1. Trajectory Pattern Mining
Fosca Giannotti
Dino Pedreschi
Mirco Nanni
Fabio Pinelli
Chris Andrews
Georgia Institute of Technology
B.S. Computer Science
5th Year Undergraduate

2. Concepts Analyze trajectory of moving objects
A 3mins B 5mins C 10mins D
Trajectory Patterns – description of frequent behavior relating to space and time
Frequent Sequence Pattern (FSP)
Determine if trajectory sequence matches any trajectory patterns in a given set
Study different methods of preparing a Temporally Annotated Sequence (TAS) for data mining

3. Trajectory Patterns (T-Patterns)
sequence of time-stamped locations
S = { ( x0, y0, t0 ) , … , ( xn, yn, tn ) }
Temporal Annotation
set of times relating to trajectories
A = { a1 , a2, … an }
Temporally Annotated Sequence
(S,A) = (x0,y0) a1 (x1,y1) a2 … an (xn,yn)

4. Neighborhood Function
Neighborhood Function N : R2 -> P (R2)
Calculates spatial containment of regions
Input point to find enclosing Region of Interest
Defines the necessary proximity to fall into a region
Parameters:
e – radius or necessary proximity of points

5. Regions of Interest (RoI)
Performing these comparisons on points is costly
A simple preprocessing step can alleviate this
Utilize the Neighborhood Function NR()
Translate each set of points into regions
Timestamp is selected from when the trajectory first entered the region
Now compare sequence of regions and timestamps using the TAS mining algorithm presented in [2].

6. Static RoI Neighborhood Function NR()
Initially receives set of R disjoint spatial regions
R regions are predefined based on prior knowledge
Each represents relevant place for processing
Static NR() simplifies problem of mining patterns
Sequence of points become grouped
Result: sequence of regions
(x,y) a1 (x’,y’) becomes X a1 Y

7. Dynamic RoI Data sets often do not possess predetermined regions
Instead need to formulate regions based on criteria of density of the trajectories
Preprocessing now must determine set R of popular regions from the data set
R is now the set of Region of Interests from used by the Neighborhood Function NR() to translate points into Regions of Interest

8. Popular Regions Grid G of n x m cells Density Threshold d
Each cell with density G(i,j) Set R of popular regions
Each region in R forms rectangular region
Sets in R are pair wise distinct
Dense cells always contained in some region in R
All regions in R have average density above d
All regions in R cannot expand without their average density decreasing below d

9. Grid Density Preparation
Split space into n x m grid with small cells
Increment cells where trajectory passes
Neighborhood Function NR() determines which surrounding cells
Regression - increment continuously along trajectory

10. Popular Regions Algorithm
Algorithm: PopularRegions( G, d )
Complexity: O ( |G| log |G| )
Iteratively consider each dense cell
For each:
Expands in all four directions
Select expansion that maximizes density
Repeat until expansion would decrease below density threshold

11. Results

12. Evaluating the T-Patterns
Compute density of each cell of grid
Compute set of RoI’s by determining Popular Regions
Translate the input trajectories into sequence of RoI’s and timestamps for the transitions
Input the trajectories and times into TAS mining algorithm[2]

13. Experiments GPS Data Performance Analysis
Fleet of 273 trucks in Athens, Greece
112,203 total points recorded
Running both static & dynamic pattern algorithms
Various parameter settings
Performance Analysis
Synthetic Data by CENTRE synthesizer
50% random & 50% predetermined

14. Pattern Mining Results
Static found: A t1 B t2 B
Dynamic found: A t1 B’ t2 B’’

15. Execution Time Results
Increase linearly with increasing number of input trajectories (both algorithms)
Grow when density threshold decreases
Static performs better with extreme threshold
Static does not perform with middle threshold

16. Additional Results
Increasing radius of spatial neighborhood obtains irregular performance and large values lead to poor execution times
Changing time tolerance (t) obtains results similar to TAS’s
Increasing the number of points in each trajectory causes linear growth of execution times

17. Works Cited
[1] Trajectory pattern mining, Fosca Giannotti, Mirco Nanni, Fabio Pinelli, Dino Pedreschi, Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining KDD. ACM, 2007.
[2] Efficient Mining of Sequences with Temporal Annotations. F. Giannotti, M. Nanni, and D. Pedreschi. In Proc. SIAM Conference on Data Mining, pages 346–357. SIAM, 2006.