This document discusses using spatial analysis and mixture of Gaussians modeling to analyze geo-tagged tweets from a city to identify hot spots and patterns in people's behavior over time and location. The goal is to empirically model the spatial density of tweets. The document describes using expectation maximization to fit the mixture of Gaussians model to synthetic and real tweet location data, and issues that arose such as model collapsing and the lack of a global maximum. BIC was used to select the number of clusters but also had limitations. Future work proposed focusing on city centers and understanding when BIC works best.

1. Spatial Analysis of
Geo-tagged Tweets
Intern : Ali Akbari
Supervisors: Mohamed Kafsi
Vincent Etter
Professors: Matthias Grossglauser
Patrick Thiran
Summer internship @ EPFL

2. GOAL
• Finding a spatial pattern for geo-tagged
tweets
– Finding hot spots of a city
– Study their temporal evolution
– People behavior
• Empirical approach using real world data

3. Data
How can we model the spatial density of tweets ?

4. Mixture of Gaussians
• Normalization and positivity require
• Linear super-position of Gaussians
1
( ) ( | , )
K
k k k
k
p x N x 

 
1
1 0 1
K
k k
k
 

  
• Mixture of distributions
1
( ) ( | )
K
k
k
p x p x k

 

5. Maximum Likelihood
• Likelihood Function
 One Gaussian
 independent and identically distributed (i.i.d.)
 Data Set
• Likelihood Function for GMM
1
( | , ) ( | , )
N
n
n
p D N x 

  
1 2 3{ , , ,..., }ND x x x x
1 1
ln ( | , , ) ln ( | , )
N K
k n k k
n k
p D N x   
 
 
   
 
 

6. Expectation Maximization
1. Initialize
2. E-Step
 Evaluate responsibilities
3. M-step
 Re-estimate parameters, using current responsibilities
4. Evaluate log-likelihood
Check for convergence.
{ , , }k k k 
( )kz

7. • E-Step
• M-Step

8. EM Evolution

9. EM on Synthetic Data
Synthetic data EM Result
10,000 Points

10. Number of Clusters
• What if the number of clusters is unknown ?
• Naive approach: maximize likelihood
• Bayesian Information Criterion (BIC)

11. EM and real data
• EM works well with synthetic data
• Different situation with real data
• EM does not always converge
Collapsing
Duplicated point (repetitions)
Outlier points

12. Duplicated Point

13. Removing Places
• More flexible hot points

14. Heat Map of Revised Data (per day)

15. Heat Map of Revised Data (per Hour)

16. BIC Result

17. EM on Real Data
• First local minimum close to extremum
Number of clusters = 25

18. Conclusion
• Finding a spatial pattern for geo-tagged
tweets
 Mixture of Gaussian
 EM
 Given Number of Clusters
 Collapsing Issue
 BIC
 No global minimum !

19. Future Work
• Vary number of clusters with respect to time
• Understand in which cases BIC works
• Focus on city center to avoid outliers

20. Thank you