1. Discrete Choice Models for Incident Prediction Donald E. Brown Calcott Professor & Chair, Department of Systems & Information Engineering

2. Systems Engineering UVA AGENDA Incident Prediction Spatial Models Preferential Point Process Models Discrete Choice Models Application for Incident Prediction Conclusions

3. Systems Engineering UVA Incident Prediction Problem Inputs –series of incidents (e.g., crimes, attacks) in an area of interest and over a fixed time interval, –(optional) doctrine or subjective behavioral descriptions of the criminals or attackers, and –Formal description of the named areas of interest and actions by friendly elements given by values of features that are known or believed to be relevant to the occurrence of the attacks or incidents Output1: The likelihood that another attack or incident occurs at specified locations within the named area of interest and within a specified time range Output2: The change that occurs in the likelihood of attack over multiple periods

4. Systems Engineering UVA Spatial Models Grid region into discrete cells Cells show measurements and are vector valued As with time, space is correlated. Points that are close in space are more similar in their measurements than far away points. x 11 x 12 x 13 x 14 x 15 x 16 x 21 x 22 x 23 x 24 x 25 x 26 x 31 x 32 x 33 x 34 x 35 x 36 x 41 x 42 x 43 x 44 x 45 x 46 x 51 x 52 x 53 x 54 x 55 x 56 x 61 x 62 x 63 x 64 x 65 x 66

5. Systems Engineering UVA Selected Literature in Spatial Modeling STARMA (Cliff, et al., 1975) Spatial Autoregression (Anselin, 1980) Spatial Point Processes (Snyder & Miller, 1991) Components of Spatial Modeling (Cressie, 1993) Spatial Scan Statistic (Kulldorff 1997) Point Patterns (Diggle, 2003) Spatial Preferential Point Processes (Liu & Brown, 2003) Discrete Choice Models for Spatial Incident Prediction (Brown & Xue, 2003)

6. Systems Engineering UVA Kernel Density Estimation –Common method for visually identifying hot spots –Implies only spatial relationship are important –As a predictive tool the method assumes that what happened yesterday will happen tomorrow.

7. Systems Engineering UVA Given a realization of a marked space-time shock point process { s  D, t  T, X s,t  }, locations, times, and feature values - where –D   is the study region or geographical space; –T   is the study horizon; –   p is the feature space Estimate transition density  n (s n+1, t n+1 | D n, T n,  n ) where –D n = {s 1, s 2, …, s n } –T n = {t 1, t 2, …, t n } and –  n = {X 1, X 2, …, X n } Preferential Point Processes

8. Systems Engineering UVA Preferential Point Processes Model Construction First decomposition - separating space and time, we model each aspect with a conditional density function Assumptions: 1. Feature space does not contain temporal features; 2. Temporal evolution does not depend on spatial evolution (not essential). Ã ( 1 ) n ( s n + 1 j D n ; t n + 1 ; T n ; Â n ) ¢ Ã ( 2 ) n ( t n + 1 j T n ) Ã n ( s n + 1 ; t n + 1 j D n ; T n ; Â n ) =

9. Systems Engineering UVA Model Components A se t o f pre f erences, ac l i que, Â ( j ) n ; j = 1 ;:::; JC orrespon d i ngse t s i nspacean d t i me, D ( j ) n ; T ( j ) n C orrespon d i ngse t s i nspacean d t i me, D ( j ) n ; T ( j ) n C orrespon d i ngse t s i nspacean d t i me, D ( j ) n ; T ( j ) n S econ dd ecompos i t i on{ran d om i za t i on & separa t ees t i ma t i ono f 1 s t & 2 n d or d er

10. Systems Engineering UVA Components of Preferential Point Process Model

11. Systems Engineering UVA Point Processes in Feature Space Each event location in space/time maps to a location in feature space Some feature values (key features) are related to the occurrence of events Cliques in key feature space define site selection preferences Models in feature space enable us to predict events outside the hot spot regions: Anticipate!

12. Systems Engineering UVA Example Applications Law enforcement –Breaking and entering analysis for Richmond, VA (Liu and Brown 2003) showed significant improvement over kernel density estimates for predicting criminal incidents Counter-Terrorism –Model developed for suicide bombings in Israel –Significant performance improvements over kernel density estimates (Brown, et al., 2004)

13. Systems Engineering UVA Richmond Application -Data Acquisition 579 completed forcible “Breaking and Entering” incidents between July 1, 1997 and Aug. 31, 1997. Feature data (100 features) –Demographic counts –Consumer expenditures –Distances to geographic landmarks Feature data are coarse –Areal census data –Errors inherent in “distance to highway” calculation

14. Systems Engineering UVA Preferential Point Process (Mixture) Training: July 7-20; Testing: following 1 week & 2 weeks.

15. Systems Engineering UVA Preferential Point Process (WPK) Training: July 7-20; Testing: following 1 week & 2 weeks.

16. Systems Engineering UVA Preferential Point Process (FPK) Training: July 7-20; Testing: following 1 week & 2 weeks.

17. Systems Engineering UVA Suicide Bombing Study Region Suicide bombing incidents were analyzed for all of Israel. To evaluate the model a smaller study region was selected in the Jerusalem area The preliminary urban model for a particular group was calculated for the area defined by the cyan box on the image to the left. This area represents most if not all of Jerusalem proper with leading edges into the West Bank.

18. Systems Engineering UVA Test Results with Later Incidents

19. Systems Engineering UVA Approaches with Explicit Decision Models: Motivation Spatial decision making - offenders choose the place of a crime based on attributes at that place (Brantingham and Brantingham 1975, Molumby 1976, Newman 1972, Repetto 1974, Scarr 1973) Journey to crime - distance to the place of the crime is important (Amir 1971, Baldwin and Bottoms 1976, Capone and Nichols 1976, LeBeau 1987, Rossmo 1993, Rossmo 1994) Spatial alternatives have three components –target attributes (e.g., protection characteristics of the victim) –location (e.g., distance to other features) –time (e.g., time from a motivating speech)

20. Systems Engineering UVA Random Utility Maximization

21. Systems Engineering UVA Modeling Criminal & Terrorist Spatial Decisions Derived from discrete choice model Alternatives are discrete spatial and temporal points The number of alternatives is very large –Depends on the size of the grid –Feature components: spatial alternatives ’ characteristics Aggregate alternatives –Decision makers are not considering all possible alternatives –“ Chunk ” alternatives using clustering –Hierarchical DCM Aggregation based on feature selection

22. Systems Engineering UVA Feature Selection Feature selection methods –Simple attribute ranking –Forwards and backward selection –Branch-and-bound selection –Clustering Example feature selection criteria –Gini index –Entropy

23. Systems Engineering UVA Hierarchical Choice

24. Systems Engineering UVA Estimation for Logistic Models

25. Systems Engineering UVA Models Considered Logistic Models –main effects –quadratic –interaction Tree-based Generalized Additive Models

26. Systems Engineering UVA Modeling Nonlinearities in the Choice Process Generalized Additive Models (GAM) provide a mechanism to model nonlinearities in the relationship between the spatial features and the probability a location is chosen for the attack. The nonlinear functions are shown as f(X i ).

27. Systems Engineering UVA Splines in GAM We use restricted cubic splines for f(X i ). Spline Components –Connection points are called knots and their number can vary depending on the data –Cubic splines fit curved data better than linear splines –Cubic spines can be made to join at the knots –Constraining the function to be linear in the tails improves performance

28. Systems Engineering UVA Geographic Information System Implementation We use multiple GIS layers (topography, transportation networks, demographics, economic features, etc. ) to construct a discrete suitability surface representation Algorithm searches over cells and scores them accordingly

29. Systems Engineering UVA Example: Terrain Suitability Slope Surface Material Vegetation Roads,Water,Obstacles  Terrain/Doctrine-based prior field

30. Systems Engineering UVA Example Applications Law Enforcement –Richmond breaking & entering data –Linear main effects model –Compared predictions on test sets –Reject hypothesis of equality in methods (p = 0.005) Counter-Terrorism –Data from asymmetric warfare attacks –GAM –Method showed significance in ROC

31. Systems Engineering UVA Richmond Choice Model Results

32. Systems Engineering UVA Asymmetric Warfare Attacks Against the U.S. Attacks take many forms –suicide bombings –improvised explosive devices –hostage taking –mortar & rocket attacks –Complex attacks The incident on the right was a suicide bombing at a police station in Iraq that occurred on February 12, 2004 & killed 47 people Hull, Bryson, “100 die in two Iraq suicide bombings,” The Age, February 12, 2004, http://www.theage.com.au/

33. Systems Engineering UVA Example Asymmetric of Warfare: IED Attacks in Iraq Major method of attacking U.S. forces in Iraq Responsible for more U.S. deaths than any other attack mode Inexpensive, easy to deploy, and deadly Picture on right shows U.S. troops with IED on March 15, 2004 Models of insurgent decision making are predictive of attacks Picture from http://www.middle- east-online.com/english/?id=9250

34. Systems Engineering UVA Example Features: F3

35. Systems Engineering UVA Example Features: F5

36. Systems Engineering UVA October Threat Surface

37. Systems Engineering UVA October Surface with Attack Points

38. Systems Engineering UVA Evaluating Predictive Models ROC Curve

39. Systems Engineering UVA DCM Evaluation with KDE Comparison of density (surface) values at actual attack points KDE and DCM were normalized to 1 Hypotheses –H 0 :  D –  K = 0 –H a :  D –  K > 0 DCM results show we can reject H 0 –Wilcoxon: p <.01 Results true for multiple DCM forms

40. Systems Engineering UVA Conclusions Process models that account for preference can perform incident prediction Discrete choice models provide explicit representations of an opponents ’ utility functions Both modeling approaches have shown good results on real data from law enforcement and terrorism Models can account for multiple decision making groups but performance has yet to be tested

41. Systems Engineering UVA Questions?