# Information Dynamics in Urban Crime

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Crime Pattern Theory and Related Perspectives

#### 2.2. Multifractal Analysis

#### 2.2.1. Multifractal Spectrum

#### 2.2.2. Information Dimension ${D}_{1}$

#### 2.3. Analysis of Observed Chaotic Data

#### 2.3.1. Taken’s Theorem

#### 2.3.2. Average Mutual Information

#### 2.3.3. False Nearest Neighbors

## 3. Materials and Methods

#### 3.1. Criminal Reports

#### 3.2. MF Time Series

**Definition.**

#### 3.3. MF-A2-OCD Method

- Generate the OCR: Given the record of urban crime complaints in a time window $\Delta t$, a temporary scale ${t}_{s}$ is defined for the construction of the OCR. Depending on the scale chosen, the report will contain T disjunctive subreports $\widehat{R}(n)$. The index n reveals the order in occurrence of the subreports over the OCR and will refer to the day, week, or month of the subreport $\widehat{R}$ within the OCR, depending on the selected scale.
- Multifractal analysis and concavity test: Given a minimum spatial scale ${L}_{min}$, multifractal analysis is executed for each of the subreports $\widehat{R}(n)$. The multifractal analysis is standardized considering for all the cases the same sizing of the support given by the maximum and minimum of the spatial coordinates of all complaints in the OCR. The concavity index of each spectrum $f(\alpha (n))$ is obtained according to Equation (28), until completing the length of the OCR M. Then, the CCI is obtained and the concavity test is verified, and if negative a new ${L}_{min}$ is chosen and the MF analysis is executed again. In practical terms it is desirable to start with a small ${L}_{min}$ and increase it until the test becomes positive, keeping in mind the possible degeneration of some multifractal spectra that should be corrected.
- Synthesis of MF time series: The signals ${D}_{0}(n)$, ${D}_{1}(n)$, ${D}_{-\infty}(n)$, ${D}_{+\infty}(n)$,${\alpha}_{min}(n)$ and ${\alpha}_{max}(n)$ are constructed from the accepted dynamic multifractal spectrum $f(\alpha (n))$. For those spectra whose concavity index is at zero, the value of MF time series can be recalculated using a larger ${L}_{min}$. However, there is no guarantee of achieving the concavity of the spectrum despite this increase, because it will depend on whether there are enough complaints in the subreports that configure objects with at least monofractal behavior. Other mechanisms can be used to fix these values, such as filling methods that preserve local statistics of the signal around problematic values [58].
- Linear processing: Linear statistics are computed over produced MF time series, such as: autocorrelation function, power spectrum, mean estimation, variance estimation, and coefficient of variation, among others. It is recommended to complement this analysis with the calculation of the signal histogram. The autocorrelation and the power spectrum make it possible to determine if there are any periodic behaviors within the signal detectable in a linear sense. These two statistics have a special link through the Wiener–Khinchin [59] theorem. The other statistics are calculated in order to have an appreciation of the overall behavior of the signal [30,60].
- Nonlinear processing: In this stage, a battery of nonlinear statistics is applied to explore the structure of the time series to reveal details of its behavior that escape the linear analysis [30,50]. Some of the statistics that can be considered here are: average mutual information, dimension of the embedded phase space, and estimation of the maximum Lyapunov exponent, among others, which are based on the theory of dynamic systems, particularly nonlinear and chaotic systems [49,55,56]. Other approaches related to the detection of chaos in time series may consulted in [61]. This analysis can be complemented from a statistical perspective with an indicator of self-similarity and predictability, such as the Hurst exponent [14,29,37].
- Characteristic scales: In addition to the results produced from previous stages, spatio-temporal scales are suggested to approximate the understanding of the phenomenon. The $CCI$ reveals the minimum scale over which the temporal consistency in the mutifracatal properties of the phenomenon in space can be judged, manifesting itself as a sequence of coherent multifractal spectra, on which an attempt has been made to minimize the effect of degeneration. Results from linear processing may reveal the conservation of a spatial multifractal characteristic that can be predictable at a certain time scale. Meanwhile, the results from nonlinear processing indicate to what extent this characteristic may be chaotic, which would limit the prediction horizons in a certain time scale.

#### 3.4. Information Scaling in Crime Reports

#### 3.5. Information Patterns in Ordered Crime Reports

#### 3.6. Research Data

## 4. Results

#### 4.1. Multifractal Analysis of Crime Subreports

#### 4.2. Cumulative Concavity Index

#### 4.3. MF ${D}_{1}$ Time Series

#### 4.4. Linear and Nonlinear Processing Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- D’Orsogna, M.R.; Perc, M. Statistical physics of crime: A review. Phys. Life Rev.
**2014**, 12, 1–21. [Google Scholar] [CrossRef] [PubMed] - Brantingham, P.J.; Brantingham, P.L. Crime Pattern Theory. In Environmental Criminology and Crime Analysis; Willian Publishing: Norwich, NY, USA, 2008; Chapter 5; pp. 78–93. [Google Scholar]
- Garnier, S.; Caplan, J.M.; Kennedy, L.W. Predicting Dynamical Crime Distribution From Environmental and Social Influences. Front. Appl. Math. Stat.
**2018**, 4, 13. [Google Scholar] [CrossRef] - Perc, M.; Donnay, K.; Helbing, D. Understanding Recurrent Crime as System-Immanent Collective Behavior. PLoS ONE
**2013**, 8, e76063. [Google Scholar] [CrossRef] [PubMed] - Helbing, D.; Brockmann, D.; Chadefaux, T.; Donnay, K.; Blanke, U.; Woolley-Meza, O.; Moussaid, M.; Johansson, A.; Krause, J.; Schutte, S.; et al. Saving Human Lives: What Complexity Science and Information Systems Can Contribute; Springer: Berlin/Heidelberg, Germany, 2014; Volume 158, pp. 735–781. [Google Scholar]
- Bettencourt, L.M.A.; Lobo, J.; Strumsky, D.; West, G.B. Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities. PLoS ONE
**2010**, 5, e13541. [Google Scholar] [CrossRef] [PubMed] - Barnum, J.D.; Caplan, J.M.; Kennedy, L.W.; Piza, E.L. The crime kaleidoscope: A cross-jurisdictional analysis of place features and crime in three urban environments. Appl. Geogr.
**2017**, 79, 203–211. [Google Scholar] [CrossRef] - Felson, M. Routine Activity Approach. In Environmental Criminology and Crime Analysis; Routledge: Abingdon, UK, 2008; pp. 70–77. [Google Scholar]
- Cornish, D.B.; Clarke, R.V. The rational choice perspective. In Environmental Criminology and Crime Analysis; Routledge: Abingdon, UK, 2008; pp. 21–45. [Google Scholar]
- Roth, R.E.; Ross, K.S.; Finch, B.G.; Luo, W.; MacEachren, A.M. Spatiotemporal crime analysis in U.S. law enforcement agencies: Current practices and unmet needs. Gov. Inf. Q.
**2013**, 30, 226–240. [Google Scholar] [CrossRef] - Ratcliffe, J.H. CrimeMapping: Spatial and Temporal Challenges. In Handbook of Quantitative Criminology; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Andresen, M.A.; Linning, S.J. The (in)appropriateness of aggregating across crime types. Appl. Geogr.
**2012**, 35, 275–282. [Google Scholar] [CrossRef] - Mohler, G.O.; Short, M.B.; Brantingham, P.J.; Schoenberg, F.P.; Tita, G.E. Self-Exciting Point Process Modeling of Crime. J. Am. Stat. Assoc.
**2011**, 106, 100–108. [Google Scholar] [CrossRef][Green Version] - Melgarejo, M.; Obregon, N. Multifractal approach to the analysis of crime dynamics: Results for burglary in San Francisco. Fractals
**2017**, 25, 1750043. [Google Scholar] [CrossRef] - Mohler, G. Marked point process hotspot maps for homicide and gun crime prediction in Chicago. Int. J. Forecast.
**2014**, 30, 491–497. [Google Scholar] [CrossRef] - Mohler, G.; Short, M. Geographic profiling from kinetic models of criminal behavior. SIAM J. Appl. Math.
**2012**, 72, 163–180. [Google Scholar] [CrossRef] - Short, M.B.; D’Orsogna, M.B.; Pasqour, V.B.; Tita, G.B.; Brantingham, P.J.; Bertozzi, A.L.; Chayes, L.B. A Statistical Model of Criminal Behavior. Math. Models Methods Appl. Sci.
**2008**, 18, 1249–1267. [Google Scholar] [CrossRef] - Grubesic, T.H.; Wei, R.; Murray, A.T. Spatial Clustering Overview and Comparison: Accuracy, Sensitivity, and Computational Expense. Ann. Assoc. Am. Geogr.
**2014**, 104, 1134–1156. [Google Scholar] [CrossRef] - Mayorga, D.; Melgarejo, M.; Obregon, N. A Fuzzy Clustering Based Method for the Spatiotemporal Analysis of Criminal Patterns. In Proceedings of the 2016 IEEE World Congress on Computational Intelligence, Vancouver, BC, Canada, 24–29 July 2016. [Google Scholar]
- Kang, H.W.; Kang, H.B. Prediction of crime occurrence from multi-modal data using deep learning. PLoS ONE
**2017**, 12, e0176244. [Google Scholar] [CrossRef] [PubMed] - Kennedy, L.W.; Caplan, J.M.; Piza, E. Risk Clusters, Hotspots, and Spatial Intelligence: Risk Terrain Modeling as an Algorithm for Police Resource Allocation Strategies. J. Quant. Criminol.
**2011**, 27, 339–362. [Google Scholar] [CrossRef] - Ohyama, T.; Amemiya, M. Applying Crime Prediction Techniques to Japan: A Comparison Between Risk Terrain Modeling and Other Methods. Eur. J. Crim. Policy Res.
**2018**, 1–19. [Google Scholar] [CrossRef] - Drawve, G. A Metric Comparison of Predictive Hot Spot Techniques and RTM. Just. Q.
**2016**, 33, 369–397. [Google Scholar] [CrossRef] - Oliveira, M.; Bastos-Filho, C.; Menezes, R. The scaling of crime concentration in cities. PLoS ONE
**2017**, 12, 1–13. [Google Scholar] [CrossRef] [PubMed] - Davies, T.P.; Bishop, S.R. Modelling patterns of burglary on street networks. Crime Sci.
**2013**, 2, 10. [Google Scholar] [CrossRef] - Malleson, N.; See, L.; Evans, A.; Heppenstall, A. Optimising an Agent-Based Model to Explore the Behaviour of Simulated Burglars. Intell. Syst. Ref. Libr.
**2014**, 52, 179–204. [Google Scholar] - Gershenson, C.; Fernández, N. Complexity and information: Measuring emergence, self-organization, and homeostasis at multiple scales. Complexity
**2012**, 18, 29–44. [Google Scholar] [CrossRef][Green Version] - Frigg, R. In What Sense is the Kolmogorov-Sinai Entropy a Measure for Chaotic Behaviour? Bridging the Gap Between Dynamical Systems Theory and Communication Theory. Br. J. Philos. Sci.
**2004**, 55, 411–434. [Google Scholar] [CrossRef] - Feder, J. Fractals; Plenum Press: New York, NY, USA, 1988. [Google Scholar]
- Abarbanel, H. Analysis of Observed Chaotic Data; Springer: New York, NY, USA, 1996. [Google Scholar]
- Lopes, R.; Betrouni, N. Fractal and multifractal analysis: A review. Med. Image Anal.
**2009**, 13, 634–649. [Google Scholar] [CrossRef] [PubMed] - Puentes, C.; Obregon, N. A deterministic geometric representation of temporal rainfall: Results for a storm in Boston. Water Resour. Res.
**1996**, 32, 2825–2839. [Google Scholar] [CrossRef] - Chen, Y. Zipf’s law, 1/f noise, and fractal hierarchy. Chaos Solitons Fractals
**2012**, 45, 63–73. [Google Scholar] [CrossRef] - Bettencourt, L.M.a. The origins of scaling in cities. Science
**2013**, 340, 1438–1441. [Google Scholar] [CrossRef] [PubMed] - Murcio, R.; Masucci, A.P.; Arcaute, E.; Batty, M. Multifractal to monofractal evolution of the London street network. Phys. Rev. E
**2015**, 92, 062130. [Google Scholar] [CrossRef] [PubMed] - Semecurbe, F.; Tannier, C.; Roux, S.G. Spatial Distribution of Human Population in France: Exploring the Modifiable Areal Unit Problem Using Multifractal Analysis. Geogr. Anal.
**2016**, 48, 292–313. [Google Scholar] [CrossRef] - Verma, A. The fractal dimension of Policing. J. Crim. Just.
**1998**, 26, 425–435. [Google Scholar] [CrossRef] - Machado Filho, A.; Da Silva, M.F.; Zebende, G.F. Autocorrelation and cross-correlation in time series of homicide and attempted homicide. Phys. A Stat. Mech. Appl.
**2014**, 400, 12–19. [Google Scholar] [CrossRef] - Brantingham, P.L.; Glasser, U.; Jackson, P.; Kinney, B.; Vajihollahi, M. Mastermind: A computational modeling and simulation of spatiotemporal aspects of crime in urban environments. In Artificial Crime Analysis Systems: Using Computer Simulations and Geographic Information Systems; Information Science Reference: Hershey, PA, USA, 2008; Chapter XIII; pp. 252–280. [Google Scholar]
- Brantingham, P.; Brantingham, P.; Vajihollahi, M.; Wuschke, K. Crime Analysis at Multiple Scales of Aggregation: A Topological Approach Patricia. In Putting Crimen in Its Place; Springer: New York, NY, USA, 2009; Chapter 4; pp. 87–107. [Google Scholar]
- Brantingham, P.L.; Brantingham, P.J. Criminality of place. Eur. J. Crim. Policy Res.
**1995**, 3, 5–26. [Google Scholar] [CrossRef] - Ratcliffe, J.H. A Temporal Constraint Theory to Explain Opportunity-Based Spatial Offending Patterns. J. Res. Crime Delinq.
**2006**, 43, 261–291. [Google Scholar] [CrossRef][Green Version] - Rey, S.J.; Mack, E.A.; Koschinsky, J. Exploratory Space-Time Analysis of Burglary Patterns. J. Quant. Criminol.
**2012**, 28, 509–531. [Google Scholar] [CrossRef] - Andresen, M.A.; Malleson, N. Testing the Stability of Crime Patterns: Implications for Theory and Policy. J. Res. Crime Delinq.
**2011**, 48, 58–82. [Google Scholar] [CrossRef] - Felson, M. Crime and Nature; SAGE Publications: Newcastle upon Tyne, UK, 2006. [Google Scholar]
- Posadas, A.N.D.; Giménez, D.; Quiroz, R.; Protz, R. Multifractal Characterization of Soil Pore Systems. Soil Sci. Soc. Am. J.
**2003**, 67, 1361. [Google Scholar] [CrossRef] - Lou, D.; Zhang, C.; Liu, H. The multifractal nature of the Ni geochemical field and implications for potential Ni mineral resources in the Huangshan-Jing’erquan area, Xinjiang, China. J. Geochem. Explor.
**2015**, 157, 169–177. [Google Scholar] [CrossRef] - Nicolis, G.; Nicolis, C. Foundations of complex systems. Eur. Rev.
**2009**, 17, 237–248. [Google Scholar] [CrossRef] - Strogatz, S.; Dichter, M. SET with Student Solutions Manual. In Nonlinear Dynamics and Chaos, 2nd ed.; Studies in Nonlinearity; Avalon Publishing: New York, NY, USA, 2016. [Google Scholar]
- Abarbanel, H.D.I.; Brown, R.; Sidorowich, J.J.; Tsimring, L.S. The analysis of observed chaotic data in physical systems. Rev. Mod. Phys.
**1993**, 65, 1331–1392. [Google Scholar] [CrossRef] - Takens, F. Detecting strange attractors in turbulence. Lecture notes in mathematics. In Dynamical Systems and Turbulence; Springer: Berlin/Heidelberg, Germany, 1981; Volume 898, pp. 366–381. [Google Scholar]
- Grassberger, P.; Procaccia, I. Measuring the Strangeness of Strange Attractors. In The Theory of Chaotic Attractors; Hunt, B.R., Li, T.Y., Kennedy, J.A., Nusse, H.E., Eds.; Springer: New York, NY, USA, 2004; pp. 170–189. [Google Scholar]
- Kennel, M.B.; Brown, R.; Abarbanel, H.D.I. Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A
**1992**, 45, 3403–3411. [Google Scholar] [CrossRef] [PubMed] - Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom.
**1985**, 16, 285–317. [Google Scholar] [CrossRef][Green Version] - Rosenstein, M.T.; Collins, J.J.; De Luca, C.J. A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D Nonlinear Phenom.
**1993**, 65, 117–134. [Google Scholar] [CrossRef][Green Version] - Maus, A.; Sprott, J.C. Evaluating Lyapunov exponent spectra with neural networks. Chaos Solitons Fractals Interdiscip. J. Nonlinear Sci. Nonequilib. Complex Phenom.
**2013**, 51, 13–21. [Google Scholar] [CrossRef] - Xiong, G.; Yu, W.; Zhang, S. Time-singularity multifractal spectrum distribution based on detrended fluctuation analysis. Phys. A Stat. Mech. Its Appl.
**2015**, 437, 351–366. [Google Scholar] [CrossRef] - Figueroa García, J.C.; Kalenatic, D.; Lopez Bello, C.A. Missing Data Imputation in Time Series by Evolutionary Algorithms. In 4th International Conference on Intelligent Computing; Springer: Berlin/Heidelberg, Germany, 2008; pp. 275–283. [Google Scholar]
- Ricker, D. Echo Signal Processing; Kluwer Academic Publishers: Norwell, MA, USA, 2003. [Google Scholar]
- Obregón, N.; Puente, C.E.; Sivakumar, B. Modeling High-Resolution Rain Rates Via a Deterministic Fractal-Multifractal Approach. Fractals
**2002**, 10, 387–394. [Google Scholar] [CrossRef] - Kodba, S.; Perc, M.; Marhl, M. Detecting chaos from a time series. Eur. J. Phys.
**2005**, 26, 205–215. [Google Scholar] [CrossRef] - Lin, Y.L.; Yen, M.F.; Yu, L.C. Grid-Based Crime Prediction Using Geographical Features. ISPRS Int. J. Geo-Inf.
**2018**, 7, 298. [Google Scholar] [CrossRef] - Tompson, L.A.; Bowers, K.J. Testing time-sensitive influences of weather on street robbery. Crime Sci.
**2015**, 4. [Google Scholar] [CrossRef] - Linning, S.J. Crime seasonality and the micro-spatial patterns of property crime in Vancouver, BC and Ottawa, ON. J. Crim. Just.
**2015**, 43, 544–555. [Google Scholar] [CrossRef] - Malleson, N.; Andresen, M.A. Spatio-temporal crime hotspots and the ambient population. Crime Sci.
**2015**, 4, 10. [Google Scholar] [CrossRef] - Sparks, C.S. Violent crime in San Antonio, Texas: An application of spatial epidemiological methods. Spat. Spatio-Temporal Epidemiol.
**2011**, 2, 301–309. [Google Scholar] [CrossRef] [PubMed] - He, L.; Páez, A.; Liu, D.; Jiang, S. Temporal stability of model parameters in crime rate analysis: Anempirical examination. Appl. Geogr.
**2015**, 58, 141–152. [Google Scholar] [CrossRef] - Hipp, J.R.; Kane, K. Cities and the larger context: What explains changing levels of crime? J. Crim. Just.
**2017**, 49, 32–44. [Google Scholar] [CrossRef][Green Version] - Phillis, Y.A.; Kouikoglou, V.S.; Verdugo, C. Urban sustainability assessment and ranking of cities. Comput. Environ. Urban Syst.
**2017**, 64, 254–265. [Google Scholar] [CrossRef] - Chillar, V.; Drawve, G. Unpacking Spatio-temporal Differences of Risk for Crime: An Analysis in Little Rock, AR. Policy J. Policy Pract.
**2018**, pay018. [Google Scholar] [CrossRef] - Caplan, J.M. Mapping the Spatial Influence of Crime and Implications for Crime Operationalization Schemes Correlates: A Comparison of Analysis and Criminal Justice Practice. Cityscape J. Policy Dev. Res.
**2011**, 13, 57–83. [Google Scholar]

**Figure 1.**The graph $f(\alpha )$ vs. $\alpha $ or multifractal spectrum. The spectrum is a concave function whose maximum coincides with the fractal dimension of the object ${D}_{0}$. Its intersection with the identity line is the information dimension ${D}_{1}$. The asymmetry of the spectrum is related to the abundance of high or low masses in the object. The wider the spectrum, the more multifractal the object is.

**Figure 2.**An illustrative example of an ordered criminal report (OCR) that covers one month of crime complaints. In this example, events corresponding to one month of criminal activity have been aggregated and then splitted into four disjunctive weekly subreports.

**Figure 3.**Proposed approach that combines multifractal analysis (MFA) and analysis of observed chaotic data (AOCD) (MF-A2-OCD) to study information dynamics in urban crime reports. The method focuses on detecting spatio-temporal scales where information production exists in crime reports given that multifractal behavior appears to be consistent.

**Figure 4.**Example of crime densities in four spatial scales when computing the multifractal spectrum of a crime subreport. Spatial patterns of crime densities change with scale, although some characteristics are preserved.

**Figure 5.**An example of how the information dimension ${D}_{1}$ is obtained from multifractal analysis of the crime subreport in Figure 4. Linear scaling of the informational entropy is observed for a limited set of scales which corresponds to where multifractal behavior appears. Multifractality is not observed in all scales since there is only a finite number of points in the crime subreport.

**Figure 6.**${D}_{1}$ estimation for subreports with the largest number of criminal complaints. Note that information production measured through the informational entropy scaled linearly with $LogL$ only in a finite set of scales. The determination coefficient ${R}^{2}$ validates the goodness of the linear regression that was used to estimate the information dimension ${D}_{1}$ (i.e., slope of the linear regression).

**Figure 7.**Multifractal spectra of subreports with the largest number of criminal complaints. The spectra show that these crime reports exhibited multifractal behavior over the scales in which the informational entropy scaled linearly with $LogL$. Both the fractal dimension ${D}_{0}$ and the information dimension ${D}_{1}$ increased, in general, with the temporal scale.

**Figure 8.**MF ${D}_{1}$ time series: (

**a**) Daily scale, generated with ${L}_{min}=1000$ m; (

**b**) Weekly scale, generated with ${L}_{min}=500$ m. MF ${D}_{1}$ series were generated from the dynamic multifractal spectra that achieved the best scores according to CCI. These series indirectly represent how spatial information production of urban crime fluctuates in time.

**Figure 9.**Processing results for ${D}_{1}(n)$, daily scale. The figure columns are organized from left to right as follows: histogram of the series, autocorrelation function, power spectrum, average mutual information, false nearest neighbors, largest Lyapunov exponent estimation, and Hurst exponent estimation.

**Figure 10.**Processing results for ${D}_{1}(n)$, weekly scale. The figure is organized from left to right as follows: histogram of the series, autocorrelation function, power spectrum, average mutual information, false nearest neighbors, largest Lyapunov exponent estimation, and Hurst exponent estimation.

**Table 1.**Some relevant features of cities and criminal reports. Four cities in North America (NA) and one city in South America (SA) are considered in this study.

Case | Area (km${}^{2}$) | Population (Billions USD) | Report Size (Complaints) | Mean Daily Complaints (Complaints per Day) | |
---|---|---|---|---|---|

Los Angeles (NA) | $50.7\times 70.8$ | 12.15 | 860.45 | 121,974 | 99 |

Chicago (NA) | $34.1\times 42$ | 8.6 | 563.18 | 57,745 | 47 |

Philadelphia (NA) | $26.7\times 31.9$ | 5.44 | 346.45 | 91,806 | 75 |

San Francisco (NA) | $13.5\times 13.9$ | 3.36 | 331.02 | 19,683 | 16 |

Bogota (SA) | $24.5\times 42.9$ | 8.08 | 159.85 | 23,577 | 19 |

**Table 2.**Cumulative concavity index (CCI) results. CCI Was computed for dynamic multifractal spectra generated with three minimal spatial scales ${L}_{min}$ and two temporal scales: daily and weekly.

${\mathit{L}}_{\mathit{min}}$ (m) | CCI (Daily Scale) | CCI (Weekly Scale) | ||||
---|---|---|---|---|---|---|

250 | 500 | 1000 | 250 | 500 | 1000 | |

Los Angeles (NA) | 0.7500 | 0.6564 | 0.9742 | 0.8039 | 0.9755 | 0.9804 |

Chicago (NA) | 0.6404 | 0.6507 | 0.9692 | 0.8798 | 0.9663 | 0.9760 |

Philadelphia (NA) | 0.2889 | 0.9602 | 0.9767 | 0.9808 | 0.9760 | 0.9760 |

San Francisco (NA) | 0.6188 | 0.9605 | 0.9757 | 0.9709 | 0.9757 | 0.9806 |

Bogota (SA) | 0.8382 | 0.8732 | 0.9244 | 0.9657 | 0.9771 | 0.9771 |

Average | 0.6273 | 0.8202 | 0.9640 | 0.9202 | 0.9741 | 0.9780 |

**Table 3.**Quantitative attributes obtained from linear signal processing. Results show that information production in urban crime exhibited considerable fluctuation, short linear memory and wide frequency content in both temporal scales.

Statistic | Mean | Std | CV | CorrLag | Specent | |||||
---|---|---|---|---|---|---|---|---|---|---|

Scale (Time) | Daily | Weekly | Daily | Weekly | Daily | Weekly | Daily | Weekly | Daily | Weekly |

Los Angeles (NA) | 1.0578 | 1.2499 | 0.1367 | 0.1339 | 0.1292 | 0.1071 | 2 | 2 | 0.3342 | 0.2121 |

Chicago (NA) | 0.9481 | 1.1616 | 0.1079 | 0.0838 | 0.1138 | 0.0722 | 4 | 2 | 0.2995 | 0.1606 |

Philadelphia (NA) | 1.0936 | 1.2301 | 0.1536 | 0.1935 | 0.1405 | 0.1573 | 2 | 2 | 0.3328 | 0.2583 |

San Francisco (NA) | 0.9252 | 1.2490 | 0.1839 | 0.1977 | 0.1988 | 0.1583 | 5 | 2 | 0.3774 | 0.2373 |

Bogota (SA) | 0.5940 | 0.8870 | 0.1985 | 0.1556 | 0.3342 | 0.1755 | 2 | 3 | 0.3820 | 0.2356 |

**Table 4.**Quantitative attributes obtained from nonlinear signal processing. Results show that information production in urban crime evolved as low-dimensional chaotic attractors that exhibited strong nonlinear memory in both temporal scales.

Statistic | AMILag | EmbD | LLE | Hurst | ||||
---|---|---|---|---|---|---|---|---|

Scale (Time) | Daily | Weekly | Daily | Weekly | Daily | Weekly | Daily | Weekly |

Los Angeles (NA) | 1 | 2 | 3 | 3 | 76.1373 | 280.0472 | 0.5913 | 0.8043 |

Chicago (NA) | 2 | 1 | 3 | 3 | 294.2945 | 698.2849 | 0.7356 | 0.8264 |

Philadelphia (NA) | 2 | 1 | 4 | 3 | 156.3149 | 705.9980 | 0.5263 | 0.8143 |

San Francisco (NA) | 3 | 2 | 3 | 3 | 53.3821 | 410.1712 | 0.6488 | 0.7790 |

Bogota (SA) | 4 | 2 | 4 | 3 | 0.7062 | 61.7643 | 0.9718 | 0.8870 |

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**MDPI and ACS Style**

Melgarejo, M.; Obregon, N. Information Dynamics in Urban Crime. *Entropy* **2018**, *20*, 874.
https://doi.org/10.3390/e20110874

**AMA Style**

Melgarejo M, Obregon N. Information Dynamics in Urban Crime. *Entropy*. 2018; 20(11):874.
https://doi.org/10.3390/e20110874

**Chicago/Turabian Style**

Melgarejo, Miguel, and Nelson Obregon. 2018. "Information Dynamics in Urban Crime" *Entropy* 20, no. 11: 874.
https://doi.org/10.3390/e20110874