INTRODUCTION
Crime prevention is one of the important roles of the police system in any
country. One of the components of crime prevention is crime rate predictions
or in other word is crime forecasting. Police will require crime forecasting
to make operational and tactical strategies in the future, as like to allocate
police patrols in the right area, install CCTV in the right place and plan other
operations. A common practice is to identify hot spots in the preceding period
based on their geographical location and assume these hot spots will persist
into the next period (Pieterse, 2006).
Some researches in crime forecasting have been done by several researchers
like Cohen and Gorr (2005), Deadman
(2003) and Gorr et al. (2003). Many methods
have been applied in crime forecasting research. Used of Naïve lag, exponential
smoothing methods and classical decomposition by Gorr et
al. (2003). Use of the decomposition method made by Jiang
and Barricarte (2011), Deadman (2003), Chen
et al. (2008) and Noor et al. (2011b)
used ARIMA model.
Explored with the use of ARIMA model for crime forecasting has not been widely
used. One of the research used ARIMA model discussed by Chen
et al. (2008). They used the ARIMA model for making shortterm forecasting
of property crime for one city of China, then compared forecasting results with
the Simple Exponential Smoothing (SES) and Holt’s twoparameter exponential
smoothing (HES). The conclusion of the research is the ARIMA model fits the
series better than SES and HES and makes higher accurate forecasting than the
other two models. ARIMA model provides forecasting result with the interval
range of forecasting value. The interval range describe the possible condition.
To improve interval forecasting accuracy of crime forecasting this study proposes combined between ARIMA model and Fuzzy alphacut method for crime forecasting process. The results could provide information about crime trends, especially the possible worst and better condition in the future. This information can help the police in making decision for operational and tactical strategies in crime prevention.
CRIME PREVENTION IN MALAYSIA
In the recent years, volumes of crime had brought serious problems, not least
in Malaysia (Menteri, 2010). Such reported in “The
Roadmap 2010”, the overall index crime rate in Malaysia increases from
746 reported crimes per 100,000 people in 2006 to 767 in 2007, a rise of nearly
3% (Menteri, 2010). The Malaysian government has taken
a step forward in “Reducing Crime” program included in Government
Transformation Program (GTP). The government target in GTP 2010 in terms of
reducing crime is achieved 5% reduction in index crime in overall reported index
crime every year and 20% reduction in street crime. As of 31 December 2010,
the target of reducing incidents of index crime by 5% and street crime by 20%
have been exceeded. For index crime, there has been a reduction of 32,297 cases
(15%) while Street Crime has dropped by 13,193 cases (35%). Even though the
index crime rate has dropped by 15%, there are still 177,520 cases reported
as of December 2010. This shows crime is still occurring (Menteri,
2011a).
The Royal Police Malaysia (RMP) in recent years has several operational and
tactical strategies in crime prevention. The operations and strategy are separation
of pedestrian walkways with railings, improved lightings, mirrors, safety alarms,
GISbased (geographic information system) crime mapping and closedcircuit televisions
or CCTVs (implemented separately under the supervision of the Ministry of Housing
and Local Government). At the center of this crime prevention effort is a Safe
City Monitoring System (SCMS) for monitoring the current crime at that time
(Menteri, 2011b). To support the government programme,
require crime forecasting system to help operational and tactical strategy.
By knowing the crime rate prediction in the future it is expected that police
can be more accurate in determining the strategy choices, for example are allocated
police patrols and install CCTV in the selected hot spot area.
To fulfill the need we propose Crime Forecasting System. In forecasting process of this system we used ARIMA model and fuzzy alphacut.
MATERIAL AND METHODS
This research is a part of the framework Crime Management System (CeMaS), shown
in Fig. 1, where this framework proposed for police to support
reducing crime program (Noor et al., 2011a).
Discussion in this paper is focused on Crime Forecasting Module. Our proposed
system consists of 3 parts, shown in Fig. 2. There is database,
operation, output. Part 1 is crime database which contain about historical crime
data, location and crime forecasting data. Part 2 describe two process. The
first process is a forecasting process used ARIMA model and fuzzy alphacut
method. The second process is crime information retrieval process. The results
of crime information retrieval process would be displayed to the decision maker,
discussed in Part 3. Wherein the information would be showed trends of crime
with visualization on graph, table and mapping.
Crime forecasting process in this study used ARIMA model and Fuzzy alphacut method. With combination between ARIMA model and Fuzzy alphacut method is expected to generate more accurate interval forecasting results with minimum error.
Autoregressive integrated moving average (ARIMA): In forecasting process, we use ARIMA model. The ARIMA model is denoted by ARIMA (p,d,q), where “p” stands for the order of the auto regressive process, ‘d’ is the order of the data stationary and ‘q’ is the order of the moving average process. In ARIMA(p, d, q) model, the future value of a variable is assumed to be a linear function of several past observations and random errors. That is, the underlying process that generates the time series has the form:
where, y_{t} and ε_{t} are the actual value and random
error at time period t, respectively φ_{i} (i = 1,2,...,q) and
φ_{j} (i = 1,2,...,q) are model parameters. p and q are integers
and often referred to as orders of the model. Random errors ε_{t}
are assumed to be independently and identically distributed with a mean of zero
and a constant variance of σ^{2 }(Zhang, 2003).
The forecasting process used Box Jenkin’s methodology for ARIMA model.
The BoxJenkins methodology includes three iterative steps (Khashei
et al., 2009):
• 
Model identification 
• 
Parameter estimation 
• 
Diagnostic checking 

Fig. 1: 
Crime management system ( CeMaS) 

Fig. 2: 
Crime forecasting module 
Before any model is identified we must consider the pattern of data. After
getting the pattern of data we follow the first step BoxJenkins methodology,
that is identification model. In this step, data transformation is often required
to make the time series stationary. Stationarity is a necessary condition in
building an ARIMA model used for forecasting. In second step several models
could be obtained. After having the tentative models the next step is checking
the model to obtain the best model. This threestep model building process is
typically repeated several times until a satisfactory model is finally selected.
Furthermore, the best model can be used for forecasting and the results of forecasting
will be stored in a database. Implementation of this process discussed by Noor
et al. (2011b). Figure 3 describes the process
of crime forecasting with BoxJenkins methodology used historical crime index
data.
Fuzzy alphacut (FAC): Alpha cuts are simply threshold levels that convert
a fuzzy set into a crisp set. The process of converting a fuzzy set to a crisp
one is called defuzzification. An alphacut A of a fuzzy number A is defined
as the set {xεRA (x)≥α}. A is completely determined by the collection
(A_{α})αε[0,1].

Fig. 3: 
The crime forecasting process used ARIMA model 
An alpha cut is the degree of sensitivity of the system to the behavior under
observation. At some point, as the information value diminishes, one no longer
wants to be "bothered" by the data. In many systems, due to the inherent limitations
of the mechanisms of observation, the information becomes suspect below a certain
level of reliability (Kumar and Schuhmacher, 2005).
The membership function is cut horizontally at a finite number of αlevels
between 0 and 1. For each αlevel of the parameter, the model is run to
determine the minimum and maximum possible values of the output. This information
is then directly used to construct the corresponding fuzziness (membership function)
of the output which is used as a measure of uncertainty. If the output is monotonic
with respect to the dependent fuzzy variable, the process is rather simple since
only two simulations will be enough for each αlevel (one for each boundary).
Otherwise, optimization routines have to be carried out to determine the minimum
and maximum values of the output for each αlevel. Figure
4 shows an illustration of the alphacut of Triangular Fuzzy Number (TFN),
where Lo and Up are the lower and upper bounds of the closed interval.
A triangular fuzzy number Ã can be written as Eq. 2
can be defined by a triplet (a,b,c). The membership function μ_{Ã
} (x) is defined as (Kaufman and Gupta, 1991):
Alternatively, by defining the interval of confidence at level α, we can
characterize the triangular fuzzy number as (Cheng, 1999):
Standard deviation (SD): Standard Deviation (SD) shows how much variation
or "dispersion" exists from the average (mean, or expected value). A low SD
indicates that the data points tend to be very close to the mean, whereas high
standard deviation indicates that the data points are spread out over a large
range of values.

Fig. 4: 
Fuzzy numbers Ã with αcuts 
Equation 4 shows formula of the SD of the sample:
RESULTS AND DISCUSSION
The model proposed is combination between ARIMA model and fuzzy alphacut method, as shown in Fig. 5. This combination is aimed to obtain more accurate interval forecasting result.
This process has six steps. The first step is making the pattern of the data and the next steps are Box Jenkin’s methodology steps. The fifth step is forecast with the best model ARIMA. The results of the ARIMA forecasting models based on actual data will be converted to fuzzy observations with lower and upper membership function. The arithmetic operation on the fuzzy alphacut, for instance the alphacut method is applied to the data through the fuzzy environmental process.
Implementation of first step until fifth step has been made on Noor
et al. (2011b). The result found ARIMA (0,1,1) is the best model
in this study with of mean square error (MSE) = 7411. Then crime forecasting
is done using this model. The forecasting result is shown in Table
1.

Fig. 5: 
Crime forecasting process with fuzzy αcut 
Table 1: 
Forecasting results with ARIMA (0,1,1) 

Table 2: 
Alpha cut results 

Table 3: 
Standard deviation results 

The forecasting values with lower and upper values became a crisp value by a triplet (a,b,c). For example used values from month 31, the lower value as a, upper value as c and forecast value as b. Then the value of a = 963.37, b = 1132.14 and c = 1300.91.
For the next step with Eq. 3 we calculated TFN with crisp values from forecasting values and αcut values 0.2 and 0.5. The result of calculation became Lo and Up bounds of the closed interval. The αcut result shows in Table 2. Standard Deviation (SD) performed after the results of αcut obtained. The SD results show in Table 3.
Table 2 shows the interval values with α = 0. 5 closer to the forecast value, better than interval values with α = 0. 2.
Table 3 shows the values of SD from lower and upper forecasting results, lower and upper with α = 0.2 and with α = 0.5. The better result with the lowest SD is lower and upper with α = 0.5. The SD with α = 0.5 nearest to the SD forecasting.
CONCLUSION
Two values of αcut presented for getting the better interval values of crime forecasting which closer to forecasting values. More closer to the forecasting value indicate more closely to actual value. The SD results with α = 0.5 is lower than SD with α = 0.2. This is indicated that SD with α = 0.5 is better than α = 0.2. The proposed study in combining fuzzy αcut with ARIMA is expected to the accuracy of interval forecasting results of the crime, especially for minimum and maximum values of forecasting. Further study is needed to strengthen the proposed combination methods. Used of various αcut values for testing combination methods will be present the better conclusion.
ACKNOWLEDGMENTS
The authors would like to thank for continuous supports given by Datuk Seri Mohd Bakri bin Mohd Zinin, Director of Criminal Investigation Department, Royal Police Malaysia (RPM).
This study was supported by a grant from Fundamental Research Grant Scheme (FRGS) Ministry of Higher Education (MOHE) Malaysia, with vot. number 59201.