Prediction of Crime/Terrorist Event Locations

National Defense and Homeland Security: Anomaly Detection

Francisco Vera, SAMSI

Outline

• Introduction

• Location space and feature space

• The model

• Feature selection

• Examples

• Evaluation/comparison of models

• Discussion

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

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

Introduction

• Proposed method look for offender’s 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

Location Space

North

East

Cops

I-40

I-85

Feature Space

Highway

Cops

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)

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

The Model

• Times:• Locations:• Features:• Transition density:

The Model

• Spatial transition density• Temporal transition density• Assumption: Temporal transition does not

depend on spatial transition

The Model

The Model

The Model

Feature Selection

Feature Selection

Feature Selection• Second paper mentions:

– Use of the correlation structure to drop variables– Principal Components

Features Selected

Example

Gaussian Mixture Model

Weighted Product Kernel

Filter Product Kernel

Terrorist Events Example

Features Selected

Distance Features Only

Logistic Regression

Logistic Regression

Combination

Evaluation/Comparison of Models

Evaluation/Comparison of Models

• 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?

)else everywherean greater th is at density ()()()(

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Crime Example

Crime Example

Crime Example

Crime Example

Terrorist Example

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