Chapter 3
Results and Discussions
This chapter presents the results of the analysis of the data and the discussion of the results.
Descriptive Analysis
This section includes the tabular and the graphical presentation of the time series data.
The table shows the distribution of the 
. The interpretations for the table can be seen in the next figures.
Table 1
Data on Crime Solution Efficiency, Crime Rate for Murder, Homicide, Physical Injury, Robbery and Theft, and Average Crime Rate from January 2000 to January 2006
|
|
CSE |
CR murder |
CR homicide |
CR phy in |
CR rob |
CR theft |
AverageCR |
|
Jan-00 |
100 |
0.26 |
0.26 |
2.56 |
0.51 |
1.02 |
4.61 |
|
Feb-00 |
96.15 |
0.51 |
0.26 |
2.05 |
0.77 |
0.51 |
4.1 |
|
Mar-00 |
86.36 |
1.02 |
0.26 |
3.84 |
0.77 |
1.54 |
7.43 |
|
Apr-00 |
100 |
0 |
0.26 |
3.07 |
0 |
1.02 |
4.36 |
|
May-00 |
97.67 |
1.02 |
0.51 |
4.36 |
0 |
1.79 |
7.69 |
|
Jun-00 |
97.06 |
1.02 |
0 |
2.56 |
0.26 |
0.51 |
4.36 |
|
Jul-00 |
91.18 |
0.51 |
0.26 |
2.31 |
0.26 |
0.26 |
3.59 |
|
Aug-00 |
97.22 |
1.02 |
0.51 |
1.02 |
1.28 |
1.28 |
5.12 |
|
Sep-00 |
96.15 |
1.02 |
0.77 |
1.54 |
1.54 |
0.77 |
5.64 |
|
Oct-00 |
84 |
0.51 |
0.26 |
3.84 |
2.05 |
1.02 |
7.69 |
|
Nov-00 |
95.38 |
0.77 |
0.51 |
4.61 |
0.77 |
1.54 |
8.2 |
|
Dec-00 |
93.48 |
0.77 |
0.51 |
3.84 |
0.26 |
0.26 |
5.64 |
|
Jan-01 |
98 |
0.51 |
0.77 |
2.31 |
0.51 |
1.02 |
5.12 |
|
Feb-01 |
97.92 |
0.51 |
0.26 |
2.05 |
0.26 |
1.02 |
4.1 |
|
Mar-01 |
100 |
1.02 |
0 |
2.31 |
1.02 |
0.26 |
4.61 |
|
Apr-01 |
95.92 |
1.28 |
0.26 |
2.56 |
1.28 |
0.51 |
5.89 |
|
May-01 |
94.12 |
0 |
0.26 |
1.28 |
0.51 |
1.28 |
3.33 |
|
Jun-01 |
90.63 |
0 |
0.77 |
2.56 |
1.02 |
0.26 |
4.61 |
|
Jul-01 |
97.83 |
1.54 |
0.51 |
2.56 |
1.28 |
0.51 |
6.41 |
|
Aug-01 |
92.31 |
0.77 |
0.51 |
0.51 |
0.77 |
1.02 |
3.59 |
|
Sep-01 |
100 |
1.79 |
0.26 |
1.02 |
0.26 |
0.77 |
4.1 |
|
Oct-01 |
100 |
0 |
0.26 |
1.54 |
0.26 |
0.51 |
2.56 |
|
Nov-01 |
93.94 |
0.51 |
0 |
2.31 |
1.02 |
0.51 |
4.36 |
|
Dec-01 |
95 |
1.28 |
0.26 |
2.56 |
0.51 |
1.02 |
5.64 |
|
Jan-02 |
86.96 |
1.02 |
0.51 |
0.51 |
0.51 |
0.51 |
3.07 |
|
Feb-02 |
94.74 |
0.26 |
0.26 |
3.33 |
1.28 |
0.51 |
5.64 |
|
Mar-02 |
92.11 |
1.28 |
0.51 |
2.31 |
1.54 |
0.77 |
6.41 |
|
Apr-02 |
100 |
0.26 |
1.28 |
2.05 |
1.79 |
2.05 |
7.43 |
|
May-02 |
100 |
1.54 |
1.02 |
1.02 |
0.51 |
1.54 |
5.64 |
|
Jun-02 |
78.95 |
0 |
0 |
2.05 |
0.26 |
2.31 |
4.61 |
|
Jul-02 |
100 |
1.02 |
0.26 |
1.02 |
0.51 |
2.31 |
5.12 |
|
Aug-02 |
100 |
1.28 |
0 |
2.56 |
0 |
1.79 |
5.64 |
|
Sep-02 |
100 |
1.28 |
0.77 |
1.28 |
0.77 |
1.28 |
5.38 |
|
Oct-02 |
100 |
0.51 |
0.26 |
0.77 |
0.77 |
1.28 |
3.59 |
|
Nov-02 |
100 |
0.77 |
0.26 |
1.79 |
0.51 |
0.51 |
3.84 |
|
Dec-02 |
95.65 |
1.54 |
0 |
1.79 |
0.26 |
0.26 |
3.84 |
|
Jan-03 |
97.67 |
0.26 |
1.02 |
3.07 |
0.51 |
2.05 |
6.92 |
|
Feb-03 |
96.55 |
0.26 |
0.26 |
1.02 |
0.51 |
1.28 |
3.33 |
|
Mar-03 |
100 |
0.26 |
0.51 |
0.77 |
0.77 |
1.54 |
3.84 |
|
Apr-03 |
100 |
1.02 |
0.51 |
0.51 |
0 |
0.77 |
2.82 |
|
May-03 |
93.75 |
0.51 |
0.26 |
0.77 |
0.26 |
0.26 |
2.05 |
|
Jun-03 |
94.44 |
0.77 |
0 |
0.51 |
0.77 |
0.77 |
2.82 |
|
Jul-03 |
94.44 |
1.02 |
0.51 |
1.54 |
0.51 |
1.28 |
4.87 |
|
Aug-03 |
96.77 |
0 |
0.26 |
2.56 |
0.77 |
1.54 |
5.12 |
|
Sep-03 |
100 |
1.02 |
0.26 |
1.79 |
0.77 |
2.31 |
6.15 |
|
Oct-03 |
97.37 |
0.51 |
0.26 |
2.31 |
0.51 |
1.28 |
4.87 |
|
Nov-03 |
100 |
0.51 |
0 |
1.79 |
0.26 |
1.54 |
4.1 |
|
Dec-03 |
91.18 |
0 |
0.26 |
2.31 |
1.54 |
1.02 |
5.12 |
|
Jan-04 |
97.22 |
1.02 |
0.26 |
2.56 |
0.77 |
0.77 |
5.38 |
|
Feb-04 |
100 |
0.51 |
0.26 |
1.79 |
0.51 |
1.79 |
4.87 |
|
Mar-04 |
100 |
0 |
0 |
2.05 |
0.51 |
0.77 |
3.33 |
|
Apr-04 |
93.1 |
0.51 |
0 |
3.59 |
1.02 |
1.02 |
6.15 |
|
May-04 |
96.55 |
0.51 |
0.26 |
2.56 |
1.02 |
1.28 |
5.64 |
|
Jun-04 |
92 |
0.51 |
0.26 |
1.54 |
0.26 |
2.56 |
5.12 |
|
Jul-04 |
94.87 |
1.02 |
0 |
1.54 |
1.54 |
1.28 |
5.38 |
|
Aug-04 |
95.83 |
0.77 |
0.26 |
2.05 |
0.51 |
1.79 |
5.38 |
|
Sep-04 |
90 |
1.02 |
0.26 |
2.31 |
1.28 |
1.02 |
5.89 |
|
Oct-04 |
100 |
0.26 |
0 |
0.51 |
0.26 |
1.79 |
2.82 |
|
Nov-04 |
94.74 |
0.26 |
0.26 |
0.77 |
0.26 |
1.54 |
3.07 |
|
Dec-04 |
100 |
0.26 |
0.77 |
2.05 |
0.77 |
2.31 |
6.15 |
|
Jan-05 |
95.24 |
0.51 |
0.26 |
2.56 |
1.28 |
0.26 |
4.87 |
|
Feb-05 |
100 |
0.26 |
0.26 |
2.31 |
0.51 |
0.77 |
4.1 |
|
Mar-05 |
100 |
0.51 |
0.51 |
1.28 |
1.02 |
0.26 |
3.59 |
|
Apr-05 |
100 |
0.51 |
0.51 |
0.77 |
1.79 |
0.77 |
4.36 |
|
May-05 |
100 |
1.02 |
0.77 |
1.54 |
0 |
2.31 |
5.64 |
|
Jun-05 |
100 |
0.26 |
0 |
1.28 |
0.77 |
0.51 |
2.82 |
|
Jul-05 |
100 |
0.26 |
0.77 |
0 |
0.26 |
0.26 |
1.54 |
|
Aug-05 |
100 |
0.26 |
0.51 |
0.51 |
0.26 |
0.26 |
1.79 |
|
Sep-05 |
100 |
0.51 |
0.26 |
0 |
0.26 |
0.51 |
1.54 |
|
Oct-05 |
100 |
0 |
0.26 |
0.77 |
0.51 |
0.51 |
2.05 |
|
Nov-05 |
100 |
0.51 |
0.26 |
1.28 |
0.26 |
0.51 |
2.82 |
|
Dec-05 |
100 |
1.28 |
0.26 |
1.79 |
0.26 |
0 |
3.59 |
|
Jan-06 |
100 |
0.77 |
0.26 |
0 |
0.26 |
0.26 |
1.54 |
|
Feb-06 |
100 |
0.51 |
0.26 |
1.02 |
0 |
0.26 |
2.05 |
|
Mar-06 |
100 |
0.51 |
0.26 |
0.51 |
0.77 |
0.77 |
2.82 |
|
Apr-06 |
94.29 |
1.02 |
0.51 |
1.54 |
1.79 |
1.54 |
6.41 |
|
May-06 |
100 |
0.51 |
0.26 |
1.79 |
0 |
0.77 |
3.33 |
|
Jun-06 |
100 |
0 |
0.26 |
0.77 |
0.26 |
0.51 |
1.79 |
|
Jul-06 |
100 |
0.26 |
0 |
0.51 |
0 |
2.82 |
3.59 |
|
Aug-06 |
100 |
0 |
0.26 |
0.51 |
0.77 |
3.07 |
4.61 |
|
Sep-06 |
97.06 |
0.26 |
0.26 |
2.82 |
1.54 |
2.56 |
7.43 |
|
Oct-06 |
95.24 |
0 |
0.26 |
1.02 |
1.28 |
0 |
2.56 |
|
Nov-06 |
100 |
0.51 |
0 |
0.51 |
0.77 |
1.02 |
2.82 |
|
Dec-06 |
97.3 |
0.51 |
0.26 |
1.02 |
0.77 |
2.56 |
5.12 |
___________________________________________________________
We can see in the figure that the lowest number of crimes solved among the reported crimes (CSE) was between the 2nd and 3rd Quarter of 2002. Also, we can see that there are cases wherein all the crimes reported were solved and these are between the 3rd quarter of 2002 to 1st quarter of 2003, 2nd quarter of 2005 to 1st quarter of 2006, and 2nd quarter of 2006 to 3rd quarter also of 2006. We can also see an increasing trend in the Crime Solution Efficiency.
Figure 2. Time Trend of Crime Solution Efficiency during First Quarter of 2000 to the Fourth Quarter of 2006
The Average Crime Rate has the highest crime rate that occurred between the 4th quarter of 2000 to 1st quarter of 2001. And the lowest crime can be seen in the 3rd quarter, 4th quarter of 2005 and the 1st quarter of 2006. The figure also shows that there is a seasonal variation with increases in the 4th quarter of each year except in the 4th quarter of 2001 in the data.
Figure 3.Time Trend of Average Crime Rate during 1st quarter 2000 to fourth quarter 2006
The figure shows that the highest crime rate for murder can be seen between the 3rd to 4th quarter of 2001 and the lowest crime rate can be seen in the 2nd quarter of 2000, between the 2nd quarter to 3rd quarter of 2001, in the 4th quarter of 2001, between the 2nd quarter to 3rd quarter of 2002, between the 3rd quarter to 4th quarter of 2003, between 4th quarter of 2003 to 1st quarter of 2004, in the 4th quarter of 2005, between the 2nd quarter to 3rd quarter of 2006, between the 3rd quarter to 4th quarter of 2006, and 4th quarter of 2006. Also it shows a decreasing trend.
Figure 4.Time Trend of Crime Rate for Murder during 1st quarter 2000 to fourth quarter 2006
Figure illustrates that the highest rate of crime for homicide was in the 2nd quarter of 2002. And the lowest crime rates were between the 2nd quarter to 3rd quarter of 2000, between the 1st quarter to 2nd quarter of 2001, between 4th quarter of 2001 to 1st quarter of 2002, between 2nd quarter to 3rd quarter of 2002, between 3rd quarter to 4th quarter of 2002, between 2nd quarter to 3rd quarter of 2003, between 4th quarter of 2003 to 1st quarter of 2004, between 1st quarter to 2nd quarter of 2004, in the 3rd and 4th quarter of 2004, between the 2nd to 3rd quarter of 2005, in the 3rd quarter of 2006, and between the 4th quarter of 2006 to 1st quarter of 2007. The graph shows irregular patterns in the movement of the crime rate of homicide.
Figure 5.Time Trend of Crime Rate Homicide during 1st quarter 2000 to fourth quarter 2006
The diagram shows that the highest crime rate for physical injury was between the 4th quarter of 2000 to the 1st quarter of 2001. Also, the lowest crime rate can be seen in the 3rd quarter of 2005, between the 3rd to 4th quarter of 2005 and the 1st quarter of 2006. The data shows seasonal variation with increases in the 4th quarter of each year.
Figure 6. Time Trend of Crime Rate Physical Injury during 1st quarter 2000 to fourth quarter 2006
Figure shows that the highest crime rate was in the 4th quarter of 2000. And the lowest crime rate were in the 2nd quarter of 2000, between 2nd to 3rd quarter of 2000, between the 3rd to 4th quarter of 2002, in the 2nd quarter of 2003, between 2nd to 3rd quarter of 2005, between 1st to 2nd quarter of 2006, between 2nd to 3rd quarter of 2006, and the 3rd quarter of 2006. The graph shows an irregular pattern showing no seasonal variation.
Figure 7. Time Trend of Crime Rate Robbery during 1st quarter 2000 to fourth quarter 2006
The figure shows that the highest crime rate for theft was between the 3rd to 4th quarter of 2006. The lowest crime rate was between the 4th quarter of 2005 to the 1st quarter of 2006 and the 4th quarter of 2006. The figure shows no seasonal variation among the time points.
Figure 8. Time Trend of Crime Rate Theft during 1st quarter 2000 to fourth quarter 2006
Time Series Analysis
With the use of SPSS, the Autocorrelation Function and the Partial Autocorrelation Function was generated.
Figure 9 show that the Autocorrelation Function (ACF) of the Average Crime Rate for Index Crimes show a sine wave pattern with spikes in lags 1, 4 and 5. The function show decays toward zero. The ACF shows MA characteristics.
Figure 9. Autocorrelation Function of the Average Crime Rate for Index Crimes
Figure 10 shows that the PACF has a spike in lag 1 with a damped sine wave pattern. The PACF shows an AR characteristic.
Figure 10. Partial Autocorrelation Function of the Average Crime Rate for Index Crimes
Autoregressive Models, Moving Average Models and mixtures of both were generated to identify a good fit for the model. The following table shows the Autoregressive Moving Average Models explored and have the qualifications to be the model for the Average Crime Rate for Index Crimes.
Table 2 presents the models generated using ARIMA. It shows that ARMA(5,0,5) model has lower AIC and Residual Variance compared to ARMA(4,0,4) which are criterions for a good fit in which the lower the AIC, SBC, and Residual Variance the better. Although the SBC of ARMA(4,0,4) is lower the other criterions overshadow the SBC. Also we can see that the Log Likelihood of ARMA(5,0,5) is higher than that of ARMA(4,0,4) which means further gives edge of having ARMA(5,0,5) as the model.
Table 2
ARMA Models for the Average Crime Rate for Index Crimes
|
Model |
Log Likelihood |
AIC |
SBC |
Residual Variance |
|
ARMA(5,0,5) |
-141.32746 |
306.65492 |
335.82472 |
1.9099897 |
|
ARMA(4,0,4) |
-142.60351 |
307.20702 |
333.946 |
1.9566938 |
Table 3 shows the coefficients that are included in the model and these are AR5, MA1 and MA2.
Therefore the model generated is ARMA(5,0,5):
where:
Table 3
Variables in the ARMA(5,0,5) model
|
|
Coefficient |
P-Value |
Included? |
|
AR1 |
-0.5488318 |
0.10323957 |
Not Included |
|
AR2 |
-0.3853758 |
0.17596546 |
Not Included |
|
AR3 |
0.1607127 |
0.54760688 |
Not Included |
|
AR4 |
0.3495610 |
0.11433213 |
Not Included |
|
AR5 or |
0.4943672 |
0.02686218 |
Included |
|
MA1 or |
-0.9368993 |
0.01218376 |
Included |
|
MA2 or |
-0.8406828 |
0.04000051 |
Included |
|
MA3 |
-0.1617000 |
0.70670683 |
Not Included |
|
MA4 |
0.2659247 |
0.48577325 |
Not Included |
|
MA5 |
0.1803848 |
0.56933875 |
Not Included |
|
Constant |
4.5048530 |
0.00000000 |
Included |
The ARMA(5,0,5) model satisfies the qualifications for stationarity and invertibility because AR(p) models are always invertible while MA(q) models are always stationary. Furthermore, which means that its roots are outside the unit circle thus satisfying the stationarity condition for AR(p) and the roots for MA(q) satisfies the invertibility condition. Therefore the model is appropriate to use for the Average Crime Rate for Index Crimes.
