Review Journal 1A simpliﬁed mathematical-computational model of .docx

Review Journal 1
A simpliﬁed mathematical-computational model of the immune
response to the yellow fever vaccine
1. This model can be improved in a way if there are more test
subjects and more variables and parameters with test data is
added. Plus the mathematical process is always improvable so if
there is an equation which is more better for this experiments
then it’s can improve the model and experiment with
development. Another try is to improve the qualitative results
obtained from our model. Additional computational
experiments, such as the effects of a) a booster dose and b) a
reduction in the population of TCD8+ naive. Also, a sensitivity
analysis will be performed to identify sensitive parameters and
to identify connections between change in parameters values
and computational results.
2. if more cases or more experiments were added then it could
be more expanded research and could improve the research more
but the similar results are achieved by the shorter experiments
so we can say this number of experiments were enough. But
there is always a room left for improvements. The second
difference between the two models is that this work reduces the
amount of equations from 19 to 10. The reduced model
described in this work considers only the main populations of
cells and molecules involved in the response to the vaccine, and
abstracts some details that are not crucial to represent the
behavior of the immune response. For example, the distinct
compartments are not represented here. Also, some populations
were not considered because no experimental data is available
to validate the simulations, such as the CD4+ T cells. In future,
more cell or molecule can be included in the model again, if its
role is important to explain or represent some behavior that the
reduced model presented in this section could not represent.
That was something in the first paper journal which was not
satisfactory for me.

3. This model can be applied to the numerous medical
applications like cancer immunity and other immune
vaccinations for the diseases or viruses in the open environment
which are lethal and are possible to cure in the future. The virus
cannot proliferate itself, it needs to infect a cell and use it as a
factory for new viruses. This is implicitly considered in the
term πvV , which represents the multiplication of the virus in
the body, with a production rate πv. The term cv1V cv2+V
denotes a non-speciﬁc viral clearance made by the innate
immune system
4. The main problem which was to solve is that the author needs
something authentic calculations to perform the experiment to
show the results of that experiment as negative and for that only
programming and developing an algorithm or a program is not
enough. It needs the proper calculations of the human body and
cells about each and every details in depth. So that’s why it was
necessary to include the calculations in the immune system
development. Previously the author wasn’t including proper
calculations so the API for python wasn’t providing the accurate
results. Because the default made up API’s would response only
to limited calculations. But he need some extensive and
complex calculations so it was necessary to include it in. in this
research mathematical model is being used as a tool to assist the
research in vaccinology and public health. With the use of
mathematical and computational models, it is possible to
experiment, in silico, different scenarios related to vaccination,
for answering important questions still open.
5. The model discussed in this paper is based on the previous
study on mathematical model which describes the human
immune response to the infection by YF virus. For this
development of the model code was built in python
programming which included libraries for solving mathematical
problems. It was scipy. It was tested on different human cells
and each of the antibody response generates the results of the
experiments and test. The model is then evaluated using distinct
scenarios, and as will be shown, it was able to qualitatively

reproduce some experimental results reported in the literature.
The results were found to be improved than in the previous
study and the conclusion for the paper is that he validated the
model with two distinct scenarios that were simulated. The ﬁrst
one simulates the immune response to the administration of the
standard dose of the 17DD-YFV. The second one simulates the
immune response to distinct doses of vaccine and then
compared the two results to find out the best.
References
Bonin, C., Fernandes, G., dos Santos, R., & Lobosco , M.
(2017). A simpliﬁed mathematical-computational model of the
immune response to the yellow fever vaccine. 7-8.
Kumar, M., A. S., MB, M. M., Vinodini , V. M., & Lakshmi , A.
K. (2018). Forecasting of Annual Crime Rate in India : A case
Study . International Conference on Advances in Computing,
Communications and Informatics, 6.
Forecasting of Annual Crime Rate in India
Case study analysis
(
June 7
, 2019
)
Question 1
In this the authors are finding ways to manipulate the best thing
and equations which will help them to precisely measure the
strategic procedures regarding criminality rate and controlling
the uprising rate of crime.
The equations are the best thing to describe the proceeding and
calculating the rate of crime which has to lessen at any cost.

After the brief and intellectual description of the equation
authors have ways to mend them along with the care and
prevention of the people’s behavior and environmental related
factors.
· ARIMA which autoregressive integrated moving average is the
good and expressive equation built up of the steps towards the
mortality rate of crime.
· The forecasting of the crime has made easy for the authors and
police departments by manipulating the findings of a case.
· Orders of crime in this have been described by China research
and the early findings of a planning. Analyses of the model and
socially interlinking of crime in plotting the analyses and data
structures of the person’s social behavior which affect the
homicide and unemployment victims. This data has now been
computerized and elevated in the database of our system.
The Box-Jenkins model proved to be the best model with the
precise and data analyses testing the best essential series of
criminal cases.
Question 2
The model Box-Jenkins has the most precise and analytical way
to evaluate the best and improving tribe to control and prevent
crime technologies.
The building of the urban areas has the greater values towards
the dynamic pursue of the technology base crime basis.
Collection of the historic data from the previous eras is the
major hindrance in fluctuating environments, the crime rate
through the social interlinking and touching the boundaries of
the extreme provision of exposure of a person personal data.
· The parameters were the historical crime related data
collection and seasonal theft varying communities produce a
great impact on this collection of the data.
· Non-stationary peak of crime rate may have an affecting way
to kill the possibilities and hope in finding the exact location of
the crime server.
The model was built on the basis and practical supervision
depending on the areas and police departments allocating in that

area.
Question 3
The model was tested based on the collected data. Providing one
of the control group which has the recent studies of the crime
rate and the other have the possible growing hub of gangs
interlinked in urban areas from the starting. By calculating
through the equation precisely, and improving data and results
stood up in front of the research team.
· Hardships do come in doing such a risky job which is equal to,
putting your hand in the mouth of an alligator and spectating the
thing which is causing suffocation.
Question 4
The results were good enough and helping towards what was
expected by the authors, crime strategy has been broken through
it and dismantle the connection networks indulging in crime.
The model was built to measure the precise analyses intruding
and indulging the different possibilities in enhancing the
equation oriented crime analyses.
· Sharing through the third parties series of cases have been
detected to control and manage the outcomes, developing
techniques will enhance the data accuracy and security.
Question 5
Logical researching based on the right way to manipulate case
studies and testing the hypothesis has a great impact on the rest
of the experimental studies. Urban areas have the highest rate of
crime, it’s all because the easy availability of data access and
approach of persons from street to street so the security must be
provided to the users. The advantage of the devices and social
interconnection has been the major outcome of the crime related
problems and being better to understand it will also mend ways
towards more capturing of pre-planned crime on scenes.
References
1. Kumar, M., S, A., & MB, M. M. (2018). Forecasting of
Annual Crime Rate in India: A case Study. Center for
Excellence in Data Engineering and Computational Modeling

Indian Institute of Information Technology and Management-
Kerala.
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
978-1-5386-5314-2/18/$31.00 ©2018 IEEE 2087
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
2088
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
2089
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
2090

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-
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2092
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>>
>> setdistillerparams
<<
/HWResolution [600 600]
/PageSize [612.000 792.000]
>> setpagedevice

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