0 50 100 150 200 300
320 340 360 380
400 Assault
0 50 100 150 200
920 930 940 950 960 970
980 Battery
0 50 100 150 200
180 200 220 240 260 280 300
320 Burglary
0 50 100 150 200
780 790 800 810 820 830
840 Domestic
0 50 100 150 200
200 250 300 350
400 Narcotic
0 50 100 150 200
48 49 50 51 52 53
54 Rare Felony
0 50 100 150 200
140 160 180 200 220 240
260 Robbery
0 50 100 150 200
1050 1100 1150 1200
1250 Theft
0 50 100 150 200
100 105 110 115
120 Sexual crime
Figure 9: Plot of the dominant DMD Mode of Temporal Data for each types of crime.
Assault Battery Burglary Domestic Narcotic
Rare Felony Robbery Theft
Sexual crime Assault
Battery
Burglary
Domestic
Narcotic
Rare Felony
Robbery
Theft
Sexual crime
Correlation of Dominant Dynamic Mode for Temporal Data
0.1403
-0.8187
-0.3227
-0.1146
-0.6835 0.1403
0.0236
0.09284
0.1362
-0.1572
-0.0568
0.5016
-0.8187
0.0236
-0.5842
-0.1507
-0.06295
-0.4775
-0.6175 -0.5842
-0.1865
-0.0104
-0.6029 -0.3227
0.09284
-0.1507
-0.1865
0.5139
-0.3922
-0.6643
0.3238
-0.1146
0.1362
-0.06295
-0.0104
0.5139
-0.2698
-0.4131
0.4815
-0.6835
-0.1572
-0.6029
-0.3922
-0.2698
-0.2707
-0.7244 -0.0568
-0.4775
-0.6643
-0.4131
-0.2707
0.06649 0.5016
-0.6175
0.3238
0.4815
-0.7244
0.06649 1
0.8401
0.825
0.5537 1
0.5469 1
0.9002 0.8401
0.5469
1
0.633
0.7005 1
1
0.9002
1
0.825
0.633
1
0.5537
0.7005
1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 10: Correlation matrix visualization of scaled mode for temporal data. In the cell, the Pearson correlation value is listed. from -1 to 1, colorize from white to blue. We can check that there are some clusters between scaled modes. Similarly for the basic data analysis case.
mode, other DMD eigenvalues have λj
<1 which means most of the modes represent decay mode.
Similar to temporal data, The dominant mode has a real DMD eigenvalue nearly close to 1 but does not exceed. This implies dominant mode effects on the decaying and oscillation of the spatio-temporal data. Thus, we can think this means each crime is spreading near the site. We more deal with the next subsection.
Spatio-temporal correlation in crimes
Now, we recover this dominant mode as column vector form to compute the correlation matrix. It’s similar to the time series case and temporal data case. In figure 12, we again have clustering between crime types; Narcotic, Theft, Sexual Crime, Burglary, and others. Note that Narcotic is close to Rare Felony and, Robbery. On the other hand, rare felony can treat other clusters from Domestic, Assault, and Battery. Now, compare this with figure 3 and 10. Narcotic crime is isolated which is similar to time series data. But, Theft is far from Assault, Battery, and Domestic while sexual crime is close to
-1 -0.5 0 0.5 1 -1
-0.5 0 0.5 1
Assault
#1
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Battery
#1
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Burglary
#1
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Domestic
#1
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Narcotic
#1
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Rare Felony
#1
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Robbery
#1
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Theft
#1
-1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
Sexual crime
#1
Figure 11: The eigenspectrum of DMD eigenvalues for sptatio-temporal data of each crime types.
Domestic. In contrast to this, clusters are similar to the temporal data case. But, some types might have different distances such as Burglary and Robbery, and Battery with others. From this result, Narcotic and Theft might be correlated with spatial characteristics which overlooked in time series correlation.
Hot-spot identification
After Applying DMD, we have the Dominant Mode of each type of crime. As we can check in figure 8 and 4, most of DMD modes are having decay movement and the dominant one has decay and (nearly) oscillation form. From this result, we connect this to criminal hot-spot concepts. Criminal hotspot has been studied since 2000s. Usually, when crimes are visualized on the map with longitude and latitude coordinates, we can check if there is some high crime intensity area. This area is called a crime hotspot.
More precisely, there is some definition of crime hotspot [16]. Many aspects of hotspots are stated But,
Assault Battery Burglary Domestic Narcotic
Rare Felony Robbery Theft
Sexual crime Assault
Battery
Burglary
Domestic
Narcotic
Rare Felony
Robbery
Theft
Sexual crime
Correlation of Dominant Dynamic Mode for Spatio-Temporal Data
0.6085
0.432
0.62
0.7426
0.5072
0.397
0.6613
0.4743
0.6625
0.7562
0.4533
0.4241 0.6085
0.6613
0.7122
0.2601
0.4474
0.5319
0.2775
0.2995
0.7122
0.4275
0.6245
0.5899
0.2456
0.3858 0.432
0.4743
0.2601
0.4275
0.6105
0.4367
0.1387
0.2029 0.62
0.6625
0.4474
0.6245
0.6105
0.5723
0.2132
0.302
0.7426
0.7562
0.5319
0.5899
0.4367
0.5723
0.5279
0.3547
0.5072
0.4533
0.2775
0.2456
0.1387
0.2132
0.5279
0.2047 0.397
0.4241
0.2995
0.3858
0.2029
0.302
0.3547
0.2047 1
0.8961
0.7945
0.8961
1
0.8794 1
0.7945
0.8794
1
1
1
1
1
1 0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 12: Visualization of the correlation matrix between dominant modes of spatio-temporal data.
Same with the previous visualization of the correlation matrix.
we are more focused on that criminal hotspot is the high intensity and occurrences of crime. We also can check that near-repeated property of crime, the near area can deal with the crime spreading. This crime hotspot is a critical point for police and government since the policy and control can be determined by this information [8]. Hotspot measuring is related toThe nearest neighbor index(NNI) which was formulated by Philip Clark and Francis Evans [23]. After then, Crime mapping software CrimeStat uses this formula to detect the crime hotspots [33]. Using this method, crime mapping software detects the hotspot using ellipse [63], Moreover, there are some more visualization of hotspot such as discrete mapping, choropleth mapping, thematic mapping and kernel density mapping [118].
Now, we connect crime hotspot with DMD mode and Amplitude. In the concept of DMD Mode, it describes the growth or decay rate or, oscillation of frequency. This can be seen as the criminal activity is suppressing and spreading corresponding to growth and decay rates respectively. Also, the oscillation move of DMD mode can be understood by repeated criminal activity. DMD amplitude is related to the measurement of how the effect on the given dynamics. So, we can treat this concept in criminal activity as high activity. This can be used for the campaign or the patrol of the government or police in real-world
applications. In figure 13, we can check where is the high intensity of sexual crime. Also, we have decay
Figure 13: Following is the dominant DMD mode of sexual crime with a hotspot area. The red dot indicates the DMD mode of the spatio-temporal data of sexual crime. Ellipse represents the hotspot area of the corresponding crime. The area of the blue ellipse contains the Chicago-Kansas city expressway and Cicero metra where many people come and go.
and (near) oscillation effect on the Dominant mode, in the ellipse area, sexual crime will be spread and happened repeatedly. inside the ellipse is the hotspot of sexual crime. Police and government need to use the resource of policy to check this area to reduce sexual crime. Also, this area contains several specific places related to people’s leisure or work. This indicates that the related residual, geographical factors are involved in sexual crime [62, 63, 71] In figure 14, we also check the hotspot of narcotics. For Narcotic related crime, similar areas are related to sexual crime. In contrast to sexual crime, we can pick more areas near the Chicago-Kansas expressway and metro stations since there are some clusters These are also related to transportation and park that many people use every day which is good for dealing with providers. We can check the hotspot of theft. The location of theft is different from sexual crime and narcotic-related crime. This must be related to environmental factors such as the hotels, markets, and cultural centers.
Figure 14: Following is the dominant DMD modes of narcotic-related crime withHotspot area. The red dot indicates the DMD mode of the spatio-temporal data of narcotic-related crime. Ellipse represents the hotspot area of narcotic-related crime. The result is similar to Figure 13
Furthermore, in figure 13 to 15, the ellipse area is drawn by visibility of the high intencity. But, we can extend this ellipse or dectecting hotspots with combining the more dedicated way to detect crime hotspot
Figure 15: Following is the dominant DMD mode of theft with Hotspot area. The red dot indicates the DMD mode of the spatio-temporal data of Thefts. Ellipse represents the hotspot area of Theft. Unlike sexual crime and narcotics-related crime, hotspot area is different. The hotspot contains Chicago, Lake, and Millennium stations. Moreover, the area has many hotels, markets, and cultural centers such as theater and business centers.
4
Conclusion
We present two results about the mathematical analysis of crime. First, we propose the dynamic model- ing of criminals using population dynamics. This model focus on the transition of criminals from mis- demeanor to felony. We use two sociological concepts, so-called,Broken-Window TheoryandPrison as Crime school. Using these, the basic model(2.1)and(2.15)are extended to the in and out of prison dynamic model(2.3),(2.4),(2.16)and(2.21). These confirm that both the Broken-Window-Effect and Prison as crime school change criminal distribution. These Modeling showed that oppression of the interaction between inmates in overcapacity of the prison is critical to controlling crimes in society. It’s different from simple thinking, extending the imprisonment period only result in a rapid increase in felony unless suppressing the contract rate of the inmate in prison is at a low level. (See Figure 10 in Chapter 2). Also, If we didn’t care about the visible little signs of the disorder, this will affect the huge increase of Felony because of Broken Window Effect as we can check in Figure 4 and 7 in Chapter 2.
These two social effects also have an impact on the allocation of police resources to control crime. When the budget of the control for felony is too high with negligence on minor crimes, this eventually leads to the excessive occurrence of major crimes due to the broken windows effect and the crime school effect (See Figure 11 in Chapter 2). Thurs, our model suggests that In order to promote the reduction of serious crimes, it is necessary to consider both sides at an appropriate level without belittling misdemeanors.
Second, we analyze the criminal patterns with a data-driven method called,Dynamic Mode Decom- position. Usually, DMD is used for fluid dynamics. Here, we propose the crime record application of DMD with enhanced algorithm 1. We treat crime record data; temporal and spatio-temporal data.
For temporal data, we suggest a consecutive approach to generate the data matrix as in Fig 4. Using this approach, we generate 1-year data to 2-year length data matrix, apply DMD and reconstruct the remaining 2 years. However, the prediction is not working in good shape since the data matrix doesn’t fit the condition for reconstruction [55]. And, we do a similar way to 4-year length data to generate a data matrix. This is a more suitable form for the DMD application since the length of the snapshot is much greater than the number of data snapshots. From that data matrix, we can get the DMD outputs.
By correlation analysis for the dominant mode of each crime type, we have that Assault-Theft is still correlated, and some types are correlated with other crimes such as Narcotic-Felony and Battery-Sexual
Crime. In the contrast to, we do the same application to spatio-temporal cases. Then, we have the result that Narcotics are very far from away others, and Theft, Sexual crime, and Burglary are similar. In sum for both cases of data with correlation analysis, we propose that narcotics, Sexual crime, Theft, Robbery, and Battery are more related to spatial characteristics. Furthermore, we detect the hotspot of crime using the dominant DMD mode. We show three examples of hotspot using the dominant DMD modes, sexual crime, narcotic-related crime, and theft. The results lead us to that hotspot areas are more specific and sharp than normal data, and each crime types are very close to environmental factors.
Appendices
I Numerical Simulation Result for Temporal DMD
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 Assault
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 Battery
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5
0.6 Burglary
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 Domestic
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5
0.6 Narcotic
0 20 40 60
0 0.1 0.2 0.3 0.4
0.5 Rare Felony
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5
0.6 Robbery
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 Theft
0 20 40 60
0 0.1 0.2 0.3
0.4 Sexsual crime
Figure 1: Normalized Singular value plot of the 2 year generating data for temporal data. We take the rankr=12 since it’s near 60% to 80% for each types.
Assault
5 10 15
0 1 2 3 4 5
6 106 Battery
5 10 15
0 1 2 3 4
5 107 Burglary
5 10 15
0 0.5 1 1.5 2 2.5 3 3.5 106
Domestic
5 10 15
0 0.5 1 1.5 2 2.5 3
3.5 107 Narcotic
5 10 15
0 1 2 3 4 5
6 106 Rare Felony
5 10 15
0 5 10 15 104
Robbery
5 10 15
0 0.5 1 1.5
2 106 Theft
5 10 15
0 1 2 3 4 5 6
7 107 Sexsual crime
5 10 15
0 1 2 3 4 5 6 7 105
Figure 2: The Dominant structure of DMD result of 2 year length of training set for temporal Data.
x−axis is numbering of DMD modes, y−axis represents the value calculated by (3.25) where p is usually take number of data snapshot. But, we can take any other number less than equal to number of snapshot since it’s sufficient to check that which mode is dominant. Here, we pick p=m/2=52. For each crime, we can check there is a one dominant DMD mode. This modes is most influential on the system of training 2 year length set. This is corresponding Red dot in Figure 6.
0 20 40 60 0
0.1 0.2 0.3 0.4 0.5
0.6 Assault
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 Battery
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5
0.6 Burglary
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 Domestic
0 20 40 60
0 0.1 0.2 0.3 0.4
0.5 Narcotic
0 20 40 60
0 0.1 0.2 0.3 0.4
0.5 Rare Felony
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5
0.6 Robbery
0 20 40 60
0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 Theft
0 20 40 60
0 0.1 0.2 0.3 0.4
0.5 Sexual crime
Figure 3: Normalized singular value plot of theX1for temporal (4 year window length) data set. Here, we truncate the singular value with rankr=10 which nearly approximated to 60% to 80% of singular values
Assault
1 2 3 4 5 6 7 8 9 10 0
1000 2000 3000 4000 5000
6000 Battery
1 2 3 4 5 6 7 8 9 10 0
2000 4000 6000 8000 10000 12000
14000 Burglary
1 2 3 4 5 6 7 8 9 10 0
1000 2000 3000 4000
Domestic
1 2 3 4 5 6 7 8 9 10 0
2000 4000 6000 8000 10000
12000 Narcotic
1 2 3 4 5 6 7 8 9 10 0
1000 2000 3000
4000 Rare Felony
1 2 3 4 5 6 7 8 9 10 0
200 400 600 800
Robbery
1 2 3 4 5 6 7 8 9 10 0
500 1000 1500 2000 2500
3000 Theft
1 2 3 4 5 6 7 8 9 10 0
0.5 1 1.5
2 104 Sexual crime
1 2 3 4 5 6 7 8 9 10 0
500 1000 1500
Figure 4: Plot of the Dominant DMD Amplitude of temporal data. Situation is same with figure 2.
More precisely,x−axis is numbering of DMD modes, y−axis represents the value of(3.25). We use p=m/2=26. Similarly to previous result, for each crime, we can check there is a one dominant DMD mode. This modes is most influential on the temporal system. This is corresponding Red dot in Figure 8.
0 20 40 60 250
300 350 400 450 500
550 Assault
DMD Data
0 20 40 60
600 700 800 900 1000 1100 1200
1300 Battery
DMD Data
0 20 40 60
100 150 200 250 300
350 Burglary
DMD Data
0 20 40 60
600 700 800 900 1000
1100 Domestic
DMD Data
0 20 40 60
150 200 250 300
350 Narcotic
DMD Data
0 20 40 60
20 30 40 50 60 70
80 Rare Felony
DMD Data
0 20 40 60
100 150 200 250
300 Robbery
DMD Data
0 20 40 60
800 1000 1200 1400
1600 Theft
DMD Data
0 20 40 60
80 100 120 140 160
180 Sexsual crime DMD Data
Figure 5: Forecasting of temporal data i.e, 4year window length case. We can easily check that fore- casting of future 1year is more good fit than 2 year training and 2 year test case. But, the forecasting of future states are not working well.From this result, for the temporal case, we need more information about the given data.
II Numerical Simulation Result for Spatio-Temporal DMD
0 100 200 300
0 0.005 0.01 0.015
0.02 Assault
0 100 200 300
0 0.005 0.01 0.015 0.02 0.025
0.03 Battery
0 100 200 300
2 4 6 8 10
12 10-3 Burglary
0 100 200 300
0 0.005 0.01 0.015 0.02
0.025 Domestic
0 100 200 300
0 0.005 0.01 0.015 0.02 0.025 0.03
0.035 Narcotic
0 100 200 300
2 3 4 5 6 7 8
9 10-3 Rare Felony
0 100 200 300
2 4 6 8 10 12
14 10-3 Robbery
0 100 200 300
0 0.01 0.02 0.03 0.04 0.05
0.06 Theft
0 100 200 300
0 0.005 0.01 0.015
0.02 Sexual crime
Figure 6: Normalized singular value plot of theX1 for spatio-temporal data set. almost types of crime with large singular value. Here, we truncate the singular value with rankr=64 which nearly approxi- mated to 30% to 35% of singular values
Assault
10 20 30
0 0.5 1 1.5 2
2.5 Battery
10 20 30
0 5 10 15 20 25
30 Burglary
10 20 30
0 0.05 0.1 0.15 0.2 0.25 0.3
Domestic
10 20 30
0 5 10 15 20 25 30
35 Narcotic
10 20 30
0 10 20 30 40
50 Rare Felony
10 20 30
0 0.005 0.01 0.015
Robbery
10 20 30
0 0.5 1 1.5 2
2.5 Theft
10 20 30
0 20 40 60 80
100 Sexual crime
10 20 30
0 1 2 3 4 5
Figure 7: Dominant Amplitude for Spatio-temporal data using (3.25). Similar to the temporal case, we can pick suitable one dominant mode. To make the results easier to see, drop-out the number of amplitude by 32 and, we use powerp=15 since mode domination already happened with that p.
Figure 8: This figure shows the mapping of the dominant DMD mode of each crime. The red dot indicates the value of dominant modes. With this result, we can detect the hotspot of each crime. To make the visualization more clear, we normalized the domain DMD mode. In addition, Narcotic, Theft, and Sexual Crimes are cut off by 0.15 and the above values treat as the same values., depending on the distribution. Similarly, Burglary also does above 0.75. Other types are cut off by 0.5.
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