Crime prevention of bus pickpocketing in Beijing, China: does air quality affect crime?

Abstract

In recent years, along with the development of the urban public transport system, bus pickpocketing crimes have garnered increasing attention. In this paper, a complete and clear structure including finding patterns based on data, applying criminological theory to explain patterns, and using empirical investigation method to verify theoretical explanation are presented. We found that temperature and season are not clearly correlated with bus pickpocketing. The AQI and PM_2.5 indices, however, demonstrated significant correlations with daily bus pickpocketing incidents: the worse the air quality, the more bus pickpocketing occurred. Then two empirical investigations were carried out to verified that crime pattern theory and rational choice theory can be used to explain the impact of air quality on bus pickpocketing crime. Furthermore, we utilized the SVM method to predict daily bus pickpocketing crime risk with an accuracy rate of 81%. The results of this paper can provide early warnings of urban bus pickpocketing and help police reduce crimes.

Appendix

Principles of SVM

When using SVM to classify two types of data, training samples are divided into “1” and “− 1” categories that are then trained to get a hyperplane that is maximally distant from the positive samples and negative samples. Test samples are plotted in a high-dimensional space to distinguish whether they are positive or negative according to an optimal separating hyperplane. This study employed a training set:

$$\{ (x_{i} ,y_{i} )[x_{i} \in R^{n} ,y_{i} \in R,i = 1,2,3 \ldots ,]\}$$

(1)

where x_i ϵ Rⁿ represents the eigenvector, and y_i ϵ {+ 1, − 1} is the category mark. In this paper, “+ 1” indicates a high crime rate, whereas “− 1” is low. Most real-world problems are not linearly separable; SVM uses the kernel technique to automatically realize nonlinear mapping on a feature space. SVM uses the kernel function Φ(x) to map the training set data x_i to a high-dimensional linear feature space to find an optimal hyperplane:

$$\varPhi (x) \cdot \omega + b = 0$$

(2)

where ω ϵ Rⁿ and x ϵ Rⁿ, b is the offset. Therefore, two samples can be separated properly with the largest distance between the two sample types. The discriminant function is

$$y(x) = {\text{sign}}[\omega \cdot \varPhi (x) + b].$$

(3)

The problem of an optimal regression hyperplane can be transformed into a programming function as follows:

$$\hbox{min} \left\{ {\frac{1}{2}||\omega ||^{2} } \right\} + C\sum\nolimits_{i = 1}^{n} {[C(\xi_{i} ) + C(\xi_{{}}^{*} )]} .$$

(4)

The constraint of the function is

$$\left\{ {\begin{array}{*{20}c} {y_{i} - \left( {\omega \cdot x_{i} } \right) - b \le \varepsilon + \xi_{i} } \\ {\left( {\omega \cdot x_{i} } \right) + b - y_{i} \le \varepsilon + \xi_{i}^{*} } \\ \end{array} \quad i = (1,2,3, \ldots ,n)} \right.,$$

(5)

where $\xi_{i} \ge 0$ and $\xi_{i}^{*} \ge 0$ are each slack variables, $c(\xi_{i} )$ is the loss function, and C is the penalty term constant. By using the Lagrangian multiplier to solve the dual form of the quadratic programming problem with linear constraints, we get:

$$\hbox{max} \left\{ {L_{\text{D}} = \mathop \sum \limits_{i = 1}^{l} a_{i} - \frac{1}{2}\mathop \sum \limits_{i = 1}^{l} \mathop \sum \limits_{j = 1}^{l} a_{i} a_{j} y_{i} y_{j} \varPhi (x_{i} ) \cdot \varPhi (x_{j} ) = \mathop \sum \limits_{i = 1}^{l} a_{i} - \frac{1}{2}\mathop \sum \limits_{i = 1}^{l} \mathop \sum \limits_{j = 1}^{l} a_{i} a_{j} y_{i} y_{j} K(x_{i} , x_{j} )} \right\}$$

(6)

with the following restrictions:

$$0 \le a_{i} \le C,\quad \mathop \sum \limits_{i = 1}^{l} a_{i} y_{i} = 0 ,$$

(7)

where $K(x_{i} , x_{j} ) = \varPhi (x_{i} ) \cdot \varPhi (x_{j} )$, called a kernel function with the discriminant function of

$$y(x) = {\text{sign}}\left[ {\omega^{*} \cdot \varPhi (x) + b^{*} } \right] = {\text{sign}}\left[ {\mathop \sum \limits_{{x_{i} \in SV}} a_{i}^{*} y_{i} K(x_{i} , x) + b^{*} } \right],$$

(8)

where $\omega^{*} ,b^{*} ,a_{i}^{*}$ are the optimal solutions.

There are several common kernel functions of the SVM method:

• (1) Linear kernel function: $Kx_{i} ,x = x \cdot x_{i}$

• (2) Polynomial kernel function: $Kx_{i} ,x = [(x \cdot x_{i} ) + 1]^{q} ,q = 1,2, \ldots ,n$

• (3) Radial basis function (RBF):$Kx_{i} ,x = \exp \left( { - \frac{{\left| {x - x_{i} } \right|^{2} }}{{2\sigma^{2} }}} \right)$

• (4) Cauchy kernel function: $Kx_{i} ,x = \tanh (v(x \cdot x_{i} ) + C)$

The RBF kernel is most often used in the above function because of its good learning ability. It is an ideal classification function for any type of sample (low-dimensional, high-dimensional, small, large, etc.). This study used the RBF kernel as well.

Principles of Naive Bayes

The probability that the quantity level of the number of bus pickpocketing on a certain day belongs to category c is:

$$P(c|d) \propto P(c)\mathop \prod \limits_{{1 \le k \le n_{d} }} P(t_{k} |c),$$

(9)

where P(c) is the prior probability that the number of bus pickpocketing on a certain day belongs to category c, $(t_{1} ,t_{2} , t_{1} , \ldots ,t_{{n_{d} }} )$ is the characteristic attribute of day d, and n_d represents the total amount of all feature attributes of d. After knowing that the number of bus pickpocketing on a certain day belongs to the prior probability of category c_i, it is necessary to find the most likely category for a certain day d. For Naive Bayes, it is the category of the estimate of maximum a posteriori (MAP). i.e. the formula (2) (where c ϵ C):

$$c_{\text{map}} = \arg \hbox{max} \,\hat{P}(c|d) = \arg \hbox{max} \hat{P}(c)\mathop \prod \limits_{{1 \le k \le n_{d} }} \hat{P}(t_{k} |c).$$

(10)

Calculating the product of the conditional probability using Eq. (2) may cause the lower bound of the floating point number to overflow, so introduce the logarithm to get the formula (3) (where c ϵ C):

$$c_{\text{map}} = \arg \hbox{max} \hat{P}(c|d) = \arg \hbox{max} \left[ {\log \hat{P}(c) + \mathop \sum \limits_{{1 \le k \le n_{d} }} \log \hat{P}(t_{k} |c)} \right].$$

(11)