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

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

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)

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

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].

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

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

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

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

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

Results

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]

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

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

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

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

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.