The red dot indicates the DMD mode of the spatio-temporal drug-related crime data. In particular, based on data that can be confirmed in real life, the statistical values related to the basic characteristics of the data and the correlations between crimes are analyzed.
Broken Window Effect
The model (2.1) is a simple linear system and one can easily find the corresponding equilibrium solution as. In this subsection we focus on the basic model (2.1) which is now called Model With Imprisonment.
Transitions according to prison density
In (i), each equilibrium of the dynamics in the normal/crowd regime appears in the same regime. Since the dynamics in one regime have no influence in the other regime, neither equilibrium is available.
Bifurcation Analysis
When β1 is small, both P?(β0) and P?(β1) are less than Kan and therefore a unique equilibrium (M?(β0),F?(β0)) is in the normal regime. In the basic model (2.1) we now consider the more group for the imprisoned criminal since without of the group.
Broken Window Effect
We also consider incarceration of criminals, we now take into account the period of imprisonment for each criminal. LetiM>0 and iF >0 should be the prison term for minor and serious offenders. The equilibrium is locally asymptotically stable either for sufficiently small values of aM and aF or for sufficiently large values of iM and iF.
Setting the right-hand side of equation (2.18) zero gives a unique stable solution (N?,M?,F?,PM?,PF?) in the positive domain. For the equilibrium point to be asymptotically stable, all the roots of the equation a4λ4+a3λ3+ a2λ2+a1λ+a0=0 must appear on the left side of the complex plane. While the last two inequalities proved to be true with many numerical tests and are likely to be true for all possible values of parameters, their rigorous proofs remain to be explored.
Since the coefficients a0,· · ·,a4 can be considered as continuous functions of aM and aF, the last two inequalities are satisfied for sufficiently small values of aMan and aF. In the same way, one can show that the last two inequalities are also true for sufficiently large values of iM and iF.
Explicit crime school effect
The population of uncaught criminals can be estimated from the number of prisoners. The following theorem shows that the distribution of criminals becomes more complex if juvenile criminals in prison can turn into potential criminals. Since potential criminals PM0 are officially considered minor criminals, an estimate of the criminal population based on the number of minor and major criminals as shown in equation (2.22) results in an underestimation of the number of minor criminals in society.
Now, for the equilibrium point to be asymptotically stable, all the roots of an equation a5λ5+a4λ4+ a3λ3+a2λ2+a1λ+a0=0 must have negative real parts, that is, appear on the left side of the complex. From the Routh-Hurwitz stability criterion, we have the following necessary and sufficient condition. Note that in the model (2.21) we have a unique stable equilibrium point, if all the parameters are positive except b=β =0.
Asymptotic behavior
How effective imprisonment is in reducing crime is arguably one of the most important issues in criminology. The disappearance of N=0 non-criminals is not quite reliable and is caused by the extreme setting of the parameters. Although we do not consider the birth and death of the population in our model for simplicity of analysis, numerical simulations show that adding the process of birth and death to the model can alleviate this situation and lead to the survival of a part of civilians under domination. of criminals in the balance.
One of the possible scenarios for this is to stigmatize people who have once committed crimes and hardly accept them as part of society. This also sends a signal of strong precaution to people that even with one minor criminal activity they can be expelled from society, leading to a low transition rate. PM−βPM PF+PM0 εdPF. PM0 +βPM PF+PM0 The system has an equilibrium solution. T-N).
Heavy rehabilitation of criminals with strict segregation eventually divides the population into some criminals and criminals. However, in order to have a higher proportion of non-criminals, we must have a relatively higher rehabilitation rate than the transition rate, i.e. k2>k3.
Bifurcation Analysis
3×105. Considering the large values of sensitivity indices, we adopt relatively small intervals such as 0≤b≤0.001 and 0≤β ≤0.002 in the following bifurcation diagrams. This implies that eliminating the environmental factors that predispose to offending is important to prevent the occurrence of more serious crimes. This implies that the effect of broken widows becomes even more important in a secure society where the ratio of major criminals is relatively low.
While repression b promotes a deterrent effect on serious crime outside prison, prison control can act as a more practical measure against crime. The bifurcation diagram in Figure 8 shows how the distribution of criminals changes with the prison contact rate β.
Police Resource Optimization
Spending a larger portion of the budget on major crime causes neglect of minor crimes, which ultimately leads to an over-incidence of major crimes due to the broken windows effect and the penal school effect. In particular, the data to be processed in the future is data in the Chicago area, which includes spatial information, that is, geographic location, type of crime and time of the crime. There is also a difference between police control activity dealing with serious crimes and that dedicated to minor crimes [13].
More precisely, in the data of each year we have these data; Case number, time of incident, Major crime types, arrest record (false and true), domestic record (false and true), and geographic data including community area, beat, district, neighborhood , block, latitude and longitude. Using this large amount of data, we consider several functions or governing equations of the system of phenomena in time evolution. In general, the dynamic system is used to analyze real-world phenomena to obtain a situation-related goal such as future state prediction, state assessment, feedback control, etc.
Even for the classic problems like fluid and turbulence, we focused on the abundant data of the phenomena due to the lack of the principles in the natural to describe the phenomena, and many techniques are related to data-driven approaches. DMD identifies spatial patterns associated with frequencies and growth, decay rate, or oscillatory motion related to the behavior of the given system.
Dynamic Mode Decomposition
For the application of Dynamical Mode Decomposition, such as fluid dynamics or nonlinear dynamics, which are typically used, the number of data points is much larger than the number of snapshots, i.e. nm. Here matrixA is a related matrix of the discrete-time system of locally approximate linear dynamics of a dynamical system of form (3.1), collected the given snapshots of data with dynamics f which may or may not be known. Then the dynamic mode decomposition is the eigendecomposition of the best-fitting linear operator A of relation X2≈AX1i.e, A=X2X1†.
With this idea, the diagonalizability of A is an important part of reconstructing or predicting the future state from the initial condition x0. From the data array definition, columns X1 and X2 share the same columns. So each column in X2 can be written as a linear combination of the columns in X1, except for the last column xm.
Also, the columns of the Urare POD matrix modes, and the columns of the orthonormal Vrare matrix. Formula (3.21) is called the exact DMD mode since these are the exact eigenvectors of the matrix A[109].
Temporal DMD
Spatio-temporal DMD
Temporal analysis
We can check whether some types of reconstruction are similar to the original data. We can also check another prediction attempt in Fig. 5 in Appendices I for temporal (4-year window length). In Fig. 3 in Appendices I, we can first check whether multiple singular values are a large fraction of singular values.
In Figure 4 in Appendices I, we can check if there is a dominant mode in the DMD modes. For the DMD eigenvalue of the dominant mode (red point in Figure 8), the DMD eigenvalue is real and less than 1. From this plot, we can check the motion of the scaled modes with respect to time.
Now finally we check the Correlation of dominant DMD modes plotted in figure 9. In figure 10 we can check that there are clusters; Domestic-Burglary-Robbery, Assault-Theft, Narcotic-Rare Crime, and Battery-Sexual Crime.
Spatio-temporal analysis
This implies dominant mode effects on the decay and oscillation of the spatiotemporal data. This can be used for government or police campaign or patrol in real world. The red dot indicates the DMD mode of the spatiotemporal sex crimes data.
Moreover, in Figures 13 to 15, the ellipse region is drawn by the visibility of the high intensity. This is a more suitable form for the DMD application, as the length of the snapshot is much larger than the number of data snapshots. 37] Jacinta M Gau and Travis C Pratt, Revisiting broken windows theory: examining the sources of the discriminant validity of perceived disorder and crime, Journal of Criminal Justice38(2010), no.
69] David McMillon, Carl P Simon and Jeffrey Morenoff, Modeling the underlying dynamics of crime spread, PloS one9(2014), no. 104] J Sooknanan, B Bhatt and DMG Comissiong, Catching a gang – a mathematical model of the spread of gangs in a population treated as an infectious disease, International Journal of Pure and Applied Mathematics83(2013), No. Thank you very much for capturing the details of the thesis and the parts I forgot in the big framework and for giving advice.
Thank you very much for your basic knowledge of the research you want to do in the future and for your important advice in life.
