0 0.2 0.4 0.6 0.8 1
CF/C
T 2
3 4 5 6 7
F*
K=1.3
0 0.2 0.4 0.6 0.8 1
CF/C
T 2
3 4 5 6 7
F*
K=1.5
(Thousand) (Thousand)
Figure 9: Bifurcation diagrams of Imprison model for allocation of police resource. pM=0.5, pF=0.2.
Changes in the security measures are likely to have a greater impact on crime [61]. Let us inves- tigate how the allocation of the police resource to control the activity of major/minor crime affects the distribution of criminals. LetcT be the total budget for security. Also letcM andcF be the budget for the control of minor crime and major crime, respectively. NotecT=cM+cF. We assume that the arrest rate is proportional to the budget used to control the crime. Then we can setaM=pMcMandaF=pFcF
where pM and pF are the police efficiency for minor and major crime, respectively. In the example, pM =0.5 and pF =0.2 are used. Figure 9 shows how the budget ratio cF/cM affects the number of major criminals. We are especially interested in finding the optimal budget ratio at which the number of major criminals is minimized, depending on the prison capacityK. To be specific, increasingKfrom 1300 to 1500 brings down the minimum number of major criminals, from 2636 to 2258.
Another in-prison measure to control crimes is the period of imprisonment for criminals. There are mixed evidence regarding the question of whether spending more time in prison increases the rehabil- itation rate [43, 59]. However, here we assume that the period of imprisonment and the rehabilitation rate is weakly positively correlated, as long as criminals are held in custody in an effectively managed facility. Let us denotew≥1 as a general sentence weight and set the period of imprisonment as
iM=w iminM and iF =w iminF (2.26)
whereiminM =0.5(year) andiminF =5(year) are the minimum period for minor and major criminals, re-
spectively. The sentence weight is also related with the rehabilitation rate. We set as
rM=0.2w+0.2 and rF =0.1w+0.1 (2.27)
so thatrMandrF slightly increases withw. Note that this agrees with Equation 2.21 whenw=1.
(a)β =0.0001 (b)β=0.001
Figure 10: Equilibrium distribution of Imprison model according to the sentence weightw(a) atβ = 0.0001 and (b) atβ =0.001.
Figure 10 shows how the sentence weight contributes to the criminal distribution in two cases: (a) withβ=0.0001 and (b)β =0.001. It is not surprising that the number of inmates increases withwin both cases since a higher sentence weight means a longer detention. More noteworthy differentiation between (a) and (b) is the change inF, the number of major criminals in society, according tow. When the transmission contact rate is as low asβ =0.0001, a higher sentence weight reduces the number of criminals. On the contrary, whenβ=0.001, assigning more sentence weight leads to increase of major crimes in society. Hence, a higher sentence weight has a positive reform effect only when the frequent contact between prisoners is effectively prohibited.
Similarly, we assume that the arrest rate is proportional to the budget used to control the crime. Then we can setaM=eMcM andaF =eFcF whereeM andeF are the police efficiency for minor and major crime, respectively. In the example,eM=1 andeF=0.5 are used. Figure 11 shows how the budget ratio cF/cT affects the number of major criminals. The minimum ofF is achieved at aroundcF/cT ≈0.76.
Spending more portion of the budget for the major crime control brings negligence on minor crimes, which eventually leads to excessive occurrence of major crimes due to the broken windows effect and the crime school effect.
Figure 11: Equilibrium distribution of Imprison model according to the allocation of police resource.
eM=1,eF =0.5
3
Spatio-temporal Analysis of Crime Data
In this chapter, we want to introduce how to analyze and predict the model of a given problem using experimental data or recorded data. First, the statistical characteristics of experimental data or recorded data will be introduced. The data dealing with is crime data, basically, we want to classify crimes and focus on the statistics for each crime. Through this, it is possible to grasp the distribution of basic crime data. In particular, the data to be dealt with in the future is data in the Chicago area, which includes spatial information, that is, geographical location, type of crime, and time of the crime. Second, As a method, we want to use a method called Dynamic Mode Decomposition (DMD). DMD is a method of making predictions and the dominant mode of given data by quickly calculating vast amounts of data through Dimension Reduction. We will use the DMD for Temporal and Spatio-temporal data, then get the DMD Mode which represents the oscillation, growth, or decay modes.
I Overview
Crime data records are accumulated [27, 52] in many years. These data of records highly related to the what factors cause a crime rate which is essential to developing measures to prevent crime in a society.
These records show that felony and misdemeanor differ in many measures such as occurrence rate, arrest rate, and rehabilitation rate. There is also a difference between the control activity of the police devoted to serious crimes and that devoted to minor crimes [13].
II Chicago crime data
First, we will introduce our data set. Our data is from the city of Chicago data from 2015 to 2019.
According to the data from 2019, there are total 260,588 crime records and 32 primal types of crimes are occurs. More precisely, in each year’s data, we have the following data; Case number, the time of the incident, Primary types of crime, arrest record (false and true), domestic record (false and true), And geographical data including community area, beat, district, ward, block, latitude, and longitude.
Here, we focus on primal types, latitude, longitude, and the time of the incident. Then, we take 9 categories from the primal types of crime; Assault, Battery, Domestic crime, Narcotic crime (Narcotics
and other narcotic violations), Rare felony (Homicide, kidnapping, arson, gambling, interference with public officer), Sexual crime (Sexual assault, sexual offense, prostitution, obscenity, offense involving children, and stalking), Theft, and Robbery. Some categories of crime are rarely appeared, and, are identified as similar categories, so we put some things together. For example, Rare Felony and Narcotics.
For reducing the data size, we emerge all the daily data by weekly data.
Figure 1: Visualization of entire crime of Chicago from 2015 to 2019.
Also, we treat the related data, the environmental factor such as weather, unemployment rate, and Dow Jones Industrial Average Index (DJIA). The weather daily data contains temperature, Sensory temperature, humidity, dew point, snow, wind speed, sea level, sunrise time, and sunset time. Here, we focus on the temperature for environmental factors. Also, we take daily DJIA data with open, close, low, and high. After the comparison, there is no difference between each case for our purpose, we take only close value and, called as closing stock market index or, closing stock index for simplicity. Lastly, we have the employment rate of Cook county of Illinois. Since it’s monthly data and about employment, we interpolate into the weekly data by simply using the step function case and taking calculations for unemployment. Thus, we have environmental factor data; temperature, humidity, close DJIA (we later called Closing stock market index for convention), and unemployment rate. We can check statistics in Table 3.1. Theft, Battery, and Domestic crime are the top 3 records in the dataset. In contrast to that, Sexual crime and Rare Felony are appeared rarely in 5 years. This might be related to each crime’s characteristics, or any other factors such as weather, politics, geography, and so on.
In figure 3, we can check that there are clusters; Narcotic, Burglary and Robbery, Sexual Crime, Rare Felony, and others. Here, Rare felony and Sexual crime are more close to Theft, Assault, Battery,
0 50 100 150 200 250 Weekly
0 500 1000 1500 0 500 1000 1500 0 500
Record
Assault Battery Burglary Domestic Narcotic Rare Felony Robbery Theft Sexsual crime
Figure 2: Time series plot for each crime type in Chicago from 2015 to 2020. x-axis is a weekly number.
y-axis number of crime occurrences.
types
statistics
Minimum Maximum Mean Median
Assault 110 509 366 372.5
Battery 383 1276 944.2 948.5
Burglary 66 367 235.3 236.5
Domestic 356 1107 815.1 810.5
Narcotic 108 580 283.0 268.0
Rare Felony 17 84 50.98 50.00
Robbery 90 301 195.1 192.0
Sexual Crime 31 222 110.7 110.0
Theft 411 1646 1172 1188
Table 3.1: Basis Descriptive statistics for City of Chicago Crime record data.
and Domestic. This correlation cluster seems reasonable since burglary and Robbery have some similar situations, Narcotic is far from any other crimes and, Domestic crime might have some more positive correlation with clustered (or closed) crimes. In figure 3.2, we can check that Temperature is positively correlated with Battery, Domestic, Theft, Felony, and Assault. More precisely, Battery and Domestic are the first and second order of positively correlated with weather. This can be thought that Battery and Domestic might be related to the person’s feeling affected by temperature. But the difference is temperature is more correlated with Felony while Dew does with Theft. For closing sotck index, only Assault is near the correlation level which implies that Assault is roughly correlated with DIJA positively.
Narcotic crime is (roughly) positively correlated with the unemployment rate since the value is near
Assault Battery Burglary Domestic Narcotic
Rare Felony Robbery Theft
Sexsual crime Assault
Battery
Burglary
Domestic
Narcotic
Rare Felony
Robbery
Theft
Sexsual crime 0.09029
-0.1296
0.6132
0.08463
0.3574 0.3135
0.05726
0.2367
0.3972 0.09029
0.3135
0.2246
-0.1245
0.2682
0.487
-0.03749 0.2246
-0.005295
0.1986
0.626
0.4583 -0.1296
0.05726
-0.1245
-0.005295
0.1362
-0.3943
-0.277
0.1172 0.6132
0.2682
0.1362
0.1471
0.5655
0.2193 0.08463
0.2367
0.1986
-0.3943
0.1471
0.4835
-0.04387 0.487
0.626
-0.277
0.5655
0.4835
0.2113 0.3574
0.3972
-0.03749
0.4583
0.1172
0.2193
-0.04387
0.2113 1
0.8202
0.7929
0.6937 0.8202
1
0.9155
0.7132
0.686 1
0.771 0.7929
0.9155
1
0.6366 1
0.7132
0.6366
1
0.771
1
0.6937
0.686
1
1
-0.2 0 0.2 0.4 0.6 0.8 1
Figure 3: Correlation matrix visualization of Crime types of Chicago data from 2015 to 2019. In the cell, the Pearson correlation value is listed. from -1 to 1, colorize by white to blue. In a row and column labels, we can check the crime types.
Temperature Closing stock Market Index Unemployment Rate
Assault 0.70719636 0.3528212548 -0.37740793
Battery 0.83311730 -0.0005395683 -0.02460227
Burglary 0.40446485 -0.5057182509 0.38805835
Domestic 0.74859551 0.0738297588 -0.08152754
Narcotic -0.02162125 -0.3973480053 0.41527976
Rare Felony 0.73901782 0.0043284679 -0.02351950
Robbery 0.32945960 -0.3313261324 0.25786004
Sexual Crime 0.27826234 0.0625426173 -0.01338349
Theft 0.72275404 0.2387604646 -0.28460504
Table 3.2: The Pearson correlation result table between time series data and environmental factors.We take the Correlation value level as 0.5. Types in the table are Pearson correlation coefficient values in order. For closing stock market index and unemployment rate, there is no crimes with a correlation coefficient exceeding 0.5. Assault is highest positively correlated with closing index value by 0.35. Last, the unemployment rate in COOK county is correlated with narcotic by 0.42.
the level. This might seems to be the connection between the job and the substance as the common perception of society found in movies, TV, and novels.
III Data analysis using Dynamic Mode Decomposition
When we observe real-world phenomena, the measurement of given phenomena is abundant. Using this huge amount of data, we consider some functions or, governing equations of the system of phenomena in the evolution of time. Dynamical System is a tool for describing phenomena with function or equation i.e, a mathematical framework. Generally, dynamical system is used for analyzing real-world phenom- ena to obtain some purpose related to the situation such as prediction of the future state, estimation of state, control of feedback, and so on. Generally, dynamical system is described by the following form
dx
dt = f(x,t,µ) (3.1)
wherex(t)∈Rn are state vectors at timet,µ is parameters of the system, and f is a vector field that describes the dynamics of the phenomena. Here, This form(3.1)is continuous in time. We also have a discrete-time version of dynamical system
xk+1=F(xk,t) (3.2)
This form(3.2)is also known as a map or, flow.
In the real-world problem, we have abundant data while the dynamics f or flowF remain elusive such as finance, epidemiology, and neuroscience. In other words, we face the problem that the dynamics of dynamical systems may not be known. We more focus on the data to drive the dynamics of given system. Even for the classical problems like fluid and turbulence, we focused on the abundant data of the phenomena because of the lack of the principles in the natural to describe the phenomena, and many techniques are related to data-driven approaches.
To identify the unknown dynamics with abundant data, regression is the most common technique that can be considered in this case. The method that will be introduced in this subsection is the dimensional reduction method related to the regression technique, called Dynamic Mode Decomposition. DMD was first introduced by Schmid [95, 96] in applied fluid dynamics. DMD identifies spatial patterns associated with frequencies, and growth, decay rate, or oscillation movement related to the behavior of the given system. The approach of DMD developed by Schmid [95, 96] is very close to Proper Orthogonal Decomposition (POD) which is the reduction of complexity in fluid dynamics. In [96], DMD algorithms are first introduced and developed by Singular Value Decomposition (SVD). Later then, Tu et al [109] developed theexactDMD. Also, DMD has a connection to the spectral analysis of Koopman Operator [92]. We will introduce each type of DMD algorithm and, propose our algorithm using to analyze the spatio-temporal crime data.