BIBLIOGRAPHY ALMERA L. CARIAS, JINKY ROSE J....
BIBLIOGRAPHY
ALMERA L. CARIAS, JINKY ROSE J. JUSTO and LIZEL A. MAXIMINO,
April 2009. Principal Component Analysis of Crime Occurrences at La Trinidad,
Benguet. Benguet State University, La Trinidad, Benguet.
Adviser : DR. SALVACION Z. BELIGAN
ABSTRACT
Specifically the study aimed to: 1) identify the different types of crimes
committed at the different barangays of La Trinidad, Benguet; 2) identify the barangay
that has the most number of crimes; 3) the month and year that has the most number of
crimes; and 4) the pattern of crime occurrences in La Trinidad, Benguet.
Across year, the most prevalent crimes that were committed are theft, robbery and
physical injuries. Of the 7 crimes committed at La Trinidad Police District Station, 5
crimes namely: murder, homicide, acts of lasciviousness, robbery, and theft occurred
with similar pattern from 2004 to 2008 and accounted 54% of the variation in the percent
occurrences. The second group of crimes which accounted 29% variability are rape and
physical injury.
Looking across month, three crime clusters was noted. The first group consisting
of acts of lasciviousness and murder were prevalent from February to November. Rape
and robbery, the second group of crimes were prevalent in the
month of May. The third group comprising theft, physical injury and homicide were
prevalent from January to December.
Barangay-wise, homicide, physical injuries, robbery, illegal possession of fire
arms, carnapping, women abuse and child abuse showed similar pattern of occurrences in
Balili and Pico. Murder, rape, theft, illegal drugs, adultery and malicious mischief were
also found to have similar pattern of occurrences in Pico and Betag.
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TABLE OF CONTENTS
Page
Bibliography
………………………………………………….
i
Abstract
………………………………………………….
i
Table of Contents
………………………………………………….
iii
INTRODUCTION
Background of the Study
………………………..……….... 1
Statement of the problem
………………………...………..
2
Objectives of the Study
………………………..…….…...
2
Significance of the Study
………………………..….……..
3
Scope and Delimitation of t he Study ……………….….…..….
3
REVIEW OF RELATED LITERATURE
Crime Occurrences
…………………………………. 4
Application of Principal Component Analysis ……………..
5
THEORETICAL FRAMEWORK
………………………………….
8
Definition of Terms
………………………………….. 14
METHODOLOGY
Data Source
…………………………………………….. 16
Data Analysis
…………………………………………...... 16
RESULTS AND DISCUSSION
Mean Number of Crime Occurrences
from 2004 to 2008
…………………………………….…. 17
iii
Component Loadings for PCA Performed
on the Occurrences of Crimes at La Trinidad,
Benguet from 2004 to 2008
………………………………….. 19
Mean Number of Crime Occurrences
by Month from 2004 to 2007 ………………………………….. 22
Component Loadings for PCA Performed
on the Occurrences of Crimes at La Trinidad,
Benguet Monthly from 2004 to 2007
…………………….... 23
Mean Number of Crime Occurrences in the
Different Barangays of La Trinidad, Benguet
from 2005 to 2008
……………………………………….. 26
Component Loadings for PCA Performed on
the Occurrences of Crimes in the Different
Barangays of La Trinidad, Benguet from
2005 to 2008
……………………………………..………. 28
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Summary
………………………………………………. 31
Conclusion
………………………………………………. 32
Recommendation ……………………………………………….. 32
LITERATURE CITED
………………………………………………. 33
APPENDICES
…………………………………………..………. 34
iv
1
INTRODUCTION
Background of the Study
Crimes are divided into felonies and misdemeanors which vary in
seriousness. A felony is a serious crime such as rape, homicide, or aggravated
assault for which punishment typically ranges for more than a year’s imprisonment
to death. A misdemeanor is a minor crime that is typically punished by less than a
year in jail. In either event, a fine maybe part of the sanction as well (Kendall,
2004).
Crimes are usually prevalent in areas inhabited by people with mixed
culture and with different values orientation as in the Philippines. Generally
speaking, crime occurrences are fewer in countries where there is a settled way of
life and a traditional respect for law.
Crime occurrences due to some factors such as economic, peer influence,
personality, culture and other factors are prevalent in urban and suburban areas
where business establishments are located.
Rampant robbery, stealing, rape and other unlawful actions are usually
prevalent in such areas where business and school transactions, and other social
activities are performed.
Because of the harm it may inflict on the innocent people victimized by
these criminal elements in the society, it is necessary that police authorities should
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come up with a study to establish crime spots in all areas where business
transactions and other social activities are being performed.
With this establishment of crime patterns in La Trinidad areas, it can be
used by the government to come up with strategies that will prevent the
occurrences of such crimes.
Hence, the authors of this study are willing to help the police authorities in
establishing crime spots in La Trinidad area by Principal Component Analysis.
Statement of the Problem
The study sought to determine the most prevalent crimes committed by
people, crime hot spots, and the period when a particular crime is being committed.
Specifically, the study intended to answer the following questions:
1. What are the different types of crimes committed at the different
barangays of La Trinidad, Benguet?
2. What barangay has the most number of crimes committed?
3. What month and year has the most number of crimes committed?
4. Where and when the crime hot spots occurred?
Objectives of the Study
With the application of the Principal Component Analysis (PCA), the study
aimed to determine the areas and period of crime hot spots in La Trinidad, Benguet.
Specifically, the study aimed to identify the following:
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1. the different types of crimes committed at the different barangays of La
Trinidad, Benguet;
2. the barangay that has the most number of crimes;
3. the month and year that has the most number of crimes; and
4. the pattern of crime occurrences in La Trinidad, Benguet.
Significance of the Study
Results of this study may give information to the society, local government
and law enforcers about the crimes being committed in a particular place. It could
also provide crime patterns so that the law enforcers and policy makers could have
an idea on what actions to take to prevent and/or lessen the incidence of different
crimes.
Scope and Delimitations
The data collection was from December 2008-February 2009. The data
were gathered from the La Trinidad Police District Office. The data collected were
crime statistics by year (2004-2008), by month (2004-2007) and by place or
barangay (2005-2008).
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REVIEW OF RELATED LITERATURE
Crime Occurrences
Seasonality of crime occurrences is valuable information to law enforcers
for crime prevention. It is important to police officers in their decision-making not
only for long reallocation of uniformed officers across precincts but also for short-
term targeting of patrols for hotspots and serial criminals (Cohen, J. et al., 2003).
As reported by Block (1984) and Wright (1996), property crimes peaked in the
months of June to December while violent crimes peaked on summer months.
Crime occurrences is a known fact that they may occur everywhere in
urban areas such as vandalism and destruction of public properties, robbery,
stealing and many others because of varied cultures and races of people in the area.
Hence, these are some issues that people should know for them to take precautions
for their personal safety such as staying away from high crime areas, transferring
to other neighborhood or avoiding walking alone at night if there is a perceived
danger on their lives. (Hoel, L. A., 1999).
Because of the seriousness and frequency of crime occurrences, the US
Federal Bureau of Investigation chose seven offenses to compromise a “Crime
Index” and serve as indicators of the nation’s crime experience. Crime Index
includes homicide, rape, robbery aggravated assault, burglary, theft/larceny, grand
theft auto and arson.
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Crime statistics showed that in 2007, a small increase in in violent crime
incidents from the previous year was noted although violent crimes were still down
by 27% from 2005. In 2007, incidents of larceny/theft increased by 25%, while
burglary dropped 9% and auto theft by 17%. In virtually every community
nationwide, larceny/theft represents the highest percentage.
In the Philippines, crime occurrences had gone up significantly. Crime
statistics reveal that the following crimes: theft, robbery and physical injuries were
prevalent in urban places like Metro Manila, Baguio City and other cities in the
country.
Is sub-urban areas like La Trinidad, Benguet, several crimes such as theft,
robbery and physical injuries were likewise committed in different degrees.
Thus, the researchers of this study were challenged to propose a study that
will determine the patterns of crime occurrences by year, season, and places in La
Trinidad, Benguet.
In finding the patterns of the occurrences of the different crimes, a
multivariate statistical technique known as Principal Component Analysis was
proposed to be used.
Application of Principal Component Analysis
Principal Component Analysis (PCA) was employed in ecologic studies
conducted by Horel,S.et.al., (2006) to determine a relationship between alcohol
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outlet densities, illicit drug use and violence. The study examined this relationship
using a sample of 439 census tracts. Neighborhood socio-cultural covariates,
alcohol outlet density, drug crime density and violent crime data were collected for
the year 2000. Four neighborhood explanatory variables were identified using
PCA. The best fitted model was selected as one considering both unstructured and
spatial dependence random effects. The results showed that drug law violation
explained a greater amount of variance in violent crime rates than alcohol outlet
densities. The analysis suggests that activity around illicit drug markets is more
strongly associated with violent crime than is alcohol density.
Cohen, J. et.al, (2003) used Principal Component Analysis in their study of
estimation of crime seasonality. It is a method of data reduction closely related to
factor analysis, to characterize the ecological structure of each spatial unit or place.
The dependent variables in their models of seasonality for each crime type are the
monthly crime counts recognizing that the spatial units in their analysis vary not
only in their seasonality but also in their relative overall levels of crimes, adding a
dummy variable for each spatial unit. Furthermore, the spatial unit dummies are
interacted with the time trend variables to allow each spatial unit to have a unique
time trend.
The concept of crime place is essential to crime pattern theory because the
characteristics of place influence the likelihood of a crime. According to Coomb et.
al., (1994), some areas are more prone to criminal activity than others. Motivation
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to perpetrate a crime tends to be person-specific, whereas opportunity tends to
relate more specifically to the characteristics of place (Eck & Weisburd, 1995). The
recognition of the concept of place in crime theory allows a new dimension to
implementing crime prevention.
Chen, D. (2000) identified aspects of the natural and built environment that
may be conducive to crime, and thereby will provide an independent determinant of
the local crime rate. The data set used includes large number of independent
variables listed under these headings of propensity and opportunity. And since each
variables listed under the headings has a high likelihood of being correlated, the
number of variables was reduced by means of Principal Component Analysis.
Using a varimax rotation with 25 iterations, the number of variables under
“propensity” was reduced to three components. The result revealed that crime
locations are not spatially random and that place characteristics influence the
decision to commit a crime. The study also showed that certain land use activities
are prone to criminal activity, including commercial and residential areas.
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THEORETICAL FRAMEWORK
Principal Component Analysis (PCA) is one of the simplest of the
multivariate methods. The object of the analysis is to take p variables X1, X2, … ,
Xp and find the combinations of these to produce indices Z1, Z2, … , Zp that are
uncorrelated.
The principal component analysis was first described by Karl Pearson
(1902). He apparently believed that this was the correct solution to some of the
problems that were of interest to biometricians at that time, although he did not
propose a practical method of calculation for more than two or three variables. He
introduced the principal component approach to parsimony and studied the problem
for the case of non-stochastic variables in a different context. The technique was
generalized by Hotelling (1933) to the case of stochastic variables. Hotelling
considered weighted sums of all the distinct random variable and attempted to find
a set of weights that would maximize the variance of the sum. The new variable
(weighted sum) is called the first principal component and oftentimes, its variance
is such a further analysis which hinge on total variance. In the other problems,
other weighted sums that are orthogonal to the first sum are also considered.
The object of the analysis is to take p variables X1, X2, … , Xp and find the
combinations of these to produce indices Z1, Z2, … , Zp that are uncorrelated. The
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lack of correlation is a useful property because it means that the indices are
measuring different ‘dimensions’ in the data.
There is always a hope that the variances of most of the indices will be as
low as to be negligible. In that case, the variation in the data set can be adequately
described by the Z variables with variances that are not negligible. Some degree of
economy is then achieved and the variation in the p original x variables is
accounted for by a smaller number of Z variables.
In general, the objectives of principal component analysis are (1) data
reduction and (2) interpretation.
A PCA starts with data on p variables observed from n individuals, as
shown in Table 1.
Table 1. Data format for principal component analysis.
Individuals X1 X2 . . . . Xp
1 X11 X12 . . . . X1p
2 X12 X22 . . . . X2p
. .
.
.
. .
.
.
. .
. .
n Xn1 Xn2 . . . . Xnp
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The first principal component which is the linear combination of the
variables X1, X2, ... , Xp, expressed as,
Z1 = a11X1 + a12X2 + ...+ a1pXp
usually accounts for the largest eigenvalue or maximum amount of variance of the
sample covariance matrix on the condition that the sum of squares of the coefficient
is equal to 1, that is
a211 + a212 +...+ a21p = 1.
Thus the variance of Z1, var (Z1), is as large as possible given this constraint
on the constants a1j. The constraint is possible given this is not done then the var
(Z1) can be increased by simply increasing any one of the a1j values. The second
principal component,
Z2 = a21X1 + a22X2 +...+ a2pXp,
is such that var(Z2) is as large as possible subject to the constraint that
a221 + a222 +...+ a22p = 1,
and also to the condition that Z1 and Z2 are uncorrelated. The third principal
component,
Z3 = a31X1 + a32X2 +...+ a3pXp,
is such that var(Z3) is as large as possible subject to the constraint that
a231 + a232+...+ a23p = 1,
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and also to the condition that Z3 is uncorrelated with Z2 and Z1. Further principal
components are defined by repeating the same process. If there are p variables then
there can be up to p principal components.
Principal component analysis just involves finding the eigenvalues of the
sample covariance matrix. The matrix is symmetric and has the form
c11
c12 … c1p
c21
c22 … c2p
.
.
.
.
.
.
.
.
.
cp1
cp2 … cpp
Where the diagonal element cii is the variance of Xi and cij is the covariance of
variables Xi and Xj.
The variances of the principal components are the eigenvalues of the matrix
C. There are p of these, some of which may be zero. Negative eigenvalues are not
possible for a covariance matrix. Assuming that the eigenvalues are ordered as
λ1 ≥ λ2 ≥ … ≥ λp ≥ 0,
then λ1 corresponds to the ith principal component.
Zi = ai1X1 + ai2X2 +...+ aipXp
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In particular, var (Zi) = λi and the constants ai1, ai2, …, aip are the elements
of the corresponding eigenvector, scaled so that a2i1 + a2i2 +…+ a2ip = 1.
An important property of the eigenvalues is that they add up to the sum of
the diagonal elements (the trace) of C. That is
λ1 + λ2 +…+ λp = c11 + c22 +…+ cpp.
If
cii is the variance of Xi and λi is the variance of Zi, this means that the
sum of the variances of the principal components is equal to the sum of the
variances of the original variables. Therefore, the principal components account for
all of the variation in the original data.
The matrix C then takes the form
1
c12 … c1p
c21
1
…
c2p
.
.
.
.
.
.
.
.
.
cp1
cp2
…
1
where
cij = cji is the correlation between Xi and Xj. In other words, the
principal component analysis is carried out on the correlation matrix. In that case,
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the sum of the diagonal terms, which is the sum of the eigenvalues, is equal to p,
the number of variables.
The steps in Principal Component Analysis
1) Start by coding the variables X1, X2, …, Xp to have zero means and unit
variances. This is usual, but is omitted in some cases.
2) Calculate the covariance matrix C. this is a correlation matrix if step 1
has been done.
3) Find the eigenvalues λ1, λ2, …, λp and the corresponding eigenvectors
д1, д2, …, дp. The coefficients of the ith principal component are then
given by дi while λi is its variance.
4) Discard any components that only accounts for a small proportion of the
variation in the data.
Hence, we shall also consider some patterned matrices and their principal
components. The component structure of a covariance or correlation matrix can
sometimes be approximated rather well by inspection of the elements and
knowledge of the characteristics roots and vectors of certain patterned matrices. We
shall now give the components of two such special matrices and a general upper
bound on the greatest characteristic root of any square matrix.
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Definition of Terms
Crime. An act that are legally forbidden by a society.
Correlation. The extent to which two or more things are related
(“correlated”) to one another.
Covariance. A measure of joint or (co-) variance of two or more variables.
Eigenvalues. A statistic used if factor analysis, canonical correlation
analysis and principal components analysis to indicate how much of the variation in
the original group of variables is accounted for by a particular factor.
Homicide. A person was killed and the accused had the intention to kill,
which is presumed.
Index Crimes/Felony. A serious crime for which punishment typically
ranges for more than a year’s imprisonment to death.
Murder. A person was killed and the killing was attended with treachery,
taking advantage of superior strength, with the aid of armed men, or employing
means to weaken the defense or of means or persons to ensure or afford impunity.
Non-Index Crimes/Misdemeanor. A minor crime that is typically punished
by less than a year in jail.
Orthogonal. Intersecting or lying at right angles. Uncorrelated variables are
said to be orthogonal because, when plotted on a graph, they form right angles to
one of the axes (if there is no variance in one of the variables).
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Principal Component Analysis. Methods for undertaking a linear
transformation of a large set correlated variables into a smaller group of
uncorrelated variables. This makes analysis easier by grouping data into more
manageable units and eliminating problems of multicollinearity.
Robbery. Unlawful taking of personal property with intent to gain by means
of violence against or intimidation of any person or force upon things.
Theft. Unlawful taking of personal property with intent to gain without the
consent of the owner without the use of violence against or intimidation of person
or force upon things.
Variance. A measure of the spread of scores in a distribution of scores, that
is, a measure of dispersion. The larger the variance, the further the individual scores
from the mean. The smaller the variance, the closer the individual scores to the
mean.
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METHODOLOGY
Data Source
Available data on crime occurrences by year, month and places were
obtained from the police records of La Trinidad Police District Office. Different
crimes committed by people and reported at the police station were included in this
study.
Data Analysis
The raw data gathered were utilized and was reduced to three tables, crime
occurrences by year, by month and by barangay. The data were transformed into
percentage to obtain uniformity. To determine the pattern of crimes, Principal
Component Analysis (PCA) was applied using the SPSS program.
Scatter plots of the different derived principal components by year, by
month and areas were produced to show the trend of the occurrences of the
different crimes.
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RESULTS AND DISCUSSION
Mean Number of Crime Occurrences
from 2004-2008
Table 2 shows the mean number of occurrences of the different crimes from
2004 to 2008 which ranged from about 2% to 33%. Theft was observed to have the
most number of occurrences (33%) followed by physical injuries and robbery
which have the same mean number of occurrences (28%). The least committed
crime from 2004 to 2008 is acts of lasciviousness having 2% mean number of
occurrence.
Table 2. Prevalence of Crimes committed by individuals from 2004 to 2008 in
La Trinidad, Benguet
Mean
Standard Error
95% Confidence Interval
Murder
2.80
0.66
0.96
4.64
Homicide
10.60
1.21
7.25
13.95
Rape
8.00
2.07
2.24
13.76
L.A.
2.20
1.02
-0.63
5.03
Physical Injuries
27.80
4.13
16.34
39.26
Robbery
27.80
3.02
19.41
36.19
Theft
33.40
5.71
17.53
49.27
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An examination of the pattern of occurrences of the different crimes from
2004 to 2008 reveals that the occurrences of murder and homicide and the
occurrences of robbery and theft were found significantly correlated. These results
suggest that murder and homicide had occurred in similar frequencies. Likewise
robbery and theft occurrences are in the same percentage. Since the occurrences of
the different crimes by year showed such pattern, then the application of principal
component to combine the crimes with similar occurrences is justified.
Table 3. Correlations among the different crime occurrences in La Trinidad,
Benguet from 2004 to 2008
Crimes
Homicide
Rape
Lascivious
Physical
Robbery
Theft
Acts
injuries
Murder
0.973**
-0.109 ns
-0.503 ns
-0.223 ns
-0.628 ns -0.629 ns
Homicide
-0.080 ns
-0.390 ns
-0.215 ns
-0.457 ns -0.458 ns
Rape
-0.497 ns
0.826 ns
0.375 ns
0.375 ns
Lascivious
-0.117 ns
0.603 ns
0.603 ns
Acts
Physical
0.512 ns
0.512 ns
Injuries
Robbery
1.000**
ns- not significant, p > .05
**- highly significant, p < .01
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Component loadings for the PCA Performed
on the Occurrences of Crimes in La Trinidad,
Benguet from 2004-2008
The Principal Component Analysis Performed on the data on the different
crimes committed in La Trinidad, Benguet from 2004 to 2008 yielded two principal
component models. The first principal component (PC1) accounted for about 54%
of the variations in the percent occurrences of the different crimes. The addition of
the second principal component yielded a model that covers about 83% of the
variations in the percent occurrences of the different crimes. The derived principal
component models are given as follows:
PC1 = -0.89(X1) - 0.78(X2) - 0.02(X3) + 0.80(X4) + 0.23(X5) + 0.84(X6)
+ 0.84(X7)
PC2 = -0.07(X1) - 0.05(X2) + 0.98(X3) - 0.428(X4) + 0.89(X5) + 0.40(X6)
+ 0.40(X7)
where:
X1- Murder; X2- Homicide; X3- Rape; X4- Lascivious Acts;
X5- Physical Injuries; X6- Robbery; X7- Theft
As shown in Table 4, the crimes that load heavily on the first principal
component are murder, homicide, acts of lasciviousness, robbery and theft. It is
further observed that the crimes that load highly on the first principal component
had a bipolar direction. Murder and homicide percent occurrences behave in a
negative direction while the occurrences of the acts of lasciviousness, robbery and
theft are in positive direction. This means that as murder and homicide occurrences
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increase, there are corresponding decrease in the occurrences of acts of
lasciviousness, robbery and theft.
The crimes that load in the second principal component are rape and
physical injuries with similar direction.
Table 4. Principal Component Loadings (unrotated, rotated) of the different
crimes committed by individuals from 2004-2008
UNROTATED ROTATED
COMPONENT
CRIMES
PC1
PC2
PC1
PC2
Murder (X1)
-0.847
0.276
-0.888
-0.072
Homicide (X2)
-0.741
0.261
-0.784
-0.045
Rape (X3)
0.365
0.912
-0.015
0.982
L.A. (X4)
0.576
-0.705
0.803
-0.428
Physical Injuries (X5)
0.552
0.737
0.226
0.893
Robbery (X6)
0.929
0.05
0.838
0.404
Theft (X7)
0.929
0.05
0.838
0.404
L.A.- Acts of Lasciviousness
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Graphically, Figure 1 shows the resulting two-dimensional map of
similarity measures. As shown in the figure, three clusters were formed which fall
in the 2nd, 3rd and 4th quadrant of the two-dimensional map. In year 2004-2005,
robbery and theft are prevalent. In year 2006 and 2008, murder and homicide are
prevalent. Rape and physical injuries were prevalent in year 2007.
2.0
2007
Rape and
Physical Injuries
1.5
1.0
.5
0.0
Robbery and
Theft
2005
2004
-.520082006
Murder and
2
Homicide
PC -1.0
-1.0
-.5
0.0
.5
1.0
1.5
PC1
Figure 1. A scatter plot of the new PC2 versus PC1
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Mean Number of Crime Occurrences from
January to December (2004-2007)
Table 5 shows that the mean number of occurrences of the different crimes
from January to December ranged from 0.75% to 11%. Theft was observed to have
the most number of occurrences (11%) followed by physical injuries and robbery.
The least committed crimes were murder and acts of lasciviousness.
Table 5. Prevalence of crimes committed by individual from January to
December (2004-2007)
Crimes
Mean
Standard
95% Confidence interval
Error
Murder
0.75
0.25
0.20
1.30
Homicide
3.17
0.53
1.99
4.34
Rape
2.92
0.43
1.96
3.87
L.A.
0.75
0.28
0.14
1.36
Physical Injuries
9.97
1.23
7.21
12.62
Robbery
9.58
0.99
7.41
11.76
Theft
11.00
1.39
7.95
14.05
*L.A. – Acts of Lasciviousness
Principal Component Analysis of Crime Occurrences
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Component Loadings for the PCA Performed
on the Occurrences of Crimes in La Trinidad,
Benguet, Monthly from 2004-2007
Performing Principal Component Analysis performed on the
monthly data of the different crimes committed in La Trinidad Benguet,
accumulated from 2004-2007, the analysis yielded three principal component
models as shown in Table 6. The first principal component (PC1) accounted for
about 29% of the variations in the percent occurrences of the different crimes. The
addition of the second principal component (PC2) yielded a model that covers
about 54% of the variations in the percent occurrences of the different crimes. And
the addition of the third principal component (PC3) produced a model that covers
about 70% of the variations in the occurrences of the different crimes.
The models are given as follows:
PC1 = - 0.68(X1) + 0.58(X2) - 0.001(X3) + 0.93(X4) + 0.42(X5) + 0.25(X6)
+ 0.05(X7)
PC2 = - 0.12(X1) - 0.26(X2) + 0.87(X3) + 0.10(X4) - 0.42(X5) + 0.72(X6)
- 0.16(X7)
PC3 = - 0.16(X1) - 0.68(X2) + 0.04(X3) - 0.14(X4) + 0.54(X5) - 0.26(X6)
+ 0.75(X7)
where: X1- Murder; X2- Homicide; X3- Rape; X4- Lascivious Acts;
X5- Physical Injuries; X6- Robbery; X7- Theft
Principal Component Analysis of Crime Occurrences
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Table 6. Principal Component Loadings (unrotated, rotated) of the different
crimes committed by individual from January to December
(2004-2007)
UNROTATED ROTATED
COMPONENT
CRIMES
PC1 PC2 PC3 PC1 PC2 PC3
Murder (X1)
-0.509
-0.356 -0.337
-0.678
-0.119
-0.158
Homicide(X2)
0.696
0.250
-0.568
0.576
-0.262
-0.684
Rape (X3)
0.295
-0.623
0.538
-1.08E-3
0.873 3.925E-2
L.A. (X4)
0.845
0.392
0.147
0.927
0.104
-0.139
P.I. (X5)
-7.51E-2 0.755
0.268
0.422
-0.422
0.540
Robbery (X6)
0.581
-0.486
0.260
0.246
0.715
-0.264
Theft (X7)
-0.374
0.438
0.509
5.487E-2
-0.158
0.750
*L.A. - Acts of Lasciviousness
*P.I.- Physical Injuries
The crimes that highly load in the first component are murder and acts of
lasciviousness in a bipolar direction. As shown in Figure 2, acts of lasciviousness
was prevalent in November while murder was prevalent in the months of February,
March, June, July and September.
Rape and robbery highly load in the second principal component and it was
observed that the two crimes were prevalent in the month of May (Figure 2).
Principal Component Analysis of Crime Occurrences
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3
May
2
Rape and Robbery
1
Oct
Aug
Jan
Dec
Apr
0
Sep
Jul
Mar
Nov
Murder
Acts of
-1
Jun
Lasciviousness
Feb
PC2 -2
-2
-1
0
1
2
3
PC1
Figure 2. A scatter plot of PC2 versus PC1
Theft, physical injuries and homicide are the crimes that highly load in the
third component in an opposite direction. Theft and physical injuries had high
occurrences in the first and last quarter of the year, respectively while homicide
was prevalent in the second quarter of the year (Figure 3).
Principal Component Analysis of Crime Occurrences
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2.0
Jan
Theft
1.5
Mar
Feb
1.0
.5
Aug
Dec
Sep
Oct
0.0
Physical Injuries
Jun
May
-.5
Homicide
Nov
-1.0
Apr
-1.5
Jul
PC3 -2.0
-2
-1
0
1
2
3
PC1
Figure 3. Scatter plot of PC3 versus PC1
Mean Number of Crime Occurrences in the
Different Barangays of La Trinidad (2005-2008)
The mean number of occurrences of the different crimes over the 16
barangays of La Trinidad, Benguet ranged from 0.25% to 6.81% (Table 7). The
Principal Component Analysis of Crime Occurrences
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27
most committed crime with 6.81% of occurrence was theft followed by robbery
then homicide. Malicious mischief was least committed.
Table 7. The Prevalence of the different crimes averaged over the 16 barangays
of La Trinidad, Benguet
Mean
Standard Err.
95% Confidence Interval
Homicide 4.81
1.43
1.77
7.86
Murder 1.50
0.41
0.63
2.37
Physical Injuries
3.19
1.03
1.00
5.37
Rape 1.56
0.39
0.74
2.39
Robbery 6.00
1.79
2.18
9.82
Theft 6.81
2.77
0.90
12.72
I.P.F.A. 1.00
0.39
0.17
1.83
Carnapping 0.38
0.20
-0.05
0.80
Illegal Drugs
0.19
0.14
-0.10
0.48
V.M.O. 1.63
1.14
-0.81
4.06
Women Abuse
1.88
0.56
0.68
3.07
Child Abuse
0.88
0.42
-0.01
1.76
Adultery 0.44
0.26
-0.11
0.99
Malicious Mischief
0.25
0.11
0.01
0.49
*IPFA- Illegal Possession of Fire Arms
*VMO- Violation of Municipal Ordinance
Principal Component Analysis of Crime Occurrences
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Component Loadings for the PCA Performed
on the Occurrences of Crimes in the Different
Barangays of La Trinidad, Benguet (2005-2008)
Performing a Principal Component Analysis on the data on the different
crimes committed in the different barangays of La Trinidad, Benguet (2005-2008),
two principal component models were extracted as shown in Table 8.
Table 8. Principal Component Loadings (unrotated, rotated) of the different
crimes committed in the different barangays of La Trinidad, Benguet
(2005-2008)
UNROTATED ROTATED
COMPONENT
CRIMES
PC1 PC2 PC1 PC2
Homicide (X1)
0.97
-0.13
0.83
0.53
Murder (X2)
0.82
0.21
0.49
0.68
Physical Injuries (X3)
0.83
-0.01
0.64
0.53
Rape (X4)
0.79
0.40
0.35
0.81
Robbery (X5)
0.94
-0.09
0.78
0.54
Theft (X6)
0.88
0.25
0.51
0.76
I.P.F.A. (X7)
0.83
-0.44
0.92
0.20
Carnapping (X8)
0.65
-0.63
0.90
-0.07
Illegal Drugs (X9)
0.50
0.71
-0.08
0.86
V.M.O. (X10)
0.95
-0.13
0.80
0.51
Women Abuse (X11)
0.95
-0.10
0.79
0.53
Child Abuse (X12)
0.91
-0.32
0.90
0.34
Adultery (X13)
0.78
0.46
0.30
0.86
Malicious Mischief (X14)
0.78
0.11
0.53
0.58
*IPFA- Illegal Possession of Fire Arms
*VMO- Violation of Municipal Ordinance
Principal Component Analysis of Crime Occurrences
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The first Principal Component (PC1) accounted for about 70% of the
variations in the percent occurrences of the different crimes. The addition of the
second Principal Component (PC2) produced a model that covers about 82% of the
variations in the percent occurrences of the different crimes. The extracted models
are shown below:
PC1 = 0.83(X1) + 0.49(X2) + 0.64(X3) + 0.35(X4) + 0.78(X5) + 0.51(X6)
+ 0.92(X7) + 0.90(X8) - 0.08(X9) + 0.80(X10) + 0.79(X11)
+ 0.90(X12) + 0.30(X13) + 0.53(X14)
PC2 = 0.53(X1) + 0.68(X2) + 0.53(X3) + 0.81(X4) + 0.54(X5) + 0.76(X6)
+ 0.20(X7) - 0.07(X8) + 0.86(X9) + 0.51(X10) + 0.53(X11) +0.34(X12)
+ 0.86(X13) + 0.58(X14)
where: X1- Homicide; X2- Murder; X3- Physical Injuries; X4- Rape; X5- Robbery;
X6- Theft; X7- Illegal Possession of Fire Arms; X8- Carnapping; X9- Illegal
Drugs; X10- Violation of Municipal Ordinance; X11- Women Abuse;
X12- Child Abuse; X13- Adultery; X14- Malicious Mischief
The crimes that load highly on the first principal component are homicide,
physical injuries, robbery, illegal possession of fire arms, carnapping, women abuse
and child abuse. The aforementioned crimes were committed with high frequency
of occurrence in Pico and Balili.
In the second principal component, the crimes that load highly are murder,
rape, theft, illegal drugs, adultery and maliscious mischief. The occurrences of the
Principal Component Analysis of Crime Occurrences
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crimes were prevalent in Pico and Betag. Other reported crimes were found to have
been less frequently committed in other barangays (Figure 4).
Principal Component Analysis of Crime Occurrences
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3
PICO
Theft
BETAG Murder Rape
Adultery
2
Illegal Drugs
Malicious Mischief
Homicide
Physical
Injuries
Robbery
Illegal
1
Possession of
Fire Arms
Carnapping
Women abuse
Child abuse
TAWANG
PUGUIS
AMBIONG
0
POBLACI
AL
WA
BE
AP N
C G
AN A
KEL
G L
CBA SH
RUZ
HO ILA
NG N
LUBAS
BINENG
2
ALNO
BALILI
PC -1
-2
-1
0
1
2
3
4
PC1
Figure 4. Scatter plot of the PC2 versus PC1
Principal Component Analysis of Crime Occurrences
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SUMMARY, CONCLUSION AND RECCOMMENDATION
Summary
Across year, the most prevalent crimes that were committed are theft,
robbery and physical injuries. Of the 7 crimes committed at La Trinidad Police
District Station, 5 crimes namely: murder, homicide, acts of lasciviousness, robbery,
and theft occurred with similar pattern from 2004 to 2008 and accounted 54% of the
variation in the percent occurrences. The second group of crimes which accounted
29% variability are rape and physical injury.
Looking across month, three crime clusters was noted. The first group
consisiting of acts of lasciviousness and murder were prevalent from February to
November. Rape and robbery, the second group of crimes were prevalent in the
month of May. The third group comprising theft, physical injury and homicide were
prevalent from January to December.
Barangay-wise, homicide, physical injuries, robbery, illegal possession of
fire arms, carnapping, women abuse and child abuse showed similar pattern of
occurrences in Balili and Pico. Murder, rape, theft, illegal drugs, adultery and
malicious mischief were also found to have similar pattern of occurrences in Pico
and Betag.
Principal Component Analysis of Crime Occurrences
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Conclusion
Based on the findings of this study, it is concluded that crime occurrences in
La Trinidad, Benguet demonstrate a pattern across year, month and place.
Across the year from 2004 to 2008, crime occurrences were prevalent in
2007.
Across the month, November had the most percentage of occurrences.
Pico, Balili and Betag are the 3 barangays with the most number of crime
occurrences.
Recommendation
The researchers recommend that there should be closer partnership between
local governments, civil society, local police and business groups. All sectors must
cooperate in order to prevent crime occurrences.
It is also recommended that security should be tightened maybe by [putting
up additional police outposts. Both local government and the police should strictly
impose laws. The barangay “tanods” should perform thei8r tasks.
To the civil society, they must be alert, precautious and must be responsible
enough.
And, using similar technique, a similar study may be conducted using more
variables and wider scope.
Principal Component Analysis of Crime Occurrences
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LITERARURE CITED
ANDERSON, ROBERT J. et.al. 1972. Sociology: The Study of Human
Relationships. Jovahovich, Inc. USA. Pp 387-393
AURELIO, JULIE M. 2007. Police Reports Dip in Metro Crime Rate. Inquirer.
CHEN, DONGMEI. 2000. Remote Sensing and Spatial Statistics as tools in Crime
Analysis. Kingston, Canada
COHEN, JENNYLYN et.al. 2003. Estimation of Crime Seasonality: A Cross-
Sectional Extension to Time Series Classical Decomposition. Pp. 2-3
HOEL, LESTER A. 1999. Public Transportation Security
HOREL, STEVEN et.al. 2006. Hierarchical Bayesian Spatial Models for Alcohol
Availability, Drug “hot spots” and Violent Crime
HORTON, PAUL B. et.al. 1988. The Sociology of Social Problems, 9th Edition.
Prentice-Hall, Inc. New Jersey, USA. Pg 77
KENDALL, DIANA. 2004. Sociology In Our Times: The Essentials, 4th Edition.
Wadsworth/ Thomson Learning Inc. USA. Pp 177, 188
MANLY, BRYAN F. J. 1994. Multivariate Statistical Methods, A Primer. St.
Edmundsbury Press. Great Britain. Pp 78-79
Principal Component Analysis of Crime Occurrences
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APPENDIX A
Request Letter to the Chief of Police
Benguet State University
College of Arts and Sciences
Math-Physics-Statistics Department
La Trinidad, Benguet
December 12, 2008
PCINSP MARIO L. MAYAMES JR
Chief of Police
La Trinidad Municipal Police Station
La Trinidad, Benguet
SIR:
We, the undersigned fourth year students taking up Bachelor of Science in Applied Statistics
at Benguet State University, are conducting a research entitled “Principal Component
Analysis of Crime Occurrences at la Trinidad, Benguet”.
In view hereof, we would like to request permission from your good office to gather crime
statistics by year, by month and by barangay for the past five years.
Thank you very much for your favorable consideration.
Respectfully yours,
(SGD) ALMERA L. CARIAS
(SGD) JINKY ROSE J. JUSTO
(SGD) LIZEL A. MAXIMINO
Researchers
Noted:
(SGD) DR. SALVACION Z. BELIGAN (SGD) DR. MA. AZUCENA B. LUBRICA
Thesis Adviser
Chairman, MPS Department
Approved:
(SGD) PROF. AUREA MARIE M. SANDOVAL
CAS Dean
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APPENDIX B
Crime Statistics by Year (2004-2008)
INDEX CRIMES
AGAINST PERSON
AGAINST PROPERTY
MURDER HOMICIDE RAPE
LA P.I.
ROBBERY
THEFT
YEAR
No. % No. % No. % No. % No. % No. % No. %
2004 2 14.29 9 16.98 5 12.50 5 45.45 32 23.02 31 22.3 54 32.34
2005 1 7.14 8 15.09 8 20.00 4 36.36 24 17.27 35 25.18 31 18.56
2006 3 21.43 10 18.87 6 15.00 -
- 21 15.11 18 12.95 20 11.98
2007 3 21.43 11 20.75 16 40.00 - - 42 30.22 31 22.3 27 16.17
2008 5 35.71 15 28.3 5 12.50 2 18.18 20 14.39 24 17.27 35 20.96
TOTAL
14 100.00 53 100.00 40 100.00 11 100.00 139 100.00 139 100.00 167 100.00
Principal Component Analysis of Crime Occurrences
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APPENDIX C
Total Variance of Crimes Committed by Individuals from 2004-2008
Rotation Sums of Squared
Initial Eigenvalues
Extraction Sums of Squared Loadings
Loading
%of
Cumulative
%of
%of
Cumulative
Component
Total
Variance
% Total
Variance Cumulative
% Total
Variance
%
1 3.763
53.756
53.756 3.763
53.756 53.756 3.504
50.062
50.062
2 2.020
28.857
82.612 2.020
28.857 82.612 2.279
32.55
82.612
3 0.965
13.784
96.396
4 0.252
3.604
100.000
5 2.32E-16
3.31E-15
100.000
6 -1.52E-16
-2.17E-15
100.000
7 -4.88E-16
-6.97E-15
100.000
Principal Component Analysis of Crime Occurrences
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APPENDIX D
Crime Statistics by Month
INDEX CRIMES
AGAINST PERSON
AGAINST PROPERTY
MURDER HOMICIDE
RAPE
LA P.I.
ROBBERY
THEFT
MONTH No.
%
No. % No.
% No.
% No. % No. % No. %
Jan - - 2 5.26 4 11.43 2 22.22 15 12.61 9 7.83 16 12.12
Feb - - 2 5.26 1 2.86 - - 15 12.61 5 4.35 13 9.85
Mar 1
11.11
1 2.63 1 2.86 - - 12 10.08 10 8.70 17 12.88
Apr 2
22.22
3 7.89 3 8.57 - - 8 6.72 10 8.70 4 3.03
May 1 11.11 2 5.26 5 14.29 1 11.11 5 4.20 18 15.65 9 6.82
Jun 2 22.22 4 10.53 1 2.86 - -
6 5.04 7 6.09 17 12.88
Jul - - 5 13.16 2 5.71 1 11.11 6 5.04 9 7.83 2 1.52
Aug - - 3 7.89 4 11.43 1 11.11 9 7.56 10 8.70 12 9.09
Sep 2
22.22
2 5.26 3 8.57 1 11.11 17 14.29 9 7.83 7 5.30
Oct - - 3 7.89 4 11.43 - - 6 5.04 11 9.57 12 9.09
Nov - - 8 21.05 2 5.71 3 33.33 13 10.92 12 10.43 11 8.33
Dec 1
11.11
3 7.89 5 14.29 - - 7 5.88 5 4.35 12 9.09
TOTAL
9 100.00 38 100.00 35 100.00 9 100.00 119 100.00 115 100.00 132 100.00
Principal Component Analysis of Crime Occurrences
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APPENDIX E
Total Variance of Crimes Committed Monthly from 2004-2007
Extraction Sums of Squared
Initial Eigenvalues
Loadings
Rotated Sums of Squared Loadings
% of
Cumulative
% of
Cumulative
% of
Cumulative
Component Total Variance
% Total
Variance
% Total
Variance
%
1 2.027 28.963
28.963 2.027
28.963
28.963 1.893
27.044
27.044
2 1.728 24.685
53.648 1.728
24.685
53.648
1.57
22.431
49.475
3 1.146 16.371
70.018 1.146
16.371
70.018 1.438
20.543
70.018
4 0.891 12.734
82.753
5 0.689 9.842
92.595
6 0.468 6.692
99.287
7
0.05
0.713
100
Principal Component Analysis of Crime Occurrences
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APPENDIX F
Crime Statistics by Barangay
Physical
Illegal Possession
Homicide Murder Injuries Rape Robbery
of Fire Arms
Theft
BARANGAY No. % No. % No. % No. % No. % No. % No. %
ALAPANG
1 1.30 - - - - 2 8.00 4 4.17 - - - -
ALNO
2 2.60 2 8.33 1 1.96 - - 1 1.04 - - 1 6.25
AMBIONG
2 2.60 1 4.17 2 3.92 1 4.00 4 4.17 1 0.92 2 12.50
BAHONG
2 2.60 1 4.17 - - 1 4.00 1 1.04 3 2.75 1 6.25
BALILI
17 22.08 3 12.50 8 15.69 2 8.00 23 23.96 11 10.09 5 31.25
BECKEL
3 3.90 3 12.50 1 1.96 - - 1 1.04 2 1.83 - -
BETAG
6 7.79 3 12.50 4 7.84 3 12.00 13 13.54 9 8.26 - -
BINENG
2 2.60 - - - - - - 1 1.04 1 0.92 - -
CRUZ
- - - - 1 1.96 1 4.00 3 3.13 2 1.83 - -
LUBAS
1 1.30 - - - - 1 4.00 2 2.08 2 1.83 1 6.25
PICO
20 25.97 6 25.00 13 25.49 6 24.00 22 22.92 45 41.28 4 25.00
POBLACION
8 10.39 - - 10 19.61 2 8.00 7 7.29 16 14.68 2 12.50
PUGUIS
4 5.19 1 4.17 3 5.88 3 12.00 2 2.08 7 6.42 - -
SHILAN
3 3.90 1 4.17 7 13.73 - - 7 7.29 3 2.75 - -
TAWANG
4 5.19 2 8.33 1 1.96 2 8.00 2 2.08 4 3.67 - -
WANGAL
2 2.60 1 4.17 - - 1 4.00 3 3.13 3 2.75 - -
TOTAL
77 100.00 24 100.00 51 100.00 25 100.00 96 100.00 109 100.00 16 100.00
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APPENDIX F Continued…
Viol. Of
Municipal
Women
Malicious
Carnapping
Illegal Drugs
Ordinance
Abuse Child
Abuse Adultery Mischief
TOTAL
BARANGAY No. % No. % No. % No. % No. % No. % No. %
ALAPANG
-
-
-
-
-
-
-
-
-
-
-
- - - 7
ALNO
1
16.67
-
-
-
-
1
3.33
-
-
-
- - - 9
AMBIONG
-
-
-
-
-
-
1
3.33
1
7.14
1
14.29 1 25.00 17
BAHONG
-
-
-
-
-
-
2
6.67
-
-
-
- - - 11
BALILI
3
50.00
-
-
10
38.46
7
23.33
5
35.71
-
- 1
25.00
95
BECKEL
-
-
-
-
-
-
1
3.33
-
-
-
- - - 11
BETAG
-
-
2
66.67
-
-
4
13.33
-
-
1
14.29 1 25.00 46
BINENG
-
-
-
-
-
-
-
-
-
-
-
- - - 4
CRUZ
-
-
-
-
-
-
1
3.33
-
-
-
- - - 8
LUBAS
-
-
-
-
-
-
-
-
-
-
-
- - - 7
PICO
1
16.67
1
33.33
16
61.54
7
23.33
5
35.71
4
57.14 1 25.00 151
POBLACION
-
-
-
-
-
-
2
6.67
-
-
-
- - - 47
PUGUIS
1
16.67
-
-
-
-
1
3.33
1
7.14
1
14.29 - -
24
SHILAN
-
-
-
-
-
-
2
6.67
1
7.14
-
14.29 - -
24
TAWANG
-
-
-
-
-
-
-
-
-
-
-
- - - 15
WANGAL
-
-
-
-
-
-
1
3.33
1
7.14
-
- - - 12
TOTAL
6
100.00 3 100.00
26
100.00
30 100.00
14 100.00
7 100.00
4 100.00
488
Principal Component Analysis of Crime Occurrences
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APPENDIX G
Total Variance of Crimes Committed by Individuals in the Different Barangays of La Trinidad,
Benguet (2005-2008)
Extracted Sums of Squared
Initial Eigenvalues
Loadings
Rotation Sums of Squared Loadings
% of
Cumulative
% of
Cumulative
% of
Cumulative
Component Total Variance
% Total
Variance
% Total
Variance
%
1 9.76 69.69
69.69
9.76
69.69 69.69
6.43
45.90
45.90
2 1.74 12.42
82.11
1.74
12.42 82.11
5.07
36.22
82.11
3
0.88
6.30
88.41
4
0.51
3.65
92.06
5
0.38
2.73
94.79
6
0.31
2.22
97.01
7
0.14
1.00
98.01
8
0.13
0.93
98.94
9
0.08
0.56
99.50
10
0.04
0.28
99.78
11
0.02
0.13
99.91
12
0.08
0.06
99.97
13
0.03
0.02
99.99
14
0.04
2.54E-03
100.00
Principal Component Analysis of Crime Occurrences
at La Trinidad, Benguet / Almera L. Carias; et al. 2009
ALMERA L. CARIAS, JINKY ROSE J. JUSTO and LIZEL A. MAXIMINO,
April 2009. Principal Component Analysis of Crime Occurrences at La Trinidad,
Benguet. Benguet State University, La Trinidad, Benguet.
Adviser : DR. SALVACION Z. BELIGAN
ABSTRACT
Specifically the study aimed to: 1) identify the different types of crimes
committed at the different barangays of La Trinidad, Benguet; 2) identify the barangay
that has the most number of crimes; 3) the month and year that has the most number of
crimes; and 4) the pattern of crime occurrences in La Trinidad, Benguet.
Across year, the most prevalent crimes that were committed are theft, robbery and
physical injuries. Of the 7 crimes committed at La Trinidad Police District Station, 5
crimes namely: murder, homicide, acts of lasciviousness, robbery, and theft occurred
with similar pattern from 2004 to 2008 and accounted 54% of the variation in the percent
occurrences. The second group of crimes which accounted 29% variability are rape and
physical injury.
Looking across month, three crime clusters was noted. The first group consisting
of acts of lasciviousness and murder were prevalent from February to November. Rape
and robbery, the second group of crimes were prevalent in the
month of May. The third group comprising theft, physical injury and homicide were
prevalent from January to December.
Barangay-wise, homicide, physical injuries, robbery, illegal possession of fire
arms, carnapping, women abuse and child abuse showed similar pattern of occurrences in
Balili and Pico. Murder, rape, theft, illegal drugs, adultery and malicious mischief were
also found to have similar pattern of occurrences in Pico and Betag.
ii
TABLE OF CONTENTS
Page
Bibliography
………………………………………………….
i
Abstract
………………………………………………….
i
Table of Contents
………………………………………………….
iii
INTRODUCTION
Background of the Study
………………………..……….... 1
Statement of the problem
………………………...………..
2
Objectives of the Study
………………………..…….…...
2
Significance of the Study
………………………..….……..
3
Scope and Delimitation of t he Study ……………….….…..….
3
REVIEW OF RELATED LITERATURE
Crime Occurrences
…………………………………. 4
Application of Principal Component Analysis ……………..
5
THEORETICAL FRAMEWORK
………………………………….
8
Definition of Terms
………………………………….. 14
METHODOLOGY
Data Source
…………………………………………….. 16
Data Analysis
…………………………………………...... 16
RESULTS AND DISCUSSION
Mean Number of Crime Occurrences
from 2004 to 2008
…………………………………….…. 17
iii
Component Loadings for PCA Performed
on the Occurrences of Crimes at La Trinidad,
Benguet from 2004 to 2008
………………………………….. 19
Mean Number of Crime Occurrences
by Month from 2004 to 2007 ………………………………….. 22
Component Loadings for PCA Performed
on the Occurrences of Crimes at La Trinidad,
Benguet Monthly from 2004 to 2007
…………………….... 23
Mean Number of Crime Occurrences in the
Different Barangays of La Trinidad, Benguet
from 2005 to 2008
……………………………………….. 26
Component Loadings for PCA Performed on
the Occurrences of Crimes in the Different
Barangays of La Trinidad, Benguet from
2005 to 2008
……………………………………..………. 28
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Summary
………………………………………………. 31
Conclusion
………………………………………………. 32
Recommendation ……………………………………………….. 32
LITERATURE CITED
………………………………………………. 33
APPENDICES
…………………………………………..………. 34
iv
1
INTRODUCTION
Background of the Study
Crimes are divided into felonies and misdemeanors which vary in
seriousness. A felony is a serious crime such as rape, homicide, or aggravated
assault for which punishment typically ranges for more than a year’s imprisonment
to death. A misdemeanor is a minor crime that is typically punished by less than a
year in jail. In either event, a fine maybe part of the sanction as well (Kendall,
2004).
Crimes are usually prevalent in areas inhabited by people with mixed
culture and with different values orientation as in the Philippines. Generally
speaking, crime occurrences are fewer in countries where there is a settled way of
life and a traditional respect for law.
Crime occurrences due to some factors such as economic, peer influence,
personality, culture and other factors are prevalent in urban and suburban areas
where business establishments are located.
Rampant robbery, stealing, rape and other unlawful actions are usually
prevalent in such areas where business and school transactions, and other social
activities are performed.
Because of the harm it may inflict on the innocent people victimized by
these criminal elements in the society, it is necessary that police authorities should
Principal Component Analysis of Crime Occurrences
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2
come up with a study to establish crime spots in all areas where business
transactions and other social activities are being performed.
With this establishment of crime patterns in La Trinidad areas, it can be
used by the government to come up with strategies that will prevent the
occurrences of such crimes.
Hence, the authors of this study are willing to help the police authorities in
establishing crime spots in La Trinidad area by Principal Component Analysis.
Statement of the Problem
The study sought to determine the most prevalent crimes committed by
people, crime hot spots, and the period when a particular crime is being committed.
Specifically, the study intended to answer the following questions:
1. What are the different types of crimes committed at the different
barangays of La Trinidad, Benguet?
2. What barangay has the most number of crimes committed?
3. What month and year has the most number of crimes committed?
4. Where and when the crime hot spots occurred?
Objectives of the Study
With the application of the Principal Component Analysis (PCA), the study
aimed to determine the areas and period of crime hot spots in La Trinidad, Benguet.
Specifically, the study aimed to identify the following:
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1. the different types of crimes committed at the different barangays of La
Trinidad, Benguet;
2. the barangay that has the most number of crimes;
3. the month and year that has the most number of crimes; and
4. the pattern of crime occurrences in La Trinidad, Benguet.
Significance of the Study
Results of this study may give information to the society, local government
and law enforcers about the crimes being committed in a particular place. It could
also provide crime patterns so that the law enforcers and policy makers could have
an idea on what actions to take to prevent and/or lessen the incidence of different
crimes.
Scope and Delimitations
The data collection was from December 2008-February 2009. The data
were gathered from the La Trinidad Police District Office. The data collected were
crime statistics by year (2004-2008), by month (2004-2007) and by place or
barangay (2005-2008).
Principal Component Analysis of Crime Occurrences
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4
REVIEW OF RELATED LITERATURE
Crime Occurrences
Seasonality of crime occurrences is valuable information to law enforcers
for crime prevention. It is important to police officers in their decision-making not
only for long reallocation of uniformed officers across precincts but also for short-
term targeting of patrols for hotspots and serial criminals (Cohen, J. et al., 2003).
As reported by Block (1984) and Wright (1996), property crimes peaked in the
months of June to December while violent crimes peaked on summer months.
Crime occurrences is a known fact that they may occur everywhere in
urban areas such as vandalism and destruction of public properties, robbery,
stealing and many others because of varied cultures and races of people in the area.
Hence, these are some issues that people should know for them to take precautions
for their personal safety such as staying away from high crime areas, transferring
to other neighborhood or avoiding walking alone at night if there is a perceived
danger on their lives. (Hoel, L. A., 1999).
Because of the seriousness and frequency of crime occurrences, the US
Federal Bureau of Investigation chose seven offenses to compromise a “Crime
Index” and serve as indicators of the nation’s crime experience. Crime Index
includes homicide, rape, robbery aggravated assault, burglary, theft/larceny, grand
theft auto and arson.
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Crime statistics showed that in 2007, a small increase in in violent crime
incidents from the previous year was noted although violent crimes were still down
by 27% from 2005. In 2007, incidents of larceny/theft increased by 25%, while
burglary dropped 9% and auto theft by 17%. In virtually every community
nationwide, larceny/theft represents the highest percentage.
In the Philippines, crime occurrences had gone up significantly. Crime
statistics reveal that the following crimes: theft, robbery and physical injuries were
prevalent in urban places like Metro Manila, Baguio City and other cities in the
country.
Is sub-urban areas like La Trinidad, Benguet, several crimes such as theft,
robbery and physical injuries were likewise committed in different degrees.
Thus, the researchers of this study were challenged to propose a study that
will determine the patterns of crime occurrences by year, season, and places in La
Trinidad, Benguet.
In finding the patterns of the occurrences of the different crimes, a
multivariate statistical technique known as Principal Component Analysis was
proposed to be used.
Application of Principal Component Analysis
Principal Component Analysis (PCA) was employed in ecologic studies
conducted by Horel,S.et.al., (2006) to determine a relationship between alcohol
Principal Component Analysis of Crime Occurrences
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6
outlet densities, illicit drug use and violence. The study examined this relationship
using a sample of 439 census tracts. Neighborhood socio-cultural covariates,
alcohol outlet density, drug crime density and violent crime data were collected for
the year 2000. Four neighborhood explanatory variables were identified using
PCA. The best fitted model was selected as one considering both unstructured and
spatial dependence random effects. The results showed that drug law violation
explained a greater amount of variance in violent crime rates than alcohol outlet
densities. The analysis suggests that activity around illicit drug markets is more
strongly associated with violent crime than is alcohol density.
Cohen, J. et.al, (2003) used Principal Component Analysis in their study of
estimation of crime seasonality. It is a method of data reduction closely related to
factor analysis, to characterize the ecological structure of each spatial unit or place.
The dependent variables in their models of seasonality for each crime type are the
monthly crime counts recognizing that the spatial units in their analysis vary not
only in their seasonality but also in their relative overall levels of crimes, adding a
dummy variable for each spatial unit. Furthermore, the spatial unit dummies are
interacted with the time trend variables to allow each spatial unit to have a unique
time trend.
The concept of crime place is essential to crime pattern theory because the
characteristics of place influence the likelihood of a crime. According to Coomb et.
al., (1994), some areas are more prone to criminal activity than others. Motivation
Principal Component Analysis of Crime Occurrences
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7
to perpetrate a crime tends to be person-specific, whereas opportunity tends to
relate more specifically to the characteristics of place (Eck & Weisburd, 1995). The
recognition of the concept of place in crime theory allows a new dimension to
implementing crime prevention.
Chen, D. (2000) identified aspects of the natural and built environment that
may be conducive to crime, and thereby will provide an independent determinant of
the local crime rate. The data set used includes large number of independent
variables listed under these headings of propensity and opportunity. And since each
variables listed under the headings has a high likelihood of being correlated, the
number of variables was reduced by means of Principal Component Analysis.
Using a varimax rotation with 25 iterations, the number of variables under
“propensity” was reduced to three components. The result revealed that crime
locations are not spatially random and that place characteristics influence the
decision to commit a crime. The study also showed that certain land use activities
are prone to criminal activity, including commercial and residential areas.
Principal Component Analysis of Crime Occurrences
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8
THEORETICAL FRAMEWORK
Principal Component Analysis (PCA) is one of the simplest of the
multivariate methods. The object of the analysis is to take p variables X1, X2, … ,
Xp and find the combinations of these to produce indices Z1, Z2, … , Zp that are
uncorrelated.
The principal component analysis was first described by Karl Pearson
(1902). He apparently believed that this was the correct solution to some of the
problems that were of interest to biometricians at that time, although he did not
propose a practical method of calculation for more than two or three variables. He
introduced the principal component approach to parsimony and studied the problem
for the case of non-stochastic variables in a different context. The technique was
generalized by Hotelling (1933) to the case of stochastic variables. Hotelling
considered weighted sums of all the distinct random variable and attempted to find
a set of weights that would maximize the variance of the sum. The new variable
(weighted sum) is called the first principal component and oftentimes, its variance
is such a further analysis which hinge on total variance. In the other problems,
other weighted sums that are orthogonal to the first sum are also considered.
The object of the analysis is to take p variables X1, X2, … , Xp and find the
combinations of these to produce indices Z1, Z2, … , Zp that are uncorrelated. The
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lack of correlation is a useful property because it means that the indices are
measuring different ‘dimensions’ in the data.
There is always a hope that the variances of most of the indices will be as
low as to be negligible. In that case, the variation in the data set can be adequately
described by the Z variables with variances that are not negligible. Some degree of
economy is then achieved and the variation in the p original x variables is
accounted for by a smaller number of Z variables.
In general, the objectives of principal component analysis are (1) data
reduction and (2) interpretation.
A PCA starts with data on p variables observed from n individuals, as
shown in Table 1.
Table 1. Data format for principal component analysis.
Individuals X1 X2 . . . . Xp
1 X11 X12 . . . . X1p
2 X12 X22 . . . . X2p
. .
.
.
. .
.
.
. .
. .
n Xn1 Xn2 . . . . Xnp
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The first principal component which is the linear combination of the
variables X1, X2, ... , Xp, expressed as,
Z1 = a11X1 + a12X2 + ...+ a1pXp
usually accounts for the largest eigenvalue or maximum amount of variance of the
sample covariance matrix on the condition that the sum of squares of the coefficient
is equal to 1, that is
a211 + a212 +...+ a21p = 1.
Thus the variance of Z1, var (Z1), is as large as possible given this constraint
on the constants a1j. The constraint is possible given this is not done then the var
(Z1) can be increased by simply increasing any one of the a1j values. The second
principal component,
Z2 = a21X1 + a22X2 +...+ a2pXp,
is such that var(Z2) is as large as possible subject to the constraint that
a221 + a222 +...+ a22p = 1,
and also to the condition that Z1 and Z2 are uncorrelated. The third principal
component,
Z3 = a31X1 + a32X2 +...+ a3pXp,
is such that var(Z3) is as large as possible subject to the constraint that
a231 + a232+...+ a23p = 1,
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and also to the condition that Z3 is uncorrelated with Z2 and Z1. Further principal
components are defined by repeating the same process. If there are p variables then
there can be up to p principal components.
Principal component analysis just involves finding the eigenvalues of the
sample covariance matrix. The matrix is symmetric and has the form
c11
c12 … c1p
c21
c22 … c2p
.
.
.
.
.
.
.
.
.
cp1
cp2 … cpp
Where the diagonal element cii is the variance of Xi and cij is the covariance of
variables Xi and Xj.
The variances of the principal components are the eigenvalues of the matrix
C. There are p of these, some of which may be zero. Negative eigenvalues are not
possible for a covariance matrix. Assuming that the eigenvalues are ordered as
λ1 ≥ λ2 ≥ … ≥ λp ≥ 0,
then λ1 corresponds to the ith principal component.
Zi = ai1X1 + ai2X2 +...+ aipXp
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In particular, var (Zi) = λi and the constants ai1, ai2, …, aip are the elements
of the corresponding eigenvector, scaled so that a2i1 + a2i2 +…+ a2ip = 1.
An important property of the eigenvalues is that they add up to the sum of
the diagonal elements (the trace) of C. That is
λ1 + λ2 +…+ λp = c11 + c22 +…+ cpp.
If
cii is the variance of Xi and λi is the variance of Zi, this means that the
sum of the variances of the principal components is equal to the sum of the
variances of the original variables. Therefore, the principal components account for
all of the variation in the original data.
The matrix C then takes the form
1
c12 … c1p
c21
1
…
c2p
.
.
.
.
.
.
.
.
.
cp1
cp2
…
1
where
cij = cji is the correlation between Xi and Xj. In other words, the
principal component analysis is carried out on the correlation matrix. In that case,
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the sum of the diagonal terms, which is the sum of the eigenvalues, is equal to p,
the number of variables.
The steps in Principal Component Analysis
1) Start by coding the variables X1, X2, …, Xp to have zero means and unit
variances. This is usual, but is omitted in some cases.
2) Calculate the covariance matrix C. this is a correlation matrix if step 1
has been done.
3) Find the eigenvalues λ1, λ2, …, λp and the corresponding eigenvectors
д1, д2, …, дp. The coefficients of the ith principal component are then
given by дi while λi is its variance.
4) Discard any components that only accounts for a small proportion of the
variation in the data.
Hence, we shall also consider some patterned matrices and their principal
components. The component structure of a covariance or correlation matrix can
sometimes be approximated rather well by inspection of the elements and
knowledge of the characteristics roots and vectors of certain patterned matrices. We
shall now give the components of two such special matrices and a general upper
bound on the greatest characteristic root of any square matrix.
Principal Component Analysis of Crime Occurrences
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Definition of Terms
Crime. An act that are legally forbidden by a society.
Correlation. The extent to which two or more things are related
(“correlated”) to one another.
Covariance. A measure of joint or (co-) variance of two or more variables.
Eigenvalues. A statistic used if factor analysis, canonical correlation
analysis and principal components analysis to indicate how much of the variation in
the original group of variables is accounted for by a particular factor.
Homicide. A person was killed and the accused had the intention to kill,
which is presumed.
Index Crimes/Felony. A serious crime for which punishment typically
ranges for more than a year’s imprisonment to death.
Murder. A person was killed and the killing was attended with treachery,
taking advantage of superior strength, with the aid of armed men, or employing
means to weaken the defense or of means or persons to ensure or afford impunity.
Non-Index Crimes/Misdemeanor. A minor crime that is typically punished
by less than a year in jail.
Orthogonal. Intersecting or lying at right angles. Uncorrelated variables are
said to be orthogonal because, when plotted on a graph, they form right angles to
one of the axes (if there is no variance in one of the variables).
Principal Component Analysis of Crime Occurrences
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Principal Component Analysis. Methods for undertaking a linear
transformation of a large set correlated variables into a smaller group of
uncorrelated variables. This makes analysis easier by grouping data into more
manageable units and eliminating problems of multicollinearity.
Robbery. Unlawful taking of personal property with intent to gain by means
of violence against or intimidation of any person or force upon things.
Theft. Unlawful taking of personal property with intent to gain without the
consent of the owner without the use of violence against or intimidation of person
or force upon things.
Variance. A measure of the spread of scores in a distribution of scores, that
is, a measure of dispersion. The larger the variance, the further the individual scores
from the mean. The smaller the variance, the closer the individual scores to the
mean.
Principal Component Analysis of Crime Occurrences
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METHODOLOGY
Data Source
Available data on crime occurrences by year, month and places were
obtained from the police records of La Trinidad Police District Office. Different
crimes committed by people and reported at the police station were included in this
study.
Data Analysis
The raw data gathered were utilized and was reduced to three tables, crime
occurrences by year, by month and by barangay. The data were transformed into
percentage to obtain uniformity. To determine the pattern of crimes, Principal
Component Analysis (PCA) was applied using the SPSS program.
Scatter plots of the different derived principal components by year, by
month and areas were produced to show the trend of the occurrences of the
different crimes.
Principal Component Analysis of Crime Occurrences
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RESULTS AND DISCUSSION
Mean Number of Crime Occurrences
from 2004-2008
Table 2 shows the mean number of occurrences of the different crimes from
2004 to 2008 which ranged from about 2% to 33%. Theft was observed to have the
most number of occurrences (33%) followed by physical injuries and robbery
which have the same mean number of occurrences (28%). The least committed
crime from 2004 to 2008 is acts of lasciviousness having 2% mean number of
occurrence.
Table 2. Prevalence of Crimes committed by individuals from 2004 to 2008 in
La Trinidad, Benguet
Mean
Standard Error
95% Confidence Interval
Murder
2.80
0.66
0.96
4.64
Homicide
10.60
1.21
7.25
13.95
Rape
8.00
2.07
2.24
13.76
L.A.
2.20
1.02
-0.63
5.03
Physical Injuries
27.80
4.13
16.34
39.26
Robbery
27.80
3.02
19.41
36.19
Theft
33.40
5.71
17.53
49.27
Principal Component Analysis of Crime Occurrences
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An examination of the pattern of occurrences of the different crimes from
2004 to 2008 reveals that the occurrences of murder and homicide and the
occurrences of robbery and theft were found significantly correlated. These results
suggest that murder and homicide had occurred in similar frequencies. Likewise
robbery and theft occurrences are in the same percentage. Since the occurrences of
the different crimes by year showed such pattern, then the application of principal
component to combine the crimes with similar occurrences is justified.
Table 3. Correlations among the different crime occurrences in La Trinidad,
Benguet from 2004 to 2008
Crimes
Homicide
Rape
Lascivious
Physical
Robbery
Theft
Acts
injuries
Murder
0.973**
-0.109 ns
-0.503 ns
-0.223 ns
-0.628 ns -0.629 ns
Homicide
-0.080 ns
-0.390 ns
-0.215 ns
-0.457 ns -0.458 ns
Rape
-0.497 ns
0.826 ns
0.375 ns
0.375 ns
Lascivious
-0.117 ns
0.603 ns
0.603 ns
Acts
Physical
0.512 ns
0.512 ns
Injuries
Robbery
1.000**
ns- not significant, p > .05
**- highly significant, p < .01
Principal Component Analysis of Crime Occurrences
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Component loadings for the PCA Performed
on the Occurrences of Crimes in La Trinidad,
Benguet from 2004-2008
The Principal Component Analysis Performed on the data on the different
crimes committed in La Trinidad, Benguet from 2004 to 2008 yielded two principal
component models. The first principal component (PC1) accounted for about 54%
of the variations in the percent occurrences of the different crimes. The addition of
the second principal component yielded a model that covers about 83% of the
variations in the percent occurrences of the different crimes. The derived principal
component models are given as follows:
PC1 = -0.89(X1) - 0.78(X2) - 0.02(X3) + 0.80(X4) + 0.23(X5) + 0.84(X6)
+ 0.84(X7)
PC2 = -0.07(X1) - 0.05(X2) + 0.98(X3) - 0.428(X4) + 0.89(X5) + 0.40(X6)
+ 0.40(X7)
where:
X1- Murder; X2- Homicide; X3- Rape; X4- Lascivious Acts;
X5- Physical Injuries; X6- Robbery; X7- Theft
As shown in Table 4, the crimes that load heavily on the first principal
component are murder, homicide, acts of lasciviousness, robbery and theft. It is
further observed that the crimes that load highly on the first principal component
had a bipolar direction. Murder and homicide percent occurrences behave in a
negative direction while the occurrences of the acts of lasciviousness, robbery and
theft are in positive direction. This means that as murder and homicide occurrences
Principal Component Analysis of Crime Occurrences
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increase, there are corresponding decrease in the occurrences of acts of
lasciviousness, robbery and theft.
The crimes that load in the second principal component are rape and
physical injuries with similar direction.
Table 4. Principal Component Loadings (unrotated, rotated) of the different
crimes committed by individuals from 2004-2008
UNROTATED ROTATED
COMPONENT
CRIMES
PC1
PC2
PC1
PC2
Murder (X1)
-0.847
0.276
-0.888
-0.072
Homicide (X2)
-0.741
0.261
-0.784
-0.045
Rape (X3)
0.365
0.912
-0.015
0.982
L.A. (X4)
0.576
-0.705
0.803
-0.428
Physical Injuries (X5)
0.552
0.737
0.226
0.893
Robbery (X6)
0.929
0.05
0.838
0.404
Theft (X7)
0.929
0.05
0.838
0.404
L.A.- Acts of Lasciviousness
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Graphically, Figure 1 shows the resulting two-dimensional map of
similarity measures. As shown in the figure, three clusters were formed which fall
in the 2nd, 3rd and 4th quadrant of the two-dimensional map. In year 2004-2005,
robbery and theft are prevalent. In year 2006 and 2008, murder and homicide are
prevalent. Rape and physical injuries were prevalent in year 2007.
2.0
2007
Rape and
Physical Injuries
1.5
1.0
.5
0.0
Robbery and
Theft
2005
2004
-.520082006
Murder and
2
Homicide
PC -1.0
-1.0
-.5
0.0
.5
1.0
1.5
PC1
Figure 1. A scatter plot of the new PC2 versus PC1
Principal Component Analysis of Crime Occurrences
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Mean Number of Crime Occurrences from
January to December (2004-2007)
Table 5 shows that the mean number of occurrences of the different crimes
from January to December ranged from 0.75% to 11%. Theft was observed to have
the most number of occurrences (11%) followed by physical injuries and robbery.
The least committed crimes were murder and acts of lasciviousness.
Table 5. Prevalence of crimes committed by individual from January to
December (2004-2007)
Crimes
Mean
Standard
95% Confidence interval
Error
Murder
0.75
0.25
0.20
1.30
Homicide
3.17
0.53
1.99
4.34
Rape
2.92
0.43
1.96
3.87
L.A.
0.75
0.28
0.14
1.36
Physical Injuries
9.97
1.23
7.21
12.62
Robbery
9.58
0.99
7.41
11.76
Theft
11.00
1.39
7.95
14.05
*L.A. – Acts of Lasciviousness
Principal Component Analysis of Crime Occurrences
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Component Loadings for the PCA Performed
on the Occurrences of Crimes in La Trinidad,
Benguet, Monthly from 2004-2007
Performing Principal Component Analysis performed on the
monthly data of the different crimes committed in La Trinidad Benguet,
accumulated from 2004-2007, the analysis yielded three principal component
models as shown in Table 6. The first principal component (PC1) accounted for
about 29% of the variations in the percent occurrences of the different crimes. The
addition of the second principal component (PC2) yielded a model that covers
about 54% of the variations in the percent occurrences of the different crimes. And
the addition of the third principal component (PC3) produced a model that covers
about 70% of the variations in the occurrences of the different crimes.
The models are given as follows:
PC1 = - 0.68(X1) + 0.58(X2) - 0.001(X3) + 0.93(X4) + 0.42(X5) + 0.25(X6)
+ 0.05(X7)
PC2 = - 0.12(X1) - 0.26(X2) + 0.87(X3) + 0.10(X4) - 0.42(X5) + 0.72(X6)
- 0.16(X7)
PC3 = - 0.16(X1) - 0.68(X2) + 0.04(X3) - 0.14(X4) + 0.54(X5) - 0.26(X6)
+ 0.75(X7)
where: X1- Murder; X2- Homicide; X3- Rape; X4- Lascivious Acts;
X5- Physical Injuries; X6- Robbery; X7- Theft
Principal Component Analysis of Crime Occurrences
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Table 6. Principal Component Loadings (unrotated, rotated) of the different
crimes committed by individual from January to December
(2004-2007)
UNROTATED ROTATED
COMPONENT
CRIMES
PC1 PC2 PC3 PC1 PC2 PC3
Murder (X1)
-0.509
-0.356 -0.337
-0.678
-0.119
-0.158
Homicide(X2)
0.696
0.250
-0.568
0.576
-0.262
-0.684
Rape (X3)
0.295
-0.623
0.538
-1.08E-3
0.873 3.925E-2
L.A. (X4)
0.845
0.392
0.147
0.927
0.104
-0.139
P.I. (X5)
-7.51E-2 0.755
0.268
0.422
-0.422
0.540
Robbery (X6)
0.581
-0.486
0.260
0.246
0.715
-0.264
Theft (X7)
-0.374
0.438
0.509
5.487E-2
-0.158
0.750
*L.A. - Acts of Lasciviousness
*P.I.- Physical Injuries
The crimes that highly load in the first component are murder and acts of
lasciviousness in a bipolar direction. As shown in Figure 2, acts of lasciviousness
was prevalent in November while murder was prevalent in the months of February,
March, June, July and September.
Rape and robbery highly load in the second principal component and it was
observed that the two crimes were prevalent in the month of May (Figure 2).
Principal Component Analysis of Crime Occurrences
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3
May
2
Rape and Robbery
1
Oct
Aug
Jan
Dec
Apr
0
Sep
Jul
Mar
Nov
Murder
Acts of
-1
Jun
Lasciviousness
Feb
PC2 -2
-2
-1
0
1
2
3
PC1
Figure 2. A scatter plot of PC2 versus PC1
Theft, physical injuries and homicide are the crimes that highly load in the
third component in an opposite direction. Theft and physical injuries had high
occurrences in the first and last quarter of the year, respectively while homicide
was prevalent in the second quarter of the year (Figure 3).
Principal Component Analysis of Crime Occurrences
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2.0
Jan
Theft
1.5
Mar
Feb
1.0
.5
Aug
Dec
Sep
Oct
0.0
Physical Injuries
Jun
May
-.5
Homicide
Nov
-1.0
Apr
-1.5
Jul
PC3 -2.0
-2
-1
0
1
2
3
PC1
Figure 3. Scatter plot of PC3 versus PC1
Mean Number of Crime Occurrences in the
Different Barangays of La Trinidad (2005-2008)
The mean number of occurrences of the different crimes over the 16
barangays of La Trinidad, Benguet ranged from 0.25% to 6.81% (Table 7). The
Principal Component Analysis of Crime Occurrences
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27
most committed crime with 6.81% of occurrence was theft followed by robbery
then homicide. Malicious mischief was least committed.
Table 7. The Prevalence of the different crimes averaged over the 16 barangays
of La Trinidad, Benguet
Mean
Standard Err.
95% Confidence Interval
Homicide 4.81
1.43
1.77
7.86
Murder 1.50
0.41
0.63
2.37
Physical Injuries
3.19
1.03
1.00
5.37
Rape 1.56
0.39
0.74
2.39
Robbery 6.00
1.79
2.18
9.82
Theft 6.81
2.77
0.90
12.72
I.P.F.A. 1.00
0.39
0.17
1.83
Carnapping 0.38
0.20
-0.05
0.80
Illegal Drugs
0.19
0.14
-0.10
0.48
V.M.O. 1.63
1.14
-0.81
4.06
Women Abuse
1.88
0.56
0.68
3.07
Child Abuse
0.88
0.42
-0.01
1.76
Adultery 0.44
0.26
-0.11
0.99
Malicious Mischief
0.25
0.11
0.01
0.49
*IPFA- Illegal Possession of Fire Arms
*VMO- Violation of Municipal Ordinance
Principal Component Analysis of Crime Occurrences
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Component Loadings for the PCA Performed
on the Occurrences of Crimes in the Different
Barangays of La Trinidad, Benguet (2005-2008)
Performing a Principal Component Analysis on the data on the different
crimes committed in the different barangays of La Trinidad, Benguet (2005-2008),
two principal component models were extracted as shown in Table 8.
Table 8. Principal Component Loadings (unrotated, rotated) of the different
crimes committed in the different barangays of La Trinidad, Benguet
(2005-2008)
UNROTATED ROTATED
COMPONENT
CRIMES
PC1 PC2 PC1 PC2
Homicide (X1)
0.97
-0.13
0.83
0.53
Murder (X2)
0.82
0.21
0.49
0.68
Physical Injuries (X3)
0.83
-0.01
0.64
0.53
Rape (X4)
0.79
0.40
0.35
0.81
Robbery (X5)
0.94
-0.09
0.78
0.54
Theft (X6)
0.88
0.25
0.51
0.76
I.P.F.A. (X7)
0.83
-0.44
0.92
0.20
Carnapping (X8)
0.65
-0.63
0.90
-0.07
Illegal Drugs (X9)
0.50
0.71
-0.08
0.86
V.M.O. (X10)
0.95
-0.13
0.80
0.51
Women Abuse (X11)
0.95
-0.10
0.79
0.53
Child Abuse (X12)
0.91
-0.32
0.90
0.34
Adultery (X13)
0.78
0.46
0.30
0.86
Malicious Mischief (X14)
0.78
0.11
0.53
0.58
*IPFA- Illegal Possession of Fire Arms
*VMO- Violation of Municipal Ordinance
Principal Component Analysis of Crime Occurrences
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The first Principal Component (PC1) accounted for about 70% of the
variations in the percent occurrences of the different crimes. The addition of the
second Principal Component (PC2) produced a model that covers about 82% of the
variations in the percent occurrences of the different crimes. The extracted models
are shown below:
PC1 = 0.83(X1) + 0.49(X2) + 0.64(X3) + 0.35(X4) + 0.78(X5) + 0.51(X6)
+ 0.92(X7) + 0.90(X8) - 0.08(X9) + 0.80(X10) + 0.79(X11)
+ 0.90(X12) + 0.30(X13) + 0.53(X14)
PC2 = 0.53(X1) + 0.68(X2) + 0.53(X3) + 0.81(X4) + 0.54(X5) + 0.76(X6)
+ 0.20(X7) - 0.07(X8) + 0.86(X9) + 0.51(X10) + 0.53(X11) +0.34(X12)
+ 0.86(X13) + 0.58(X14)
where: X1- Homicide; X2- Murder; X3- Physical Injuries; X4- Rape; X5- Robbery;
X6- Theft; X7- Illegal Possession of Fire Arms; X8- Carnapping; X9- Illegal
Drugs; X10- Violation of Municipal Ordinance; X11- Women Abuse;
X12- Child Abuse; X13- Adultery; X14- Malicious Mischief
The crimes that load highly on the first principal component are homicide,
physical injuries, robbery, illegal possession of fire arms, carnapping, women abuse
and child abuse. The aforementioned crimes were committed with high frequency
of occurrence in Pico and Balili.
In the second principal component, the crimes that load highly are murder,
rape, theft, illegal drugs, adultery and maliscious mischief. The occurrences of the
Principal Component Analysis of Crime Occurrences
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crimes were prevalent in Pico and Betag. Other reported crimes were found to have
been less frequently committed in other barangays (Figure 4).
Principal Component Analysis of Crime Occurrences
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31
3
PICO
Theft
BETAG Murder Rape
Adultery
2
Illegal Drugs
Malicious Mischief
Homicide
Physical
Injuries
Robbery
Illegal
1
Possession of
Fire Arms
Carnapping
Women abuse
Child abuse
TAWANG
PUGUIS
AMBIONG
0
POBLACI
AL
WA
BE
AP N
C G
AN A
KEL
G L
CBA SH
RUZ
HO ILA
NG N
LUBAS
BINENG
2
ALNO
BALILI
PC -1
-2
-1
0
1
2
3
4
PC1
Figure 4. Scatter plot of the PC2 versus PC1
Principal Component Analysis of Crime Occurrences
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SUMMARY, CONCLUSION AND RECCOMMENDATION
Summary
Across year, the most prevalent crimes that were committed are theft,
robbery and physical injuries. Of the 7 crimes committed at La Trinidad Police
District Station, 5 crimes namely: murder, homicide, acts of lasciviousness, robbery,
and theft occurred with similar pattern from 2004 to 2008 and accounted 54% of the
variation in the percent occurrences. The second group of crimes which accounted
29% variability are rape and physical injury.
Looking across month, three crime clusters was noted. The first group
consisiting of acts of lasciviousness and murder were prevalent from February to
November. Rape and robbery, the second group of crimes were prevalent in the
month of May. The third group comprising theft, physical injury and homicide were
prevalent from January to December.
Barangay-wise, homicide, physical injuries, robbery, illegal possession of
fire arms, carnapping, women abuse and child abuse showed similar pattern of
occurrences in Balili and Pico. Murder, rape, theft, illegal drugs, adultery and
malicious mischief were also found to have similar pattern of occurrences in Pico
and Betag.
Principal Component Analysis of Crime Occurrences
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Conclusion
Based on the findings of this study, it is concluded that crime occurrences in
La Trinidad, Benguet demonstrate a pattern across year, month and place.
Across the year from 2004 to 2008, crime occurrences were prevalent in
2007.
Across the month, November had the most percentage of occurrences.
Pico, Balili and Betag are the 3 barangays with the most number of crime
occurrences.
Recommendation
The researchers recommend that there should be closer partnership between
local governments, civil society, local police and business groups. All sectors must
cooperate in order to prevent crime occurrences.
It is also recommended that security should be tightened maybe by [putting
up additional police outposts. Both local government and the police should strictly
impose laws. The barangay “tanods” should perform thei8r tasks.
To the civil society, they must be alert, precautious and must be responsible
enough.
And, using similar technique, a similar study may be conducted using more
variables and wider scope.
Principal Component Analysis of Crime Occurrences
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LITERARURE CITED
ANDERSON, ROBERT J. et.al. 1972. Sociology: The Study of Human
Relationships. Jovahovich, Inc. USA. Pp 387-393
AURELIO, JULIE M. 2007. Police Reports Dip in Metro Crime Rate. Inquirer.
CHEN, DONGMEI. 2000. Remote Sensing and Spatial Statistics as tools in Crime
Analysis. Kingston, Canada
COHEN, JENNYLYN et.al. 2003. Estimation of Crime Seasonality: A Cross-
Sectional Extension to Time Series Classical Decomposition. Pp. 2-3
HOEL, LESTER A. 1999. Public Transportation Security
HOREL, STEVEN et.al. 2006. Hierarchical Bayesian Spatial Models for Alcohol
Availability, Drug “hot spots” and Violent Crime
HORTON, PAUL B. et.al. 1988. The Sociology of Social Problems, 9th Edition.
Prentice-Hall, Inc. New Jersey, USA. Pg 77
KENDALL, DIANA. 2004. Sociology In Our Times: The Essentials, 4th Edition.
Wadsworth/ Thomson Learning Inc. USA. Pp 177, 188
MANLY, BRYAN F. J. 1994. Multivariate Statistical Methods, A Primer. St.
Edmundsbury Press. Great Britain. Pp 78-79
Principal Component Analysis of Crime Occurrences
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APPENDIX A
Request Letter to the Chief of Police
Benguet State University
College of Arts and Sciences
Math-Physics-Statistics Department
La Trinidad, Benguet
December 12, 2008
PCINSP MARIO L. MAYAMES JR
Chief of Police
La Trinidad Municipal Police Station
La Trinidad, Benguet
SIR:
We, the undersigned fourth year students taking up Bachelor of Science in Applied Statistics
at Benguet State University, are conducting a research entitled “Principal Component
Analysis of Crime Occurrences at la Trinidad, Benguet”.
In view hereof, we would like to request permission from your good office to gather crime
statistics by year, by month and by barangay for the past five years.
Thank you very much for your favorable consideration.
Respectfully yours,
(SGD) ALMERA L. CARIAS
(SGD) JINKY ROSE J. JUSTO
(SGD) LIZEL A. MAXIMINO
Researchers
Noted:
(SGD) DR. SALVACION Z. BELIGAN (SGD) DR. MA. AZUCENA B. LUBRICA
Thesis Adviser
Chairman, MPS Department
Approved:
(SGD) PROF. AUREA MARIE M. SANDOVAL
CAS Dean
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APPENDIX B
Crime Statistics by Year (2004-2008)
INDEX CRIMES
AGAINST PERSON
AGAINST PROPERTY
MURDER HOMICIDE RAPE
LA P.I.
ROBBERY
THEFT
YEAR
No. % No. % No. % No. % No. % No. % No. %
2004 2 14.29 9 16.98 5 12.50 5 45.45 32 23.02 31 22.3 54 32.34
2005 1 7.14 8 15.09 8 20.00 4 36.36 24 17.27 35 25.18 31 18.56
2006 3 21.43 10 18.87 6 15.00 -
- 21 15.11 18 12.95 20 11.98
2007 3 21.43 11 20.75 16 40.00 - - 42 30.22 31 22.3 27 16.17
2008 5 35.71 15 28.3 5 12.50 2 18.18 20 14.39 24 17.27 35 20.96
TOTAL
14 100.00 53 100.00 40 100.00 11 100.00 139 100.00 139 100.00 167 100.00
Principal Component Analysis of Crime Occurrences
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APPENDIX C
Total Variance of Crimes Committed by Individuals from 2004-2008
Rotation Sums of Squared
Initial Eigenvalues
Extraction Sums of Squared Loadings
Loading
%of
Cumulative
%of
%of
Cumulative
Component
Total
Variance
% Total
Variance Cumulative
% Total
Variance
%
1 3.763
53.756
53.756 3.763
53.756 53.756 3.504
50.062
50.062
2 2.020
28.857
82.612 2.020
28.857 82.612 2.279
32.55
82.612
3 0.965
13.784
96.396
4 0.252
3.604
100.000
5 2.32E-16
3.31E-15
100.000
6 -1.52E-16
-2.17E-15
100.000
7 -4.88E-16
-6.97E-15
100.000
Principal Component Analysis of Crime Occurrences
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APPENDIX D
Crime Statistics by Month
INDEX CRIMES
AGAINST PERSON
AGAINST PROPERTY
MURDER HOMICIDE
RAPE
LA P.I.
ROBBERY
THEFT
MONTH No.
%
No. % No.
% No.
% No. % No. % No. %
Jan - - 2 5.26 4 11.43 2 22.22 15 12.61 9 7.83 16 12.12
Feb - - 2 5.26 1 2.86 - - 15 12.61 5 4.35 13 9.85
Mar 1
11.11
1 2.63 1 2.86 - - 12 10.08 10 8.70 17 12.88
Apr 2
22.22
3 7.89 3 8.57 - - 8 6.72 10 8.70 4 3.03
May 1 11.11 2 5.26 5 14.29 1 11.11 5 4.20 18 15.65 9 6.82
Jun 2 22.22 4 10.53 1 2.86 - -
6 5.04 7 6.09 17 12.88
Jul - - 5 13.16 2 5.71 1 11.11 6 5.04 9 7.83 2 1.52
Aug - - 3 7.89 4 11.43 1 11.11 9 7.56 10 8.70 12 9.09
Sep 2
22.22
2 5.26 3 8.57 1 11.11 17 14.29 9 7.83 7 5.30
Oct - - 3 7.89 4 11.43 - - 6 5.04 11 9.57 12 9.09
Nov - - 8 21.05 2 5.71 3 33.33 13 10.92 12 10.43 11 8.33
Dec 1
11.11
3 7.89 5 14.29 - - 7 5.88 5 4.35 12 9.09
TOTAL
9 100.00 38 100.00 35 100.00 9 100.00 119 100.00 115 100.00 132 100.00
Principal Component Analysis of Crime Occurrences
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APPENDIX E
Total Variance of Crimes Committed Monthly from 2004-2007
Extraction Sums of Squared
Initial Eigenvalues
Loadings
Rotated Sums of Squared Loadings
% of
Cumulative
% of
Cumulative
% of
Cumulative
Component Total Variance
% Total
Variance
% Total
Variance
%
1 2.027 28.963
28.963 2.027
28.963
28.963 1.893
27.044
27.044
2 1.728 24.685
53.648 1.728
24.685
53.648
1.57
22.431
49.475
3 1.146 16.371
70.018 1.146
16.371
70.018 1.438
20.543
70.018
4 0.891 12.734
82.753
5 0.689 9.842
92.595
6 0.468 6.692
99.287
7
0.05
0.713
100
Principal Component Analysis of Crime Occurrences
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APPENDIX F
Crime Statistics by Barangay
Physical
Illegal Possession
Homicide Murder Injuries Rape Robbery
of Fire Arms
Theft
BARANGAY No. % No. % No. % No. % No. % No. % No. %
ALAPANG
1 1.30 - - - - 2 8.00 4 4.17 - - - -
ALNO
2 2.60 2 8.33 1 1.96 - - 1 1.04 - - 1 6.25
AMBIONG
2 2.60 1 4.17 2 3.92 1 4.00 4 4.17 1 0.92 2 12.50
BAHONG
2 2.60 1 4.17 - - 1 4.00 1 1.04 3 2.75 1 6.25
BALILI
17 22.08 3 12.50 8 15.69 2 8.00 23 23.96 11 10.09 5 31.25
BECKEL
3 3.90 3 12.50 1 1.96 - - 1 1.04 2 1.83 - -
BETAG
6 7.79 3 12.50 4 7.84 3 12.00 13 13.54 9 8.26 - -
BINENG
2 2.60 - - - - - - 1 1.04 1 0.92 - -
CRUZ
- - - - 1 1.96 1 4.00 3 3.13 2 1.83 - -
LUBAS
1 1.30 - - - - 1 4.00 2 2.08 2 1.83 1 6.25
PICO
20 25.97 6 25.00 13 25.49 6 24.00 22 22.92 45 41.28 4 25.00
POBLACION
8 10.39 - - 10 19.61 2 8.00 7 7.29 16 14.68 2 12.50
PUGUIS
4 5.19 1 4.17 3 5.88 3 12.00 2 2.08 7 6.42 - -
SHILAN
3 3.90 1 4.17 7 13.73 - - 7 7.29 3 2.75 - -
TAWANG
4 5.19 2 8.33 1 1.96 2 8.00 2 2.08 4 3.67 - -
WANGAL
2 2.60 1 4.17 - - 1 4.00 3 3.13 3 2.75 - -
TOTAL
77 100.00 24 100.00 51 100.00 25 100.00 96 100.00 109 100.00 16 100.00
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APPENDIX F Continued…
Viol. Of
Municipal
Women
Malicious
Carnapping
Illegal Drugs
Ordinance
Abuse Child
Abuse Adultery Mischief
TOTAL
BARANGAY No. % No. % No. % No. % No. % No. % No. %
ALAPANG
-
-
-
-
-
-
-
-
-
-
-
- - - 7
ALNO
1
16.67
-
-
-
-
1
3.33
-
-
-
- - - 9
AMBIONG
-
-
-
-
-
-
1
3.33
1
7.14
1
14.29 1 25.00 17
BAHONG
-
-
-
-
-
-
2
6.67
-
-
-
- - - 11
BALILI
3
50.00
-
-
10
38.46
7
23.33
5
35.71
-
- 1
25.00
95
BECKEL
-
-
-
-
-
-
1
3.33
-
-
-
- - - 11
BETAG
-
-
2
66.67
-
-
4
13.33
-
-
1
14.29 1 25.00 46
BINENG
-
-
-
-
-
-
-
-
-
-
-
- - - 4
CRUZ
-
-
-
-
-
-
1
3.33
-
-
-
- - - 8
LUBAS
-
-
-
-
-
-
-
-
-
-
-
- - - 7
PICO
1
16.67
1
33.33
16
61.54
7
23.33
5
35.71
4
57.14 1 25.00 151
POBLACION
-
-
-
-
-
-
2
6.67
-
-
-
- - - 47
PUGUIS
1
16.67
-
-
-
-
1
3.33
1
7.14
1
14.29 - -
24
SHILAN
-
-
-
-
-
-
2
6.67
1
7.14
-
14.29 - -
24
TAWANG
-
-
-
-
-
-
-
-
-
-
-
- - - 15
WANGAL
-
-
-
-
-
-
1
3.33
1
7.14
-
- - - 12
TOTAL
6
100.00 3 100.00
26
100.00
30 100.00
14 100.00
7 100.00
4 100.00
488
Principal Component Analysis of Crime Occurrences
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APPENDIX G
Total Variance of Crimes Committed by Individuals in the Different Barangays of La Trinidad,
Benguet (2005-2008)
Extracted Sums of Squared
Initial Eigenvalues
Loadings
Rotation Sums of Squared Loadings
% of
Cumulative
% of
Cumulative
% of
Cumulative
Component Total Variance
% Total
Variance
% Total
Variance
%
1 9.76 69.69
69.69
9.76
69.69 69.69
6.43
45.90
45.90
2 1.74 12.42
82.11
1.74
12.42 82.11
5.07
36.22
82.11
3
0.88
6.30
88.41
4
0.51
3.65
92.06
5
0.38
2.73
94.79
6
0.31
2.22
97.01
7
0.14
1.00
98.01
8
0.13
0.93
98.94
9
0.08
0.56
99.50
10
0.04
0.28
99.78
11
0.02
0.13
99.91
12
0.08
0.06
99.97
13
0.03
0.02
99.99
14
0.04
2.54E-03
100.00
Principal Component Analysis of Crime Occurrences
at La Trinidad, Benguet / Almera L. Carias; et al. 2009
Document Outline
- Principal Component Analysis of Crime Occurrences at La Trinidad,
Benguet.
- BIBLIOGRAPHY
- ABSTRACT
- TABLE OF CONTENTS
- INTRODUCTION
- REVIEW OF RELATED LITERATURE
- THEORETICAL FRAMEWORK
- METHODOLOGY
- RESULTS AND DISCUSSION
- SUMMARY, CONCLUSION AND RECCOMMENDATION
- LITERARURE CITED
- APPENDIX