DYNAMIC ANALYSIS OF VEHICLE ROBBERIES AND THEFTS: AN APPROACH WITH SLIDING WINDOWS

Objective: This article aims to jointly analyse the time series of the rates of stolen vehicles and stolen vehicles daily in Salvador, Bahia, Brazil, using DFA and DCCA methods, both with the sliding windows approach. Theoretical Framework: Salvador, the capital of the state of Bahia and the geographic space of the research, has the second largest fleet of motor vehicles in the northeast region of Brazil and the eighth when compared to other municipalities in Brazil. Method: The DFA and ρDCCA with Sliding Windows were used. The DFA is a statistical method that estimates autocorrelation in non-stationary time series on different time scales. Results and Discussion: Through exploratory data analysis, some properties were identified, such as positive asymmetry, stationarity, and nonstationarity depending on the year and crime assessed, as well as inverse fluctuation over the years between the average rates of stolen and stolen vehicles. The sliding windows approach identified greater relative variability around the average vehicle theft rate from 2004 to 2015 for w= 365 and from 2004 to 2016 for w = 1000 and a higher frequency of persistent autocorrelation (αDFA >0.50) (w=365 and w=1000). While the level of cross-correlation varied qualitatively between positive (ρDCCA (n) >0)


INTRODUCTION
Crime is a socio-economic phenomenon that occurs in various locations around the world, in large urban centres (Grubesic & Mack, 2008;Saraiva et al., 2021;van Dijk et al., 2021) and is also considered one of the main contemporary problems (Santos & Kassouf, 2012).
However, depending on the geographic space and/or period, its manifestation occurs with different magnitudes.For example, the recent study conducted by van Dijk et al. (2021) in 166 countries proved that crime is more prevalent in African and Latin American countries, followed by Asian countries.According to the same authors, European, North American, and Oceanic countries have a prevalence of crime at intermediate or low levels.
As a methodological option, this investigation will analyse crime statistics over time, specifically crimes against property and vehicle theft.According to Santos & Kassouf (2012) and Machado et al. (2017), vehicle theft and vehicle robbery crimes are generally used as an indicator of crime, mainly due to the economic value of the stolen property and low underreporting, since the victim needs to recover the vehicle and/or calling the insurance 4 company must generally register the crime with a police authority through a police report.
According to Mao et al. (2018), crimes related to vehicle theft provide direct uninsured financial losses, the opportunity cost of the time needed to deal with the crime, the opportunity cost of the temporary unavailability of a vehicle and the costs psychological aspects of victimisation.
Vehicle subtraction (vehicle robbery and/or vehicle theft) has been evaluated by researchers, considering different methodologies in the statistical analysis.In this sense, it is possible to cite: The research by Cohen et al. (1985) that estimated the statistical difference between two procedures to calculate the rates of car thefts and robberies over 27 years in the United States; The study by Tseng et al. (2023)  Here the crimes of theft and vehicle robbery will be supported by an area of knowledge called Time Series Analysis (an area of statistics that studies and models variables as a function of time) (Box & Jenkins, 1976;Morettin & Toloi, 2006;da Silva Filho, 2014).To characterise the fluctuations of time series with the sliding windows approach (dynamic analysis), descriptive analysis (mean and relative variation around the mean) will be used, the Detrended Fluctuation Analysis-DFA method, designed by Peng et al. (1994), which is capable of estimating autocorrelation on different n time scales of non-stationary series and the cross-correlation coefficient ρDCCA implemented by Zebende (2011), which estimates the cross-correlation level of two-time series also in non-stationary regime.
It is possible to cite the works (da Silva Filho et al., 2021;Machado Filho et al., 2014;Machado et al., 2017;da Silva Filho, 2014) that evaluated crime on different time scales in series with trend and seasonality characteristics.
To contribute to studies related to the analysis and modelling of criminal data, this article aims to jointly analyse the time series of thefts and robberies of vehicles (both crimes against property) daily in Salvador, Bahia, Brazil in the period of January 2004 to December 2022 with the sliding windows approach.In the next section, the methodology will be described (Section 2); in the subsequent section, the results and discussions will be displayed (Section 3); and, finally, the conclusions will be presented (Section 4).

CHARACTERISTICS OF SALVADOR
Salvador (Figure 1), the geographic area of the research, is the capital of the State of Bahia with 2,418,005 inhabitants; according to the 2020 demographic census, it is also the most populous municipality in the Northeast region and the fifth municipality most populous in the country with a demographic density of 3,486.96inhabitants per Km 2 (IBGE, 2022).In terms of work and income, Salvador recorded a monthly salary of 3.2 minimum wages in 2021, among workers with a formal contract, leaving the municipality in fourth position in relation to the state of Bahia and ninety-sixth position in the national ranking.The information mentioned in this section reveals the importance of the capital of Bahia on the national scene, thus justifying its choice as the geographic space for the research.

DATA
The historical series of robberies and vehicle thefts in Salvador from January 2004 to December 2022 was acquired from the Public Security Secretariat of the state of Bahia (SSP-BA, 2022).To conduct joint modelling between the theft and vehicle theft time series, the data were paired (series recorded on the same date) (see Figure 2).

7
The vehicle theft indicator, in this research, is denoted as the sum of all theft occurrences (theft, for oneself or for others, of someone else's movable property) in which a land-based motor vehicle was taken: passenger car, taxi, pickup truck or unladen truck, public transport vehicle, motorcycle, scooter, etc. (Brazilian Penal Code, art.155).Vehicle theft is the sum of all incidents of robbery (robbery of another's movable property, for oneself or for another, through serious threat or violence against the person, or after having, by any means, reduced it to the impossibility of resistance) in from which the following were subtracted: land motor vehicle without transported cargo: passenger car, pickup truck, empty truck, public transport vehicle, motorcycle, scooter, etc. (Brazilian Penal Code, art.157) (Machado et al., 2017).In the subsequent section, the properties of the DFA method, the ρDCCA coefficient, and aspects of the sliding windows procedure will be presented.

DFA AND ρDCCA -SLIDING WINDOWS
DFA is a statistical method capable of estimating autocorrelation in non-stationary time series on different time scales.The DFA method is based on the fluctuation function,   (), and calculated by the following algorithm (Peng et al.,1994): 1 -Consider a time series, {()}, where  = 1, … , , and  is the length of the time series.We integrate () to obtain 2 -The integrated series () is divided into boxes of equal length  (time scale).
3 -For each box of size , we adjust (), using a polynomial function of order , which represents the trend in the box.The  coordinate of the fit line in each box is denoted by   (), as we use a polynomial fit of order , we denote the algorithm by DFA-.
4 -The integrated signal () is subtracted by the local trend   () in each bin (of length ).
5 -For a given box of size , the root mean square of the fluctuation, (), for this integrated and detrended signal is given by 6 -This calculation is repeated for a wide range of scales (boxes of size ) to provide a relationship between   and the box size .Here  varied from 4 <  <  4 .
For the DFA method, if long-range autocorrelation appears, then   ~   .Thus, using the aforementioned steps, it is possible to estimate the autocorrelation exponent α of the DFA method with the following interpretations (da Silva Filho et al., 2021;Walleczek, 2000;Santos et al., 2022;Filho et al., 2023): α > 0.5 long-range persistent behaviour; α < 0.5 long-range anti-persistent behavior, anti-correlated signal; α = 0.5 no autocorrelation (no memory); α ≅ 1.0 denotes time series with pink noise; α > 1.0 non-stationary series with random walks; α ≅ 1.5 it is possible to characterize the time series as Brownian noise.
According to da Silva Filho et al. (2021), using the exponent of the DFA method it is possible to estimate the degree to which variations in the past series can influence variations in the future series.And, due to its application in different areas of knowledge (da Silva Filho et al., 2021;Santos et al., 2022;Heneghan & McDarby, 2000;Mirzayof & Ashkenazy, 2010;Toledo et al., 2022) , some authors have evaluated statistical properties of DFA in different aspects (Carbone et al., 2004;Kristoufek, 2010).For more details on the DFA algorithm and properties, see (da Silva Filho et al., 2021;Filho et al., 2023;Filho et al., 2023;Kristoufek, 2010) .
If the researcher is interested in measuring cross-correlation of two time series  and  of the same size  , in a non-stationary regime, a coefficient proposed by Zebende (2011) can be used which is based on DFA (Peng et al., 1994) and Detrended Cross-Correlation Analysis (DCCA) (Podobnik & Stanley, 2008).To apply the crosscorrelation coefficient ρDCCA(), it is necessary to follow the following steps (Zebende, 2011;Filho et al., 2023;Toledo et al., 2022 (2) And to quantify the level of cross-correlation between two time series, in a nonstationary regime, with the coefficient ρDCCA() it is necessary to use the following equation (Zebende, 2011;Filho et al., 2023;Toledo et al., 2022;Zebende & da Silva Filho, 2018;Zebende et al., 2020): (3) In the equation 3  2   () denotes the detrended covariance function of the DCCA method (Podobnik & Stanley, 2008) between the time series () and () while   () represents the detrended autocorrelation function of the DFA method of the  series and   () of the  series.Another relevant characteristic of the coefficient ρDCCA() is the fact that it is possible to measure cross-correlation on different time scales of size  (Zebende, 2011;Filho et al., 2023).
Like other cross-correlation coefficients (Pearson, 1896), the values of the coefficient ρDCCA() have a range of variations between − 1.0 ≤ ρDCCA() ≤ 1.0 In this case, ρDCCA() = 0 means that there is no cross-correlation between the series  and , if ρDCCA() ≠ 0 the correlation can be classified as positive or negative.
And, unlike the Pearson correlation coefficient (Pearson, 1896), the ρDCCA(n) in its calculation considers the order of the pairs of the observations of the time series  and  (da Silva Filho, 2014).Furthermore, there is a generalisation of ρDCCA(n) idealised by Zebende & da Silva Filho (2018), which can be applied to more than two non-stationary time series, known in the literature as multiple cross-correlation coefficient   2 (for more details see (Zebende & da Silva Filho, 2018;Brito et al., 2019;da Silva Filho et al., 2021;Guedes et al., 2021;Guedes et al., 2022).
Recently, researchers have modelled the DFA method (da Silva Filho et al., 2021;Santos et al., 2022;Anjos, 2013;dos Anjos et al., 2015) and the coefficient ρDCCA(n) (Guedes & Zebende, 2019;Guedes et al., 2021;Guedes, 2019) with the sliding windows procedure.Here, sliding windows consists of a statistical procedure that can be repeated systematically on one or more time series of size .The objective of this methodology is to

Figure 3
Illustration of the DFA procedure with sliding windows.Graph "A" presents the original ARFIMA series (2,d,1) process with p(0.9;-0.5);d=0.49and q=0.5 (see (Guedes et al., 2021)) and the graph "B" shows the time series of DFA exponents based on the sliding window procedure.Results generated with the R language with the package called SlidingWindow developed by (Guedes et al., 2021).
Graph B in Figure 3 presents a time series of exponents of the DFA method for a window of size =1000, in this way, it is possible to have a dynamic analysis of the autocorrelation of the series object of study (da Silva Filho et al., 2021;Santos et al., 2022;Guedes, 2019).

Figure 4
Representation of the ρDCCA procedure with sliding windows (w=1000).
The number of  interactions of the sliding windows approach for a time series of size  (see Figure 3 and Figure 4) is defined by the following expression (Guedes & Zebende, 2019): According to (Guedes & Zebende, 2019;Guedes et al., 2021), for example, for pairs of time series of the same size  = 5000 and a window of  = 1000 it is possible to obtain a time series from ρDCCA(n) with 4001 points.Just like the works of (Guedes & Zebende, 2019;Guedes et al., 2021;Ferreira, 2021), in this research the time scale of size  for ρDCCA(n) varied from 4 <  <  4 .
In the next section, the results and discussions will be presented so that it is possible to characterize the fluctuations in the time series of crimes evaluated by this research, considering properties related to variability, normality, trend, autocorrelation and crosscorrelation.

EXPLORATORY DATA ANALYSIS
To evaluate the fluctuations in robberies and vehicle thefts recorded daily by the Public Security Secretariat of Bahia (SSP-BA, 2022) in the municipality of Salvador from January 2004 to December 2022, the daily crime rate was first calculated: In the equation 5   denotes the number of vehicles stolen or stolen in each period and   the vehicle fleet in Salvador for the year analyzed.
To meet the objective of this research, descriptive statistics were initially carried out on the series provided by the Public Security Secretariat of Bahia (SSP-BA, 2022), in order to measure crime fluctuations considering some descriptive measures (mean, standard deviation, asymmetry, kurtosis, coefficient of variation, data normality (Shapiro et al., 1968;Gavrilov & Pusev, 2014) and stationarity (Kwiatkowski et al., 1992;Trapletti & Hornik, 2019) throughout the period (see Figure 5).As a complementary analysis, descriptive measures of the daily series of crimes from year to year were defined (mean, coefficient of variation, coefficient of asymmetry and KPSS stationarity test) (Figure 6).

Figure 6
Annual descriptive statistics of the rates of robbery and theft vehicles per 100 thousand vehicles in Salvador, Bahia, Brazil between 2004and 2022. Source: (SSP-BA, 2022).
As mentioned previously, the descriptive results reported in this section are typical of a time series of non-stationary behaviour and converged in these aspects in relation to research on crime (Machado Filho et al., 2014;Machado et al., 2017;da Silva Filho 201;Oliveira et al., 2020).
An increasing trend in vehicle robbery rates relative to the vehicle theft rate was observed in the period 2008 to 2016 (see Figure 5-A and Figure 5-C).According to Machado et al., (2017), this is worrying because vehicle theft is characterized by a threat to the stolen vehicle's transporter and/or owner.From 2016 onwards, there was a reversal of this trend, when the theft rate showed an increasing trend and the robbery rate decreased.This reversal of the trend was due to criminals' greater dominance of technology.

DESCRIPTIVE STATISTICS -SLIDING WINDOWS
In order to dynamically analyse the historical series of the rates of robbery and theft vehicles in Salvador, sliding descriptive statistics were performed for a window of 365 days ( =365) (1 year) and 1000 days (=1000) as a function of the mean and the coefficient of relative variation (Figure 7).The ( =1000) window was used in this and subsequent sections with the purpose of dynamically identifying the behaviour of fluctuations in the time series under study with a greater number of points.

Figure 7
Average and coefficient of variation using the sliding windows approach (w =365 and w =1000) of the rates of theft vehicles and robbed vehicles per 100 thousand vehicles in Salvador, Bahia, Brazil between 2004and 2022. Source: (SSP-BA, 2022).For a window of  =1000 days, the time series of the αDFA exponents of crime rates were persistent (100%) for the entire period with the highest average for vehicle theft (0.80) (Figure 8-B), lower asymmetry (As=0.92)and lower relative variability (CV=20%) in vehicle robbery (Figure 8-D).With these findings, it is possible to state that the fluctuations of the αDFA exponent for  =1000 were more stable when compared to  =365 days of the vehicle robbery variable.

Figure 8
The behaviour of the exponent αDFA with the sliding window approach in the time series of the rates of robbery and theft vehicles in Salvador-BA, in the period between the years 2004 and 2022 for a window of size w =365 and w =1000 days.Source: (SSP-BA, 2022).
Notes: Here SD=standard deviation, CV=coefficient of variation and As=coefficient of asymmetry; The vertical and horizontal lines in yellow denote the exponent αDFA=0.50.
With the results presented in Figure 8, we can state that, on average, if there is a tendency for growth or decrease in the fluctuations in the rates of robbery and theft of vehicles, this behaviour tends to occur, in the long term, with greater intensity in the vehicle robbery time series.
Using the sliding windows methodology ( =365 and  =1000), in the time series of robbery and theft vehicle rates, it was also possible to verify αDFA close to 1 (pink noise), which denotes a transition between stationary and non-stationary behaviour (Figures 8 -A, B, C and D).Also αDFA exponents greater than 1, mainly for  =1000 from the year 2019 onwards (which means non-stationary persistent behaviour, with random paths) (see (Santos et al., 202;Filho et al., 2023;Ferreira, 2021;Walleczek, 2000)).
As mentioned previously, the fluctuations of the αDFA exponent, in general, showed smaller relative variations around the average exponent for a window of  =1000 (Figure 8 According to (Ferreira, 2021), this result may be related to the fact that DFA is more appropriate for time series with a size greater than or equal to 1000 ( 1000).And this statistical property of DFA was proven in the research of (Kristoufek, 2010).
The results found here are in accordance with the characteristics of time series relating to crime: predominance of persistent behaviour and non-stationarity.And such results converged with the research of (da Silva Filho et al., 2021;Machado Filho et al., 2014;Machado et al., 2017;da Silva Filho, 2014;Filho, 2009).

ρDCCA(n) -SLIDING WINDOWS
In this section, the series of vehicle robbery and vehicle theft crime rates will be jointly analyzed using the coefficient ρDCCA(n), with the aim of quantifying the levels of cross-correlations between the two historical series, with an approach of sliding windows, through heat graphs.
Table 1 Interpretation of the coefficient ρDCCA(n) according to the levels of cross-correlations.
Using Figure 9, it is possible to characterise the cross-correlation according to the coefficient ρDCCA for a window of  =365 (Figure 9-A) and for  =1000 days (Figure 9-B) between the rates of robbed and stolen vehicles daily in Salvador-BA, from 2004 to 2022, for different time scales ().
In general, the analysis of  Behavior of the ρDCCA coefficient with the sliding windows approach (w =365-graphic A and w =1000-graphic B) of the rates of daily robbery and theft vehicles in Salvador-BA, 2004to 2022Source: (SSP-BA, 2022).Data processed by the authors.
As in the research by (Guedes et al., 2021;Ferreira, 2021), by methodological choice, the results of the cross-correlation coefficient ρDCCA(n) were presented with the sliding windows approach ( =365 and  =1000) for specific size scales (), as a complementary analysis (see Figure 10 and Figure 11).22

Figure 10
The behaviour of the ρDCCA coefficient with the sliding windows approach (w =365), with specific time scales (n) of the time series of the rates of robbery and theft vehicles in Salvador-BA, 2004 to 2022 (n=7-graph A, n=90-graph B andn=30-graph C).Here As=Asymmetry, CV=(Standard deviation/mean )x100.Source: (SSP-BA, 2022).
In Figure 10 it can be seen that the smallest variabilities are present on the smaller scales ( =7 (Figure 10-A) and  =30 (Figure 10-C)) and the greatest variability on the scale corresponding to ninety days (Figure 10-B))(CV=266.22%),being also the only one that obtained negative asymmetry (concentration of ρvalues DCCA above average).
Highlight for the year 2020 with ρDCCA=1 for all scales evaluated (Figure 10) and in the periods between the years 2015 and 2016 higher negative correlations for  = 90 (Figure 9 -B).

Figure 11
Behaviour of the ρDCCA coefficient with the sliding windows approach (w =1000) with specific time scales (n) of the time series of the rates of robbery and theft vehicles in Salvador-BA-2004 to 2022 (n=7-graphic A, n=90-graphic B, n=30-graphic C andn=250-graphic D).Source: (SSP-BA, 2022).
In figure 11 there was a similar result to that found in figure 10 (Guedes et al., 2021;Ferreira, 2021), which found greater variations, as a function of time, in the ρDCCA coefficient also on the largest time scales.
Regarding ρDCCA in the time series of theft and vehicle theft, our results converge with the research by (Machado et al., 2017), who also evaluated vehicle theft and vehicle robbery, in the following aspects: quantitative and qualitative variability in the cross-correlation values; inversion of correlation values in some evaluated periods.
It is important to highlight that, to date, the use of the sliding windows approach with ρDCCA, specifically with criminal data, has not been found in the literature, which reveals the innovative nature of the present research.

CONCLUSION
In this research, we investigated the properties of fluctuations in the daily rates of robbed and theft vehicles in Salvador-BA from January 2004 to December 2022 using the sliding windows approach (  =365 and  =1000).The approach was conducted with descriptive statistics, DFA and ρDCCA.
The descriptive and exploratory analysis of the data identified an inverse fluctuation between the rates of robbed and stolen vehicles in certain periods.And in addition, typical behaviour of non-stationary time series.The results of the sliding descriptive statistics were convergent to those obtained in the exploratory analysis of annual data and qualitatively the descriptive measures (mean and coefficient of variation) tended in the same direction, but with less variability in the window size  =1000.
The DFA with sliding windows ( =365 and  =1000) found a predominance of persistent behaviour (αDFA>0.50) in the time series rates of robbed and stolen vehicles with lower variability in the =1000 window, corroborating some studies that used DFA to measure autocorrelation for the same window sizes (see (da Silva Filho et al., 2021;Ferreira, 2021).
The cross-correlation with the sliding windows approach, between the series object of this research, showed quantitative and qualitative variability depending on the temporal scale (n), window size (=365 and  =1000) and year assessed.Highlighting the window of size  =1000 which presented, in general, less variation in the fluctuations of the ρDCCA time series.
The findings presented here converged with the research on crime by (da Silva Filho et al., 2021;Machado Filho et al., 2014;Machado et al., 2017;da Silva Filho, 2014;Filho, 2009) in some aspects, mainly related to autocorrelation and cross-correlation.
The main limitation of this research is related to the use of administrative records relating to the crime (occurrence report).And according to (Kahn, 2000), not all crimes that occur in society are recorded.However, the crimes studied in this research are those with one of the lowest underreporting statistics, due to the need to register the vehicle theft with a police authority to enable, for example, claims with the insurance company.Therefore, the results presented in this research are as close to reality as possible.
used a set of data mining techniques to predict vehicle thefts based on the behavior of the perpetrators and other variables; The investigation by Sundt (2022) that analyzed an interrupted time series of robbery vehicles in California from 2011 to 2018 with the aim of replicating previous research that found a moderate to strong association between the time series of stolen automobiles and adoption of the law realignment of public security; The Brazilian Public Security yearbook as a descriptive analysis of robbery and thefts of vehicles by federation unit in the year 2022 (FBSP, 2023); The article by Mao et al. (2018) used a geographic information system, regression model and time series analysis in order to identify spatial-temporal patterns in vehicle theft in a location in Shanghai, China, among others.However, despite the importance of statistics on crimes related to vehicle subtraction, to date, no studies have been identified in the literature that have evaluated vehicle robbery and vehicle theft in conjunction with the sliding window procedure, which reveals the innovative nature of the present research.

Figure 1
Figure 1 displays Salvador's location map with relevant information: airport, railway section, drainage, road system, urban area, and body of water.According to data from the Ministry of Transport (SENATRAN, 2022), Salvador in 2022 had a fleet of 1,006,292 registered vehicles (20.59% of Bahia's fleet), occupying the first position in the state of Bahia and eighth position in the ranking national (0.87% of Brazil's fleet).In relation to the municipalities that make up the northeast region of Brazil, the vehicle fleet in the capital of Bahia in the year 2022 occupied the second position (5.95% of the fleet in the northeast), second only to Fortaleza, the capital of Ceará.Salvador's vehicle fleet from 2004 to 2022 recorded an average growth rate of 4.46% (geometric mean).

Figure 1
Figure 1Location map of the municipality ofSalvador, Bahia, Brazil.

Figure 2
Figure 2Original daily series of the variables studied, in the period between 2004 and 2022.Source: (SSP-BA, 2022).

Dynamic
Analysis of Vehicle Robberies and Thefts: An Approach with Sliding Windows ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.7 | p.1-30 | e08202 | 2024.11have a dynamic view of a statistical measure (mean, standard deviation, asymmetry, kurtosis, autocorrelation, cross-correlation, coefficient of variation, among others) as a function of time () for a window of size ()(Guedes & Zebende, 2019; da Silva Filho et   al., 2021;Santos et al., 2022).For a better understanding of the sliding windows approach, consider figure 3, which illustrates this procedure for the exponent of the DFA method.

Figure 5
Figure 5 Descriptive statistics of the rates of stolen and stolen vehicles per 100 thousand vehicles in Salvador, Bahia, Brazil from January 2004 to December 2022.Source: (SSP-BA, 2022).Notes: The yellow dotted lines in graphs A, B, C and D denote the series averages in the period; Data processed by the authors.
From 2006 onwards, except for 2020, a clear inverse trend was identified in the fluctuations in the average rates of vehicle thefts and vehicle thefts recorded in Salvador (Figure6-A).A similar result was also found in the time series of crime rate variation coefficients (Figure6-B).In relation to the rate asymmetry coefficient, there was a predominance of positive asymmetry in the time series of the present research (Figure6-C).In the period from 2004 to 2022, the crime rates in this research were characterized as stationary (p-value>0.05) and non-stationary (p-value<0.05),according to the KPSS test (Figure6-D).

Figure 7
Figure 7 illustrates the behaviour of the mean and coefficient of relative variation of the crimes under study for a moving window of =365 and =1000 days.Using the sliding description for both windows, it was possible to observe convergent behaviour with that obtained in the exploratory analysis of annual data: An inversion in the average rate fluctuations (Figure 7-A and 7-C ); Greater relative variability in vehicle theft rates until approximately the second half of 2016, and after this period greater relative variability in vehicle robbery rates (Figure 7-B and 7-D).The qualitative analysis of the descriptive measures (mean and coefficient of relative variation) tended towards the same direction in both windows ( =365 and  =1000), -B and Figure 8-D) than for a window of  =365 (Figure 8-A and Figure 8-C).
Figure 9 allows us to verify greater variability in the fluctuations of ρDCCA in the window of  =365 (Figure 9-A) in relation to the window of size  =1000 (Figure 9-B).In this aspect, the findings presented in Figure 9-A and Figure 9-B reveal the complexity of the behaviour of fluctuations in the series covered by this research.Specifically for  =365 days (Figure 9-A) from 2004 to 2010 there was a predominance of positive cross-correlation (ρDCCA>0) varying qualitatively from very weak to moderate, and from 2010 to 2019, it varied -0.80<ρDCCA<1 (very weak to very strong) and from 2019 onwards a greater occurrence of positive correlation (ρDCCA>0) varying -0.20<ρDCCA<0.80(weak, moderate and strong).The cross-correlation for  =1000 (Figure 9-B) between the time series of crimes that are the subject of this research was characterized, more frequently from 2006 to 2015, as a positive correlation (ρ DCCA>0) with variation field 0<ρDCCA<0.60,with some islands denoted as negative correlation (ρDCCA<0).Still, in relation to Figure 9-B, it was also possible to identify a clear presence of a negative correlation between the years 2015 and 2018 for all scales evaluated (4 <n < 250) and from the period 2019 to the end of the series ρDCCA varying -0.20<ρDCCA<0.60(weak, moderate and strong).

Figure 9
Figure 9 , that is, greater variability in the fluctuations of ρDCCA on the largest scales (n > 30) and between the years 2015 and 2018 occurrence of higher negative correlation values for these scales (Figure 11-B and Figure 11-D).The results illustrated in Figures 10 and 11 for specific scales converge in relation to the research by

Finally
, given what was presented in this research, we are offering those interested in the topic another way to analyze data related to crime, specifically the indicators of robbed and stolen vehicles.In this way, providing information that can guide the development of public policies and more effective strategies to combat and reduce crimes studied by public and private agents.