1. ST Data-warehouse for trajectories Some preliminary ideas S. Orlando, R. Orsini, A. Raffaetà, A. Roncato

2. Requirements and Starting points  Trajectories arrive in streams, as triples (ID, SpatialPos, TemporalPos) to insert information associated with them in our data warehouse, spatial and temporal dimensions must be discretized to fit our cube model For example, we can think of considering two Spatial and one Temporal dimensions  What are the main approaches present in the literature to deal with ST aggregates?  Which are the aggregates that we would like to compute on trajectories?  Can ST aggregates in literature be applied to our case?

3. Main approaches in the literature  I.F. Vega Lopez, R.T. Snodgrass, B. Moon. ST Aggregation Computation: A Survey. IEEE TKDE, 17:2, 2005 Aggregates computed on partitions, obtained by grouping on attributes Simple or sliding window aggregates No moving objects  Y. Tao, D. Papadias. Historical ST Aggregation, ACM TOIS, 23:1, 2005 Main focus is on index data structures Typical aggregates are distributive  F aggr (S 1  S 2 ) = F aggr (S 1 ) op F aggr (S 2 )  S 1  S 2 =  Partially consider moving objects  Others?

4. The cube model: an example The pollution density data: X t 5 4 3 X 5 4 3 5 3 4 44 t Dx Dt + in this ST area the pollution is 5; + in this ST area the pollution is 4; + in this ST area the pollution is 3;

5. Problems of space-driven structures Discretization problems: X t 54 t 44 44 4?5 5 5 X

6. Data-driven structures Each region is the “original” rectangle X t 5 4 t 5 4 X R1R1 R2R2

7. Problems with data-driven structures Intersectiong regions count twice?? X t 54 t 54 X 2323 Partially overlapping query counts as a whole

8. The cube model for trajectories The number of objects: X t X t Dx Dt 2 2 1 1 + a steady object (constant x); + a forward moving object (increasing x); + a backward moving object (decreasing x);

9. Problems of cube model Discretization problems with trajectories : X tt X A fast object is in 4 “places” at the same moment 1 111

10. Problems of cube model Discretization problems with trajectories : X tt X We don’t know what happens between the 2 points 1 1 ?? Should we interpolate and how?

11. Different kinds of queries  Queries computed by using only the given attributes  Queries computed by a pre-calculation which can involve more than one “close” subcubes (ST properties not explicitly given but computed)  Queries computed by considering the whole trajectory hence by using not only close subcubes  Not distributive queries

12. First kind of queries  ST density of objects Number of objects in a fixed area and in a given time interval Area and temporal intervals depend on the granularity of our cube  To compute such aggregates  We need only info related to the presence/absence of objects in the given ST element  Thus, we forget IDs and other spatio-temporal information (speed, distance etc.)

13. Problems of cube model Discretization problems with trajectories : X tt X A fast object is in 4 “places” at the same moment 1 111

14. Second kind of queries  Total distance or average distance  Number of objects moving towards East  Number of objects which change direction

15. Third kind of queries  Number of objects which have covered a certain distance  Number of objects which are back to the starting point  Difference between the going and back  The aggregation used to solve such a kind of queries should be recomputed changing the parameter

16. Fourth kind of queries  Shape of the average trajectory  Compute the median

17. Topological queries With ID: enter, leave, cross, stay within, bypass X t Enter: before out; now inLeave: before in; now outStay within: before and now inCross: before out; now out; region touchedBypass: not touched

18. Left-in and Right-in Without ID we can compute the following queries: left-in (passing the left borderline inward), right-in (passing the right borderline inward); left-out (passing the left borderline outward), right-out (passing the right borderline outward) X t left-in = enter from left + cross from left left-in+right-in ≠ enter

19. How to compute left-in, right-in Problems on computing in: 1)The aggregate is on left-in and right-in not directly on in; 2)The associative function to compute left-in (right-in) is a left projection (right projection) function: does the commercial products provide these functions? Let S and S’ be left-in S  S’ = left(left-in S, left-in S’) = left-in S right-in S  S’ = right(right-in S, rigth-in S’) = right-in S’ SS’

20. Cross (1) Without ID we cannot compute: cross X t X t From aggregate data it is impossible to distinguish the two above cases (???)

21. Cross (2) Cross cannot be computed from cube-cross X t X t 11 1 1 S cube-cross = 2 on shaded area, while cross = 0

22. Navigational queries Considering derived information: speed (max, avg, min), heading, traveled distance, covered area. Are these computable from aggregates? Speed is of type 2; Heading is of type 3; Traveled distance is of type 2; Covered area is of type 3;