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Forecasting of Annual Crime Rate in India:A case Study

Manish Kumar, Athulya S, Mary Minu MB, Vidya Vinodini M D, Aiswaria Lakshmi K G, Anjana S, Manojkumar TK*

Center for Excellence in Data Engineering and Computational Modeling Indian Institute of Information Technology and Management-Kerala (IIITM-K)

Abstract— Crime forecasting has been made possible by

criminological theories and developments in computational techniques/data analytics further improved the technologies of forecasting. It helps the various departments of the police to make a decision and strategy to prevent the crime. This paper focuses on forecasting the annual crime rate in India using the Time Series Models such as Auto-Regressive Integrated Moving Average (ARIMA) and Exponential Smoothing. Source of data is from the National Crime Record Bureau of India. As a part of modeling, data is divided into training data for the years 1953 to 2008 and test data for the years 2009 to 2013. By examining the model, it’s clear that the forecast values are within the 95% confidence interval of the test data and accuracy measurements are also significant. Hence the time series model suitable for crime forecasting.

Keywords—crime forecasting; ARIMA Model; Exponential Smoothing Model

I. INTRODUCTION India is having high population growth rates compared with

the rest of the world. India’s population is estimated to be around one billion. The high population density, combined with other factors such as lack of jobs, poverty, and illiteracy will result in a higher violence rate. The crime and violence rate vary from state to state. States like Uttar Pradesh, Bihar etc records high crime rates according to 2017 statistics. Like other counties increase in crime rate is a major concern in India also. From the reports of National Crime Record Bureau (NCRB), states that most of crime incidents recorded is in urban area [1]. In India, crime rate (case reported per lakh population) has increased from 166.7 to 215.5 in years from 1953 to 2013. By analyzing the data, crime rates got highly fluctuated in the years 1970-2005. The statistics indicate that crime rate in India is steadily increasing for the past 8-9 years.

The increasing trends in urban crime lead to various other violations of law and make life harder. Crime forecasting will help in analyzing crime rates and one can take preventive steps for reducing the number of crimes. Time series is reported to be one of the best tools for analyzing time series data and providing proper insights into various dependent factors of the series. Since Autoregressive Integrated Moving Average (ARIMA) model was created by Box and Jenkins, it has been effectively utilized as a part of forecasting economic,

marketing, production, social issues etc[2]. This model has the advantage of exact forecasting over short-term for the series. This paper used Box-Jenkins Methodology to model Annual Crime Rate Time Series (ACRTS) by using ARIMA models and Exponential smoothing model [3]. The ARIMA model could provide forecasting results with upper limits, lower limits and forecasted values, which means any realization within the interval between upper limits and lower limits will be accepted.

The efficient approach to identify and analyze patterns and trends in crime can be made by applying crime analysis and prevention. With the increasing advent of technologies in crime, data analysts may help the police officers to speed up the process of solving crimes.

II. METHODS The review of the literature for this work outlines the

papers that deals with the technique carried out for forecasting crime rates, their challenges and to take remedial actions on it etc. Utilizing time series model to make short-term forecasting of crime is a new research field appearing recently. Peng Chen et.al gave a proportional study for the forecasting crime using the ARIMA model, in this paper ARIMA is used to make short-term forecasting of property crime for one city of China. They used the ARIMA model for making short-term forecasting of property crime for one city of China, then compared forecasting results with the Simple Exponential Smoothing (SES) and Holt’s two-parameter exponential smoothing (HES). Noted that the ARIMA model has best fitting and forecasting accuracy than other two models [4]. Arye Rattner presents an attempt in social indicators and crime rate forecasting, to use macrodynamic social indicators in a time series analysis of three crime categories-homicide; property and; robbery offenses in Israel. By analyses, models are created for the earlier findings in the relationship between homicide and unemployment, and density of population and property offenses [5].

Shrivastav and Ekata validate the applicability of the ARIMA model in crime forecasting. Here they utilized the crime data of Gujarat State pertaining to counterfeiting of currency. The authors focused to plot the matter-of-facts which need to be undertaken to use Box Jenkins ARIMA time series models [3]. N.Mohamad Noor proposed crime forecasting using Autoregressive Integrated Moving Average (ARIMA) model and fuzzy alpha-cut method. This combination is expected to generate more accurate forecasting result with a

minimum error. The results will help the authorities in making the right decision in crime prevention strategies [6]. Research based on crime forecasting has a prominent role in the forecasting world. Many forecasting methods have been applied in this field as Naïve lag, exponential smoothing, decomposition method and ARIMA model [7]. Using a time series model to predict the values for future reference is relatively rare in the world. The outcomes make an impact for the right decision in taking preventions against crime. The results could give data about crime trends particularly the conceivable most evidently bad and better condition afterward. This data can help the police in settling on choice for operational and strategic procedures in crime prevention.

III. THEORETICAL FRAMEWORK

A. Data Source and Software The data were collected from National Crime Bureau of

India (www.ncrb.gov.in). Statistical tool R was used for time series modeling [1].

B. Box-Jenkins Model The basic idea of Box-Jenkins approach in modeling a time

series is summarized in Fig 1. Box-Jenkins consists of three phases [8]. The following phases are

• Identification: Selects the best model based on analysis of series.

• Testing and Estimation: In this stage the parameters are estimated and they are used for forecasting and then residuals are evaluated. Then these residuals are examined for validation of mode.

If following conditions are satisfied, we may proceed further for forecasting using that particular data set.

a. These residuals should be stationary.

b. The residuals should have zero autocorrelation coefficients between them.

c. It is expected that these residuals are normally distributed.

If all these are satisfied, use that mode for forecasting out of sample. Otherwise go to first phase and select another mode.

Fig. 1. Box Jenkins Methodology

C. Exponential Smoothing

This method is based on the principle that recent values have more weight and it decreases as the observation gets older. There is a variety of exponential methods but all they have in

common is recent values are given relatively more weight than older observations [9].

1) Single Exponential Smoothing

In Single Exponential Smoothing, forecasting is done using previous period forecast value and adjusts it using forecasting error. = + ( − ) (1)

value lies between 0 and 1

So the equation involves a basic principle of negative feedback. The past forecast error is used to correct the next forecast. An alternative way of writing equation is = + (1 − ) (2) On solving, we get = + (1 − )[ + (1 − ) ] = + (1 − ) + (1 − )

Here it is clear that has less weightage than and weightage is decreasing exponentially.

But if there is trend in the series, the forecast will lag behind the trend because there is no parameter which can be used to match trend.

2) Holt Linear Method

Holt Linear Method, which is an extension of Single Exponential Smoothing that allows to forecast data with trend given by Holt in 1957 [10].

Holts linear exponential smoothing using two parameters and . = + (1 − )( + ) (3) = ( − ) + (1 − ) (4) = + (5) denotes the estimate of level time t, denotes the estimate of trend at time t.

Equation (3) adjust level at time ( ) directly for trend of previous period and adding it to last smoothed value . This helps to estimate the trend and bring the level to approximate level of current data. Equation (4) update the trend on the difference between last smoothed value and it is appropriate because it will show the trend in previous and some randomness can be smoothed by using . This is similar to Single Exponential Smoothing but used for updating the trend. Equation (5) is used to forecast for future values. This method is sometimes called Double Exponential Method [11]. This method is very good for the data that have trend and very

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useful but they will lack for modeling the time series when there will be seasonality with trend.

3) Holt Winters Method

This method occurs when seasonality comes into effect. Holt’s method was extended by Winters to capture seasonality directly. In fact, there are two types of seasonality additive or multiplicative [12]. So we have two types of Holt Winters method for seasonality as follows :-

a) Multiplicative Seasonality = ⁄ + (1 − )( + ) (6) = ( − ) + (1 − ) (7) = ⁄ + (1 − ) (8) = ( +) ( )

b) Additive Seasonality = ( − ) + (1 − )( + ) (10) = ( − ) + (1 − ) (11) = ( − ) + (1 − ) (12) = + + (13)

Where, denotes level at time t denotes trend denotes seasonality

denotes forecasted values

D. ARIMA MODEL Auto Regression Integrated Moving Average(ARIMA)

model has been studied extensively. They were popularized by George Box and Gwilym Jenkins in early 1970s. The ARIMA model is the most general class of model for forecasting a time series like other method[13]. It requires only historic time series data. Normally the ARIMA model is denoted by the ARIMA (p, d, q)

p is number of autoregressive term

d is number of non-seasonal difference

q is number of lagged forecast error

Now if d=0, then ARMA (p, q) is known as stationary model that is it can be used for only stationary series. While in ARIMA if d > 0 that is non-stationary model. If underlying time series is nonstationary, then difference method is used to make it stationary. The order of difference determines the value of I. I (0) means original time series is stationary.

I (1) means first order differenced series is stationary. The equation for ARIMA model is given as

(1 − θ B)(1 − B)Y = c + (1 − θ B)E (14) And after differencing the series ARMA is applied on the

time series. In ARIMA model, AR (Auto Regressive) component represent the memory of process for preceding observation. So p represents the auto regressive component in ARIMA (p, d, q). If other components are zero then it is represented by AR (p) and equation is given by Y = c + ∅ Y +⋯⋯⋯⋯+ ∅ Y + E (15)

Where ∅ represent the magnitude of relationship. If p is 0 then it means that there is no relationship between

adjacent terms.

The lag of forecast error is called Moving Average. These represent the memory for random shock. q represent the number of moving average component. Then it is represented by MA (q) and equation is given by Y = c + E − θ E − θ −⋯⋯⋯⋯θ E (16)

Where θ, represents the magnitude of relationship. IV. MODELING ACRTS

In this section, the Annual Crime Rate Time Series (ACRTS) has been modeled by using Time Series models. ARIMA model and exponential smoothing method is used to model the CRTS.

A. ARIMA Modeling The Box – Jenkins Methodology is used to build a model.

In Box-Jenkins Methodology there are three steps which were followed to built ARIMA model and Exponential Smoothing for CRTS

1) Data analysis and Test for stationary-ACRTS

Fig.2 represents the Time Series plot of training data of ACRTS. It is clear from the plot that the series is non- stationary. We confirmed the non-stationary by calculating Augmented Dickey-Fuller(ADF).. The ADF test performed on data shows the p-value 0.5413 which strongly suggests that the series is non-stationary i.e. the mean is not constant. To make data stationary first order difference is performed and hence the series became stationary. Fig.3 shows the first order differenced time series of the annual crime rate.

Fig. 2. ARCTS for Training Data

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Fig. 3. ACR First Order Differenced Time Series.

The ACF and PACF plots are used to identify the models. Fig.4 and Fig.5 are the plots of ACF and PACF of differenced series. The order of difference to make series stationary is 1 it implies the model will be ARIMA (p, 1, q). On basis of the correlogram of ACF and PACF the value of p and q are chosen. By considering several models as shown in TABLE 1, the model which produce minimum AIC and BIC is chosen. Hence the model ARIMA (0, 1, 0) with minimum BIC and model ARIMA (2, 1, 2) with minimum AIC were taken for analysis. While comparing the residuals and accuracy the best fit model to forecast ACRTS is ARIMA (2, 1 ,2).

Fig. 4. ACF of ACR Differenced Time Series

Fig. 5. PACF Plot of ACR Differenced Time Series

TABLE I. COMPARISON OF ARIMA MODELS

2) Testing and Diagnostics Testing is the second phase of the Box-Jenkins

Methodology. In this phase after fitting the model, the residuals are tested for verification of model fitting. The residual should follow these tests.

• There should be no correlation between the residuals i.e. residual should be independent of each other.

• The residual should follow white noise.

• The residual should be normally distributed.

Fig.6 shows the plot of ACF of residuals and it is clear that there is no spike in this plot which means there is no correlation between the residuals [14]. For residuals to be white noise model the Box –Ljung test is applied on the residuals. The null hypothesis for this test is that the series follows a white noise model and p-value for the Box-Ljung test was obtained suggesting that it accept the null hypothesis which means residuals follow the white noise models. To check the normality of residuals there is the test called Jarque-Bera test. In this test null hypothesis in the series follows the normal distribution. The p-value obtained is high suggesting that to accept the null hypothesis means residuals are normally distributed [15]. So by testing the residuals, it concluded that ARIMA (212) can be used for forecasting.

Fig. 6. ACF of Residual

3) Forecasting

After verifying and testing it is clear that the ARIMA (212) can be used for forecasting the test data (2008-2013). All observed test data lie between the 95% confidence interval forecasted by the ARIMA (212). Fig.7 shows the actual and forecasted value by ARIMA (212)

Fig. 7. Forecasted value vs actual value

ARIMA MODEL AIC BIC

010 399.68 401.69

110 401.38 405.39

111 402.27 408.29

112 398.73 406.76

211 396.93 404.96

212 394.76 404.79

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4) Accuracy Accuracy refers to “goodness of fit”, which in turn refers

how well the forecasting model is able to reproduce data that are already known [16]. There are many standard statistical measures used for measurement of accuracy.

• MAE-Mean Absolute Error (the mean value of absolute errors)

• MASE-Mean Absolute Squared Error (the mean value of errors square)

• MAPE-Mean Absolute Percentage Error.

So the best model have low values for these measurements. The measurements for ARIMA(212) are given below in Table II

TABLE II. ACCURACY MEASUREMENT FOR ARIMA (212) MODEL

Data Set MAE MAPE MASE

Training Set 5.92 3.43 0.86

Test Set 12.93 6.33 1.88

B. Exponential Smoothing Modeling To model ACRTS using exponential smoothing model, the

following steps were used.

1) Data analysis and Model selection

By analyzing the trend of ACRTS in Fig 2, it is quite difficult to summarize, as its trend shows both increasing and decreasing[14]. Thus to make result more accurate Holt Linear method is introduced which is an extension of single exponential smoothing. Thus the Holt Linear method provided the best fit for ACRTS.

2) Initialization and Estimation of parameter

Initialization of Holt Winters function is made using R with the value of gamma equal to false (means there is no gamma in Holt linear Model) and the optimal value of coefficient α and β is given as is

α = 0.943

β = 0.181

These values indicate the dependence of value on previous data. Fig 8 shows the fitted value of data by Holt linear model. It is clear that fitted values are following pattern of observed values.

Fig. 8. Holt linear modelling vs Train data

3) Testing and Diagnostics

ACF plot for residuals of Holt linear model is shown in Fig.9 .It is clear that there is no autocorrelation between the residuals.

Fig. 9. ACF residual of Holt linear Model

Hence Holt Linear models can be used as a best fit model for forecasting the ACRTS (2008-2013). Fig 10 shows the forecasted value vs. test data. All observed test data lie between the 95% confidence interval forecasted by the Holt Linear.

Fig. 10. Forecasted value vs Test Data

4) Accuracy Accuracy measurements of Holt Linear methods are given below in Table III. The values of measurements are significant. It concludes that Holt Linear method is numerically significant for modeling ACRTS.

TABLE III. MEASUREMENT FOR HOLT LINEAR MODEL

Data Set MAE MAPE MASE

Training Set 7.48 4.36 1.09

Test Set 9.23 4.50 1.34

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V. RESULT The Annual Crime Rate in India for the years 2014-2018

are evaluated by Holt Linear Method is shown in Table IV.

TABLE IV. FORECASTED VALUE BY HOLT LINEAR METHOD

Year Point Forecast Lo80 Hi80 Lo95 Hi95 2014

161.2

153.4

202.5

98.1

224.3

2015

160.2

153.3

207.5

87.5

232.9

2016

159.3

149.6

213.3

76.6

242.0

2017

158.3

148.1

219.1

65.2

251.4

2018

157.3

147.1

225.2

53.5

261.1

Table V contains the value forecasted by ARIMA (212)

TABLE V. FORECASTED VALUE BY ARIMA (2 1 1)

Year Point Forecast Lo80 Hi80 Lo95 Hi95 2014

190.8

146.8

234.8

123.5

258.1

2015

192.4

141.8

243.1

114.9

270.0

2016

194.1

136.5

251.6

106.0

282.1

2017

195.7

131.0

260.4

96.7

294.6

2018

197.3

125.3

269.3

87.1

307.4

VI. CONCLUSION AND FUTURE WORK This paper concluded that time series model can be applied

for crime forecasting. The result obtained from both the models conclude that they are significant for forecasting all test data which are lying between a 95% confidence interval and accuracy measurements for training data shows that they are numerically significant. In future, we are trying to analyze crime against women, children so that we can predict how much police strength is convenient to decrease the crime rate.

VII. REFERENCE

[1] Official webportal of National Crime Records Bureau http://ncrb.gov.in/

[2] Box, George EP, and David A. Pierce. "Distribution of residual autocorrelations in autoregressive-integrated moving average time series models." Journal of the American statistical Association 65.332 (1970): 1509-1526.

[3] Shrivastav, Anand Kumar. "Applicability of Box Jenkins ARIMA model in crime forecasting: A case study of counterfeiting in Gujarat state." International Journal of Advanced Research in Computer Engineering & Technology (IJARCET) 1.4 (2012): pp-494.

[4] Chen, Peng, Hongyong Yuan, and Xueming Shu. "Forecasting crime using the arima model." Fuzzy Systems and Knowledge Discovery, 2008. FSKD'08. Fifth International Conference on. Vol. 5. IEEE, 2008

[5] Rattner, Arye. "Social indicators and crime rate forecasting." Social Indicators Research 22.1 (1990): 83-95

[6] Noor, Noor Maizura Mohamad, et al. "Crime forecasting using ARIMA model and fuzzy alpha-cut." Journal of Applied Sciences 13.1 (2013): 167-172

[7] Groff, Elizabeth R., and Nancy G. La Vigne. "Forecasting the future of predictive crime mapping." Crime Prevention Studies 13 (2002): 29-58.

[8] Loftin, Colin, and David McDowall. "The police, crime, and economic theory: An assessment." American Sociological Review (1982): 393-401.

[9] Williams, Billy, Priya Durvasula, and Donald Brown. "Urban freeway traffic flow prediction: application of seasonal autoregressive integrated moving average and exponential smoothing models." Transportation Research Record: Journal of the Transportation Research Board 1644 (1998): 132-141.

[10] Gorr, Wilpen, Andreas Olligschlaeger, and Yvonne Thompson. "Short- term forecasting of crime." International Journal of Forecasting 19.4 (2003): 579-594

[11] Flaxman, Seth R. A General Approach to Prediction and Forecasting Crime Rates with Gaussian Processes. Heinz College Technical Report, 2014. URL https://www. ml. cmu. edu/research/dap- papers/dap_flaxman. pdf, 2014

[12] Gorr, Wilpen, Andreas Olligschlaeger, and Yvonne Thompson. "Assessment of crime forecasting accuracy for deployment of police." International Journal of Forecasting (2000): 743-754

[13] Alwee, Razana, et al. "Hybrid support vector regression and autoregressive integrated moving average models improved by particle swarm optimization for property crime rates forecasting with economic indicators." The Scientific World Journal 2013 (2013)

[14] Mutangi, Kudakwashe. "Time Series Analysis of Road Traffic Accidents in Zimbabwe." International Journal of Statistics and Applications 5.4 (2015): 141-149.

[15] Ott, R. Lyman, and Micheal T. Longnecker. An introduction to statistical methods and data analysis. Nelson Education, 2015

[16] McClusky, Howard Y. "An Introduction to Statistical Method." (1925): 549-551

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