Information Dynamics in Urban Crime
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
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
3.3. MF-A2-OCD Method
- Generate the OCR: Given the record of urban crime complaints in a time window , a temporary scale is defined for the construction of the OCR. Depending on the scale chosen, the report will contain T disjunctive subreports . 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 within the OCR, depending on the selected scale.
- Multifractal analysis and concavity test: Given a minimum spatial scale , multifractal analysis is executed for each of the subreports . 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 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 is chosen and the MF analysis is executed again. In practical terms it is desirable to start with a small 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 , , , , and are constructed from the accepted dynamic multifractal spectrum . For those spectra whose concavity index is at zero, the value of MF time series can be recalculated using a larger . 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 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 Time Series
4.4. Linear and Nonlinear Processing Results
5. Discussion
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Case | Area (km) | Population (Billions USD) | Report Size (Complaints) | Mean Daily Complaints (Complaints per Day) | |
---|---|---|---|---|---|
Los Angeles (NA) | 12.15 | 860.45 | 121,974 | 99 | |
Chicago (NA) | 8.6 | 563.18 | 57,745 | 47 | |
Philadelphia (NA) | 5.44 | 346.45 | 91,806 | 75 | |
San Francisco (NA) | 3.36 | 331.02 | 19,683 | 16 | |
Bogota (SA) | 8.08 | 159.85 | 23,577 | 19 |
(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 |
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 |
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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Melgarejo, M.; Obregon, N. Information Dynamics in Urban Crime. Entropy 2018, 20, 874. https://doi.org/10.3390/e20110874
Melgarejo M, Obregon N. Information Dynamics in Urban Crime. Entropy. 2018; 20(11):874. https://doi.org/10.3390/e20110874
Chicago/Turabian StyleMelgarejo, Miguel, and Nelson Obregon. 2018. "Information Dynamics in Urban Crime" Entropy 20, no. 11: 874. https://doi.org/10.3390/e20110874