ST Data-warehouse for trajectories

1. ST Data-warehousefor 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 • Faggr(S1 S2) = Faggr(S1) op Faggr(S2) • S1 S2 =  • Partially consider moving objects • Others?

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

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

6. Data-driven structures Each region is the “original” rectangle t t R1 R2 5 5 4 4 X X

7. Problems with data-driven structures Intersectiong regions count twice?? t 2 3 t 2 3 5 4 5 4 X X Partially overlapping query counts as a whole

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

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

10. Problems of cube model Discretization problems with trajectories : t t 1 ? ? 1 X X We don’t know what happens between the 2 points 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 : t t 1 1 1 1 X X A fast object is in 4 “places” at the same moment

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 t Enter: before out; now in Leave: before in; now out Stay within: before and now in Cross: before out; now out; region touched Bypass: not touched X

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) t left-in+right-in ≠ enter left-in = enter from left + cross from left X

19. How to compute left-in, right-in • Problems on computing in: • The aggregate is on left-in and right-in not directly on in; • 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’ S S’

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

21. Cross (2) Cross cannot be computed from cube-cross t t 1 1 1 1 S X X 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;