Crime Prediction Using Regression and Resources Optimization Bruno Cavadas1,2 , Paula Branco3,4 , and S´ergio Pereira5 1 2

Instituto de Investiga¸ca ˜o e Inova¸ca ˜o em Sa´ ude, Universidade do Porto, Portugal Instituto de Patologia e Imunologia Molecular da Universidade do Porto, Portugal 3 LIAAD - INESC TEC 4 DCC - Faculdade de Ciˆencias - Universidade do Porto 5 ALGORITMI Centre, University of Minho, Portugal [email protected] [email protected] [email protected]

Abstract. Violent crime is a well known social problem affecting both the quality of life and the economical development of a society. Its prediction is therefore an important asset for law enforcement agencies, since due to budget constraints, the optimization of resources is of extreme importance. In this work, we tackle both aspects: prediction and optimization. We propose to predict violent crime using regression and optimize the distribution of police officers through an Integer Linear Programming formulation, taking into account the previous predictions. Although some of the optimization data are synthetic, we propose it as a possible approach for the problem. Experiments showed that Random Forest performs better among the other evaluated learners, after applying the SmoteR algorithm to cope with the rare extreme values. The most severe violent crime rates were predicted for southern states, in accordance with state reports. Accordingly, these were the states with more police officers assigned during optimization. Keywords: violent crime, prediction, smoteR, regression, optimization

1

Introduction

Violent crime is a severe problem in society. Its prediction can be useful for the law enforcement agents to identify problematic regions to patrol. Additionally, it can be a valuable information to optimize available resources ahead of time. In the United States of America (USA), according to the Uniform Crime Reports (UCR) published by the Federal Bureau of Investigation (FBI) [1], violent crimes imply the use of force or threat of using force, such as rape, murder, robbery, aggravated assault, and non-negligent manslaughter. In 2013, it was reported 1,163,146 violent crimes, with an average of 367.9 per 100k inhabitants. This was equivalent to one violent crime every 27.1 seconds. In 2012, according to the United States Department of Labor [2], there were 780,000 police officers

and detectives in the USA, with a median salary of $56,980 per year. Therefore, the optimization of police officers can be useful to optimize costs, while guaranteeing the safety of the population. In this paper, the contributions are twofold. Firstly, we propose to predict the violent crime per 100k population using regression. To the best of our knowledge, this is the first time that such problem is tackled in this way. Moreover, we preprocess the data using smoteR algorithm to improve predictions on the most critical values: the extreme high. Having the predictions, we also propose an Integer Linear Programming formulation for the optimization of police officers distribution across states. This distribution takes into account the crime severity, population, density and budget of the states. The remaining of the paper is organized as follows. In Section 2 a brief survey on related work is presented. Materials and methods are exposed in Section 3, including the description of the data set, the prediction-related procedures and the optimization scheme. Then, in Section 4, results are presented and discussed, while in Section 5 the main conclusions are pointed out.

2

Related work

Crime prediction has been extensively studied throughout the literature due to its relevance to society. These studies employ diverse machine learning techniques to tackle the crime forecasting problem. Nath [3] combined K-means clustering and a weighting algorithm, considering a geographical approach, for the clustering of crimes according to their types. Liu et al. [4] proposed a search engine for extracting, indexing, querying and visualizing crime information using spatial, temporal, and textual information and a scoring system to rank the data. Shah et al. [5] went a step further and proposed CROWDSAFE for real-time and location-based crime incident searching and reporting, taking into account Internet crowd sourcing and portable smart devices. Automatic crime prediction events based on the extraction of Twitter posts has also been reported [6]. Regarding the UCI data set used in this work, Iqbal et al. [7] compared Naive Bayesian and decision trees methods by dividing the data set into three classes based on the risk level (Low, medium and high). In this study, decision trees outperformed Naive Bayesian algorithms, but the pre-processing procedures were rudimentary. Somayeh Shojaee et al. [8], applied a more rigorous data processing methodology for a binary class and applied the usage of two different feature selection methods to a wider range of learning algorithms (Naive Bayesian, decision trees, support vector machine, neural networks and K-Nearest neighbors). In these studies no class balancing methodologies were employed. Other approaches such as the fuzzy association rule mining [9] and case-based editing [10] have also been performed. After prediction, optimization of resources can be achieved by several strategies. Donovan et al. [11] used integer linear programming for the optimization of fire-fighting resources, solving one of the most commonly constrains faced by fire

managers. The same strategy was used by Caulkins et al. [12] in the optimization of software system security measures given a fixed budget. Regarding the problem of police officer optimization, Mitchell [13] used a P-median model to determine the patrol areas in California, while Daskin [14] applied a Backup Coverage Model to maximize the number of areas covered. More recently, Li et al. [15] relied on the concept of “crime hot-spots” to create a cross entropy approach to produce randomized optimal patrol routes.

3

Materials and Methods

3.1

Data Set Description

The Communities and Crime Unnormalized Data Set 1 provides information on several crimes in the USA, combining socio-economical and law enforcement data from 90’ Census, 1990 Law Enforcement Management and Admin Stats survey and the 1995 FBI UCR. It includes 2215 examples, 124 numeric and 1 nominal attribute. It also contains 4 non-predictive attributes with information about the community name, county, code and fold. Among the several possible target variables we chose the number of violent crimes per 100k population. 3.2

Prediction

We started by pre-processing the data set. The violent crime is our target variable, thus we removed all the other 17 possible target variables contained in the data set. We also eliminated all the examples that had a missing value on our target variable and removed all the attributes that had more than 80% of missing values. The data set contained four non-predictive attributes, which we have also eliminated. Finally, we have removed one more example that still had a missing value, and have normalized all the remaining attributes. Although this problem was previously tackled as a classification task, we opted for addressing it as a regression task. This is an innovative aspect of our proposal and this choice is also based on the fact that we will use the numeric results obtained with the predictions for solving an optimization problem. Therefore, it makes sense to use a continuous variable throughout the work, instead of discretizing the target variable and latter recovering a numeric value. Another challenge involving this data set is the high number of attributes. To address this problem we have applied the same feature selection scheme with two different percentages. The scheme applies a hierarchical clustering analysis, using the Pearson Correlation Coefficient. This step removes a percentage of the features less correlated with the target variable. Then, a Random Forest (RF) learner is applied to compute the remaining features importance based on the impact in the Mean Squared Error. A percentage of the most important features provided is selected. Two different sets of features were selected by applying 1

available at UCI repository in https://archive.ics.uci.edu/ml/datasets/ Communities+and+Crime+Unnormalized.

different percentages in the previous scheme. In one of the pre-processed data we aimed at obtaining 50% of the original features and in the other the goal was to select only 30% of the original features. This way we obtained two data sets with 52 and 32 features corresponding to 50% and 30% percentages. In our regression problem we are interested in predicting the number of violent crimes per 100k inhabitants. However, we are more concerned with the errors made in the higher values of the target variable, i.e., the consequences of missing a high value of violent crimes by predicting it as low are worst than the reverse type of error. The extreme high values of the violent crime variable are the most important and yet the less represented in the data set. When addressed as a classification problem, this is clearly a problem with imbalanced classes, where the most important class has few examples. SmoteR algorithm is a proposal to address this type of problems within regression which was presented in [16,17]. This proposal uses the notion of utility-based Regression [18] and relies on the definition of a relevance function. The relevance function expresses the user preferences regarding the importance assigned to the target variable range. Ribeiro [18] proposes automatic methods for estimating the relevance function of the target variable. We have used those methods because they correspond to our specific concerns: the extreme rare values are the most important. The essential idea of SmoteR algorithm is to balance the data set by under-sampling the most frequent cases and over-sampling the rare extreme examples. The over-sampling strategy generates new synthetic examples by interpolating existing rare cases. More details can be obtained in [16,17]. The motivation for applying this procedure is to force the learning systems to focus on the rare extreme cases which would be difficult to achieve in the original imbalanced data. Our experiences included several variants of smoteR which were applied to the two pre-processed data sets. The smoteR variants used in the experiences included all combinations of the following parameters: under-sampling percentage 50% and 100%; over-sampling percentage 200% and 400%; number of neighbours 5. For the prediction task we have used three learning algorithms: Support Vector Machines (SVM), RF and Multivariate Adaptive Regression Splines (MARS). More details on the experimented parameters and the evaluation are described in Section 4.1. 3.3

Optimization through Integer Linear Programming

Given the predicted violent crime per 100k population, we propose to optimize the distribution of available police officers by state. We present our proposal as a proof of concept, since more detailed data and insight into the problem would be needed to implement a more realistic solution. Given that the number of officers by state is an integer quantity, it is used Integer Linear Programming. To solve the optimization problem it was applied the Branch-and-bound algorithm. Problem formulation We considered as resources a certain amount of police officers to freely distribute by the states of the USA. The optimization takes

into account the predictions on violent crime per 100k population to assign more officers by the states with more violent criminality. This assignment is constrained by an ideal number of officers that each state would like to receive and the available budget. However, every state should receive a minimum amount of officers to guarantee the security of its citizens. In the data set, the instances are defined by communities, with several of them belonging to the same state. Since we wanted to distribute officers by state, it was calculated the mean violent crime predictions by state. The optimization problem was defined as, maximize subject to

m X i=1 m X

si xi xi = N ; xi ≤ Hi ;

i=1

xi ≥ fi Hi ; ci xi ≤ Bi ; xi ∈ N

where i ∈ {1, ..., m} indexes each of the m states, with m = 46, xi is the number of officers to distribute by state, si is the violent crime predictions by state, Hi is the ideal number of officers by state, fi is a fraction on the ideal number of officers that each state accepts as the minimum, ci is the cost that each state should pay for each officer, and Bi is the available budget for each state. The ideal number of officers was defined in function of the violent crime prediction of the state and the population (number of citizens), since bigger populations, with more violent crime, have higher demands regarding police officers. To this end, the violent crime predictions were scaled (ssi ) to the interval [vl , vh ]. This way, it acts as a proportion on the population. However, since some populations have millions of citizens, this value was divided by 100 to get more realistic estimates for the ideal number of officers. So, ssi pi (1) 100 where pi is the real population of the state i. It was defined that the minimum number of officers should be a fraction on the ideal number, taking into account the crime predictions. Defining a lower (lb ) and an upper (ub ) bound for the fraction, the previously scaled violent crime predictions are linearly mapped to the interval [lb , ub ]. Knowing that it is in the interval [vl , vh ], the fraction on the ideal number of officers is calculated as, Hi =

si − vl (ub − lb ) + lb (2) vh − v l Budget was defined in function of the population and its density. Such definition is based on the intuition that a small and less dense population needs less budget and officers than a highly dense and big population. However, the population numbers are several orders of magnitude higher than density, which would make the effect of density negligible. So, we have rescaled both population fi =

and density to the range [0, 100] (psi and dsi ). Moreover, the budget for each state is a part of the total national budget (BT ). So, Bi was calculated as (ds + a · psi ) BT Bi = Pmi i=1 dsi + a · psi

(3)

where a > 0 is a parameter to tune the weight of the density and population over the budget calculation.

4

Experimental Analysis

We have divided our problem, and analysis, into two sub-problems: prediction and optimization. In this section, we describe the tools, metrics, and evaluation methodology for each sub-problem. Then we focus in each sub-problem results. 4.1

Experimental Setup

Prediction The main goal of our experiments is to select one of the two preprocessed data sets, a smoteR variant (in case it has a positive impact) and a model (among SVM, RF and MARS) to apply in the optimization task. The experiments were conducted with R software. Table 1 summarizes the learning algorithms that were used and the respective parameter variants. All combinations of parameters were tried for the learning algorithms, which led to 4 SVM variants, 6 RF variants and 8 MARS variants. We started by splitting each data set in train and test sets, approximately corresponding to 80% and 20% of the data. The test set was held apart to be used in the optimization, after predicting its crime severities. This set was randomly built with stratification and with the condition of including at least one example for each possible state of the USA. In imbalanced domains, it is necessary to use adequate metrics since traditional measures are not suitable for assessing the performance. Most of these specific metrics, such as precision and recall, exist for classification problems. The notions of precision and recall were adapted to regression problems with nonuniform relevance of the target values by Torgo and Ribeiro [19] and Ribeiro [18]. We will use the framework proposed by these authors to evaluate and compare our results. More details on this formulation can be obtained in[18].

Table 1: Regression algorithms, parameter variants, and respective R packages. Learner MARS SVM Random Forest

Parameter Variants nk = {10, 17}, degree = {1, 2}, thresh = {0.01, 0.001} cost = {10, 150}, gamma = {0.01, 0.001} mtry = {5, 7}, ntree = {500, 750, 1500}

R package earth [20] e1071 [21] randomForest [22]

All the described alternatives were evaluated according to the F-measure with β = 1, which means that the same importance was given to both precision and recall scores. The values of F1 were estimated by means of 3 repetitions of a 10-fold Cross Validation process and the statistical significance of the observed paired differences was measured using the non-parametric pairwise Wilcoxon signed-rank test. Optimization In the optimization sub-problem the objective was to assign to each state a certain amount of police officers, given the total budget, the total number of available officers, and the violent criminality predictions. The optimization was carried out in R software, with the package “lpSolve”. The values for the population and the density are real values, obtained from the estimates for 2014 [23]. However, the total budget, the number of available police officers, and the individual cost of the officers by state were defined by us. Although they are not real values, they serve as proof of concept. The cost of each officer by state was chosen randomly, and uniformly, from the interval [5, 15] once, then the same values were used in all experiments. Additionally, the values for ub and lb were set to 0.12 and 0.08, while vl and vh were set to 0.125 and 0.7, respectively. 4.2

Results and Discussion

Prediction We started by examining the results obtained with all the parameters selected for the two pre-processed data sets, the three types of learners and the smoteR variants. All combinations of parameters were tested by means of 3 repetitions of a 10-fold cross validation process. Figure 1 shows these results. We have also analysed the statistical significance of the differences observed in the results. Table 2 contains the several p-values obtained when comparing the SmoteR variants and the different learners, using the non-parametric pairwise Wilcoxon signed rank test with Bonferroni correction for multiple testing. The p-value for the differences between the two data sets (with 30% and 50% of the features) was 0.17. Therefore we chose the data set with less features to continue to the optimization problem. This was mainly because of: i) the non statistical significant differences and ii) the smaller size of the data (less features can explain well the target variable, so we chose the most efficient alternative).

Table 2: Pairwise Wilcoxon signed rank test with Bonferroni correction for the SmoteR strategies (left) and the learning systems (right). Strategies S.o2.u0.5 S.o2.u1 S.o4.u0.5 S.o4.u1

none S.o2.u0.5 S.o2.u1 S.o4.u0.5 1.3e-14 < 2e-16 1 2.3e-16 1 1 < 2e-16 0.18 1 1

Learners svm rf rf

Instituto de Investiga¸ca ˜o e Inova¸ca ˜o em Sa´ ude, Universidade do Porto, Portugal Instituto de Patologia e Imunologia Molecular da Universidade do Porto, Portugal 3 LIAAD - INESC TEC 4 DCC - Faculdade de Ciˆencias - Universidade do Porto 5 ALGORITMI Centre, University of Minho, Portugal [email protected] [email protected] [email protected]

Abstract. Violent crime is a well known social problem affecting both the quality of life and the economical development of a society. Its prediction is therefore an important asset for law enforcement agencies, since due to budget constraints, the optimization of resources is of extreme importance. In this work, we tackle both aspects: prediction and optimization. We propose to predict violent crime using regression and optimize the distribution of police officers through an Integer Linear Programming formulation, taking into account the previous predictions. Although some of the optimization data are synthetic, we propose it as a possible approach for the problem. Experiments showed that Random Forest performs better among the other evaluated learners, after applying the SmoteR algorithm to cope with the rare extreme values. The most severe violent crime rates were predicted for southern states, in accordance with state reports. Accordingly, these were the states with more police officers assigned during optimization. Keywords: violent crime, prediction, smoteR, regression, optimization

1

Introduction

Violent crime is a severe problem in society. Its prediction can be useful for the law enforcement agents to identify problematic regions to patrol. Additionally, it can be a valuable information to optimize available resources ahead of time. In the United States of America (USA), according to the Uniform Crime Reports (UCR) published by the Federal Bureau of Investigation (FBI) [1], violent crimes imply the use of force or threat of using force, such as rape, murder, robbery, aggravated assault, and non-negligent manslaughter. In 2013, it was reported 1,163,146 violent crimes, with an average of 367.9 per 100k inhabitants. This was equivalent to one violent crime every 27.1 seconds. In 2012, according to the United States Department of Labor [2], there were 780,000 police officers

and detectives in the USA, with a median salary of $56,980 per year. Therefore, the optimization of police officers can be useful to optimize costs, while guaranteeing the safety of the population. In this paper, the contributions are twofold. Firstly, we propose to predict the violent crime per 100k population using regression. To the best of our knowledge, this is the first time that such problem is tackled in this way. Moreover, we preprocess the data using smoteR algorithm to improve predictions on the most critical values: the extreme high. Having the predictions, we also propose an Integer Linear Programming formulation for the optimization of police officers distribution across states. This distribution takes into account the crime severity, population, density and budget of the states. The remaining of the paper is organized as follows. In Section 2 a brief survey on related work is presented. Materials and methods are exposed in Section 3, including the description of the data set, the prediction-related procedures and the optimization scheme. Then, in Section 4, results are presented and discussed, while in Section 5 the main conclusions are pointed out.

2

Related work

Crime prediction has been extensively studied throughout the literature due to its relevance to society. These studies employ diverse machine learning techniques to tackle the crime forecasting problem. Nath [3] combined K-means clustering and a weighting algorithm, considering a geographical approach, for the clustering of crimes according to their types. Liu et al. [4] proposed a search engine for extracting, indexing, querying and visualizing crime information using spatial, temporal, and textual information and a scoring system to rank the data. Shah et al. [5] went a step further and proposed CROWDSAFE for real-time and location-based crime incident searching and reporting, taking into account Internet crowd sourcing and portable smart devices. Automatic crime prediction events based on the extraction of Twitter posts has also been reported [6]. Regarding the UCI data set used in this work, Iqbal et al. [7] compared Naive Bayesian and decision trees methods by dividing the data set into three classes based on the risk level (Low, medium and high). In this study, decision trees outperformed Naive Bayesian algorithms, but the pre-processing procedures were rudimentary. Somayeh Shojaee et al. [8], applied a more rigorous data processing methodology for a binary class and applied the usage of two different feature selection methods to a wider range of learning algorithms (Naive Bayesian, decision trees, support vector machine, neural networks and K-Nearest neighbors). In these studies no class balancing methodologies were employed. Other approaches such as the fuzzy association rule mining [9] and case-based editing [10] have also been performed. After prediction, optimization of resources can be achieved by several strategies. Donovan et al. [11] used integer linear programming for the optimization of fire-fighting resources, solving one of the most commonly constrains faced by fire

managers. The same strategy was used by Caulkins et al. [12] in the optimization of software system security measures given a fixed budget. Regarding the problem of police officer optimization, Mitchell [13] used a P-median model to determine the patrol areas in California, while Daskin [14] applied a Backup Coverage Model to maximize the number of areas covered. More recently, Li et al. [15] relied on the concept of “crime hot-spots” to create a cross entropy approach to produce randomized optimal patrol routes.

3

Materials and Methods

3.1

Data Set Description

The Communities and Crime Unnormalized Data Set 1 provides information on several crimes in the USA, combining socio-economical and law enforcement data from 90’ Census, 1990 Law Enforcement Management and Admin Stats survey and the 1995 FBI UCR. It includes 2215 examples, 124 numeric and 1 nominal attribute. It also contains 4 non-predictive attributes with information about the community name, county, code and fold. Among the several possible target variables we chose the number of violent crimes per 100k population. 3.2

Prediction

We started by pre-processing the data set. The violent crime is our target variable, thus we removed all the other 17 possible target variables contained in the data set. We also eliminated all the examples that had a missing value on our target variable and removed all the attributes that had more than 80% of missing values. The data set contained four non-predictive attributes, which we have also eliminated. Finally, we have removed one more example that still had a missing value, and have normalized all the remaining attributes. Although this problem was previously tackled as a classification task, we opted for addressing it as a regression task. This is an innovative aspect of our proposal and this choice is also based on the fact that we will use the numeric results obtained with the predictions for solving an optimization problem. Therefore, it makes sense to use a continuous variable throughout the work, instead of discretizing the target variable and latter recovering a numeric value. Another challenge involving this data set is the high number of attributes. To address this problem we have applied the same feature selection scheme with two different percentages. The scheme applies a hierarchical clustering analysis, using the Pearson Correlation Coefficient. This step removes a percentage of the features less correlated with the target variable. Then, a Random Forest (RF) learner is applied to compute the remaining features importance based on the impact in the Mean Squared Error. A percentage of the most important features provided is selected. Two different sets of features were selected by applying 1

available at UCI repository in https://archive.ics.uci.edu/ml/datasets/ Communities+and+Crime+Unnormalized.

different percentages in the previous scheme. In one of the pre-processed data we aimed at obtaining 50% of the original features and in the other the goal was to select only 30% of the original features. This way we obtained two data sets with 52 and 32 features corresponding to 50% and 30% percentages. In our regression problem we are interested in predicting the number of violent crimes per 100k inhabitants. However, we are more concerned with the errors made in the higher values of the target variable, i.e., the consequences of missing a high value of violent crimes by predicting it as low are worst than the reverse type of error. The extreme high values of the violent crime variable are the most important and yet the less represented in the data set. When addressed as a classification problem, this is clearly a problem with imbalanced classes, where the most important class has few examples. SmoteR algorithm is a proposal to address this type of problems within regression which was presented in [16,17]. This proposal uses the notion of utility-based Regression [18] and relies on the definition of a relevance function. The relevance function expresses the user preferences regarding the importance assigned to the target variable range. Ribeiro [18] proposes automatic methods for estimating the relevance function of the target variable. We have used those methods because they correspond to our specific concerns: the extreme rare values are the most important. The essential idea of SmoteR algorithm is to balance the data set by under-sampling the most frequent cases and over-sampling the rare extreme examples. The over-sampling strategy generates new synthetic examples by interpolating existing rare cases. More details can be obtained in [16,17]. The motivation for applying this procedure is to force the learning systems to focus on the rare extreme cases which would be difficult to achieve in the original imbalanced data. Our experiences included several variants of smoteR which were applied to the two pre-processed data sets. The smoteR variants used in the experiences included all combinations of the following parameters: under-sampling percentage 50% and 100%; over-sampling percentage 200% and 400%; number of neighbours 5. For the prediction task we have used three learning algorithms: Support Vector Machines (SVM), RF and Multivariate Adaptive Regression Splines (MARS). More details on the experimented parameters and the evaluation are described in Section 4.1. 3.3

Optimization through Integer Linear Programming

Given the predicted violent crime per 100k population, we propose to optimize the distribution of available police officers by state. We present our proposal as a proof of concept, since more detailed data and insight into the problem would be needed to implement a more realistic solution. Given that the number of officers by state is an integer quantity, it is used Integer Linear Programming. To solve the optimization problem it was applied the Branch-and-bound algorithm. Problem formulation We considered as resources a certain amount of police officers to freely distribute by the states of the USA. The optimization takes

into account the predictions on violent crime per 100k population to assign more officers by the states with more violent criminality. This assignment is constrained by an ideal number of officers that each state would like to receive and the available budget. However, every state should receive a minimum amount of officers to guarantee the security of its citizens. In the data set, the instances are defined by communities, with several of them belonging to the same state. Since we wanted to distribute officers by state, it was calculated the mean violent crime predictions by state. The optimization problem was defined as, maximize subject to

m X i=1 m X

si xi xi = N ; xi ≤ Hi ;

i=1

xi ≥ fi Hi ; ci xi ≤ Bi ; xi ∈ N

where i ∈ {1, ..., m} indexes each of the m states, with m = 46, xi is the number of officers to distribute by state, si is the violent crime predictions by state, Hi is the ideal number of officers by state, fi is a fraction on the ideal number of officers that each state accepts as the minimum, ci is the cost that each state should pay for each officer, and Bi is the available budget for each state. The ideal number of officers was defined in function of the violent crime prediction of the state and the population (number of citizens), since bigger populations, with more violent crime, have higher demands regarding police officers. To this end, the violent crime predictions were scaled (ssi ) to the interval [vl , vh ]. This way, it acts as a proportion on the population. However, since some populations have millions of citizens, this value was divided by 100 to get more realistic estimates for the ideal number of officers. So, ssi pi (1) 100 where pi is the real population of the state i. It was defined that the minimum number of officers should be a fraction on the ideal number, taking into account the crime predictions. Defining a lower (lb ) and an upper (ub ) bound for the fraction, the previously scaled violent crime predictions are linearly mapped to the interval [lb , ub ]. Knowing that it is in the interval [vl , vh ], the fraction on the ideal number of officers is calculated as, Hi =

si − vl (ub − lb ) + lb (2) vh − v l Budget was defined in function of the population and its density. Such definition is based on the intuition that a small and less dense population needs less budget and officers than a highly dense and big population. However, the population numbers are several orders of magnitude higher than density, which would make the effect of density negligible. So, we have rescaled both population fi =

and density to the range [0, 100] (psi and dsi ). Moreover, the budget for each state is a part of the total national budget (BT ). So, Bi was calculated as (ds + a · psi ) BT Bi = Pmi i=1 dsi + a · psi

(3)

where a > 0 is a parameter to tune the weight of the density and population over the budget calculation.

4

Experimental Analysis

We have divided our problem, and analysis, into two sub-problems: prediction and optimization. In this section, we describe the tools, metrics, and evaluation methodology for each sub-problem. Then we focus in each sub-problem results. 4.1

Experimental Setup

Prediction The main goal of our experiments is to select one of the two preprocessed data sets, a smoteR variant (in case it has a positive impact) and a model (among SVM, RF and MARS) to apply in the optimization task. The experiments were conducted with R software. Table 1 summarizes the learning algorithms that were used and the respective parameter variants. All combinations of parameters were tried for the learning algorithms, which led to 4 SVM variants, 6 RF variants and 8 MARS variants. We started by splitting each data set in train and test sets, approximately corresponding to 80% and 20% of the data. The test set was held apart to be used in the optimization, after predicting its crime severities. This set was randomly built with stratification and with the condition of including at least one example for each possible state of the USA. In imbalanced domains, it is necessary to use adequate metrics since traditional measures are not suitable for assessing the performance. Most of these specific metrics, such as precision and recall, exist for classification problems. The notions of precision and recall were adapted to regression problems with nonuniform relevance of the target values by Torgo and Ribeiro [19] and Ribeiro [18]. We will use the framework proposed by these authors to evaluate and compare our results. More details on this formulation can be obtained in[18].

Table 1: Regression algorithms, parameter variants, and respective R packages. Learner MARS SVM Random Forest

Parameter Variants nk = {10, 17}, degree = {1, 2}, thresh = {0.01, 0.001} cost = {10, 150}, gamma = {0.01, 0.001} mtry = {5, 7}, ntree = {500, 750, 1500}

R package earth [20] e1071 [21] randomForest [22]

All the described alternatives were evaluated according to the F-measure with β = 1, which means that the same importance was given to both precision and recall scores. The values of F1 were estimated by means of 3 repetitions of a 10-fold Cross Validation process and the statistical significance of the observed paired differences was measured using the non-parametric pairwise Wilcoxon signed-rank test. Optimization In the optimization sub-problem the objective was to assign to each state a certain amount of police officers, given the total budget, the total number of available officers, and the violent criminality predictions. The optimization was carried out in R software, with the package “lpSolve”. The values for the population and the density are real values, obtained from the estimates for 2014 [23]. However, the total budget, the number of available police officers, and the individual cost of the officers by state were defined by us. Although they are not real values, they serve as proof of concept. The cost of each officer by state was chosen randomly, and uniformly, from the interval [5, 15] once, then the same values were used in all experiments. Additionally, the values for ub and lb were set to 0.12 and 0.08, while vl and vh were set to 0.125 and 0.7, respectively. 4.2

Results and Discussion

Prediction We started by examining the results obtained with all the parameters selected for the two pre-processed data sets, the three types of learners and the smoteR variants. All combinations of parameters were tested by means of 3 repetitions of a 10-fold cross validation process. Figure 1 shows these results. We have also analysed the statistical significance of the differences observed in the results. Table 2 contains the several p-values obtained when comparing the SmoteR variants and the different learners, using the non-parametric pairwise Wilcoxon signed rank test with Bonferroni correction for multiple testing. The p-value for the differences between the two data sets (with 30% and 50% of the features) was 0.17. Therefore we chose the data set with less features to continue to the optimization problem. This was mainly because of: i) the non statistical significant differences and ii) the smaller size of the data (less features can explain well the target variable, so we chose the most efficient alternative).

Table 2: Pairwise Wilcoxon signed rank test with Bonferroni correction for the SmoteR strategies (left) and the learning systems (right). Strategies S.o2.u0.5 S.o2.u1 S.o4.u0.5 S.o4.u1

none S.o2.u0.5 S.o2.u1 S.o4.u0.5 1.3e-14 < 2e-16 1 2.3e-16 1 1 < 2e-16 0.18 1 1

Learners svm rf rf