Using Big Data and Agent-Based Modeling to Understand ...files.meetup.com/2215331/dsdc2015.pdf ·...
35
Using Big Data and Agent-Based Modeling to Understand Social Media Diffusion William Rand in collaboration with David Darmon, Jimpei Harada, Jared Sylvester, PK Kannan, Arjuna Ariyaratne, and Michelle Girvan
Transcript of Using Big Data and Agent-Based Modeling to Understand ...files.meetup.com/2215331/dsdc2015.pdf ·...
Using Big Data and Agent-Based Modeling to Understand Social Media Diffusion
William Rand
in collaboration withDavid Darmon, Jimpei Harada, Jared Sylvester, PK Kannan,
Arjuna Ariyaratne, and Michelle Girvan
– First research center focused on the application of complex systems to business and management science
–Working toward a better understanding of emergent patterns arising from nonlinear interactions of individual components
– Examines tools such as: machine learning, agent-‐based modeling, social network analysis, geographic information systems, systems dynamic modeling, and big data tools
Complex Systems and Data Science
http://www.rhsmith.umd.edu/ccb/
Supporters
The Beauty of Big Data
• Individuals leave their footprints (and fingerprints) in the digital sand.
• These traces represent digital / social signals of human behavior.
• We can use these signals to explore the complexity of individuals and digital environments.
The Challenge of Big Data
• What do we do with all this data?
• Aggregation of the data eliminates the richness of it
• The real benefit is when the data is used at the individual level
• How to make sense of that individual-level information?
Agent-Based Modeling
• Agent-Based Modeling provides a way to understand individual-level interactions
• Traditionally, agent-based models use simple rules derived from theory
• If we could create ABMs directly from big data we would have an individual-level detailed model derived directly from digital traces
A Solution
• Use Machine Learning to derive individual-level rules for agents automatically.– Causal State Model (CSM)
• Then use these rules to construct an agent-based model.
• Use this model to forecast individual behavior.
Understanding the Predictive Power of Computational Mechanics and Echo State Networks in Social Media
David Darmon, Jared Sylvester, Michelle Girvan, and William Rand
ter.ps/compmech
Predicting Twitter Engagement Using ML
Opportunity: Social Media is a great marketing channel.Challenge: However, there is a lot of noise, and its not
apparent when users are paying attention.Solution: Understand when users are active.Goal: Create a tool that can accurately predict when
different segments will post content.
Motivation
Twitter users embedded in a 15k user follower network.
Statuses of all users collected over 7 weeks.
Select 3k subset of most frequently tweeting users.
The Dataset
Predicting Twitter Engagement Using ML
2013-08-22 12:54:06 Is Your Gmail Social? How to Use Gmail Daily to Build an Engaged Community | Socially Sorted http://t.co/WVprF4bLPa via 2013-08-22 13:11:22 Facebook's Embedded Posts Now Available to Everyone http://t.co/Cc0q3cSTt62013-08-22 13:14:06 The Credible Hulk http://t.co/q17VrcSdBs2013-08-22 13:29:02 25 Things You Didn’t Know About Ninjas http://t.co/CcJz92sRyy2013-08-22 13:32:59 Twitter Users: Revoke and Reestablish Third Party App Access Now http://t.co/qL9LbnGO6e via @ShellyKramer2013-08-22 13:48:46 10 Brilliant Facebook Marketing Tactics to Increase Reach and Engagement http://t.co/Z4YsudlJ35 via @kathikruse2013-08-22 14:17:11 Google Now Adds Cards for NCAA Football Scores, Concert Tickets, Car Rentals and More http://t.co/BG6Yz2zDyU2013-08-22 15:18:03 What is the NSA Really Up To? [COMIC] http://t.co/hakoq1mFNX2013-08-22 15:39:04 6 Things Every Good Business Blog MUST Have http://t.co/KRGzZRCrbm
Tweet TextTimestamp
User: DanielZeevi
The Setup
Predicting Twitter Engagement Using ML
Bin (in time) Twitter data, giving a discrete time series for each user v at time t:
The Setup
0 1 0 1 1 1 0 1 1 1… …
— user v doesn’t tweet— user v tweets
X(v , t) = 0
X(v , t) = 1
Time
Predicting Twitter Engagement Using ML
Creating Epsilon Machines (Shalizi and Crutchfield, 2001; Crutchfield, 1994)
Assume was generated by a conditionally stationary stochastic process.
Explicitly learn the predictive distribution by grouping together pasts x that give equivalent predictions.
{Xi}Ni=1
P (Xi |X i�1i�L = x)
The Computational Mechanics Framework
A B
C
1
0
1
01
Causal State Model (CSM) built for each user usingCausal State Splitting Reconstruction (CSSR)
Begin with one state and divide as necessary.
Call the model learned the
causal state model (CSM)
for each user. CSM has the unique property of being “maximally predictive and minimally complex”
Learn this state-space representation of the process using
Causal State Splitting Reconstruction (CSSR).
The Computational Mechanics Framework
User: OcioNet
User: HadiJayaPutra
●●●●●●●●●●
●●●
●
●●●
●
●
●●
●●●●●
●●
●
●
●
●
●●●●●●●●
●●●●
●●●●●
●●●
●
●
●●
●●
●●●
●
●●●●●●
●●●●●●
●
●●
●●●●●●●●●
●●●●
●
●●●●●●●●●●●●
●
●●●●
●●●
●●●
●
●●●●
●
●
●
●●●
●
●●●
●●●●●
●●
●
●
●●●●
●●●●●
●●●
●
●●
●
●●
●
●●
●●
●
●●●●●●
●
●
●
●
●●●●●●●●
●●
●●●●
●●
●●●●
●●●●●
●
●
●
●
●
●●
●
●●
●●●
●
●
●●
●●
●
●●
●●●●●●
●●
●
●
●
●●●●
●
●
●
●●
●
●
●●●●●●●●
●
●●
●
●
●
●●●
●
●
●
●
●●●●
●●●
●
●
●●
●
●●●●
●
●
●●
●
●
●●
●
●●●●●●●
●
●●
●
●●
●●
●●
●
●●
●●
●
●●
●●
●●●●●●
●●●
●●●●●●●
●
●
●
●
●
●
●
●●●●●
●●
●
●
●
●●●●●●●●●●●
●●●●
●
●
●●●●
●
●●●●●●
●●
●
●
●●●●
●
●●●●●●
●●●●
●
●●●
●●●●●●
●●●●●●●
●
●●●
●●●
●●
●
●●●●
●●●
●
●
●●●●●●
●
●●●
●●
●●●●●●
●●●●●●●●●●●
●●●●
●●●
●●●
●
●
●
●
●●●
●
●●
●
●●
●
●●●●●●●●●●●●●
●
●●●●
●
●●
●
●●
●●●●
●●
●●
●●●●●●●●
●●●●●
●●●●●●
●
●●
●●●●●
●
●●
●●
●●
●●●
●●
●●●●●●●●●
●
●●
●
●●●●
●
●●●●●●●●
●●●●●
●
●●●●●
●
●●●
●
●
●
●●
●●
0 100 200 300 400 500 600
010
0AnCS
M
●●●
●●
●
●●●
●
●
●●●
●●
●●●
●
●
●●
●●●●
●
●●
●●●●●●●●
●●●●●
●●●●
●
●
●
●
●
●
●
●●●●●
●
●●
●●●
●
●
●●●
●
●●●●
●●●
●●
●
●
●●
●
●
●●●●
●●●●●●
●●
●
●●
●●●
●●●●●
●
●
●●
●●●●●●●●
●●
●
●
●●
●
●
●
●
●
●●
●●●
●●●●●●●●●
●●●
●●●●
●
●●●
●
●●
●●
●
●
●●
●
●
●●●●
●●●●●●●●
●
●
●
●
●●
●
●
●
●●●
●●●●●
●
●●
●●●●●●●
●
●
●
●
●
●●●●
●
●●●●
●●●●
●●●●
●●●
●
●
●●
●
●
●
●●●●
●
●
●●●●●●●
●
●
●●●●
●
●●●●●●●
●●●
●●●
●
●●●●
●●●●●
●
●
●
●
●
●
●
●
●●●
●●
●
●●●●
●●●●
●
●●●
●
●●
●
●
●
●
●●●●
●
●
●
●
●
●●●●●
●
●●●●
●●●
●●
●
●●
●●
●●●●
●●●
●
●
●●●●●●●●●●
●●●
●●●●
●
●
●●●●●●●●
●●
●●●●●●●●●●●
●
●●●●
●
●
●●●●
●
●
●
●●●●●
●●●●●●
●●●●●
●●
●
●●
●
●
●
●●●●●●
●
●●●
●
●
●●●●●
●
●●●●●●●●
●●●
●●
●●●●
●
●●
●●●
●●●
●
●●●
●●●●
●●●
●●●●
●
●
●●●●●●●
●●
●●●●●●●●
●●●●●
●
●●●●●●●
●●●●●
●
●●●●
●
●●●●●●●●●
●●
●
●●
●●●●●●●●●
●
●●●●●
●
●●●●●●●●●
●●
●
●●●●●
●●●
●
●
●
●●●
●
●●●●●●●
●●●●●
0 100 200 300 400 500 600
010
0AnCS
M
●●●●●●●
●
●●●
●●
●
●
●●●●
●●●
●
●●●
●●
●●●●
●●●●
●
●●
●●●
●●
●
●
●●●
●●●●●●●●●●●●●●
●
●●
●
●
●●
●●
●
●●●
●
●
●●
●●●●●●
●●
●
●●●
●
●
●●
●
●●
●●●
●
●
●
●
●●●●
●●●●●
●
●●
●
●●●●
●
●●
●●
●●
●●
●
●●●
●
●●●●
●
●●
●
●
●
●●
●
●
●
●●●●
●
●●●
●●●
●●
●
●●
●●
●
●
●●
●
●●●●
●●
●
●
●
●●
●
●
●●●
●
●
●
●
●
●●
●●●
●●
●●
●
●●●●●●
●●●●●
●
●●
●
●●●●●●
●●
●
●
●●
●●
●●
●●
●●●●●
●●●
●●●●
●
●●
●●●●
●
●●●●
●●●●●●●●●●●●
●●
●
●●
●
●●●●●
●●
●●●●
●●
●
●
●●●●
●●●
●●●●
●●●●●
●●
●
●●●
●
●
●
●
●●●
●●
●
●
●●●
●
●
●
●●●●●●●●●
●●●
●
●
●●●
●●●●
●●●●●●
●●●●●●
●●
●●●●●●
●●●
●
●●
●
●●
●●
●
●●
●●
●●
●
●
●
●
●●●
●
●●●●
●
●●
●
●●●●●
●●●●●●●●●●●●●
●
●
●●●●●●
●
●●●●●●●
●●●●●●
●●
●
●
●●
●●●
●
●●●●
●
●●●
●
●●
●
●●●
●
●●●●
●●●
●
●●●●●●●●●●
●●
●●●●
●●
●
●●●●●
●●●
●
●●●●
●●●●
●●●
●●●●●
●●●●●
●
●
●●
●
●●●●●●●●
●●
●●
●●●
●●
●
●●
●●●●●●
●●
●●●●●●
●
●●
●
●
●●
●●
●
●
●
●
●
●
●
●●●
●
●●●●●
●
●●
●
●
0 100 200 300 400 500 600
010
0AnCS
M
●
●
●●
●●
●
●
●●●
●
●●●●●●
●
●
●●●
●●●●●●●●●●
●
●
●●●●●●
●●●●●
●
●●●
●
●●●●
●●●●●
●
●
●
●
●
●
●●●●
●●●●
●●
●
●
●●
●
●●
●
●●
●●
●●●●●●●
●●
●●●●●●
●
●●●
●●●
●
●
●
●●●●
●●
●
●
●●●●●●
●●
●
●●
●●●
●
●●
●●●●
●
●●
●
●●●●
●●●●
●●●●●●●●●
●
●
●●●●
●
●
●●●●
●
●
●●●
●●●●●●
●
●●●●●
●
●●●●
●
●●
●●
●
●●●●●
●
●
●●●
●●
●
●
●●●
●
●●
●●
●●
●●
●
●●●●
●
●●
●●●
●●●
●●●●
●
●●●●●
●●●●●
●●●●●●●●
●
●●●●●●
●
●●●
●
●
●●●
●●●
●
●
●
●●●●●
●
●●●
●
●●●●●
●
●●●
●
●●●●●
●
●●●
●
●●●
●
●●●●●●
●
●
●
●
●●
●
●
●●●●
●
●●●
●●
●
●
●
●
●●●●●
●●
●
●
●●
●●
●
●
●●●●
●
●●●
●
●●
●
●●●●●
●●●●●●
●●●
●
●
●●●●●●●
●●●●
●●●●●●●●●
●●●●●●●
●●●●●●●●
●●
●●●●●
●●●●
●●
●●●
●●●●●●●●
●
●●
●
●
●●
●●●●
●
●
●●
●●●●●●●
●●
●●
●●●
●●●
●
●
●●
●●●●●
●●
●
●
●●●
●●●●●●
●●
●
●
●●
●
●
●●●●●●
●
●
●●●●
●●●●●●●●●●●
●●
●
●●●
●●●●
●
●
●
●
●
●●●
●
●●●
●
●●●●●●●●●●
●●●●●
●
●●●●●
●
●
●
●
●●
●
●●
●
●
●
●●●
●●
●●●●
0 100 200 300 400 500 600
010
0
Time (∆ = 600 seconds)
AnCS
M
Xn(u
2)X
n(u
2)X
n(u
2)
n (∆ = 600 seconds)
Xn(u
2)
Xn(u
3)X
n(u
3)X
n(u
3)
n (∆ = 600 seconds)
Xn(u
3)
Xn(u
4)X
n(u
4)X
n(u
4)
n (∆ = 600 seconds)
Xn(u
4)
Xn(u
2685)
Xn(u
2685)
Xn(u
2685)
n (∆ = 600 seconds)
Xn(u
2685)
Xn(u
1)X
n(u
1)X
n(u
1)
n (∆ = 600 seconds)
Xn(u
1)
...
+ + + + +
Xn(u1) Xn(u2) Xn(u3) Xn(u4) Xn(u2685)
An
Testing Procedure
• Build model for each user separately
• Training: 45 days
• Testing: 4 days
• Look back 10 steps
• Predict ahead 1 step
• 0-1 Loss
• Compare to “majority vote” baseline
Predicting Twitter Engagement Using ML
CSM vs. ESN
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
● ●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●●●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●●
●●●●
●
●
●
● ●● ●●
●
●
●●●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●●
●● ●●
●●●
●
●
●
●●●
●
●
●
●
●
●
●●
●
●
●●
●●
●
●
●●●
●
●●
●
●●●
●
●●
●
●●
●
●
●●
●
●
●
● ●●
●
●●
●
●●
●●
●
●
●
●●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●●
●●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●●●
●
●
●●
●
●
●
●
●
●
●
●●
●
●●●●●●●
●
●
● ●●
●
●
●
●
●●
●
●
●
●
●
● ●●●●
●
●●●
●
●
●
●●●●●●●
●
●
●
●
●
●●
●
●●●●●
●
●
●
●●●●●
●
●● ●
●
●
●
●
●●●●
●
●● ●●
●
●●
●●
●
●●●●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●●●●●
●
●
●
●
●●●●●●
●●
●
●
●●●
●
●
●●●●● ●●
●
●
●
●●
●
●●●
●
●
●
●
●●●
●
●
●
●●●●
●
●
●●
●●
●
●
●●●●
●
●
●
●
●
●●●
●
●●
●
● ●●
●
●
●●
●
●●●
●
●●●●●●●●●
●
●●●
●
●●
●
●
●
●●
●
●
●
●●●●●●
●
●
●
●
●
●
●●●
●
●●●●●
●
●●●●●●●
●
●●●●●●●●●
●
●●
●●
●
●●●●
●
●
●
●
●●
●
●
●
●
●●●●●●●●
● ●●●●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●
●●●
●
●●●
●
●
●●●●
●
●
●●●●●
●
●●●●●●
●
●●●●●●●●●●
●
●●●●●●●●●●●●●●●
●●●●
●●
●
●●●●
●
●
●
●
●
●
●
●●●
●
●●●●●●
●
●●●
● ●
●
●
●
●
●●●●●●●
●
●
●●●●●●●
●
●● ●●
●
●●●
●
●●●
●●●
●●●●
●
●●
●
●
●
●
●●●●●●
●●●●
●
●●
●
●
●
●
●
●●●●●
●●●●
●●
●
●
●●
●
●
●
●●●
●
●●●
●
●
●●●●●●
●
●●
●
●●
●
●
●
●
●
●
●
●
●●●●●●●●●
●
●●●●●●
●
●●●●●●
●
●●●●●●●●
●
●●●●●●●●●●
●
●
●
●
●●●●●●●●●
●
●●
●
●●●●●●
●
●●●●●●●●●●●●●
●
●
●
●
●●●
●
●●●●●
●
●●●●●●●●●●
●
● ●
●
●
●●● ●●
●
●
●●
●●●●●●
●●
●
●
●
●
●
●
●
●
●●●
●●●● ●●●●●●●●●
●●
●●●●●●●●●●●●●●●●●●
●
●●●●●●
●
●●●
●
●
●
●●●●●●
●
●
●
●●●
●
●
●●●●
●
●●
●
●
●●
●
●
●●●
●
●
●●●●●
●
●
●
●●●●●●●● ●●●
●
●●●
●
●●●●
●
●
●●
●
●●●
●
●
●
●●●●
●
●●●●
●
● ●●
●●●●●●
●
●●● ●
●
●
●●●●●●●●●●
●
●●●● ●
●
●●
●
●●●●●
●●●●
● ● ●●●● ●●● ●●
●
●●●
●●●●●●
●●●●●●●
●
●●●
●
●●
●
●●●●●●●●●●●
●
●
●●
●
●●●●●●●●
●
●●●●●●●●● ●●●●●
●
●
●
●●●●
●
●●●●●●●●
●
●●●●●●●●●
●
●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●
●
●●●●●●●●●●●●●
●
●●●●●●●●●● ●●●●●●●●
●
●
●●
●
●
●
●●●●●
●
●
●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●● ●●●●
●●●
●
●
●
●●●●●●●● ●●●●●●●
●
●●● ●●●●●●●●●●●●●●●●●
●
●
●●●●●●●●● ●●●●●●● ●
●
●● ●●
●
●
●
●
●●
●●●●●●●●●● ●●●● ●●
●
● ●●●●
●
●●●●●●●●● ●●●●●●●●
●●●●●●
●
●●●●●●●●●●
● ●●●●●●●●●●
●
●●
●
●●●
●
●
●●●●●●●
●
●●●●●●●●
●
●●●
●
●●●●●●
●
●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●
●
●●●●●
●
●
●
●●●●●●●●●●● ●●●●
●
●●●●●●●
●
●●●●
●
●●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●●●●
●
●●●●●●●●
●●●●●●●●●● ●●●●●●●●●●●●●●
●
●●
●
●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●
●
●●●●●●●●●●●●●
●
●
●
●●●
●
●
●
●
●●●●
●
●
●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●
●
●
●●
●
●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●●●●● ●●●●●●●
●
●
●●●●●●●
●
●●●●●●●●● ●●
●
●●●●
●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●●●●●●●
●
●●●●●●●●●●●●●●●●● ●●●●●●●●●
●
●●
●
●●●●●●●●●● ●●● ●
●
●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●● ●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●
●
●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●
●
●●●●●●●●●●●●●●●
●
●●●
●
●●●●●●●●●●●●●●●●
●
●●●●●●●●●● ●●●●●
●
●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●
●
●●●●
●
●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●●●●●●●
●
●●●●●●● ●●●
●
●●●
0.0 0.2 0.4 0.6
0.0
0.2
0.4
0.6
CSM Improvement
ESN
Impr
ovem
ent
CSM vs. ESN
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
● ●●
●●
●
●
●
●
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●●●
● ●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●●
●
●
●●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●●
●
●
●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
● ●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●●
●
●●
●
●
●
●●
●●●● ●
●
●●
●
●
●
●
●
●
●● ●●●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●●
●
●
●
●
●
● ●
●
● ●●●
●
●
●
●●
●
●
●
●
●
●
●●
● ●●
●
●
●
●
●●
●
●
●
●
●
● ●●
●
●●
●
●
●●
●●●
●
●
●
●●●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●● ● ●
●
●
●
●●●
●
● ●
●●●
● ●
●
● ●
●
●●
●●● ●
●
●
●
●
●
●●● ●●●● ●
●
●
●
●
●
●
● ●●
●●●
●
●
●
●●
●●
●●●● ●
●
●
●●●
●●
●
●
●
●
●
●●
●
●
● ●
●
●●
●
●
●●●
●
●
●
● ●●● ●
●
●●●●
● ●●
●
●●
●●
●
●
●●
●
●●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
● ●● ●●
●
●
●●● ●● ●●●●
●
●
●
●
●●
●
●
●● ●
●
●●
●
●●●
●●
●
●
●
●
●●
●
● ●●
●
● ●
●
● ● ●●●
●
●
●
●
●●●
●
●
●
●● ●
●
●
●
●●
●
●●●●
●
●● ●
●
●
●
● ● ●● ●● ● ●
●
●
●
●●●
● ●●●● ● ●●
●
●●
●●● ●● ●
●●●
●
●
●● ●● ●●
●
●●
●●●●●
●● ●● ●
●
● ●●●●
●
●
●
●● ●●
● ●●●● ●
● ●
●
●
●
●●● ●●●
●● ●
●
●●●
●●● ●●●●
● ●●
●
●
● ●
●
●● ●
●
●
●
●
●●●
●
●
●
●●●
●●
●
● ●
●
●●
●
● ● ●●
●
●
●
●●
● ●●
●
● ● ●
●
● ●●
●
●●
●
●● ●●● ●● ●●●● ●●
●
●● ●●●
●
●
●
●
●●
●
●
●
●●● ● ●●
●
●
●
●
●
●
● ●●
●● ● ●
●●
●
● ●● ● ● ●●
●
● ●●●● ●● ●●
●
●● ● ●
●●●● ●
●
●● ● ●●●
●
●
●
● ●●● ●● ●●
●
●●● ●
●
●●
●●
●
●
●● ●●
●
● ●
●
●●●●
●
●● ●
●●
●●●●
●
●●●
●●●
●
● ●● ●●●
●
● ●●● ● ●● ● ●●
●
●● ●● ● ●● ● ●● ●●●● ●
●
● ● ●●
●
●
● ● ●●●
● ●
●
●
●
●
● ●●
●
● ●●●● ●
●
●● ●●
●
●
●●
●
●
●
●●●
●●●●● ●●●
●● ●
●
●●
●
●●● ●●
●
●●
●
●●● ● ●● ●
●
●
●
●
●●
●
● ●● ●●●
●
●● ●● ●●●
●
●
●
● ●●●● ● ●●●●
●
●●
● ●● ●
●
●●● ●
●
●●●
●●
● ●● ●● ●
●
● ●●
●●
●●●
●
●
●
●
●
●● ●● ●●● ●●
●
● ●●● ● ●
●
● ●● ●● ●● ● ●● ● ● ● ●●
●
●● ●● ● ●● ● ●●
●●
●
●
●
● ●●●● ●● ●
●
●●●
● ● ●●●●
●
● ●● ●
●
●●● ●●●●●
●
●
●
● ●●●
●
●● ●●
●
●●● ● ●●● ●● ●●
●
●
●●●●
●●●
●●
●
●●
●● ● ●●●
●●
●
●
●
●
●
●●
●
● ●● ●●●●●
● ●●●●●● ●
●●
●●● ● ●● ● ●● ●●●● ●● ●
●●
●
● ● ● ●●●●●●
●●
●
●
●●● ●● ●●●●
●●●
●
●● ●● ●
●● ●●● ●●
●
●●●●
●
●
●● ●●●
●
●
●
● ●● ●●● ●●
● ●●
●
●●●
● ●●● ●
●
●
●●
●
●● ● ●
●
●●●● ●●● ●●●●
●
●●
● ● ●●● ●● ●●●
●●
●
●●●● ●●● ●●
●
●
●●●●
●● ● ●●
● ●●●●●
●●●●
●
●
●●●
●
●
●
●●
●
●
●
●●
●
● ● ●● ●● ● ●● ●●● ●●●
●
● ●
●
●●
● ●● ● ●● ●●●●
●
●●●
●● ● ●● ● ●●
●●
●● ●● ● ●●
●●● ●●●
●
●
●
●
●
●●
●
●● ● ●●●● ●
●
● ● ● ●●●●●●
●●●●
●
●● ●● ●●
● ●● ●●●● ●● ●●● ●●● ●● ●● ●
●
● ●● ●● ●● ● ●●●● ●●● ●● ●●
● ● ●● ●●
●
●● ●●● ●● ●●● ●●●● ●●
●●● ●● ●●●
●
●●●● ● ●●
●
● ● ●●●●● ● ●●● ● ●●● ● ●● ● ●●●● ●●
●●● ●● ●●
●●
●●
●
●
●
●●● ●●
●
●
●● ● ●● ●● ● ●●● ●●
●●
● ●●● ●●● ●● ●●●● ●● ●●
●● ●● ●● ●●● ●● ●●●● ●● ●●● ● ●●
●●
●●● ● ●●● ●●
●
● ●
●
● ●●●
●
● ● ●●●● ●● ●●
●●●● ●● ● ●● ●●
●●●● ●●● ● ●●●● ●● ● ● ●●
●
●●● ●●● ●● ●
●
● ●●● ●●●
● ●●
●●
●
●●
●
●
●●● ●● ●●● ● ●●
●
●●
●
●●● ●●
●●●
●
●
● ● ●●●● ●●
●●● ●● ●●● ●● ●● ●●●●● ●● ●●●● ● ●●
● ●● ●●●● ●
●
●● ● ●●
●●●
●
●
● ●● ●●●● ●● ●●●●● ●●●
●●●
●
● ●● ●● ●
●
● ● ● ●●● ● ●●● ●● ●●●
●
●● ●●●●● ●● ●● ●● ● ● ●●●●
●
● ●● ● ●● ●● ● ●● ●●
●●●●
●
●● ● ●●●● ●● ●● ● ●●●●●● ● ●●●● ●● ●● ●●● ●●● ●●●● ●●●
● ●
●● ●●
●
●● ●● ●● ●●●● ●● ●●●● ●●
●
● ●●●
●●●● ●● ●●●●● ●
●
● ●
●
● ●●● ●● ●●● ●●●
●●● ●● ● ●●● ●● ● ●● ●● ●●● ●●● ●● ●● ●●● ● ●●● ●●●●●●
●
● ●●● ●● ●● ●● ● ●●●● ● ● ●●● ●●●● ●
●
●
●
●●
● ●●● ●● ●●
● ●
● ●●● ●●● ●●● ●● ●● ●●● ● ●●
●● ●●
●●● ●●● ●● ●● ●●● ● ●● ● ●● ●● ●●
●
● ● ●●●●●●●● ●● ●● ●● ●●●●● ●● ●●● ●●● ●● ●● ●●
●●● ●●
● ●● ●●● ●● ●● ●●● ●● ●●●●●●● ●●
●● ●●● ●●
●
●
●● ●● ●●●●●●● ●●●● ●●
●
●
●
●● ●●
●
●
●
●● ●●● ●● ● ●● ●●● ●● ●●●● ●●●●●● ●● ●● ● ●● ●●● ●● ● ●● ●●●●
●
●● ●●● ●● ●
●
●●● ● ●●● ● ●●
● ● ●●●●●
●● ●●● ●● ●●● ●●●● ●● ●●● ● ● ●●
● ●●
●● ●●● ●● ●●●● ● ●● ●●●●●● ●●● ●●●
●
●●●●● ●● ●● ●●●●● ●●●● ● ●● ●●● ●●●●●● ●●● ● ●● ●● ●●●● ●● ●●●●● ●●● ●●●●●●●● ● ●● ●
●
●● ●●●● ●●● ●●●●●
●
● ●● ●●●● ●●● ●●●●● ●●● ● ●● ● ●●● ● ●●● ●● ●●● ● ●●● ●●●
● ●●●●●●●
● ●● ●●●● ●●●●● ●●● ●●●●●● ●●●● ●●●● ●● ●● ●●●● ●● ●● ●●●● ●●●●●● ●●● ●●● ●●
●● ●● ●●●●●● ●
●●
●
●●
●
●●●● ●
●
●● ●● ●● ●● ●● ●●●●● ●● ●●●● ●● ●●● ●●●● ●●●● ●● ● ●●●●● ●●● ● ●●●● ●●●● ●● ●●●●
●
●●●● ●●●●● ● ●● ● ●●● ●●●●●● ●● ●● ●● ●● ●● ●● ●●● ●● ●●●●● ●●●● ● ●● ●● ●●●●●● ●● ●●●●
●●●●● ●●● ●●●●● ●●● ●●●●● ●●●● ●● ●●● ● ●●● ●●● ●● ●●● ●● ●
●
●●● ●● ● ●● ●●● ●● ●●●●●● ●● ●●●● ●●● ●●●● ●●● ●●●●●●
●● ●●● ●●● ●●●● ●●● ●●● ●● ●● ●●●● ●●● ●● ●●●
●●●● ●●●● ●●●● ●●●●●●●● ● ●●● ●● ●●● ●
●●● ●● ●● ●●●● ● ●●●●● ●●●● ●●●●
●● ●●● ●● ●●
●
● ●●●● ●●
●
●●● ●●●
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
Tweet Rate
Impr
ovem
ent
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●● ●●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
● ●●●
●
●
●●
●
●
●●●
●
●
●
●●
●
●●●
●
●
●●
●
●●●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●●●●
●
●
●
●●
●
●
●●
●
●
●●
●●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●●
●
●
●
●●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
● ●
●
●
●
●
●
●
●
●●
●● ●
●
●●
●
●●●●
●●
●
●
●
● ●●●●
●
●
●●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●
● ●●●●
● ●
●
●
●
● ●●
●
●
●
●
●●
●●
●
●
●●
●●
●●
●●●
●
●●
●
● ●●●
●●●
●●
●
●
●●
●
●
●
● ●●
●
●●
●
●●
●●
●
●
●
●●
●●
●
●
●
●
●
●● ●
●
●
●
●
●
●●
●
●
●●
●●
●●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●● ●●
●
●
●●
●
●
●
●
●
●
●
● ●
●
● ● ●●●●●
●
●
● ●●
●
●
●
●● ●
●
●
●
●
●
●●●●●
●
●● ●
●
●
●
●● ●
● ●●●
●
●
●
●●
●●
●
●●●●
●●
●
●
●●
● ●●
●
● ●●●
●●
●
●● ●●●
● ●●●
●
●●
●●●
● ●● ●
●
●●
●
●
●
●
●
●
●●
●●●
●
●●● ●●
●
●
●
●
● ●●●●●
●●
●
●
●●●
●
●
●●●●
●● ●
●
●
●
● ●
●
●●●
●
●
●
●
●●●
●
●
●
●● ●●●
●
●●
●●
●
●
● ● ●●
●
●
●
●
●● ●●
●
● ●
●
● ●●
●
●
●●●●●
●
●
● ●● ●●●● ●●
●
●● ●
●●●
●
●
●
●●
●
●
●
●●● ● ●●
●
●
●●●
●
● ●●
●
● ● ●● ●
●
● ●● ● ● ●●
●
●●●
●● ●● ●●
●
●●●
●
●
●●● ●
●
●
●
●
●●
●
●●
●
● ●●● ●●
●●●●
●● ●●●
●●
●
●
●
●●
●
●
●
● ●
●
●
●●●
●
●● ●
●
●
●●●●
●
●● ●
●●●
●
● ●● ●●●
●
● ●●● ● ●●●
●●
●
●● ●● ● ●● ● ●●●
●●● ●● ● ● ●●●
●● ● ●●
●
●●
●
●
●
●
● ●●
●
● ●●●● ●
●
●●●
● ●
●
●
●
●
● ●●
●● ● ●
●
●
● ●●● ●● ●
●
●● ●●
●
● ●●
●
●●●
●●●●
●● ●
●
● ●
●
●
●
●
● ●● ●●
●●●
● ●
●
●●●
●
●
●
●
●●●●●
●●●●●
●●
●
●●●
●
●●● ●
●
●●●
●
●
● ●● ●● ●
●
● ●
●
●●●
●
●●
●
●
●
●
●● ●● ●●● ● ●
●
● ●●● ●●
●
●●● ●● ●
●
● ●● ● ● ● ●●
●
●●●
●● ●● ● ●
●
●●
●
●
●● ●●●● ●● ●
●
●●●
● ● ●● ●●
●
● ●● ●●●●● ●●●●●
●
●
●
●
●●●
●
●● ●● ●
●
●● ● ●●● ●● ●●
●
●●●●
●● ●●●
●
●
●●
●● ● ●●●●●
●
●
●●
●
●●
●
● ●●
●●●● ●● ●●●●●● ●
●●
●●● ● ●● ●●● ●●●● ●● ●● ●
●
●●
● ●●●●
●●●
●
●
●
●●● ●● ●
●
●●
●●●
●●
●●● ●
●
● ●
●
●●
●
●
●●
●●●
●
●● ●●●
●
●●
● ●● ●●● ●●● ●●
●
●●●●
●●● ●
●
●
●●
●
●● ●●
●
●
●●● ●●
●●
●●●
●●●●
● ●●● ●
●
●●● ●
●
●
● ●●
● ●●●●●
●
●●●●●
●●
● ●
●
● ●●●
●●
●●●●● ●●●● ●●● ●●
●
●●●
●●● ● ●●
●● ● ●● ●●
●●●●
●
● ●
●
● ●● ●● ● ●● ●●●●
●
●●
●
●● ●●
● ● ●●
●
●● ● ●● ● ●● ●●● ●●●
●
●
●
●●●
●
●
●● ● ●●●● ●
●
●● ● ●●●●● ●
●
●●●
●
●● ●● ●●● ●● ●●●●
●●
●●●
●●
● ●● ●● ●
●
●
●● ●● ●● ● ●●●● ●●● ●● ●● ● ● ●●●●
●
●●
●●● ●● ●●● ●●●● ●● ●●● ●● ●●●●
●●●● ● ●●
●
● ● ●●●●● ● ●●● ● ●
●
● ● ●● ● ●●●● ●● ●●● ●● ●●
●
●
●●
●
●
●
●●● ●●
●
●
●● ● ●● ●● ● ●●● ●●
●●
● ●●● ●●● ●● ●●●● ●● ● ● ●● ●● ●● ●●● ●● ●●●● ●● ●●●●
● ● ●● ●●● ● ●●
● ●●●● ●●●
●●●
●
●
●
●●●● ●● ●● ●●●● ●● ●●
● ●●●●●● ●●● ● ●●●●
●● ● ● ●
●●
●●● ●●● ●● ● ●● ●●● ●● ●
●●●●●
●
●
●
●
●●
●● ●● ●●● ● ●● ●●● ● ●●
●
●●●●●
●
●● ● ●●●● ●● ●●●●● ●●
●●● ●● ●●●
●● ●● ●●●
● ●●
●● ●● ●●●● ● ●●
●
● ●●
●●●●
●
● ●● ●●●●●
● ●●●●● ●●●
●●●
●
● ●● ●● ●
●
●●
● ●●● ● ●●● ●● ●●● ● ●● ●●●●● ●● ●● ●●
●
● ●●●●
●
●
●
● ● ●●
●● ● ●● ● ●●●●●●
●● ● ●●●●●
● ●● ●
●
●●
●●● ● ●●●● ●● ●● ●●● ●●● ●●●● ●●●
● ●
●● ●●●
●● ●● ●● ●●
●● ●● ●●●● ●●●● ●● ●●●●● ●● ●●●
●
● ●
●
● ●●
● ●●● ●● ●●● ●● ● ●●● ●●
● ●●● ●● ● ●● ●● ●●● ●●● ●● ●● ●●● ● ●●● ●
●●●●●
●● ●●● ●● ●● ●●
●
●●●● ● ● ●●● ●●●●
●
●
●●
●●
●
●
●
●
●● ●●
●●
● ●●● ●●
●●●● ●● ●● ●●● ● ●● ●● ●
●●●● ●●● ●● ●● ●●● ● ●● ● ●●
●●
●●
●
● ●●●●●●●●● ●● ●● ●● ●●●●● ●● ●
●● ●●● ●● ●● ●●●●● ●●● ●● ●●● ●● ●● ●●● ●● ●●
●
●
●●● ●● ●● ●●● ●●●
●
●● ●● ●●●●●●● ●●●● ●● ●●
●
●● ● ●
●
●
●
●● ●●● ●● ● ●● ●●● ●● ●●●● ●●●●●● ●● ●●
● ●● ●●● ●● ● ●● ●●●●●
●● ●●● ●● ●
●
●●● ● ●●● ● ●●● ● ●●●● ● ●● ●●● ●● ●●●
●●
●
● ●● ●●● ●●
●●● ● ●●
●
●●● ●●
●●●● ● ●● ●●●●●● ●●● ●●●
●
●●●●● ●● ●● ●●●●● ●●●● ● ●● ●●● ●●●●●● ●●● ● ●● ●● ●●●● ●● ●●●●● ●●● ●●●●●●●● ● ●● ●●●● ●●
●● ●●● ●●●●●
●
● ●● ●●●● ●●● ●●●●● ●●● ● ●● ● ●●● ● ●●● ●● ●●●
●
●●● ●●●
● ●●●●●●● ● ●● ●●●● ●●●●● ●●● ●●●●●● ●
●
●● ●●●● ●● ●●●
●●● ●●
●● ●
●
●● ●●●●●● ●●●
●●● ●●
●
● ●● ●●●●●● ● ●●●●●
●
● ●●● ● ●●● ●● ●● ●● ●● ●●●●● ●● ●●●● ●● ●
●
● ●●●● ●●●● ●● ● ●●●●● ●●● ● ●●●● ●●●
● ●● ●●●●
●
●●●●●
●●●●
● ●● ● ●●● ●●●●●● ●● ●●
●● ●● ●● ●● ●●● ●● ●●●●● ●●●● ● ●● ●● ●●●●●● ●● ●●●
●●●●●● ●●● ●●●●● ●●● ●●●●● ●●●● ●● ●●● ● ●●● ●●● ●● ●●● ●● ●●●●● ●● ● ●● ●●● ●● ●●●●●● ●● ●●●● ●●● ●●●● ●●● ●●●●●●
●● ●●● ●●● ●●●
●
●●● ●●● ●● ●● ●●●● ●●● ●● ●
●●● ●●● ●●●● ●●●● ●●●●●●●● ● ●●● ●● ●●● ●●●● ●● ●● ●●●● ● ●●●●● ●●●● ●●●●
●
●
●●● ●● ●●
●
● ●●●● ●●●●●●
●●●
●
●
CSMESN
User: DanielZeevi
Base Rate:
CSM Rate: ESN Rate:
0.45060.9477 0.9419
Forecasting High Tide:Predicting Times of Elevated
Activity in Online Social MediaDavid Darmon, Jimpei Harada, William Rand,
and Michelle Girvanter.ps/hightide
Motivation
▪ Many individuals, firms, and organizations have interest in spreading information via online social networks.
▪ Spreading a message on social media requires:■ Influence: Position in social network.■ Persuasiveness: Success at retransmission.■ Timing: Success at timing transmission.
●●●
●●
●
●
●●●●●
●
●
●
●●●
●
●●
●●●●
●●●●●●●
●●
●
●●
●
●●●
●●
●
●●
●●●
●●
●●●
●●●
●●
●●●
●●
●
●●
●
●●
●●●●●●●●●●●
●
●
●●●
●●●●●●●●●●●●
●●
●
●●●●●●
●
●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●
●●
●
●●●●
●●
●●●
●
●●
●
●
●●
●●
●●●
●●●●●●●●
●
●
●
●●
●
●●
●
●●●
●●●
●
●
●●●
●
●●●
●●
●
●●●
●●●●●
●●
●●
●●●●
●
●
●●●
●
●●●●
●
●●
●●
●●
●
●
●●
●●●
●
●●●●●●●●●●●●●
●●
●●●●●●
●●●
●●●
●
●●
●●●●●●
●
●●●
●●●
●
●
●
●
●●
●
●
●●●
●
●
●
●●
●
●●●
●
●●
●●●●●●●
●
●
●
●●●●
●
●●
●●
●
●●●●
●●●●●●
●
●●
●●
●
●●
●
●●●
●●●●
●
●
●●●
●●
●
●
●
●●
●
●
●
●●●
●●●
●●
●
●●●●
●●●
●●●●●●●
●●●●●●●
●●●●
●
●●●
●
●●●●
●
●●●●●
●●
●
●
●●●●●●
●
●
●
●●
●
●●●●
●●
●
●●●
●
●
●
●●
●●●●
●●
●
●●
●●●●
●
●●●●
●●●●●
●
●●●●●●●
●
●●●●●●●●
●●
●●●●●
●
●●●
●●●●
●●●
●●●
●●●
●●●●●●
●●●●●
●●●●
●●
●●
●
●●●●●
●
●●●●
●
●●●●●
●
●●●●●
●●●
●●
●●
●●●●
●●●
●●●●
●●●
●●●●●●●
●●
●●
●●●●●●
●●●
●●●●●●
●
●●●
●●●●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●●
●
●●
●
●
●
●●●
●
●
●●●
●
●●
●●
●
●●●●●
●
●
●
●●
●
●
●●●
●●
●●●●●●●●●●
●●●
●
●●
●●
●●
●●●
●
●
●●●●●●●●●●●●●●●
●
●
●
●●●●●
●●●●●●●●●●●●●
●●●●●
●●
●●
●
●
●●●●●●●
●●
●●
●●
●
●●
●
●●
●
●●
●●●●
●
●
●●●●●●●●
●
●●
●●●●●
●●●●●
●
●●
●
●
●●●●●
●
●●●
●●●●
●●●
●●●●
●●●●
●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●
●
●●●
●
●●●
●●●●
●●
●●
●●
●●●●●
●
●
●●●●●●●●●●
●●
●●
●
●●●
●
●●
●●●
●●●●
●●●●●
●●●●●
●
●
●●●
●
●●●●●●●
●●●●●●●●●●
●●
●
●●
●●●●
●
●
●●●
●
●
●
●
●
●
●●
●
●
●●
●●●
●●●
●●
●●●●●●●●●●●
●●
●●●●●●●●●●●●
●
●●●
●
●
●●●●
●●●●
●●
0 200 400 600 800 1000
015
0
An
●
●
●
●●●●●
●●
●
●●●●
●●●●●
●●●●●
●●●
●●
●
●●●●●
●●●●
●
●
●●●●●●●●
●
●●
●●●●
●
●
●●●
●
●
●●
●
●
●
●
●●●●
●
●●
●●●
●
●
●
●
●
●●
●●●●
●
●●●
●
●●●●●
●●
●●
●●
●●●●●●●●●●●●●
●●●
●●●●●●●●●●●●●●●
●
●
●●●●●
●●●●●●
●●
●
●
●
●●
●
●●●
●●
●●●●●
●
●
●●
●
●
●●
●●●●●
●
●●
●●
●
●●●●●
●
●
●●
●●
●
●
●
●●●●●●●
●●
●
●
●
●
●
●●●●●●●●●●
●●●
●●●●●
●●●●
●
●●
●
●●●●●●●●
●●
●●●●●
●
●●●●●●●●●●●●
●●●
●●
●●●●●
●
●●●
●
●●
●●●●●
●
●
●●
●
●
●
●●●
●
●●●
●
●●
●●●
●
●
●
●
●
●
●
●
●●●
●●●●●●
●
●
●●●●
●
●●●●●
●
●●
●
●
●●●●●●
●●●
●●●●
●
●
●
●
●
●
●●
●
●●●●●●
●●
●●●●●●
●●●
●●
●
●●●●●●
●●●●
●●●●●●●●
●●●●●●●●
●●
●
●●
●
●●
●●●●
●
●
●
●
●
●
●
●●
●●●●●●
●●●
●
●●●
●●●
●
●
●●●●●●●
●●
●
●
●
●●●●●●●●●
●●●●●●
●●
●
●
●
●●
●●
●
●●
●
●●
●
●●
●
●●
●
●●●●
●●●
●●
●●●●
●●
●
●
●
●●
●
●
●
●
●●●●●●●
●●
●●●●●●●
●
●●
●
●
●●●●●●
●●●●●●●●
●●
●●
●
●●●●
●
●●
●●
●●●●
●●●
●●●
●
●
●●
●
●●●
●
●
●
●●●●●●
●
●●●
●●
●
●
●●
●●●
●
●
●
●
●●
●●●●
●
●
●
●●●●
●
●●
●●●
●
●
●●●●●●●
●●●●
●●●●●●●●●
●●●
●●●
●●●●
●●
●●●●●●●●●●●●●
●●●●●
●
●
●
●
●●
●●●●●●●●●●
●
●●
●
●●●●●●
●
●
●●●●
●
●●
●●●●
●
●●
●●●
●●●●●●
●●
●●
●●●●●
●
●
●●
●
●●
●
●
●●●●●●●●●●●●●●●●●●●
●●●●●●●
●
●●
●●●●
●
●●
●●
●
●
●
●●●
●●
●●●
●●●●●
●
●●●●●●●●●●●●●●
●
●
●●
●●
●
●●●
●●●●●●●
●●●
●
●
●
●●●
●●●●●●●●●
●●
●
●●●●●●●●
●
●●
●●
●
●
●●
●●●●
●
●
●
●
●●
●
●●●●
●
●●●●●●●●●●●
●●
●
●
●
●●
●
●●●
●●●
●●●●●
●●●●
●●●
●●●●●
●
●
●
●
●●●
●●●
●
●●●●
●
●●●●
●
●●●
●●●●●●●●●
●●●●●
●●
●●●●
●●●●●●●●●
●●●
●
●●●●●
●●●●
●
0 200 400 600 800 1000
015
0
An
●
●
●
●
●●
●
●
●●●●
●
●
●
●●●
●
●
●●●●●●●
●●●
●●
●
●●
●
●
●●●●
●
●
●●●
●
●
●●●
●
●
●
●
●
●
●
●
●●
●●
●
●
●●
●
●
●
●●●●●●
●
●●●●●
●●
●
●●●
●●●●●●●●●●●●●●●
●●●●●●
●
●
●
●●●●●●
●●●●●
●●●●●●
●
●●●●
●●
●
●
●
●●
●●
●
●
●●
●●
●
●
●●●●
●
●●●●
●
●●●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●●●●●●●
●
●
●
●
●
●
●●
●●
●
●●●●
●●●●
●
●●
●
●
●
●
●●●●●
●●●
●●●●●●●●
●●●
●●●●●●
●●●●
●●
●●●●●●●●●●●
●
●
●●
●●
●
●
●●●●
●●
●
●
●●
●●●
●●
●●●●●
●
●●●●●
●
●●●●●
●
●
●
●
●●●●●●●
●●●●●●●
●
●
●●
●●
●
●●
●
●●
●●
●
●●●●●●●●●●
●
●●●
●
●
●●●
●●●●
●
●●●
●●●●●
●
●
●●●
●
●
●
●
●●
●
●●●●●●●●
●●●●●●●●●●
●●●●●●●●
●●
●●●●●
●●●●●
●●●
●●
●●●●●●●●●
●
●●
●
●●●●●
●●●
●●●
●
●●
●
●
●●●●●●●●
●●●
●
●●
●
●●●
●●●●●
●
●
●
●
●
●●●●●●
●
●
●●●●●
●
●
●
●●●●
●●●●●●
●
●●
●
●●●●●●
●●●
●●●●
●
●●●
●●●●●●
●
●●●●●●●●●●●●●●●●●●
●
●
●
●●
●
●●●
●●●
●●●●
●●●
●
●●
●●●
●●
●
●●
●●
●●
●●●
●
●
●
●
●
●
●●●●
●
●●
●●
●
●●●●
●●
●
●
●
●
●●
●●
●●
●
●●●
●
●●
●●
●●●
●●
●●●
●
●●●●●
●●●●●●
●
●●●●●●●
●
●●
●●●●
●
●●●●
●●●
●●●●●●●●●
●●●●
●●
●●●●●●
●●●●●
●●●●
●●●●●●●
●●●●●
●●●●●
●●●●●●●●●●●
●●
●●●
●●
●
●●
●
●
●
●
●●●●●
●●●●
●
●
●●
●
●●●●
●●●●●●●●
●
●●
●
●
●●
●
●
●
●●●●●
●
●
●●●●
●●●
●●●●●●●●●●●
●●●●●●●●●
●●
●
●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●
●
●
●
●●●●●●●
●●●
●
●
●
●
●●●
●●
●●
●
●●●●
●
●●●●●●●
●●
●●●●●
●
●●●●
●●
●●
●●●
●●●●
●●
●
●
●
●
●●●●●●
●●
●
●●●
●
●
●●
●
●●●●●●●●
●
●●
●●●
●
●
●●
●
●
●●●●●●●●●●●
●
●●
●●●
●
●●●●●
●
●●●●●●●●●
●●●●●●●●●
●
●
●●
●●●
●●
●●
0 200 400 600 800 1000
015
0
An
●●●
●
●●●●
●
●●●●
●
●
●
●
●
●
●●
●
●●
●●
●●
●
●●●●
●●●●●
●●●
●
●●●●
●●●●●
●
●
●
●
●●
●●
●●●●●
●
●●●
●
●
●
●
●●●
●●
●
●●●●●
●
●
●
●
●●●●●
●
●
●●●●●●●●●●●●●●●●●●●
●●
●●●●●
●
●●
●●●●
●
●
●●●
●●●●●●●
●
●
●
●
●
●●●●●
●
●●
●
●
●●
●
●
●
●●
●
●
●●●
●●●
●
●
●●●●●●
●
●●
●●
●
●
●
●
●
●●●
●
●
●●
●●●●●●
●
●●
●●
●●●●●●●●
●
●
●
●●●
●
●
●
●●●●●●●
●●●●●
●
●
●
●
●
●
●
●
●
●●●
●
●●●
●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●
●●
●
●
●●
●
●●
●●
●●
●●●
●
●
●
●●●●●●●●
●
●
●●
●●
●●●
●●
●●●●●●●
●
●●●
●●●●●
●
●●
●
●●●●●
●●●●
●●●
●
●●
●
●
●●●
●
●●●
●●
●●
●
●
●●●●
●
●
●
●
●
●
●●
●●●●●●
●●
●
●●
●
●●●●●●●●●●●●●●●
●●●●
●
●●●
●●●●
●
●
●
●●●●●
●
●●
●
●●
●
●
●●●
●
●
●
●
●
●
●●●
●
●●
●
●●●●●
●
●●
●
●●●
●
●
●
●●●
●●●●●●
●●●
●●
●●
●
●●●●●●
●
●
●
●●●●●●
●
●
●●
●
●●●●●●
●●●●
●●
●●
●
●
●●●
●
●●
●●●●●
●
●
●
●
●●
●●●●●●●
●●●
●
●
●
●●●●●
●
●●●●
●
●
●●
●●
●●●
●
●
●
●
●
●
●●●●●
●●
●
●
●●
●●
●●●●
●
●
●
●
●
●
●
●●●●●
●
●●●●●●
●
●
●
●
●●●
●
●●●●●●●●●
●●●
●●
●●●●●●●●
●
●●●
●●
●
●●
●
●
●●
●●●
●●●●●●
●●
●
●●●●●●●
●
●
●●●●●●●
●
●●
●
●●●●
●
●●●●●
●●●●●
●●●
●●●●●●
●●●●●●●●●●●●●●●●●
●●
●●
●
●●●
●
●
●●
●●●●●
●
●●●
●
●●●●●
●
●
●●
●
●
●●●
●
●●●
●
●
●●●●
●
●●●
●●
●●
●●
●●●●
●
●●●
●
●
●
●●
●●
●●
●●
●●
●
●●●●
●
●●●
●
●
●●●
●●●
●●●
●●●
●
●●
●
●●
●●●●●●●●●
●
●●
●●●●●●
●
●●●●●●●●●●●
●
●●●
●
●●●●
●●
●
●●●●
●●●●●●●●
●●
●
●●●
●●●●
●
●
●●●
●●●
●
●
●
●●●
●
●
●
●●●●●
●●●●●●
●
●●
●●
●
●
●●
●●●●
●
●●●
●
●
●
●●●
●●●
●●●●●●●
●
●●
●
●
●●●●●●●●●●
●●●●●
●●
●●●●●●●●●●●●●●●●●
●●
●●
●
●
●●
●●●
●
●
●
0 200 400 600 800 1000
015
0
Time (∆ = 600 seconds)
An
Motivation
▪ The ‘optimal’ time varies from day-to-day andweek-to-week.
▪ Can we do better than seasonality by incorporating more information about the user activity?
▪ Compare three models:■ Seasonality
■ Prediction based on day-of-week+time-of-day.
■ Aggregate Autoregressive Model■ Prediction based on overall ‘excitement.’
■ Aggregation-of-Individuals■ Prediction based on individual ‘excitement.’
Methodology — Seasonality
▪ We see that the number of users active showsweek-to-week seasonality.
▪ We estimate the seasonality using an average across W weeks:
▪ Take our seasonality prediction for time n as the value of this estimator at time n:
sn =1
W
X
j2{0,1,...,W�1}
An+jT .
ASn = sn.
Model 1 — Seasonality
▪ We see that the number of users active showsweek-to-week seasonality.
▪ We estimate the seasonality using an average across W weeks:
▪ Take our seasonality prediction for time n as the value of this estimator at time n:
sn =1
W
X
j2{0,1,...,W�1}
An+jT .
ASn = sn.
Model 2 — Aggregate Autoregressive
▪ An extension to seasonality allows the residuals about the seasonality to exhibit memory.
▪ Can think of this as ‘excitement’ in the Twitterverse.
▪ Ex. The capture of Osama bin Laden.
●●●
●●
●
●
●●●●●
●
●
●
●●●
●
●●
●●●●
●●●●●●●
●●
●
●●
●
●●●
●●
●
●●
●●●
●●
●●●
●●●
●●
●●●
●●
●
●●
●
●●
●●●●●●●●●●●
●
●
●●●
●●●●●●●●●●●●
●●
●
●●●●●●
●
●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●
●●
●
●●●●
●●
●●●
●
●●
●
●
●●
●●
●●●
●●●●●●●●
●
●
●
●●
●
●●
●
●●●
●●●
●
●
●●●
●
●●●
●●
●
●●●
●●●●●
●●
●●
●●●●
●
●
●●●
●
●●●●
●
●●
●●
●●
●
●
●●
●●●
●
●●●●●●●●●●●●●
●●
●●●●●●
●●●
●●●
●
●●
●●●●●●
●
●●●
●●●
●
●
●
●
●●
●
●
●●●
●
●
●
●●
●
●●●
●
●●
●●●●●●●
●
●
●
●●●●
●
●●
●●
●
●●●●
●●●●●●
●
●●
●●
●
●●
●
●●●
●●●●
●
●
●●●
●●
●
●
●
●●
●
●
●
●●●
●●●
●●
●
●●●●
●●●
●●●●●●●
●●●●●●●
●●●●
●
●●●
●
●●●●
●
●●●●●
●●
●
●
●●●●●●
●
●
●
●●
●
●●●●
●●
●
●●●
●
●
●
●●
●●●●
●●
●
●●
●●●●
●
●●●●
●●●●●
●
●●●●●●●
●
●●●●●●●●
●●
●●●●●
●
●●●
●●●●
●●●
●●●
●●●
●●●●●●
●●●●●
●●●●
●●
●●
●
●●●●●
●
●●●●
●
●●●●●
●
●●●●●
●●●
●●
●●
●●●●
●●●
●●●●
●●●
●●●●●●●
●●
●●
●●●●●●
●●●
●●●●●●
●
●●●
●●●●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●●
●
●●
●
●
●
●●●
●
●
●●●
●
●●
●●
●
●●●●●
●
●
●
●●
●
●
●●●
●●
●●●●●●●●●●
●●●
●
●●
●●
●●
●●●
●
●
●●●●●●●●●●●●●●●
●
●
●
●●●●●
●●●●●●●●●●●●●
●●●●●
●●
●●
●
●
●●●●●●●
●●
●●
●●
●
●●
●
●●
●
●●
●●●●
●
●
●●●●●●●●
●
●●
●●●●●
●●●●●
●
●●
●
●
●●●●●
●
●●●
●●●●
●●●
●●●●
●●●●
●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●
●
●●●
●
●●●
●●●●
●●
●●
●●
●●●●●
●
●
●●●●●●●●●●
●●
●●
●
●●●
●
●●
●●●
●●●●
●●●●●
●●●●●
●
●
●●●
●
●●●●●●●
●●●●●●●●●●
●●
●
●●
●●●●
●
●
●●●
●
●
●
●
●
●
●●
●
●
●●
●●●
●●●
●●
●●●●●●●●●●●
●●
●●●●●●●●●●●●
●
●●●
●
●
●●●●
●●●●
●●
0 200 400 600 800 1000
015
0
An
●
●
●
●●●●●
●●
●
●●●●
●●●●●
●●●●●
●●●
●●
●
●●●●●
●●●●
●
●
●●●●●●●●
●
●●
●●●●
●
●
●●●
●
●
●●
●
●
●
●
●●●●
●
●●
●●●
●
●
●
●
●
●●
●●●●
●
●●●
●
●●●●●
●●
●●
●●
●●●●●●●●●●●●●
●●●
●●●●●●●●●●●●●●●
●
●
●●●●●
●●●●●●
●●
●
●
●
●●
●
●●●
●●
●●●●●
●
●
●●
●
●
●●
●●●●●
●
●●
●●
●
●●●●●
●
●
●●
●●
●
●
●
●●●●●●●
●●
●
●
●
●
●
●●●●●●●●●●
●●●
●●●●●
●●●●
●
●●
●
●●●●●●●●
●●
●●●●●
●
●●●●●●●●●●●●
●●●
●●
●●●●●
●
●●●
●
●●
●●●●●
●
●
●●
●
●
●
●●●
●
●●●
●
●●
●●●
●
●
●
●
●
●
●
●
●●●
●●●●●●
●
●
●●●●
●
●●●●●
●
●●
●
●
●●●●●●
●●●
●●●●
●
●
●
●
●
●
●●
●
●●●●●●
●●
●●●●●●
●●●
●●
●
●●●●●●
●●●●
●●●●●●●●
●●●●●●●●
●●
●
●●
●
●●
●●●●
●
●
●
●
●
●
●
●●
●●●●●●
●●●
●
●●●
●●●
●
●
●●●●●●●
●●
●
●
●
●●●●●●●●●
●●●●●●
●●
●
●
●
●●
●●
●
●●
●
●●
●
●●
●
●●
●
●●●●
●●●
●●
●●●●
●●
●
●
●
●●
●
●
●
●
●●●●●●●
●●
●●●●●●●
●
●●
●
●
●●●●●●
●●●●●●●●
●●
●●
●
●●●●
●
●●
●●
●●●●
●●●
●●●
●
●
●●
●
●●●
●
●
●
●●●●●●
●
●●●
●●
●
●
●●
●●●
●
●
●
●
●●
●●●●
●
●
●
●●●●
●
●●
●●●
●
●
●●●●●●●
●●●●
●●●●●●●●●
●●●
●●●
●●●●
●●
●●●●●●●●●●●●●
●●●●●
●
●
●
●
●●
●●●●●●●●●●
●
●●
●
●●●●●●
●
●
●●●●
●
●●
●●●●
●
●●
●●●
●●●●●●
●●
●●
●●●●●
●
●
●●
●
●●
●
●
●●●●●●●●●●●●●●●●●●●
●●●●●●●
●
●●
●●●●
●
●●
●●
●
●
●
●●●
●●
●●●
●●●●●
●
●●●●●●●●●●●●●●
●
●
●●
●●
●
●●●
●●●●●●●
●●●
●
●
●
●●●
●●●●●●●●●
●●
●
●●●●●●●●
●
●●
●●
●
●
●●
●●●●
●
●
●
●
●●
●
●●●●
●
●●●●●●●●●●●
●●
●
●
●
●●
●
●●●
●●●
●●●●●
●●●●
●●●
●●●●●
●
●
●
●
●●●
●●●
●
●●●●
●
●●●●
●
●●●
●●●●●●●●●
●●●●●
●●
●●●●
●●●●●●●●●
●●●
●
●●●●●
●●●●
●
0 200 400 600 800 1000
015
0
An
●
●
●
●
●●
●
●
●●●●
●
●
●
●●●
●
●
●●●●●●●
●●●
●●
●
●●
●
●
●●●●
●
●
●●●
●
●
●●●
●
●
●
●
●
●
●
●
●●
●●
●
●
●●
●
●
●
●●●●●●
●
●●●●●
●●
●
●●●
●●●●●●●●●●●●●●●
●●●●●●
●
●
●
●●●●●●
●●●●●
●●●●●●
●
●●●●
●●
●
●
●
●●
●●
●
●
●●
●●
●
●
●●●●
●
●●●●
●
●●●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●●●●●●●
●
●
●
●
●
●
●●
●●
●
●●●●
●●●●
●
●●
●
●
●
●
●●●●●
●●●
●●●●●●●●
●●●
●●●●●●
●●●●
●●
●●●●●●●●●●●
●
●
●●
●●
●
●
●●●●
●●
●
●
●●
●●●
●●
●●●●●
●
●●●●●
●
●●●●●
●
●
●
●
●●●●●●●
●●●●●●●
●
●
●●
●●
●
●●
●
●●
●●
●
●●●●●●●●●●
●
●●●
●
●
●●●
●●●●
●
●●●
●●●●●
●
●
●●●
●
●
●
●
●●
●
●●●●●●●●
●●●●●●●●●●
●●●●●●●●
●●
●●●●●
●●●●●
●●●
●●
●●●●●●●●●
●
●●
●
●●●●●
●●●
●●●
●
●●
●
●
●●●●●●●●
●●●
●
●●
●
●●●
●●●●●
●
●
●
●
●
●●●●●●
●
●
●●●●●
●
●
●
●●●●
●●●●●●
●
●●
●
●●●●●●
●●●
●●●●
●
●●●
●●●●●●
●
●●●●●●●●●●●●●●●●●●
●
●
●
●●
●
●●●
●●●
●●●●
●●●
●
●●
●●●
●●
●
●●
●●
●●
●●●
●
●
●
●
●
●
●●●●
●
●●
●●
●
●●●●
●●
●
●
●
●
●●
●●
●●
●
●●●
●
●●
●●
●●●
●●
●●●
●
●●●●●
●●●●●●
●
●●●●●●●
●
●●
●●●●
●
●●●●
●●●
●●●●●●●●●
●●●●
●●
●●●●●●
●●●●●
●●●●
●●●●●●●
●●●●●
●●●●●
●●●●●●●●●●●
●●
●●●
●●
●
●●
●
●
●
●
●●●●●
●●●●
●
●
●●
●
●●●●
●●●●●●●●
●
●●
●
●
●●
●
●
●
●●●●●
●
●
●●●●
●●●
●●●●●●●●●●●
●●●●●●●●●
●●
●
●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●
●
●
●
●●●●●●●
●●●
●
●
●
●
●●●
●●
●●
●
●●●●
●
●●●●●●●
●●
●●●●●
●
●●●●
●●
●●
●●●
●●●●
●●
●
●
●
●
●●●●●●
●●
●
●●●
●
●
●●
●
●●●●●●●●
●
●●
●●●
●
●
●●
●
●
●●●●●●●●●●●
●
●●
●●●
●
●●●●●
●
●●●●●●●●●
●●●●●●●●●
●
●
●●
●●●
●●
●●
0 200 400 600 800 1000
015
0
An
●●●
●
●●●●
●
●●●●
●
●
●
●
●
●
●●
●
●●
●●
●●
●
●●●●
●●●●●
●●●
●
●●●●
●●●●●
●
●
●
●
●●
●●
●●●●●
●
●●●
●
●
●
●
●●●
●●
●
●●●●●
●
●
●
●
●●●●●
●
●
●●●●●●●●●●●●●●●●●●●
●●
●●●●●
●
●●
●●●●
●
●
●●●
●●●●●●●
●
●
●
●
●
●●●●●
●
●●
●
●
●●
●
●
●
●●
●
●
●●●
●●●
●
●
●●●●●●
●
●●
●●
●
●
●
●
●
●●●
●
●
●●
●●●●●●
●
●●
●●
●●●●●●●●
●
●
●
●●●
●
●
●
●●●●●●●
●●●●●
●
●
●
●
●
●
●
●
●
●●●
●
●●●
●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●
●●
●
●
●●
●
●●
●●
●●
●●●
●
●
●
●●●●●●●●
●
●
●●
●●
●●●
●●
●●●●●●●
●
●●●
●●●●●
●
●●
●
●●●●●
●●●●
●●●
●
●●
●
●
●●●
●
●●●
●●
●●
●
●
●●●●
●
●
●
●
●
●
●●
●●●●●●
●●
●
●●
●
●●●●●●●●●●●●●●●
●●●●
●
●●●
●●●●
●
●
●
●●●●●
●
●●
●
●●
●
●
●●●
●
●
●
●
●
●
●●●
●
●●
●
●●●●●
●
●●
●
●●●
●
●
●
●●●
●●●●●●
●●●
●●
●●
●
●●●●●●
●
●
●
●●●●●●
●
●
●●
●
●●●●●●
●●●●
●●
●●
●
●
●●●
●
●●
●●●●●
●
●
●
●
●●
●●●●●●●
●●●
●
●
●
●●●●●
●
●●●●
●
●
●●
●●
●●●
●
●
●
●
●
●
●●●●●
●●
●
●
●●
●●
●●●●
●
●
●
●
●
●
●
●●●●●
●
●●●●●●
●
●
●
●
●●●
●
●●●●●●●●●
●●●
●●
●●●●●●●●
●
●●●
●●
●
●●
●
●
●●
●●●
●●●●●●
●●
●
●●●●●●●
●
●
●●●●●●●
●
●●
●
●●●●
●
●●●●●
●●●●●
●●●
●●●●●●
●●●●●●●●●●●●●●●●●
●●
●●
●
●●●
●
●
●●
●●●●●
●
●●●
●
●●●●●
●
●
●●
●
●
●●●
●
●●●
●
●
●●●●
●
●●●
●●
●●
●●
●●●●
●
●●●
●
●
●
●●
●●
●●
●●
●●
●
●●●●
●
●●●
●
●
●●●
●●●
●●●
●●●
●
●●
●
●●
●●●●●●●●●
●
●●
●●●●●●
●
●●●●●●●●●●●
●
●●●
●
●●●●
●●
●
●●●●
●●●●●●●●
●●
●
●●●
●●●●
●
●
●●●
●●●
●
●
●
●●●
●
●
●
●●●●●
●●●●●●
●
●●
●●
●
●
●●
●●●●
●
●●●
●
●
●
●●●
●●●
●●●●●●●
●
●●
●
●
●●●●●●●●●●
●●●●●
●●
●●●●●●●●●●●●●●●●●
●●
●●
●
●
●●
●●●
●
●
●
0 200 400 600 800 1000
015
0
Time (∆ = 600 seconds)
An
Model 3 — Aggregation of Individuals
▪ The number of individuals active is the aggregation of individual activities.
▪ Previous work showed that many users exhibit stereotypical / predictable behavior.
▪ Predict each user u separately and then aggregate these predictions to get the overall prediction for AnAn.
▪ Use a hidden state representation, called an !-machine, to model the user’s behavior.
●●●●●●●●●●
●●●
●
●●●
●
●
●●
●●●●●
●●
●
●
●
●
●●●●●●●●
●●●●
●●●●●
●●●
●
●
●●
●●
●●●
●
●●●●●●
●●●●●●
●
●●
●●●●●●●●●
●●●●
●
●●●●●●●●●●●●
●
●●●●
●●●
●●●
●
●●●●
●
●
●
●●●
●
●●●
●●●●●
●●
●
●
●●●●
●●●●●
●●●
●
●●
●
●●
●
●●
●●
●
●●●●●●
●
●
●
●
●●●●●●●●
●●
●●●●
●●
●●●●
●●●●●
●
●
●
●
●
●●
●
●●
●●●
●
●
●●
●●
●
●●
●●●●●●
●●
●
●
●
●●●●
●
●
●
●●
●
●
●●●●●●●●
●
●●
●
●
●
●●●
●
●
●
●
●●●●
●●●
●
●
●●
●
●●●●
●
●
●●
●
●
●●
●
●●●●●●●
●
●●
●
●●
●●
●●
●
●●
●●
●
●●
●●
●●●●●●
●●●
●●●●●●●
●
●
●
●
●
●
●
●●●●●
●●
●
●
●
●●●●●●●●●●●
●●●●
●
●
●●●●
●
●●●●●●
●●
●
●
●●●●
●
●●●●●●
●●●●
●
●●●
●●●●●●
●●●●●●●
●
●●●
●●●
●●
●
●●●●
●●●
●
●
●●●●●●
●
●●●
●●
●●●●●●
●●●●●●●●●●●
●●●●
●●●
●●●
●
●
●
●
●●●
●
●●
●
●●
●
●●●●●●●●●●●●●
●
●●●●
●
●●
●
●●
●●●●
●●
●●
●●●●●●●●
●●●●●
●●●●●●
●
●●
●●●●●
●
●●
●●
●●
●●●
●●
●●●●●●●●●
●
●●
●
●●●●
●
●●●●●●●●
●●●●●
●
●●●●●
●
●●●
●
●
●
●●
●●
0 100 200 300 400 500 6000
100
AnCS
M
●●●
●●
●
●●●
●
●
●●●
●●
●●●
●
●
●●
●●●●
●
●●
●●●●●●●●
●●●●●
●●●●
●
●
●
●
●
●
●
●●●●●
●
●●
●●●
●
●
●●●
●
●●●●
●●●
●●
●
●
●●
●
●
●●●●
●●●●●●
●●
●
●●
●●●
●●●●●
●
●
●●
●●●●●●●●
●●
●
●
●●
●
●
●
●
●
●●
●●●
●●●●●●●●●
●●●
●●●●
●
●●●
●
●●
●●
●
●
●●
●
●
●●●●
●●●●●●●●
●
●
●
●
●●
●
●
●
●●●
●●●●●
●
●●
●●●●●●●
●
●
●
●
●
●●●●
●
●●●●
●●●●
●●●●
●●●
●
●
●●
●
●
●
●●●●
●
●
●●●●●●●
●
●
●●●●
●
●●●●●●●
●●●
●●●
●
●●●●
●●●●●
●
●
●
●
●
●
●
●
●●●
●●
●
●●●●
●●●●
●
●●●
●
●●
●
●
●
●
●●●●
●
●
●
●
●
●●●●●
●
●●●●
●●●
●●
●
●●
●●
●●●●
●●●
●
●
●●●●●●●●●●
●●●
●●●●
●
●
●●●●●●●●
●●
●●●●●●●●●●●
●
●●●●
●
●
●●●●
●
●
●
●●●●●
●●●●●●
●●●●●
●●
●
●●
●
●
●
●●●●●●
●
●●●
●
●
●●●●●
●
●●●●●●●●
●●●
●●
●●●●
●
●●
●●●
●●●
●
●●●
●●●●
●●●
●●●●
●
●
●●●●●●●
●●
●●●●●●●●
●●●●●
●
●●●●●●●
●●●●●
●
●●●●
●
●●●●●●●●●
●●
●
●●
●●●●●●●●●
●
●●●●●
●
●●●●●●●●●
●●
●
●●●●●
●●●
●
●
●
●●●
●
●●●●●●●
●●●●●
0 100 200 300 400 500 600
010
0AnCS
M
●●●●●●●
●
●●●
●●
●
●
●●●●
●●●
●
●●●
●●
●●●●
●●●●
●
●●
●●●
●●
●
●
●●●
●●●●●●●●●●●●●●
●
●●
●
●
●●
●●
●
●●●
●
●
●●
●●●●●●
●●
●
●●●
●
●
●●
●
●●
●●●
●
●
●
●
●●●●
●●●●●
●
●●
●
●●●●
●
●●
●●
●●
●●
●
●●●
●
●●●●
●
●●
●
●
●
●●
●
●
●
●●●●
●
●●●
●●●
●●
●
●●
●●
●
●
●●
●
●●●●
●●
●
●
●
●●
●
●
●●●
●
●
●
●
●
●●
●●●
●●
●●
●
●●●●●●
●●●●●
●
●●
●
●●●●●●
●●
●
●
●●
●●
●●
●●
●●●●●
●●●
●●●●
●
●●
●●●●
●
●●●●
●●●●●●●●●●●●
●●
●
●●
●
●●●●●
●●
●●●●
●●
●
●
●●●●
●●●
●●●●
●●●●●
●●
●
●●●
●
●
●
●
●●●
●●
●
●
●●●
●
●
●
●●●●●●●●●
●●●
●
●
●●●
●●●●
●●●●●●
●●●●●●
●●
●●●●●●
●●●
●
●●
●
●●
●●
●
●●
●●
●●
●
●
●
●
●●●
●
●●●●
●
●●
●
●●●●●
●●●●●●●●●●●●●
●
●
●●●●●●
●
●●●●●●●
●●●●●●
●●
●
●
●●
●●●
●
●●●●
●
●●●
●
●●
●
●●●
●
●●●●
●●●
●
●●●●●●●●●●
●●
●●●●
●●
●
●●●●●
●●●
●
●●●●
●●●●
●●●
●●●●●
●●●●●
●
●
●●
●
●●●●●●●●
●●
●●
●●●
●●
●
●●
●●●●●●
●●
●●●●●●
●
●●
●
●
●●
●●
●
●
●
●
●
●
●
●●●
●
●●●●●
●
●●
●
●
0 100 200 300 400 500 600
010
0AnCS
M
●
●
●●
●●
●
●
●●●
●
●●●●●●
●
●
●●●
●●●●●●●●●●
●
●
●●●●●●
●●●●●
●
●●●
●
●●●●
●●●●●
●
●
●
●
●
●
●●●●
●●●●
●●
●
●
●●
●
●●
●
●●
●●
●●●●●●●
●●
●●●●●●
●
●●●
●●●
●
●
●
●●●●
●●
●
●
●●●●●●
●●
●
●●
●●●
●
●●
●●●●
●
●●
●
●●●●
●●●●
●●●●●●●●●
●
●
●●●●
●
●
●●●●
●
●
●●●
●●●●●●
●
●●●●●
●
●●●●
●
●●
●●
●
●●●●●
●
●
●●●
●●
●
●
●●●
●
●●
●●
●●
●●
●
●●●●
●
●●
●●●
●●●
●●●●
●
●●●●●
●●●●●
●●●●●●●●
●
●●●●●●
●
●●●
●
●
●●●
●●●
●
●
●
●●●●●
●
●●●
●
●●●●●
●
●●●
●
●●●●●
●
●●●
●
●●●
●
●●●●●●
●
●
●
●
●●
●
●
●●●●
●
●●●
●●
●
●
●
●
●●●●●
●●
●
●
●●
●●
●
●
●●●●
●
●●●
●
●●
●
●●●●●
●●●●●●
●●●
●
●
●●●●●●●
●●●●
●●●●●●●●●
●●●●●●●
●●●●●●●●
●●
●●●●●
●●●●
●●
●●●
●●●●●●●●
●
●●
●
●
●●
●●●●
●
●
●●
●●●●●●●
●●
●●
●●●
●●●
●
●
●●
●●●●●
●●
●
●
●●●
●●●●●●
●●
●
●
●●
●
●
●●●●●●
●
●
●●●●
●●●●●●●●●●●
●●
●
●●●
●●●●
●
●
●
●
●
●●●
●
●●●
●
●●●●●●●●●●
●●●●●
●
●●●●●
●
●
●
●
●●
●
●●
●
●
●
●●●
●●
●●●●
0 100 200 300 400 500 600
010
0
Time (∆ = 600 seconds)
AnCS
M
Xn(u
2)X
n(u
2)X
n(u
2)
n (∆ = 600 seconds)
Xn(u
2)
Xn(u
3)X
n(u
3)X
n(u
3)
n (∆ = 600 seconds)
Xn(u
3)
Xn(u
4)X
n(u
4)X
n(u
4)
n (∆ = 600 seconds)
Xn(u
4)
Xn(u
2685)
Xn(u
2685)
Xn(u
2685)
n (∆ = 600 seconds)
Xn(u
2685)
Xn(u
1)X
n(u
1)X
n(u
1)
n (∆ = 600 seconds)
Xn(u
1)
...
+ + + + +
Xn(u1) Xn(u2) Xn(u3) Xn(u4) Xn(u2685)
An
Forecasting High Tide
▪ Question: How well can we predict that the activity level at a future time n will be at or abovequantile p* of activity for day d?
●●●
●●
●
●
●●●●●
●
●
●
●●●
●
●●
●●●●
●●●●●●●
●●
●
●●
●
●●●
●●
●
●●
●●●
●●
●●●
●●●
●●
●●●
●●
●
●●
●
●●
●●●●●●●●●●●
●
●
●●●
●●●●●●●●●●●●
●●
●
●●●●●●
●
●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●
●●
●
●●●●
●●
●●●
●
●●
●
●
●●
●●
●●●
●●●●●●●●
●
●
●
●●
●
●●
●
●●●
●●●
●
●
●●●
●
●●●
●●
●
●●●
●●●●●
●●
●●
●●●●
●
●
●●●
●
●●●●
●
●●
●●
●●
●
●
●●
●●●
●
●●●●●●●●●●●●●
●●
●●●●●●
●●●
●●●
●
●●
●●●●●●
●
●●●
●●●
●
●
●
●
●●
●
●
●●●
●
●
●
●●
●
●●●
●
●●
●●●●●●●
●
●
●
●●●●
●
●●
●●
●
●●●●
●●●●●●
●
●●
●●
●
●●
●
●●●
●●●●
●
●
●●●
●●
●
●
●
●●
●
●
●
●●●
●●●
●●
●
●●●●
●●●
●●●●●●●
●●●●●●●
●●●●
●
●●●
●
●●●●
●
●●●●●
●●
●
●
●●●●●●
●
●
●
●●
●
●●●●
●●
●
●●●
●
●
●
●●
●●●●
●●
●
●●
●●●●
●
●●●●
●●●●●
●
●●●●●●●
●
●●●●●●●●
●●
●●●●●
●
●●●
●●●●
●●●
●●●
●●●
●●●●●●
●●●●●
●●●●
●●
●●
●
●●●●●
●
●●●●
●
●●●●●
●
●●●●●
●●●
●●
●●
●●●●
●●●
●●●●
●●●
●●●●●●●
●●
●●
●●●●●●
●●●
●●●●●●
●
●●●
●●●●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●●
●
●●
●
●
●
●●●
●
●
●●●
●
●●
●●
●
●●●●●
●
●
●
●●
●
●
●●●
●●
●●●●●●●●●●
●●●
●
●●
●●
●●
●●●
●
●
●●●●●●●●●●●●●●●
●
●
●
●●●●●
●●●●●●●●●●●●●
●●●●●
●●
●●
●
●
●●●●●●●
●●
●●
●●
●
●●
●
●●
●
●●
●●●●
●
●
●●●●●●●●
●
●●
●●●●●
●●●●●
●
●●
●
●
●●●●●
●
●●●
●●●●
●●●
●●●●
●●●●
●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●●
●
●●●
●
●●●
●●●●
●●
●●
●●
●●●●●
●
●
●●●●●●●●●●
●●
●●
●
●●●
●
●●
●●●
●●●●
●●●●●
●●●●●
●
●
●●●
●
●●●●●●●
●●●●●●●●●●
●●
●
●●
●●●●
●
●
●●●
●
●
●
●
●
●
●●
●
●
●●
●●●
●●●
●●
●●●●●●●●●●●
●●
●●●●●●●●●●●●
●
●●●
●
●
●●●●
●●●●
●●
0 200 400 600 800 1000
015
0
An
●
●
●
●●●●●
●●
●
●●●●
●●●●●
●●●●●
●●●
●●
●
●●●●●
●●●●
●
●
●●●●●●●●
●
●●
●●●●
●
●
●●●
●
●
●●
●
●
●
●
●●●●
●
●●
●●●
●
●
●
●
●
●●
●●●●
●
●●●
●
●●●●●
●●
●●
●●
●●●●●●●●●●●●●
●●●
●●●●●●●●●●●●●●●
●
●
●●●●●
●●●●●●
●●
●
●
●
●●
●
●●●
●●
●●●●●
●
●
●●
●
●
●●
●●●●●
●
●●
●●
●
●●●●●
●
●
●●
●●
●
●
●
●●●●●●●
●●
●
●
●
●
●
●●●●●●●●●●
●●●
●●●●●
●●●●
●
●●
●
●●●●●●●●
●●
●●●●●
●
●●●●●●●●●●●●
●●●
●●
●●●●●
●
●●●
●
●●
●●●●●
●
●
●●
●
●
●
●●●
●
●●●
●
●●
●●●
●
●
●
●
●
●
●
●
●●●
●●●●●●
●
●
●●●●
●
●●●●●
●
●●
●
●
●●●●●●
●●●
●●●●
●
●
●
●
●
●
●●
●
●●●●●●
●●
●●●●●●
●●●
●●
●
●●●●●●
●●●●
●●●●●●●●
●●●●●●●●
●●
●
●●
●
●●
●●●●
●
●
●
●
●
●
●
●●
●●●●●●
●●●
●
●●●
●●●
●
●
●●●●●●●
●●
●
●
●
●●●●●●●●●
●●●●●●
●●
●
●
●
●●
●●
●
●●
●
●●
●
●●
●
●●
●
●●●●
●●●
●●
●●●●
●●
●
●
●
●●
●
●
●
●
●●●●●●●
●●
●●●●●●●
●
●●
●
●
●●●●●●
●●●●●●●●
●●
●●
●
●●●●
●
●●
●●
●●●●
●●●
●●●
●
●
●●
●
●●●
●
●
●
●●●●●●
●
●●●
●●
●
●
●●
●●●
●
●
●
●
●●
●●●●
●
●
●
●●●●
●
●●
●●●
●
●
●●●●●●●
●●●●
●●●●●●●●●
●●●
●●●
●●●●
●●
●●●●●●●●●●●●●
●●●●●
●
●
●
●
●●
●●●●●●●●●●
●
●●
●
●●●●●●
●
●
●●●●
●
●●
●●●●
●
●●
●●●
●●●●●●
●●
●●
●●●●●
●
●
●●
●
●●
●
●
●●●●●●●●●●●●●●●●●●●
●●●●●●●
●
●●
●●●●
●
●●
●●
●
●
●
●●●
●●
●●●
●●●●●
●
●●●●●●●●●●●●●●
●
●
●●
●●
●
●●●
●●●●●●●
●●●
●
●
●
●●●
●●●●●●●●●
●●
●
●●●●●●●●
●
●●
●●
●
●
●●
●●●●
●
●
●
●
●●
●
●●●●
●
●●●●●●●●●●●
●●
●
●
●
●●
●
●●●
●●●
●●●●●
●●●●
●●●
●●●●●
●
●
●
●
●●●
●●●
●
●●●●
●
●●●●
●
●●●
●●●●●●●●●
●●●●●
●●
●●●●
●●●●●●●●●
●●●
●
●●●●●
●●●●
●
0 200 400 600 800 1000
015
0
An
●
●
●
●
●●
●
●
●●●●
●
●
●
●●●
●
●
●●●●●●●
●●●
●●
●
●●
●
●
●●●●
●
●
●●●
●
●
●●●
●
●
●
●
●
●
●
●
●●
●●
●
●
●●
●
●
●
●●●●●●
●
●●●●●
●●
●
●●●
●●●●●●●●●●●●●●●
●●●●●●
●
●
●
●●●●●●
●●●●●
●●●●●●
●
●●●●
●●
●
●
●
●●
●●
●
●
●●
●●
●
●
●●●●
●
●●●●
●
●●●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●●●●●●●
●
●
●
●
●
●
●●
●●
●
●●●●
●●●●
●
●●
●
●
●
●
●●●●●
●●●
●●●●●●●●
●●●
●●●●●●
●●●●
●●
●●●●●●●●●●●
●
●
●●
●●
●
●
●●●●
●●
●
●
●●
●●●
●●
●●●●●
●
●●●●●
●
●●●●●
●
●
●
●
●●●●●●●
●●●●●●●
●
●
●●
●●
●
●●
●
●●
●●
●
●●●●●●●●●●
●
●●●
●
●
●●●
●●●●
●
●●●
●●●●●
●
●
●●●
●
●
●
●
●●
●
●●●●●●●●
●●●●●●●●●●
●●●●●●●●
●●
●●●●●
●●●●●
●●●
●●
●●●●●●●●●
●
●●
●
●●●●●
●●●
●●●
●
●●
●
●
●●●●●●●●
●●●
●
●●
●
●●●
●●●●●
●
●
●
●
●
●●●●●●
●
●
●●●●●
●
●
●
●●●●
●●●●●●
●
●●
●
●●●●●●
●●●
●●●●
●
●●●
●●●●●●
●
●●●●●●●●●●●●●●●●●●
●
●
●
●●
●
●●●
●●●
●●●●
●●●
●
●●
●●●
●●
●
●●
●●
●●
●●●
●
●
●
●
●
●
●●●●
●
●●
●●
●
●●●●
●●
●
●
●
●
●●
●●
●●
●
●●●
●
●●
●●
●●●
●●
●●●
●
●●●●●
●●●●●●
●
●●●●●●●
●
●●
●●●●
●
●●●●
●●●
●●●●●●●●●
●●●●
●●
●●●●●●
●●●●●
●●●●
●●●●●●●
●●●●●
●●●●●
●●●●●●●●●●●
●●
●●●
●●
●
●●
●
●
●
●
●●●●●
●●●●
●
●
●●
●
●●●●
●●●●●●●●
●
●●
●
●
●●
●
●
●
●●●●●
●
●
●●●●
●●●
●●●●●●●●●●●
●●●●●●●●●
●●
●
●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●
●●●●●
●
●
●
●●●●●●●
●●●
●
●
●
●
●●●
●●
●●
●
●●●●
●
●●●●●●●
●●
●●●●●
●
●●●●
●●
●●
●●●
●●●●
●●
●
●
●
●
●●●●●●
●●
●
●●●
●
●
●●
●
●●●●●●●●
●
●●
●●●
●
●
●●
●
●
●●●●●●●●●●●
●
●●
●●●
●
●●●●●
●
●●●●●●●●●
●●●●●●●●●
●
●
●●
●●●
●●
●●
0 200 400 600 800 1000
015
0
An
●●●
●
●●●●
●
●●●●
●
●
●
●
●
●
●●
●
●●
●●
●●
●
●●●●
●●●●●
●●●
●
●●●●
●●●●●
●
●
●
●
●●
●●
●●●●●
●
●●●
●
●
●
●
●●●
●●
●
●●●●●
●
●
●
●
●●●●●
●
●
●●●●●●●●●●●●●●●●●●●
●●
●●●●●
●
●●
●●●●
●
●
●●●
●●●●●●●
●
●
●
●
●
●●●●●
●
●●
●
●
●●
●
●
●
●●
●
●
●●●
●●●
●
●
●●●●●●
●
●●
●●
●
●
●
●
●
●●●
●
●
●●
●●●●●●
●
●●
●●
●●●●●●●●
●
●
●
●●●
●
●
●
●●●●●●●
●●●●●
●
●
●
●
●
●
●
●
●
●●●
●
●●●
●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●
●●
●
●
●●
●
●●
●●
●●
●●●
●
●
●
●●●●●●●●
●
●
●●
●●
●●●
●●
●●●●●●●
●
●●●
●●●●●
●
●●
●
●●●●●
●●●●
●●●
●
●●
●
●
●●●
●
●●●
●●
●●
●
●
●●●●
●
●
●
●
●
●
●●
●●●●●●
●●
●
●●
●
●●●●●●●●●●●●●●●
●●●●
●
●●●
●●●●
●
●
●
●●●●●
●
●●
●
●●
●
●
●●●
●
●
●
●
●
●
●●●
●
●●
●
●●●●●
●
●●
●
●●●
●
●
●
●●●
●●●●●●
●●●
●●
●●
●
●●●●●●
●
●
●
●●●●●●
●
●
●●
●
●●●●●●
●●●●
●●
●●
●
●
●●●
●
●●
●●●●●
●
●
●
●
●●
●●●●●●●
●●●
●
●
●
●●●●●
●
●●●●
●
●
●●
●●
●●●
●
●
●
●
●
●
●●●●●
●●
●
●
●●
●●
●●●●
●
●
●
●
●
●
●
●●●●●
●
●●●●●●
●
●
●
●
●●●
●
●●●●●●●●●
●●●
●●
●●●●●●●●
●
●●●
●●
●
●●
●
●
●●
●●●
●●●●●●
●●
●
●●●●●●●
●
●
●●●●●●●
●
●●
●
●●●●
●
●●●●●
●●●●●
●●●
●●●●●●
●●●●●●●●●●●●●●●●●
●●
●●
●
●●●
●
●
●●
●●●●●
●
●●●
●
●●●●●
●
●
●●
●
●
●●●
●
●●●
●
●
●●●●
●
●●●
●●
●●
●●
●●●●
●
●●●
●
●
●
●●
●●
●●
●●
●●
●
●●●●
●
●●●
●
●
●●●
●●●
●●●
●●●
●
●●
●
●●
●●●●●●●●●
●
●●
●●●●●●
●
●●●●●●●●●●●
●
●●●
●
●●●●
●●
●
●●●●
●●●●●●●●
●●
●
●●●
●●●●
●
●
●●●
●●●
●
●
●
●●●
●
●
●
●●●●●
●●●●●●
●
●●
●●
●
●
●●
●●●●
●
●●●
●
●
●
●●●
●●●
●●●●●●●
●
●●
●
●
●●●●●●●●●●
●●●●●
●●
●●●●●●●●●●●●●●●●●
●●
●●
●
●
●●
●●●
●
●
●
0 200 400 600 800 1000
015
0
Time (∆ = 600 seconds)
An
Num
ber o
f Use
rs R
etw
eetin
gA n
a : FTrued (a) = p⇤
In =
⇢1 : FTrue
d(n) (An) > p⇤
0 : otherwise
Forecasting High Tide
▪ For prediction, use a historical distribution based on the seasonality for day d:
▪ For a predicted activity level , one of predict whether the activity is above the quantile p* using:
FHistd (a) =
1
N�
N�dX
n=N�(d�1)+1
⇥AS
n a⇤.
ˆIn(p) =
⇢1 : FHist
d(n)(ˆAn) > p
0 : otherwise
An {ASn, A
ARn , AAI
n },
Results
SeasonalityAutoregressiveAgg. of Ind.
Year Seasonality Autoregressive Agg. of Individuals
2011 0.778 0.773 0.8252012 0.720 0.773 0.771
Area Under the ROC Curve (AUC):
Attribution Modeling with Machine LearningDavid Darmon, Arjuna Ariyaratne,
William Rand, PK Kannan, and Michelle Girvan
Click on Email Promotion Link
Click on TripAdvisor Link
No conversion No conversion Converts
Day 1 Day 8
Click on Paid Search Link
Day 1
Purchase Funnel
Firm.com Firm.com Firm.com
Conclusions
• Machine Learning plus Agent-Based Modeling provides a robust and powerful solution to harnessing the power of big data
• This method can provide predictive power that exceeds traditional aggregate methods
• Limitation: Requires high-resolution time series data
Future Work
• Construction of a software platform that would automatically generate agent-based models in a uniform way
• Implementation on GPGPU architectures
• Tests of the long-range predictive power of these models
