discrete choice models for incident prediction donald e. brown calcott professor & chair,...

Discrete Choice Models for Incident Prediction

Donald E. Brown

Calcott Professor & Chair, Department of Systems & Information Engineering

Systems Engineering UVA

AGENDA

• Incident Prediction

• Spatial Models

• Preferential Point Process Models

• Discrete Choice Models

• Application for Incident Prediction

• Conclusions


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


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.

x11 x12 x13 x14 x15 x16

x21 x22 x23 x24 x25 x26

x31 x32 x33 x34 x35 x36

x41 x42 x43 x44 x45 x46

x51 x52 x53 x54 x55 x56

x61 x62 x63 x64 x65 x66


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)


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. 5 10 15 20 25 30

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2030

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• Given a realization of a marked space-time shock point process {s D, t T, Xs,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(sn+1 ,tn+1| Dn, Tn, n) where

– Dn = {s1 , s2 , …, sn}

– Tn = {t1 , t2 , …, tn} and

– n = {X1 , X2 , …, Xn}

Preferential Point ProcessesPreferential Point Processes


Preferential Point Processes Model ConstructionPreferential 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 (sn+1jDn ;tn+1;Tn ;Ân) ¢Ã(2)n (tn+1jTn)

Ãn(sn+1;tn+1jDn ;Tn ;Ân) =


Model Components

A set of preferences, a clique, Â(j )n ; j = 1;:: : ; J

Corresponding sets in spaceand time, D(j )n ;T (j )n



Second decomposition { randomization &

separateestimation of 1st & 2nd order


Components of Preferential Point Process Model

F A set of preferences, a clique, Â(j )n ; j = 1;:: : ; JF Corresponding sets in spaceand time, D(j )

n ;T (j )nF Second decomposition {

randomization & separateestimation of1st & 2nd order spatial e®ects

Ã(1)n (sn+1jDn;Ân;tn+1;Tn) =®¢Ã(11)n (X̂ n+1jÂn)P Jj =1

Ã(12)n (sn+1jD(j )n ;T (j )n ;tn+1)P r(X n+1 2 Â

(j )n )

AssumesF Static set of preferences in the timehorizonF \Quasi-Markov" point patterns in feature space


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

0 t1 t2 t3 t4 t5 t6 t7 TTimeAxis

. . .

f1

f2

fi

fp

...

.

..FeatureSpace

x1 x3

x4

x2x5

x6

x7

GeographicSpace

s1

s2s1 s3

s4

s2 s5

s6 s7

• Cliques in key feature space define site selection preferences

• Models in feature space enable us to predict events outside the hot spot regions: Anticipate!


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)


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


Preferential Point Process (Mixture)

• Training: July 7-20; Testing: following 1 week & 2 weeks.


Preferential Point Process (WPK)



Preferential Point Process (FPK)



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.


Test Results with Later Incidents


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)


Random Utility Maximization

F RandomUtility Maximization (McFadden, 1973)F U(d;si ) = V(d;si ) +"(d;si )

I Individual d, location, siI V (d;si ) is deterministicI "(d;si ) is stochastic

F With choiceset A, common assumptions lead toP (ai jA;d) =

exp(V (d;si ))Pa j 2 A

exp(V (d;sj ))


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


Feature Selection

• Feature selection methods– Simple attribute ranking– Forwards and backward selection– Branch-and-bound selection– Clustering

• Example feature selection criteria– Gini index– Entropy


Hierarchical Choice

F Decision makers do not considerall theelements of A

F A is aggregated into a smaller set MF Decision maker has someprobability

of evaluating elements of MF Choice fromM is

P (ai ) = P r fUi ¸ Uj +lnP r f j 2 M g;8j 6= igP r fai 2 M g


Estimation for Logistic Models

F Likelihood functionL(~̄;M ;~s) =

Qsi 2D

Qj 2M P (j )I f si =j g

F Globally concaveF Estimates areasymtotically Gaussian


Models Considered

• Logistic Models– main effects– quadratic– interaction

• Tree-based

• Generalized Additive Models


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(Xi).


Splines in GAM

• We use restricted cubic splines for f(Xi).

• 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


Geographic Information System Implementation

•••

Layer=1

Layer=2

Layer=L

•••

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


Example: Terrain Suitability

Slope Surface Material

Vegetation Roads,Water,ObstaclesTerrain/Doctrine-based prior field


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


Richmond Choice Model Results


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/


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


Example Features: F3


Example Features: F5


October Threat Surface


October Surface with Attack Points


Evaluating Predictive Models

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

ROC for IED Models

False Positive

Tru

e P

osi

tive

InteractionDensityGAM

ROC Curve


DCM Evaluation with KDE

• Comparison of density (surface) values at actual attack points

• KDE and DCM were normalized to 1

• Hypotheses

– H0: D – K = 0– Ha : D – K > 0

• DCM results show we can reject H0

– Wilcoxon: p < .01

• Results true for multiple DCM forms


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


Questions?

discrete choice models for incident prediction donald e. brown calcott professor & chair,...

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