with discrete model Criminals are agents on a lattice that has attractiveness field

12 The Short et al Model (2008) Starts with discrete model Criminals are agents on a lattice that has attractiveness field Every time step, criminals move to neighboring lattice spaces based on attractiveness.

13 The Short et al Model (2008) Starts with discrete model Criminals are agents on a lattice that has attractiveness field Every time step, criminals move to neighboring lattice spaces based on attractiveness. Then decide to burgle or not based off attractiveness.

14 The Short et al Model (2008) Starts with discrete model Criminals are agents on a lattice that has attractiveness field Every time step, criminals move to neighboring lattice spaces based on attractiveness. Then decide to burgle or not based off attractiveness. Crimes are self-exciting; cause attractiveness at lattice point and neighbors to increase.

15 The Short et al Model (2008) (cont.) Then, taking limits as grid spacing and time steps go to zero, we get system of PDEs The Model ρ t = A t = η A A + ρa + A 0 ( ( ρ A A) ρ ) A A ρa + A A 0 (1a) (1b) A = attractiveness at a point ρ = criminal density at a location A 0 = a constant, background level of attractiveness A = the spatially homogeneous equilibrium solution for A.

16 Lévy Flights Short et al model assumes Brownian motion (and thus distance traveled normally distributed)

17 Lévy Flights Short et al model assumes Brownian motion (and thus distance traveled normally distributed) We use Lévy Flights

18 Lévy Flights Short et al model assumes Brownian motion (and thus distance traveled normally distributed) We use Lévy Flights Power law distribution of step sizes (P(k) k (2s+1), 0 < s 1)

19 Lévy Flights Short et al model assumes Brownian motion (and thus distance traveled normally distributed) We use Lévy Flights Power law distribution of step sizes (P(k) k (2s+1), 0 < s 1) Fractal-like motion; reflects many scales of human movement.

20 Implementation: Discrete We change the transition probability as follows: p i i+1 (t) = A i+1 (t) A i+1 (t) + A i 1 (t) (Brownian) p i j (t) = A j (t) i j (2s+1) Σ k Z,k i A k (t) i k (2s+1) (Lévy)

21 Implementation: Continuous From before we had: A t = η A A + ρa + A 0, ( ( ρ ρ t = A A) ρ ) A A ρa + A A 0.

22 Implementation: Continuous From before we had: A t = η A A + ρa + A 0, ( ( ρ ρ t = A A) ρ ) A A ρa + A A 0. Using our new transition probabilities and taking limits as before we get: A t = η A A + ρa + A 0 (No change) ( ( ρ t = A s ρ ) ρ ) A A s A ρa + A A 0,

23 Implementation: Continuous From before we had: A t = η A A + ρa + A 0, ( ( ρ ρ t = A A) ρ ) A A ρa + A A 0. Using our new transition probabilities and taking limits as before we get: where A t = η A A + ρa + A 0 (No change) ( ( ρ t = A s ρ ) ρ ) A A s A ρa + A A 0, s A = c s c s is a constant and 0 < s 1. A(y) A(x) dy, (2) y x 2s+1

24 Fractional Calculus The operator s is the Riesz Derivative,

25 Fractional Calculus The operator s is the Riesz Derivative, Generalization of the second derivative such that lim s 1 s A = 2 A x. 2

26 Fractional Calculus The operator s is the Riesz Derivative, Generalization of the second derivative such that lim s 1 s A = 2 A x. 2 But note that 1 2 A A x

27 Fractional Calculus The operator s is the Riesz Derivative, Generalization of the second derivative such that lim s 1 s A = 2 A x. 2 But note that 1 2 A A x Is a non-local operator. Leads to super-diffusion. Degree of non-locality controlled by s.

28 Fractional Calculus The operator s is the Riesz Derivative, Generalization of the second derivative such that lim s 1 s A = 2 A x. 2 But note that 1 2 A A x Is a non-local operator. Leads to super-diffusion. Degree of non-locality controlled by s. Fourier Transform has nice property: F x q { s A} = q 2s Â

29 Numerical Solutions: Spectral Method of Lines Discretize in space and the calculate derivatives in Fourier space. This turns into ODE in time.

30 Numerical Solutions: Spectral Method of Lines Discretize in space and the calculate derivatives in Fourier space. This turns into ODE in time. We make use of the fact that F x q { s A} = q 2s Â.

31 Numerical Solutions: Spectral Method of Lines Discretize in space and the calculate derivatives in Fourier space. This turns into ODE in time. We make use of the fact that F x q { s A} = q 2s Â. Then use MATLAB s stiff ODE solver.

32 Numerical Solutions: Spectral Method of Lines Discretize in space and the calculate derivatives in Fourier space. This turns into ODE in time. We make use of the fact that F x q { s A} = q 2s Â. Then use MATLAB s stiff ODE solver. We used to use a forward Euler spectral method, but that could not handle much of the parameter space.

33 Examples Figure: Single Hot-Spot.

34 Examples Figure: Four Hot-Spots, D = 1, s = 1, ε = 0.05.

35 Examples Figure: No Hot-Spot, D = 1, s = 0.5, ε = 0.05.

36 Examples Figure: Oscillating Hot-Spots, s = 0.7, η = 0.1, ρ = 0.4, A = 0.12.

37 Linear Stability of Homogeneous Equilibrium The homogeneous equilibrium does not change from Short et al. We get A = A 0 + B and ρ = B A 0 + B. (3)

38 Linear Stability of Homogeneous Equilibrium The homogeneous equilibrium does not change from Short et al. We get A = A 0 + B and ρ = B A 0 + B. (3) Perturb from the homogeneous equilibrium as follows and plug into linearized equations: A(x, t) = A + δ A e σt e ik x, (4a) ρ(x, t) = ρ + δ ρ e σt e ik x. (4b)

39 Linear Stability of Homogeneous Equilibrium The homogeneous equilibrium does not change from Short et al. We get A = A 0 + B and ρ = B A 0 + B. (3) Perturb from the homogeneous equilibrium as follows and plug into linearized equations: A(x, t) = A + δ A e σt e ik x, (4a) ρ(x, t) = ρ + δ ρ e σt e ik x. (4b) Solving the resulting eigenvalue problem results in this condition for instability: there exists k such that η k 2s+2 k 2s (3ρ 1) + ηa k 2 + A < 0. (5)

40 Stability (cont.) The first attempt to solve Eq. (5): the condition for instability is equivalent to, there exists k such that ρ > 1 ( 1 + η k 2 ) ( 1 + A ) 3 k 2s. (6)

41 Stability (cont.) The first attempt to solve Eq. (5): the condition for instability is equivalent to, 1 ( ρ > inf 1 + η k 2 ) ( 1 + A ) k 3 k 2s. (6)

42 Stability (cont.) The first attempt to solve Eq. (5): the condition for instability is equivalent to, ρ > where k is a root of 1 ( 1 + η k 2) ( 1 + A ) 3 k 2s, (6) k 2s+2 + A(1 s) k 2 As η = 0. (7)

43 Stability (cont.) The second attempt to solve Eq. (5): the condition for instability is equivalent to, there exists k such that A < 3ρ k 2s 1 + η k 2 k 2s. (8)

44 Stability (cont.) The second attempt to solve Eq. (5): the condition for instability is equivalent to, A < sup k 3ρ k 2s 1 + η k 2 k 2s. (8)

45 Stability (cont.) The second attempt to solve Eq. (5): the condition for instability is equivalent to, A < 3ρ k 2s 1 + η k 2 k 2s, (8) where k is a root of the equation η 2 s k 4 + η(3ρ(1 s) + 2s) k 2 + s(1 3ρ) = 0. (9)

46 Stability (cont.) The second attempt to solve Eq. (5): the condition for instability is equivalent to, A < 3ρ k 2s 1 + η k 2 k 2s, (8) where k is a root of the equation η 2 s k 4 + η(3ρ(1 s) + 2s) k 2 + s(1 3ρ) = 0. (9)

47 Stability (cont.) For ρ > 1 3, this generates condition for linear instability for the system: ( 3ρ(1 s) 2s + ) s ( W 3ρ(1 + s) W A < A (ρ, η, s) 2ηs 3ρ(1 s) + W (10) where W = 3ρ(3ρ(1 s) 2 + 4s). ),

48 Stability (cont.) For ρ > 1 3, this generates condition for linear instability for the system: ( 3ρ(1 s) 2s + ) s ( W 3ρ(1 + s) W A < A (ρ, η, s) 2ηs 3ρ(1 s) + W (10) where W = 3ρ(3ρ(1 s) 2 + 4s). When s = 1, the above inequality reduces to Aη + 1 < 3ρ, (11) which agrees with the result from Short et al. ),

49 Stability (cont.) For ρ > 1 3, this generates condition for linear instability for the system: ( 3ρ(1 s) 2s + ) s ( W 3ρ(1 + s) W A < A (ρ, η, s) 2ηs 3ρ(1 s) + W (10) where W = 3ρ(3ρ(1 s) 2 + 4s). When s = 1, the above inequality reduces to Aη + 1 < 3ρ, (11) which agrees with the result from Short et al. Has bifurcations in s, so changing degree of Lévy Flight alters stability. ),

50 Changing Stability with Varying Parameter (a) A stable regime Figure: Different possibilities of the effect of fractional diffusion.

51 Changing Stability with Varying Parameter (a) An unstable regime Figure: Different possibilities of the effect of fractional diffusion.

52 Changing Stability with Varying Parameter (a) Fractional diffusion leads to stability (b) Fractional diffusion leads to stability Figure: Different possibilities of the effect of fractional diffusion.

53 Changing Stability with Varying Parameter (a) Fractional diffusion leads to unstability (b) Fractional diffusion leads to unstability Figure: Different possibilities of the effect of fractional diffusion.

54 Changing Stability with Varying Parameter (a) Fractional diffusion leads (b) Fractional diffusion leads to to stability and then instability stability and then instability Figure: Different possibilities of the effect of fractional diffusion.

55 Changing Stability with Varying Parameter Remark. The fixed points k of σ( k ) with respect s are given by 6(ρ 1 3 k 1 = 1, k 2 = ) + 3A. (12) 2η

56 Changing Stability with Varying Parameter (a) A regime in which k 2 < k 1 = 1 Figure: Different possibilities of the effect of fractional diffusion.

57 Numerical Verfication We need to automate a hot-spot detection.

58 Numerical Verfication We need to automate a hot-spot detection. The variance and its derivative.

59 Numerical Verfication We need to automate a hot-spot detection. The variance and its derivative. The difference of the maximum of the solution in the final frame from A as a percentage of A.

60 Numerical Verfication We need to automate a hot-spot detection. The variance and its derivative. The difference of the maximum of the solution in the final frame from A as a percentage of A. The difference between the maximum and the minimum.

61 Numerical Verfication We need to automate a hot-spot detection. The variance and its derivative. The difference of the maximum of the solution in the final frame from A as a percentage of A. The difference between the maximum and the minimum. An ensemble method.

62 Numerical Verfication (cont.) Figure: A parameter analysis with a bifurcation curve. Fix ρ and η.

63 Numerical Verfication (cont.) Figure: A hot-spot formation when A = 13, s = 0.8.

64 Numerical Verfication (cont.) Figure: A parameter analysis with a bifurcation curve. Fix ρ and η.

65 Numerical Verfication (cont.) Figure: No hot-spot formation when A = 21, s = 0.6

66 Numerical Verfication (cont.)

67 Hot-spot Shape Are the hotspots any different under the two regimes?

68 Hot-spot Shape Are the hotspots any different under the two regimes? Yes and no. Attractiveness hotspots do not change, and first order approximations from Kolokolnikov et al. still work very well.

69 Hot-spot Shape Are the hotspots any different under the two regimes? Yes and no. Attractiveness hotspots do not change, and first order approximations from Kolokolnikov et al. still work very well. But distribution of criminals changes.

70 Hotspot Shape (cont.) Figure: The inner region and the outer region of a hotspot.

71 Hotspot Shape (cont.) Let x = εy, and v = ρ/a 2.

72 Hotspot Shape (cont.) Let x = εy, and v = ρ/a 2. In the inner region ( x < ε), A ε 1 v 1/2 0 w(y), (12) v v 0, (13) where v 0 and v 1 are constants, and w = 2 sech y (same as Kolokolnikov et al.).

73 Hotspot Shape (cont.) Let x = εy, and v = ρ/a 2. In the inner region ( x < ε), A ε 1 v 1/2 0 w(y), (12) v v 0, (13) where v 0 and v 1 are constants, and w = 2 sech y (same as Kolokolnikov et al.). In the outer region (ε x l), we have where A = α + o(1), (14a) v = h 0 (x) + o(1), (14b) s h 0 (x) = ζ = α γ D 0 α 2 < 0, 0 < x l, (h 0) x (±l) = 0, (15)

74 Future Work Analyze the perturbation near hot-spots.

75 Future Work Analyze the perturbation near hot-spots. Study the dynamics of K-hot-spots with Lévy Flights. Brownian based model. Lévy based model.

76 Future Work Analyze the perturbation near hot-spots. Study the dynamics of K-hot-spots with Lévy Flights. Brownian based model. Add the police. Lévy based model.

77 Future Work Analyze the perturbation near hot-spots. Study the dynamics of K-hot-spots with Lévy Flights. Brownian based model. Lévy based model. Add the police. Improve the numerical simulation, especially hotspot detector.

78 Future Work Analyze the perturbation near hot-spots. Study the dynamics of K-hot-spots with Lévy Flights. Brownian based model. Lévy based model. Add the police. Improve the numerical simulation, especially hotspot detector. Analyze weakly-nonlinear stability.

79 Acknowledgments Thanks... Theodore Kolokolnikov Scott McCalla UCLA and REU Program Organizers Harvey Mudd College Mathematics Department for funding Tum Nestor Guillen for creating the Nonlocal Equations Wiki (

80 References M. Short, M. DOrsogna, P. Brantingham, and G. Tita. Measuring and modeling repeat and near-repeat burglary effects. Journal of Quantitative Criminology, 25: , /s M.B. Short, P.J. Brantingham, A.L. Bertozzi, and G.E. Tita. Dissipation and displacement of hotspots in reaction-diffusion models of crime. Proceedings of the National Academy of Sciences of the United States of America, 107: , 2010.