Self-Exciting Point Processes and Their Applications to ...

Self-Exciting Point Processes and Their
Applications to Civilian Deaths in Iraq

Erik Lewis

November 4, 2010

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

self-exciting, inhibitory, Poisson

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

self-exciting, inhibitory, Poisson
The model has a branching structure that corresponds to
background events and offspring events.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

self-exciting, inhibitory, Poisson
The model has a branching structure that corresponds to
background events and offspring events.
We will focus on estimating the background rate better

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

self-exciting, inhibitory, Poisson
The model has a branching structure that corresponds to
background events and offspring events.
We will focus on estimating the background rate better
Results

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

self-exciting, inhibitory, Poisson
The model has a branching structure that corresponds to
background events and offspring events.
We will focus on estimating the background rate better
Results
We apply the model to insurgency in Iraq

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

self-exciting, inhibitory, Poisson
The model has a branching structure that corresponds to
background events and offspring events.
We will focus on estimating the background rate better
Results
We apply the model to insurgency in Iraq
We propose three different models for the background rate

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

self-exciting, inhibitory, Poisson
The model has a branching structure that corresponds to
background events and offspring events.
We will focus on estimating the background rate better
Results
We apply the model to insurgency in Iraq
We propose three different models for the background rate
We analyze two locations in Iraq, Karkh and Mosul

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Introduction

Point process models are used in seismology, epidemiology,
neuroscience, ecology, etc.

self-exciting, inhibitory, Poisson
The model has a branching structure that corresponds to
background events and offspring events.
We will focus on estimating the background rate better
Results
We apply the model to insurgency in Iraq
We propose three different models for the background rate
We analyze two locations in Iraq, Karkh and Mosul

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Point Process

A point process N is a random measure on a complete
separable metric space S that takes values on N ∪ ∞.

We typically characterize a point process N by prescribing its
conditional intensity λ(t), which represents the infinitesimal
rate at which events are expected to occur around a particular
time t, given the history of the process, Ht, up to time t:

λ(t) = lim E [N[t, t + ∆t)|Ht]
∆t →0 ∆t

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Point Process

More simply stated for our terms, we may view a realization of
N as a list of times t1, t2, ...., tn at which events occur (or
perhaps coordinates in (x,y,t,m) space depending on the
context).
Throughout this talk we will assume that ti ∈ [0, T ] for all
i = 1, 2, . . . , n where T is the largest ti .

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Self-exciting Point Processes

A point process N is self-exciting if
Cov[N(t1, t2), N(t2, t3)] > 0

for any t1 < t2 < t3. This means that if an event occurs, another
event becomes more likely locally in time and space. This is not
true in a Poisson process. It has independent increments so
Cov[N(t1, t2), N(t2, t3)] = 0.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Hawkes Process

The general form for a Hawkes process 1 is

t (1)

λ(t) = µ + k0 g (t − tk )dZ (u)

−∞

= µ + k0 g (t − tk )

t >tk

where Z is the normal counting measure. Many choices for g , the

triggering density, have been tested in earthquake modeling. We
use an exponential distribution 2:

λ(t) = µ + k0 we−w(t−tk ) (2)

t >tk

1“Spectra of some self-exciting and mutually exciting point processes.”,

Hawkes, 1971
2“Self-exciting point process modeling of crime.”, Mohler, Short,

Brantingham, Schoenberg, Tita, submitted

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Maximum Likelihood Estimation

The log-likelihood function for a process with conditional intensity

λ(t) is:

nT

log(L) = log(λ(ti )) − λ(t)dt

i=1 0

We want to find µ, k0 and w such that the likelihood function is

maximized: n T

max log(λ(ti )) − λ(t)dt
µ,k0,w i =1
0

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Akaike Information Criterion

To compare two models, we use Akaike’s Information
Criterion (AIC).
The AIC is equal to 2k − 2 log(L) where k is the number of
parameters of the model and L is the maximum value of the
log-likelihood function.
A smaller AIC value implies a better model.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Data

We obtained our data from Iraq Body Count, an organization
dedicated to recording civilian deaths in Iraq since the military
intervention began. We have 15,977 entries occuring from March
20, 2003 to December 31, 2007. With each event, we have the
time it happened down to the day as well as how many people died
and the city it occurred in.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Data

To get an idea of what the data looks like, we display a histogram
of all the events occurring in Iraq below with 50 bins.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Simplifying Assumptions

We consider an event as an atom, not taking into account the
number of deaths per event
We take the beginning of the time window for when the event
occurred
We subdivide the data into smallest known regions to get as
much spatial resolution as possible

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

What should the background rate look like?

We want to capture the trend starting somewhere around a year
into the data:

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

What should the background rate look like?

We decide to incorporate a non-stationary background rate µ(t) as
an alternative model:

Step Function
Piecewise-Linear
Nonparametric

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Step Function

The simplest choice for a non-stationary µ(t) is a step function
with a few jumps at well chosen points t1 and t2

λ(t) = µstep(t) + k0 we−w(t−tk )

tk <t

where 
µ1
 for 0 ≤ t ≤ t1
µstep(t) = µ2 for t1 < t ≤ t2
µ3 for t2 < t ≤ T

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Piecewise Linear Model

We try to capture the increase with a constant background rate,
and then allowing µ to increase linearly:

µpl (t) = µ1 for 0 ≤ t ≤ tc
µ2(t − tc ) + µ1 for tc < t ≤ T

where we choose tc a priori. Although this is better in some of the
cities we consider, we would like to allow more flexibility.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Smoothing

For our last model, we use variable bandwidth kernel smoothing as
a guess for the background rate:

µs (t) = 1n 1 t − ti
K di
n i =1 di

where K t − ti − ti 2
di 2di 2
= √1 exp − t
2π

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Smoothing

This leads us to a new modified background rate:

λ(t) = pnµs (t) + (1 − p)k0 we−w(t−tk ) (3)

tk <t

where µs (t) is a kernel smoothing of all events normalized to 1 and
n is the number of events.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Karkh

This is a map of Baghdad, with the district of Karkh colored brown
in the center

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Karkh

Model Parameter Estimates for Karkh No SE AIC
3,330.6
µ µˆ kˆ0 wˆ −1 Hawkes AIC 3,331.1
µstep (t) 1,624.7
µpl (t) 0.060 0.959 19.23 906.0 855.0
µsm (t)
0.053, 0.1430, 0.053 0.935 19.23 905.6

0.0552, 0.0001 0.915 18.59 905.7

0.6368 0.3643 15.08 829.6

For the second row, t1 = 661 and t2 = 1, 385 while for the third

row tc = 400. For the fourth row, we use the 200-th nearest
neighbors and bmin = 80. Note that in the last row, µˆ is pˆ and kˆ0
is actually (1 − pˆ)kˆ0.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Karkh

A histogram of all events in Karkh (left). The estimated fit of the
data for the smoothed background rate model λˆ(t) (right). The

smoothed background rate µˆsm(t) is plotted on the right as well
for reference.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Mosul

Mosul is located in the north, circled here in red.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Mosul

Parameter Estimates for Mosul

Model µˆ kˆ0 wˆ −1 Hawkes AIC No SE AIC
3,370.4
µ 0.0533 1.0024 58.82 2,570.5 2,611.2
µstep 2,626.9
µpl 0.0969, 0.4169, 0.5639 0.7123 68.28 2,558.4 2,551.8
µsm
0.0950, 0.0008 0.7354 49.02 2,570.4

0.6344 0.3856 41.08 2,545.1

For the second row, t1 = 1, 050 and t2 = 1, 350 while for the third
row tc = 975. For the fourth row, we use the 200-th nearest
neighbors and bmin = 150.

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Mosul

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Conclusion

Our results suggest changes in the background rate over
smaller timescales than other applications of a hawkes process

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Conclusion

Our results suggest changes in the background rate over
smaller timescales than other applications of a hawkes process
A nonparametric approach for the background rate seems to
outperform other models

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Conclusion

Our results suggest changes in the background rate over
smaller timescales than other applications of a hawkes process
A nonparametric approach for the background rate seems to
outperform other models
Further work includes better nonparametric estimates of the
background rate and the triggering density at the same time

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Conclusion

Our results suggest changes in the background rate over
smaller timescales than other applications of a hawkes process
A nonparametric approach for the background rate seems to
outperform other models
Further work includes better nonparametric estimates of the
background rate and the triggering density at the same time
Thanks to George Mohler, Jeff Brantingham and Andrea
Bertozzi

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D

Conclusion

Our results suggest changes in the background rate over
smaller timescales than other applications of a hawkes process
A nonparametric approach for the background rate seems to
outperform other models
Further work includes better nonparametric estimates of the
background rate and the triggering density at the same time
Thanks to George Mohler, Jeff Brantingham and Andrea
Bertozzi

Erik Lewis Self-Exciting Point Processes and Their Applications to Civilian D