1. Prediction of Crime/Terrorist Event Locations National Defense and Homeland Security: Anomaly Detection Francisco Vera, SAMSI

2. Outline Introduction Location space and feature space The model Feature selection Examples Evaluation/comparison of models Discussion

3. Introduction Talk based on two papers –Criminal incident prediction using a point- pattern-based density model By Hua Liu and Donald Brown –Spatial forecast methods for terrorist events in urban environments By Donald Brown, Jason Dalton, and Heidi Hoyle Same modeling approach in both papers

4. Introduction Hot spots: Criminal events tend to cluster in space. Traditional methods look for clusters in space –Only coordinates, dates and times are used –Poor performance –Unable to predict new hot spots Terrorist events are rare, do not cluster in space

5. Introduction Proposed method look for offenders preferences in crime site selection –Instead of looking at the coordinates, look at the features of crime locations Demographic, social, economic Distance to key features –Closest police station –Closest highway –Closest convenience store

6. Location Space North East Cops I-40 I-85

7. Feature Space Highway Cops

8. Location Space and Feature Space Transform observations from location space to feature space Look for clusters in the feature space Fit a density in feature space For each coordinate, the likelihood of an event is the density of the transformed coordinate (from location to feature)

9. Advantages Better performance (issues with comparison) Ability to predict new hot spots Terrorist events do not cluster in location space, but they do in feature space

10. The Model Times: Locations: Features: Transition density:

11. The Model Spatial transition density Temporal transition density Assumption: Temporal transition does not depend on spatial transition

12. The Model

15. Feature Selection

17. Second paper mentions: –Use of the correlation structure to drop variables –Principal Components

21. Features Selected

22. Example

23. Gaussian Mixture Model

24. Weighted Product Kernel

25. Filter Product Kernel

26. Terrorist Events Example

27. Features Selected

28. Distance Features Only

29. Logistic Regression

31. Combination

32. Evaluation/Comparison of Models

33. The reasoning: Percentile scores should be larger at event points Evaluate percentile scores at all event point and average. Best model has highest average percentile score Is this good?

34. Crime Example

38. Terrorist Example

39. Discussion Feature space has advantages over location space The Model: Decomposition of the transition density Feature selection: Correlations, principal components, Gini index Evaluation/comparison of models: Percentile score Paper: Detecting local regions of change in high- dimensional or terrorist point processes, by Michael Porter and Donald Brown