Dynamic Analysis of Criminal Behavior: An Application of Empirical Mode Decomposition

Our objective is to measure the permanent and transitory components of criminality in Brazilian states by using the methodology proposed by At and Chappe (2005). The empirical strategy used follows the Empirical Mode Decomposition method (EMD), proposed by Huang et al. (1998). Based on a sample collected using the Mortality Information System (SIM) from DATASUS, the decomposition process was carried out for the 27 Brazilian states from 1996 to 2015. The results of the decomposition for criminality show that the choice for crime occurs, for the largest part, due to permanent elements, which is a predictor of future crimes over time. The decomposition of criminality into these two types of components establishes some evidence regarding criminal behavior that can serve as reference for policy makers, since the implications of the results found raise questions about the policies to confront and reduce crime.


Introduction
Works on crime are relevant for their nature and the extent to which this issue touches the lives of people, as well as the development of countries since the negative implications of crime disproportionately affect individuals who are most vulnerable (Gramckow, Greene, Marshall, & Barã o, 2016). Soares and Naritomi (2010) point out that the proportion of crimes is noticeably greater in the Latin American and the Caribbean (LAC) regions when compared to the rest of the world. Specifically in Brazil, the growing problem of crime is responsible for bringing insecurity and fear to the entire population. According to data from the DATASUS Mortality Information System (SIM), the homicide rate per 100,000 inhabitants grew by 14% between 1996 and 2015, with a large part of this growth occurring in regions where the crime was not considerable, as in the case of the North and the Northeast.
The research focused on the problem of crime usually approaches the determinants and the costs of crime to society through the cost-benefit analysis featured in Becker (1968), as can be seen in Fajnzylber, Lederman, and Loayza (1998); Araújo Junior and Fajnzylber (2010); Lisboa and Andrade (2000); Resende and Andrade (2011). These works analyze the effects on crime of socioeconomic variables such as income inequality, education, unemployment, urbanization etc. Also, other authors discuss the deterrent effects on crime, such as Lott Jr. (1992); Levitt (1995); Santos (2009) ;Justus, Kahn, and Cerqueira (2016).
the cost in the future by the option for criminal behavior at the present moment.
As a result, one can observe that there are dynamic effects that explain the choice of the criminal agent that is not taken into consideration by a static model. Among some works that use dynamic models, it is possible to mention Leung (1995); O'Flaherty (1998); Mocan, Billups, and Overland (2005); Lee and McCrary (2017).
Choosing to commit a crime reflects the state of nature in which the potential criminal is. Thus, it can be argued that a change in this state will result in alterations in the expected value of the crime, as there will be periods in which the agent will make the option to become a criminal, while in others that individual may abstain from illegal activities (At & Chappe, 2005). Therefore, considering the existence of components that are permanent (long term), transitory (short term), or both, criminality is rarely discussed in the literature. The identification of these components leads to the need for a more adequate approach to government policies since a structural problem cannot be dealt with only short-term measures.
In this sense, this work aims to identify the permanent and transitory components for a series of violent criminality in the states of Brazil. For that purpose, this approach follows the theoretical model proposed by At and Chappe (2005). In the empirical strategy, the Empirical Mode Decomposition -EMD method, proposed by Huang, Shen, Long, Wu, Shih, Zheng, and Liu (1998) will be used. The goal is to find evidence as to which aspects of crime dynamics can be composed of structural issues that require a greater effort for remediation or solution.
In addition to this introductory section, this work features four more sections. The next two sections present the intertemporal model of crime choice and the empirical strategy is discussed. Section four features the empirical results for the decomposition of violent crime. Lastly, the concluding remarks are presented.

Theoretical Model of Crime: An Intertemporal Approach to the Individual's Choice
In the model proposed by At and Chappe (2005), each individual is identified by the benefit they would achieve by making the choice, b ∈ [0, ]. The option for the crime generates damages, , and by definition, < ; in other words, it is socially acceptable some type of offense. Moreover, the model has an infinite number of periods, = 0, 1, … and a continuous discount rate . At = 0, individuals must decide whether to commit or abstain from crime. Individuals who commit the crime are subject to a probability of being detected, , and fine, . If they are caught, the game ends. However, if individuals decide to postpone the option for crime, then the benefit increases at the rate, that is, it becomes ( − ) .
Considering first the decision of individuals to commit the crime immediately, or later, the strategies of the individuals are clearly outlined as follows. First, individuals can decide not to commit the crime. The expected return is then zero. Second, individuals can decide to commit the crime at t = 0. In this case, the expected result is: Third, individuals can decide to wait. Therefore, this strategy generates an expected return: Note that when > , waiting longer to commit the crime would always be a better policy and the optimum would not exist. Consequently, from now on, the case is considered when < . There will be a critical benefit above which individuals will commit the crime immediately, and below which they will commit the crime later.
The critical benefit is defined by the probability of detection, the sanction level, and the growth rates, which defines the critical benefit as ̅ = − . If ∈ (0, ̅ ) individuals commit the crime in * = 1 [ ( − ) ] > 0. If ∈ [ ̅ , +∞) individuals commit the crime at t = 0.
If the benefit is greater than the critical benefit ̅ , then it is ideal to commit the crime immediately, since the benefit to postponing is less than its cost due to the discount, . If the benefit is less than the critical benefit ̅ , then individuals should expect to increase their benefit sufficiently. The discounted benefit increases by one factor ( − ) , while the expected discounted penalty increases by a smaller factor − . In other words, there are situations in which it is preferable to postpone the crime, even if the gain from the option exceeds the punishment expected in the first period.
return to its trajectory. In other words, transitory choices are made according to the state of nature in the period of decision making, given that at some point they may dissipate, leaving a structural component (trend) that is perpetuated throughout the periods.

Empirical Strategy
As could be seen above, the action of committing or abstaining from a crime is considered as an intertemporal process of choice, where the main objective of the criminal agent is to maximize the expected return of the criminal act. This decision-making process, therefore, can be seen as a volatile element, which changes according to the current state of nature. Thus, criminal behavior can be altered by permanent or transitory conditions, which modify an individual's decision to commit a crime.
The condition which is always present and motivates the individual to commit a crime, regardless of the state of nature in which that person is in, is defined as the permanent factor. On the other hand, the condition for criminal behavior that occurs in some periods, but not in others (i.e. anticipating or abstaining from the option of committing a crime), can be defined as fluctuations around a trend, a transitory factor.
The decomposition of the criminality series can be obtained following the recommendations of Engelen (2004) and At and Chappe (2005), where the decision to commit the criminal act can be implemented at different times in the life cycle of an agent (for example, conditions for deciding on criminal behavior is more frequent during the juvenile phase). According to Gottfredson and Hirschi (1986), the chance of committing a crime usually increases until the end of adolescence, and then decreases. Furthermore, Elliott, Huizinga, and Ageton (1985), point out that it is in this stage of life where the highest rate of antisocial behavior is observed.
The understanding that this decision obeys cyclical movements and, thus, may reflect higher intensities according to the state of nature, implies that a series of criminality presents a composition distributed in elements in which the proportions fluctuate along with its trend. This condition is considered in this work, where the specific components for the series of criminality will be defined and empirically identified.
The time series decomposition in this work will follow the proposal of (Huang et al., 1998). The Hilbert-Huang Transform (HHT) is a method that decomposes, through the Empirical Mode Decomposition (EMD), the signal for a given time series in Intrinsic Mode Functions (IMFs). The EMD is a method that can be applied to non-stationary and non-linear data, ensuring that this method obtains an advantage over other methods. Kožić and Sever (2014) point out some advantages and limitations that are related to the EMD technique. One of the advantages is the fact that the technique is self-adaptive, allowing the data to respond to its characteristics, enabling a high degree of clarity, transparency, and intuition; while one of the limitations is the lack of an adequate theoretical basis.

Empirical Mode Decomposition
The Empirical Mode Decomposition is a method that empirically identifies all the behavior of intrinsic oscillations through the temporal characteristics of the signal and, subsequently, decomposes the series accordingly. The calculation presented by Huang et al. (1998) consists of the creation of an envelope defined by local maxima and minima of a finite set of oscillations and subsequent subtraction of the average of those envelopes from the initial set. Unlike the Fourier method and wavelets, EMD does not require any convolution of the signal with a predefined function, and its decomposition process is entirely data-driven.
Each oscillation is derived from the data and is referred to as an IMF. These functions are called intrinsic because each represents the signal change in the time series. An IMF must meet the following conditions: 1). The number of extremes and the number of crossings through the origin must be equal or differ by a maximum of 1.
2). At any point, the average of the envelopes defined by local maxima and local minima must be zero.
IMFs are obtained through a process that makes use of local extremes to remove oscillations, starting with those most frequent. Thus, given a time series ( ), t = 1, 2,..., T, the process decomposes it into a finite number of intrinsic mode functions represented by ( ), = 1,..., n, plus one residue ( ). At the end of the decomposition process, the original time series can be reconstructed as: with ( ) being the permanent component of the series.
The systematic method for extracting intrinsic functions is called the sifting process. This process can be seen in Figure 1 below. The decomposition process comprises the following steps: 1). Initialize the original time series for the residue 0 ( ) = ( ) and set the IMF index k = 1.
3). When the residue ( ) is a constant or a monotonic slope that contains only one extreme, the process is interrupted; otherwise, the decomposition of step 2 continues, defining = + 1. As shown in Figure 1, the selection algorithm identifies and extracts the oscillatory component with the highest local frequency in the data, leaving a partial residue. The successive application of the algorithm on the sequence of partial residues completely decomposes the time series into a set of IMFs plus one residue, represented as: Where ( ) are the intrinsic mode fuctions and ( ) = − ∑ ( ) =1 is the residue, and k is the number of extracted IMFs.
Equation (4) shows that a given time series, , can be divided into two components: ( ) is the component of the criminal act that is created in each period, being understood as a short-term element; ( ) corresponding to the part of the crime that is passed on for each period, which is the structural component of the series. It is believed that the first component is the deviation from the permanent component; that is, the fraction of crime that is generated in the short term and that dissipates itself over time. The second component, therefore, would be ijef.ccsenet.org International Journal of Economics and Finance Vol. 13, No.4;2021 the core of the series, the portion that persists over time and, for that reason, must be addressed through medium and long-term policies.
It is worth mentioning that a measure of transitory/permanent proportion will be presented with the results of the decomposition. When such an index is less than 1, the permanent component is more important, and when it approaches 1, both are equally important.

Database
To measure the permanent and transitory components of crime, data on the homicide rate per 100 thousand inhabitants were used. These data were collected from the DATASUS Mortality Information System (SIM). The decomposition process for the 27 Brazilian states was carried out individually, from 1996 to 2015.

Results of Decomposition for Crime
This subsection presents information about which states had a greater permanent or transitory component and how these respective components behave over time. The results of the decomposition process for the Brazilian states are featured in Table 1 Vol. 13, No.4;2021 Regarding the oscillations that occur around the choices that change over time, it is possible to note that, on average, the largest transitory component corresponds to the State of Roraima, with a value of 4.40. This state reached, in 1999, a maximum of 11.90 for the transitional component, with a minimum of 0.08 in 2010. This result indicates that the expected return on crime changes over the periods in higher proportion in the state of Roraima, which may be above the critical benefit at times, while in others, it is below. This is similar to the model proposed by At and Chappe (2005), where the choice to postpone a crime can be optimal, given that the benefit of waiting for increases sufficiently in the future.
Results of the transitory/permanent ratio are shown in Figure 2 below. As discussed in the review of the theoretical model, there are periods when the agent can make the option for criminal activity, while in others the individual may abstain, due to factors that can change the economic calculation and, consequently, the expected crime value (At & Chappe, 2005). In this regard, Figure 2 displays the relative importance of the permanent and transitory component in the criminality of Brazilian states.  These results present evidence that has not been widely discussed in the literature. The permanent component of crime is that part of the individual's choice that does not change over time, which means that, for states where there is a positive variation in the permanent component, long term policies may be more appropriate for a consistent decrease in the homicide rate, so that its trajectory may be altered. On the other hand, the negative variation observed in some states possibly suggests that a combination of long and short-term policies, such as firearms control, demographic control, and improvements in the job market, may have contribu0t0ed to reverse the growth trend. ijef.ccsenet.org International Journal of Economics and Finance Vol. 13, No.4;2021 Figure 3. Variation of permanent and transitory components, by State (1996State ( /2015 Based on the panorama presented in Figure 3, it is possible to build a ranking to identify the direction of the homicide rate, depending on the variation in the importance of the permanent and transitory component of crime for each state.
For the states that show negative variation in the proportion measure, it is possible to see in Figure 4 that there was an increase in homicide rates over the periods analyzed, except for the states of Amapá  Thus, the literature highlights that structural factors in an individual's life, such as income per capita, income inequality, education, family structure, among others, can permanently alter the economic calculation when deciding on opting to commit criminal activities because factors such as these are related to increases in the opportunity cost of committing a crime (Preston, 1982;Fajnzylber, Lederman, & Loayza, 1998).
Transitory elements, which are occasional changes in the opportunity cost and which cause the decision for committing a crime to be postponed, are associated with the deterrence effect. These elements can be, for example, an ostensive patrolling action or the use of cameras for monitoring streets. According to Levitt (1995), depending on the critical situation of each state, the deterrence effect can be used as an emergency and quick combat measure. Still regarding this effect, Engelen (2014) argues that the increase in these costs for the criminal agent may not be permanent, and the decision to abstain from committing a crime simply represents a postponement of this decision until the ideal moment.
The decomposition of crime into permanent and transitory elements is, thus, relevant for policymakers, as the implications of the results achieved by the decomposition raise questions about the policies to confront and reduce crime. Through the use of information from different compositions of choices presented for the Brazilian states, it is possible to discuss and elaborate a political program to fight crime more adequately, directing financial and human resources in accordance with the nature of the problem.

Conclusion
This work analyzed the social problem of criminality through a time series approach. The objective was to decompose the permanent and transitory components of the crime rate to understand the formation of the dynamic process of choice for the option of committing a criminal act (At & Chappe, 2005). The empirical strategy followed the EMD method of (Huang et al., 1998) Regarding the results of the decomposition, it was found that the choice to commit a crime occurs, to a greater extent, on a permanent basis, and that this component had a negative variation in the Southeast region of Brazil, while in the Northeast region there was an increase. The result that the permanent component prevails over the transitory, paired with the fact that crime has increased in a greater proportion in the North and Northeast regions of Brazil, are indications that crime will hardly stop growing until there is a change in the social structure of these regions.
The decomposition of criminality is still not widely discussed in the literature, thus there are questions to be researched and answered in future works, particularly related to long-term policies, as well as specific initiatives to fight crime. Therefore, identifying how the dynamic choice process behaves in other categories of crimes, in addition to verifying if there is any type of contagion among the different types of crime, may be considered as future topics to further this analysis.