Applying Crime Prediction Techniques to Japan: A Comparison Between Risk Terrain Modeling and Other Methods

Abstract

In recent years, the field of crime prediction has drawn increasing attention in Japan. However, predicting crime in Japan is especially challenging because the crime rate is considerably lower than that of other developed countries, making the development of a statistical model for crime prediction quite difficult. Risk terrain modeling (RTM) may be the most suitable method, as it depends mainly on the environmental factors associated with crime and does not require past crime data. In this study, we applied RTM to cases of theft from vehicles in Fukuoka, Japan, in 2014 and evaluated the predictive performance (hit rate and predictive accuracy index) in comparison to other crime prediction techniques, including KDE, ProMap, and SEPP, which use past crime occurrences to predict future crime. RTM was approximately twice as effective as the other techniques. Based on the results, we discuss the merits of and drawbacks to using RTM in Japan.

Appendix

Description for ProMap

For ProMap, the bandwidth and cell size were set similarly to KDE. We estimated the intensity of crime in a location (s) on date (t), using Eq. 1.1, as formulated by Johnson et al. (2009).

$$ \widehat{\lambda}\left(t,s\right)=\sum \limits_{c_i\le \tau \cap {e}_i\le \nu}\left(\frac{1}{\left(1+{c}_i\right)}\right)\frac{1}{\left(1+{e}_i\right)} $$

(1.1)

where τ is the spatial bandwidth, ν is the temporal bandwidth, c_i is the number of cells between each location of crime i within the spatial, e_i is the time elapsed for each occurrence time of crime i within the temporal bandwidth bandwidth and the cell.

To obtain c_i in Eq. 1.1, as shown in Fig. 4, a plurality of concentric circles (radius 12.5 m + 25 m*n, where n = 0, 1, 2, …, 10) were drawn. Although the radius of the largest circle is 262.5 m at the maximum, 250 m was set as the upper limit in order to exclude points outside of it. Equal weighting was given to the points existing within the same distance range (as shown in Fig. 4, X₁ = 0, X₂ = 1, X₃ = 2). As in previous studies (Bowers et al. 2004; Johnson et al. 2007, 2009), the number of weeks from the date of occurrence was used for e_i. The number of days was divided by 7; no rounding was performed.

Fig. 4

Value of c_i of ProMap in the current study

Description for SEPP

For SEPP, the prediction was carried out using Eq. 1.2 with reference to a series of studies by Mohler (Mohler et al. 2011; Mohler 2014, 2015). The intensity of crime in a location (x,y) on date (t) is described as:

$$ \lambda \left(t,x,y\right)=\sum \limits_{\left\{k:{t}_k<t\right\}}\mu \left(x-{x}_k,y-{y}_k\right)+\sum \limits_{\left\{k:{t}_k<t\right\}}g\left(t-{t}_k,x-{x}_k,y-{y}_k\right) $$

(1.2)

where t-t_k is the numbers of dates between a prediction target period and each occurrence date of a crime event, x-x_k, y-y_k are the instance between each target cell and each crime event.

We estimated μ and g using Eqs. 1.3 and 1.4.

$$ \mu \left(x,y\right)=\frac{\alpha }{T}\sum \limits_{t_i<t}\frac{1}{2{\pi \eta}^2}\times \mathit{\exp}\left(-\frac{{\left(x-{x}_i\right)}^2+{\left(y-{y}_i\right)}^2}{2{\eta}^2}\right) $$

(1.3)

$$ g\left(t,x,y\right)=\theta \omega exp\left(-\omega \left(t-{t}_i\right)\right)\times \frac{1}{2{\pi \sigma}^2}\mathit{\exp}\left(-\frac{{\left(x-{x}_i\right)}^2+{\left(y-{y}_i\right)}^2}{2{\sigma}^2}\right) $$

(1.4)

where:

$$ \alpha =\frac{\sum_{i=1}^{N_i}{\sum}_{j=1}^{N_j}{p}_{ij}^b1}{N_i} $$

(1.5)

$$ {\upeta}^2=\frac{\sum_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{i}}}{\sum}_{\mathrm{j}=1}^{{\mathrm{N}}_{\mathrm{j}}}{p}_{ij}^b\left({\left(x-{x}_i\right)}^2+{\left(y-{y}_i\right)}^2\right)}{2{\sum}_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{i}}}{\sum}_{\mathrm{j}=1}^{{\mathrm{N}}_{\mathrm{j}}}{p}_{ij}^b} $$

(1.6)

$$ \theta =\frac{\sum_{i=1}^{N_i}{\sum}_{j=1}^{N_j}{p}_{ij}1}{N_j-{\sum}_{j=1}^{N_j}\exp \left(-\omega \left(T-{t}_j\right)\right)1} $$

(1.7)

$$ \omega =\frac{\sum_{i<j}{p}_{ij}}{\sum_{i<j}\left({t}_j-{t}_i\right){p}_{ij}} $$

(1.8)

$$ {\sigma}^2=\frac{\sum_{i=1}^{N_i}{\sum}_{j=1}^{N_j}{p}_{ij}\left({\left(x-{x}_i\right)}^2+{\left(y-{y}_i\right)}^2\right)}{2{\sum}_{i=1}^{N_i}{\sum}_{j=1}^{N_j}{p}_{ij}} $$

(1.9)

In these equations, p_ij indicates the probability of event i (offspring) being triggered by another event j (parent). These i and j events are called triggering pairs. Additionally, p^b_ij is defined as the probability that event i occurs independently from other events. Thus, i is called a background event (equal to the parent events). We divided events into triggering pairs and background events using the EM algorithm described by Rosser and Cheng (2016). First, we set initial values of p_ij following Eq. 1.8 (here, we set α as 1.5 and β as 500) and obtained the probability matrix P₀ (upper triangular matrix). Based on those values, we distinguished parents and their offspring.

$$ {p}_{ij}=\mathit{\exp}\left(-\alpha \left({t}_j-{t}_i\right)\right)\mathit{\exp}\left(-\frac{{\left({x}_j-{x}_i\right)}^2+{\left({y}_j-{y}_i\right)}^2}{2{\beta}^2}\right) $$

(1.10)

We then calculated parameters following Eqs. 1.5–1.11. The ω and σ² were computed using triggering pairs. Once we obtained the parameters, we updated p_ij following Eq. 1.11 and repeated these steps until ||P_n-P_n-1|| was smaller than 1 × 10⁻⁷.

$$ {p}_{ij}=\theta \omega exp\left(-\alpha \omega \left({t}_j-{t}_i\right)\right)\times \frac{1}{2{\pi \sigma}^2}\mathit{\exp}\left(-\frac{{\left({x}_j-{x}_i\right)}^2+{\left({y}_j-{y}_i\right)}^2}{2{\sigma}^2}\right) $$

(1.11)

Fig. 5

Prediction maps using KDE (top left), ProMap (top right), and SEPP (bottom) from July to December