A Stochastic-Statistical Residential Burglary Model with Finite Size Effects

Abstract

Transience of spatio-temporal clusters of residential burglary is well documented in empirical observations, and could be due to finite size effects anecdotally. However a theoretical understanding has been lacking. The existing agent-based statistical models of criminal behavior for residential burglary assume deterministic-time steps for arrivals of events. To incorporate random arrivals, this article introduces a Poisson clock into the model of residential burglaries, which could set time increments as independently exponentially distributed random variables. We apply the Poisson clock into the seminal deterministic-time-step model in Short et al. (Math Models Methods Appl Sci 18:1249–1267, 2008). Introduction of the Poisson clock not only produces similar simulation output, but also brings in theoretically the mathematical framework of the Markov pure jump processes, e.g., a martingale approach. The martingale formula leads to a continuum equation that coincides with a well-known mean-field continuum limit. Moreover, the martingale formulation together with statistics quantifying the relevant pattern formation leads to a theoretical explanation of the finite size effects. Our conjecture is supported by numerical simulations.

Appendix

To prove Theorem 2.1, we compute the infinitesimal means and variances of $\left < \left (B^{\ell }( t), n^{ \ell }(t)\right ), \phi ^{ \ell } \right >$ for fixed ℓ [1, 12, 16, 34, 36, 43, 47, 58, 59, 66, 68].

We first study related random variables. We assume that the Poisson-clock advances at t⁻, and analyze the transition at time t. Conditioned on (B^ℓ(s, t⁻), n^ℓ(s, t⁻)), we observe that $E^\ell _{\mathbf {s}}(t)$, $\mathbf {s} \in \mathcal S$, is a family of independently identically distributed Binomial random variables with the parameters n^ℓ(s, t⁻) and p^ℓ(s, t⁻), that is

$$\displaystyle \begin{aligned}\displaystyle \mathbb P \left(E^\ell_{\mathbf{s}}(t) = i \middle | B^\ell_{\mathbf{s}} ( t^-) , n^\ell _{\mathbf{s}} ( t^-) \right )= {n^\ell_{\mathbf{s}} (t^-)\choose i } \left ( p^\ell_{\mathbf{s}}(t^-) \right ) ^ i (1- p^\ell_{\mathbf{s}}(t^-) )^{n_{\mathbf{s}}^\ell (t^-)- i},\end{aligned} $$

(42)

where $i =0, 1, 2, \ldots , n^\ell _{\mathbf {s}}(t^-).$ Hence we have

$$\displaystyle \begin{aligned} &\mathbb E \left[E^\ell_{\mathbf{s}}(t) \middle | B^\ell_{\mathbf{s}} (t^-), n^\ell _{\mathbf{s}} (t^-) \right ]= p^\ell_{\mathbf{s}}(t^-) n^\ell _{\mathbf{s}} (t^-), {} \end{aligned} $$

(43)

$$\displaystyle \begin{aligned} &\mbox{Var} \left(E^\ell_{\mathbf{s}} (t)\middle | \left ( B^\ell _{\mathbf{s}} (t^-) , n^\ell_{\mathbf{s}} (t^-) \right) , \forall \mathbf{s} \in \mathcal S\right) = n ^\ell_{\mathbf{s}} (t^-) p ^\ell_{\mathbf{s}} (t^-) \left [1 - p ^\ell_{\mathbf{s}} (t^-)\right]. {}\end{aligned} $$

(44)

Let $J^\ell _{\mathbf {s}, j}(t)$, $j=1, 2, ..., n^\ell _{\mathbf {s}}(t^-)$ be a family of independently distributed Bernoulli random variables assuming 1 with probability $1-p^\ell _{\mathbf {s}}(t^-)$. If the j-th agent at site s chooses not to burglarize, then $J^\ell _{\mathbf {s}, j}(t)$ assumes 1. This implies that

(45)

We also note that by our construction $E ^{\ell }_{\mathbf {s}} (t) = \sum _{j =1} ^{j=n^\ell _{\mathbf {s}} (t^-)} \left (1-J^\ell _{\mathbf {s}, j} (t) \right )$.

Let $\varPhi ^\ell _{\mathbf {s}, j}(t)$, $j= 1, \ldots , n^\ell _{\mathbf {s}} (t^-)$ be a family of independently identically distributed random variables assuming $\phi _{{\mathbf {s}}^\prime }$ with probability ${A^\ell _{{\mathbf {s}}^\prime }}/{T^\ell _{\mathbf {s}}}$, for every s^′, s^′ ∼s. Then we have

$$\displaystyle \begin{aligned} \mathbb E \left [ \varPhi^\ell_{\mathbf{s}, j}(t) \middle| \left ( B^\ell_{\mathbf{s}} (t^-), n^\ell_{\mathbf{s}} (t^{-} )\right ), \forall \mathbf{s} \in \mathcal S ^\ell \right ] =\sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \phi^\ell _{{\mathbf{s}}^\prime} \frac{A^\ell_{{\mathbf{s}}^\prime} (t^-) }{T^\ell_{\mathbf{s}} (t^-)},\end{aligned} $$

(46)

$$\displaystyle \begin{aligned}\mathbb E \left [ \left (\varPhi^\ell_{\mathbf{s}, j}(t)\right)^2 \middle| \left ( B^\ell_{\mathbf{s}} (t^-), n^\ell_{\mathbf{s}} (t^{-} )\right ), \forall \mathbf{s} \in \mathcal S ^\ell \right ] = \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \left(\phi^\ell _{{\mathbf{s}}^\prime}\right)^2 \frac{A^\ell_{{\mathbf{s}}^\prime} (t^-)}{T^\ell_{\mathbf{s}} (t^-)}. \end{aligned} $$

(47)

Let the number of replaced criminals on site s at time t be $\xi ^\ell _{\mathbf {s}}(t)$. Then $\xi ^\ell _{\mathbf {s}}(t)$ is a family of Bernoulli random variables assuming 1 with probability Γℓ²∕D. Hence we have

$$\displaystyle \begin{aligned} \mathbb E \left [ \xi^\ell_{\mathbf{s}}(t) \middle| \left ( B^\ell_{\mathbf{s}} (t^-), n^\ell_{\mathbf{s}} (t^{-} )\right ) \right ] = \frac{{\varGamma} \ell ^2} {D}, \end{aligned} $$

(48)

$$\displaystyle \begin{aligned} \mbox{Var} \left( \left (\xi^\ell_{\mathbf{s}}(t) \right) ^2 \middle| \left ( B^\ell_{\mathbf{s}} (t^-), n^\ell_{\mathbf{s}} (t^{-} )\right ) \right ) = \frac{{\varGamma} \ell ^2 }{D} (1-\frac{ {\varGamma} \ell ^2 }{D}). \end{aligned} $$

(49)

Because the decision to burglarize is (conditionally) independent from the decision to move for each of the burglars, $J^\ell _{\mathbf {s}, j}(t)$ and $\varPhi ^\ell _{\mathbf {k}, h}(t)$ are (conditionally) independent for any choices of s, k, j, and h. This implies that

(50)

$$\displaystyle \begin{aligned} & \mbox{Var} \left (J^\ell_{\mathbf{s}, j}(t) \varPhi^\ell_{\mathbf{s}, j}(t) \middle | \left ( B^\ell_{\mathbf{s}} (t^-), n^\ell_{\mathbf{s}} (t^{-} )\right ) , \forall \mathbf{s}\in \mathcal S ^\ell \right ) \notag\\ & = \mathbb E\left [\left (J^\ell_{\mathbf{s}, j}(t) \varPhi^\ell_{\mathbf{s}, j}(t) \right )^2 \middle | \left ( B^\ell_{\mathbf{s}} (t^-), n^\ell_{\mathbf{s}} (t^{-} )\right ) , \forall \mathbf{s}\in \mathcal S ^\ell \right] \notag \\ &\quad - \left [ \mathbb E \left[J^\ell_{\mathbf{s}, j}(t) \varPhi^\ell_{\mathbf{s}, j}(t) \middle | \left ( B^\ell_{\mathbf{s}} (t^-), n^\ell_{\mathbf{s}} (t^{-} )\right ) , \forall \mathbf{s}\in \mathcal S ^\ell \right ] \right]^2 \notag\\ &= \mbox{(by (A.4), (A.5), (A.6), and (A.4))} \notag\\ &= \left[1-p^\ell_{\mathbf{s}}(t^-) \right] \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \left(\phi^\ell _{{\mathbf{s}}^\prime} \right)^2 \frac{A^\ell_{{\mathbf{s}}^\prime}(t^{-} ) }{T^\ell_{\mathbf{s} } (t^{-} )} -\left [1-p^\ell_{\mathbf{s}}(t^-) \right] ^2 \left [\sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \phi^\ell _{{\mathbf{s}}^\prime} \frac{A^\ell_{{\mathbf{s}}^\prime}(t^{-} ) }{T^\ell_{\mathbf{s} } (t^{-} ) } \right]^2,{} \end{aligned} $$

(51)

Right after the Poisson clock advances we have the following transition:

$$\displaystyle \begin{aligned} \sum_{\mathbf{s} \in \mathcal S ^\ell} n^\ell_{\mathbf{s}} (t)\phi ^{\ell}_{\mathbf{s}} & = \sum_{\mathbf{s} \in \mathcal S ^\ell} \sum_{j=1} ^{n^\ell_{\mathbf{s} }(t^-)} \left [1- J^\ell _{\mathbf{s}, j}(t)\right ] \varPhi^\ell_{\mathbf{s}, j} (t)+ \sum_{\mathbf{s} \in \mathcal S ^\ell} \xi ^\ell _{\mathbf{s}} (t) \phi ^\ell _{\mathbf{s}}. {} \end{aligned} $$

(52)

With the above random variables we compute the infinitesimal means and variances. In the computational steps we will drop the superscript ℓ for simplicity.

We compute the infinitesimal mean for $\left < B^{\ell }(t^-), \phi ^{\ell } \right >$. From (6) we have

$$\displaystyle \begin{aligned}\displaystyle &\mathcal{G} _1 \left ( \left< B ^\ell (t^-), \phi ^{\ell} \right>, \left< n^\ell (t^-) , \phi ^{\ell} \right> \right) \notag\\ &=\frac{{D}}{\ell ^{2} } \mathbb E \left [ \ell^2 \sum _{\mathbf{s} \in \mathcal{S } } \left [ B_{\mathbf{s}} (t)- B_{\mathbf{s}} (t^-)\right ] \phi _{\mathbf{s}} \middle| (B (t^-), n (t^-)) \right ]\notag\\ &=\mbox{by (A.2)} \notag \\ &= {{D}} \sum _{\mathbf{s} \in \mathcal{S} } \left [ \left (1 - \frac{\omega \ell^2}{{D}} \right)\frac{\eta \ell ^2}{4} \varDelta B_{\mathbf{s}}(t^-) - \frac{\omega \ell^2}{{D}} B_{\mathbf{s}} (t^-) + \theta p _{\mathbf{s}}(t^-) n _{\mathbf{s}} (t^-) \right ] \phi _{\mathbf{s}} , {} \end{aligned} $$

(53)

which implies (11).

We compute the infinitesimal variance of $\left < B^{\ell } (t^-), \phi ^{\ell } \right >$:

$$\displaystyle \begin{aligned} &\mathcal{V}^\ell_1 \left (\left< \left (B^{\ell}( t), n^{ \ell}(t)\right), \, \phi^{ \ell} \right>\right ) \notag\\ & = \lim_{\delta t \rightarrow 0} \frac{1}{\delta t}{\mathbb E \left [ \left ( \left< B (\delta t+ t^-), \phi \right> -\left< B (t^-), \phi \right> \right) ^2 \middle | (B (t^-), n (t^-)) \right ] } \notag \\ &=\frac{D}{\ell ^{2}} \mathbb E \left [ \ell^4 \left (\sum_{\mathbf{s} \in \mathcal S } B _{\mathbf{s}} (t^-)\phi _{\mathbf{s}} - \sum_{\mathbf{s} \in \mathcal S } B _{\mathbf{s}} (t) \phi _{\mathbf{s}} \right )^2 \middle| \left ( B (t^-), n (t^{-}) \right ) \right ]\notag \\ &={D}\ell ^{ 2} \mathbb E \left [ \sum_{\mathbf{s} \in \mathcal S } B _{\mathbf{s}} (t^-)\phi _{\mathbf{s}} - \sum_{\mathbf{s} \in \mathcal S } B _{\mathbf{s}} (t) \phi _{\mathbf{s}} \middle| \left ( B (t^-), n (t^{-}) \right ) \right ]^2 \notag \\ &\quad \quad + {D}\ell ^{ 2} \mbox{Var} \left [ \sum_{\mathbf{s} \in \mathcal S } B_{\mathbf{s}} (t^-)\phi _{\mathbf{s}} - \sum_{\mathbf{s} \in \mathcal S } B _{\mathbf{s}} (t) \phi _{\mathbf{s}} \middle| \left ( B (t^-), n (t^{-}) \right ) \right ] \notag \\ & := J_1+J_2 . {} \end{aligned} $$

(54)

For J₁, from (53) we have

$$\displaystyle \begin{aligned} J_1 &= \frac{ \ell ^{2} }{D} \mathcal{G} _1^2 \left ( \left< B (t^-), \phi \right>, \left< n (t^-) , \phi \right> \right) . \end{aligned} $$

(55)

Then for J₂, we apply the independence of E_s(t) for distinct $\mathbf {s} \in \mathcal S$ and with (44) we obtain

(56)

This together with (55) and (54) implies (13).

We compute the infinitesimal mean for $\left < n^{\ell }( t^-), \phi ^{ \ell } \right >$. From (52) we have

$$\displaystyle \begin{aligned} &\mathcal{G} _2 \left ( \left< \left (B ^{\ell} (t^-), n^{\ell} (t^-)\right) , \phi^{\ell} \right> \right) \notag\\ & =\frac{D}{\ell ^{ 2}} \mathbb E \left [ \ell^2 \sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t)\phi _{\mathbf{s}} - \ell^2\sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t) \phi _{\mathbf{s}} \middle| (B (t^-), n (t^-)) \right ] \notag\\ & = {D} \sum_{\mathbf{s} \in \mathcal S } \mathbb E \left [\sum_{j=1} ^{n _{\mathbf{s} }(t^-)} \left [1- J _{\mathbf{s}, j} (t)\right ] \varPhi_{\mathbf{s}, j} (t) + \xi _{\mathbf{s}} (t) \phi _{\mathbf{s}} - n _{\mathbf{s}} (t^-) \phi _{\mathbf{s}} \middle| (B (t^-), n (t^-)) \right ] .{} \end{aligned} $$

(57)

This together with (45), (46), and (48) implies

$$\displaystyle \begin{aligned} &\mathcal{G} _2 \left ( \left< \left (B ^{\ell} (t^-), n ^{\ell}(t^-)\right) , \phi ^{\ell} \right> \right) \notag \\ & = {D} \sum_{\mathbf{s} \in \mathcal S } \left [ n _{\mathbf{s}} (t^-) \left [ 1 - p _{\mathbf{s}}(t) \right] \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \phi _{{\mathbf{s}}^\prime} \frac{A _{{\mathbf{s}}^\prime}(t^-)}{T _{\mathbf{s}}(t^-) }+ \frac{{\varGamma} \ell^2}{{D}} \phi _{\mathbf{s}} - n _{\mathbf{s}} (t^-) \phi _{\mathbf{s}}\right] \notag\\ & = {D} \sum_{\mathbf{s} \in \mathcal S } \left[ \phi _{\mathbf{s}} A _{\mathbf{s}}(t^-) \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \frac{n _{{\mathbf{s}}^\prime} (t^-) \left [ 1 - p _{{\mathbf{s}}^\prime }(t) \right] } {T _{{\mathbf{s}}^\prime }(t^-) }+ \frac{{\varGamma} \ell^2}{{D}} \phi _{\mathbf{s}} - n _{\mathbf{s}} (t^-) \phi _{\mathbf{s}}\right], {} \end{aligned} $$

(58)

which implies (12).

We compute the infinitesimal variance of $\left < n^{\ell } (t^-), \phi ^{\ell } \right >$

$$\displaystyle \begin{aligned} &\mathcal{V}^\ell_2 \left (\left< \left (B^{\ell}( t), n^{ \ell}(t)\right), \, \phi^{ \ell} \right>\right ) \notag \\ &= \lim_{\delta t \rightarrow 0} \frac{1}{\delta t}{\mathbb E \left [ \left ( \left< n (\delta t+t^-), \phi \right> -\left< n (t^-), \phi \right> \right) ^2 \middle | (B (t^-), n (t^-)) \right ] } \notag \\ &={D}\ell ^{ 2} \mathbb E \left [ \sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t^-)\phi _{\mathbf{s}} - \sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t) \phi _{\mathbf{s}} \middle| \left ( B (t^-), n (t^{-}) \right ) \right ]^2 \notag \\ &\quad \quad \quad + {D}\ell ^{ 2} \mbox{Var} \left ( \sum_{\mathbf{s} \in \mathcal S } n_{\mathbf{s}} (t^-)\phi _{\mathbf{s}} - \sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t) \phi _{\mathbf{s}} \middle| \left ( B (t^-), n (t^{-}) \right ) \right ) \notag \\ &:=J_3 + J_4. {} \end{aligned} $$

(59)

For J₃ we have

$$\displaystyle \begin{aligned} J_3= \frac{\ell^2}{D} \mathcal{G} _2 ^2 \left ( \left< B (t^-), \phi \right>, \left< n (t^-) , \phi \right> \right) . \end{aligned} $$

(60)

For J₄, with the independence of the related random variables, we obtain

$$\displaystyle \begin{aligned} J_4 &= {D}\ell ^{ 2} \sum_{\mathbf{s} \in \mathcal S } \mbox{Var} \left ( \sum_{j=1} ^{n _{\mathbf{s} }(t^-)} J _{\mathbf{s}, j} \varPhi_{\mathbf{s}, j}(t) + \xi _{\mathbf{s}} (t) \phi _{\mathbf{s}} - n _{\mathbf{s}} (t^-) \phi _{\mathbf{s}} \middle| \left ( B (t^-), n (t^{-}) \right ) \right ) \notag \\ &= D \ell ^{ 2} \sum_{\mathbf{s} \in \mathcal S } \mbox{Var} \left (\xi _{\mathbf{s}} (t) \phi _{\mathbf{s}} \middle| \left ( B (t^-), n (t^{-}) \right ) \right ) \notag \\ & \quad +D \ell ^{ 2} \sum_{\mathbf{s} \in \mathcal S } {n _{\mathbf{s} }(t^-)} \mbox{Var} \left ( J _{\mathbf{s}, j} (t)\varPhi_{\mathbf{s}, j}(t) \middle| \left ( B (t^-), n (t^{-}) \right ) \right ) \notag \\ & := J_{4, 1} + J_{4, 2} . {} \end{aligned} $$

(61)

For J_4,1, by (49) we have

$$\displaystyle \begin{aligned} J_{4, 1}= \ell^{ 4} {{\varGamma} } \left(1- \frac{{\varGamma} \ell^2}{{D}}\right) \sum_{\mathbf{s} \in \mathcal S } \phi ^2 _{\mathbf{s}} . \end{aligned} $$

(62)

For J_4,2, by (50) and (51) we have

$$\displaystyle \begin{aligned} J_{4, 2} & = {D}\ell ^{ 2} \sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t^-) p_{\mathbf{s}}(t^-) \left [1-p _{\mathbf{s}}(t^-) \right ] \left [ \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \phi _{{\mathbf{s}}^\prime} \frac{A_{{\mathbf{s}}^\prime}(t^-)}{T_{\mathbf{s}} (t^-)} \right] ^2 \notag \\ & \quad + {D}\ell ^{ 2} \sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t^-) \left [ 1-p _{\mathbf{s}}(t^-) \right ] \left [ \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \phi^2 _{{\mathbf{s}}^\prime}\frac{A_{{\mathbf{s}}^\prime}(t^-)}{T_{\mathbf{s}}(t^-)}- \left ( \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \phi _{{\mathbf{s}}^\prime} \frac{A_{{\mathbf{s}}^\prime}(t^-)}{T_{\mathbf{s}} (t^-)} \right) ^2 \right] \notag \\ & := J_{4, 1,1} + J_{4, 1, 2} . {} \end{aligned} $$

(63)

We simplify J_4,1,2 as follows

$$\displaystyle \begin{aligned} J_{4, 1, 2} &= {D}\ell ^{ 2} \sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t^-) \left [ 1-p _{\mathbf{s}}(t^-)) \right ] \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \frac{A_{{\mathbf{s}}^\prime}(t^-)}{T_{\mathbf{s}}(t^-)} \left [\phi _{{\mathbf{s}}^\prime} - \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \phi _{{\mathbf{s}}^\prime} \frac{A_{{\mathbf{s}}^\prime}(t^-)}{T_{\mathbf{s}} (t^-)} \right]^2 \notag \\ &= {D}\ell ^{ 2} \sum_{\mathbf{s} \in \mathcal S } n _{\mathbf{s}} (t^-) \left [ 1-p _{\mathbf{s}}(t^-) \right] \sum_{\substack{{\mathbf{s}}^\prime \\ {\mathbf{s}}^\prime \sim \mathbf{s}} } \frac{A_{{\mathbf{s}}^\prime}(t^-)}{T_{\mathbf{s}}(t^-)} \left [\sum_{\substack{{\mathbf{s}}^{\prime \prime} \\ {\mathbf{s}}^{\prime\prime} \neq {\mathbf{s}}^\prime \\ {\mathbf{s}}^{\prime \prime}\sim \mathbf{s} } } \left (\phi _{{\mathbf{s}}^\prime} - \phi _{{\mathbf{s}}^{\prime \prime}} \right)\frac{A_{{\mathbf{s}}^{\prime \prime}}(t^-)}{T_{\mathbf{s}} (t^-)} \right]^2 . {} \end{aligned} $$

(64)

This together with (59)–(63) implies (14).

With the infinitesimal means and variances we apply Theorem (1.6), [14] or Theorem 3.32, [51], to arrive at (9), and apply Exercise 3.8.12 of [6], Lemma A 1.5.1, [44], or Proposition B.1 in [64]³ to obtain (10). To conclude the proof of Theorem 2.1 is completed.