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 Spatio-Temporal Data

 Uncertain Spatio-Temporal Data

 Modeling Uncertain Spatio-Temporal Data

 Indexing Uncertain Spatio-Temporal Data

 Probabilistic Pruning

 Experiments

Agenda

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What is (certain) Spatio-Temporal Data?

3

time

x y

 Trajectory data from

 vehicles

 mobile users

 animals

 ...

 Can be represented treating time as an additional spatial

dimension

(4)

What is (certain) Spatio-Temporal Data?

4

location space

time space

ts te

 A spatio-temporal database stores triples (oid, time, loc)

 In the best case, this allows to look up the location of an object at any time

(5)

What is (certain) Spatio-Temporal Data?

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location space

time space

Q

ts te

 A spatio-temporal database stores triples (oid, time, loc)

 In the best case, this allows to look up the location of an object at any time

 Allows to answer queries such as

„Return objects that intersects some spatial window within some time interval“.

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What is Uncertain Spatio-Temporal Data?

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 In most applications, this data is not complete

Delays between GPS signals

RFID sensors located only in certain locations

Wireless sensor nodes sending infrequently to preserve power

Geo-application check-ins location space

time space

ts ta tb te

? ? ?

? ?

? ? ? ?

Q

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What is Uncertain Spatio-Temporal Data?

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 Existing works

Bound the set of possible

(location,time) pairs of an object between observations by using spatio-temporal approximations (diamonds)

e.g. by modeling knowledge about maximum speed

location space

time space

ts ta tb te

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What is Uncertain Spatio-Temporal Data?

8

 Existing works

Bound the set of possible

(location,time) pairs of an object between observations by using spatio-temporal approximations (diamonds)

e.g. by modeling knowledge about maximum speed

Allows to make statements like

„its possible that o intersects some query window Q“

But how likely is this event?

What is the probability of the object traveling through Q?“

location space

time space

Q

ts ta tb te

(9)

Modeling Uncertain Spatio-Temporal Data

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 Using a more powerful model [1]

 The position of an object o at some time t is a random variable

 The trajectory of o follows a stochastic process, i.e. a family

of random variables o(t) loc

ation space

time space

ta

large probability small

probability

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Modeling Uncertain Spatio-Temporal Data

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 Using a more powerful model [1]

 The position of an object o at some time t is a random variable

 The trajectory of o follows a stochastic process, i.e. a family of random variables o(t)

 Given a predicate

, the event that o satisfies

is a random event.

location space

time space

ta

Q

large probability small

probability

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Markov Chain Model for UST data

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 Assumes discrete state space S and discrete time space T

 Given the position of an object o at time t=i, the position at t=i+1 is conditionally independent of t=i-1

 Transition probabilities stored in a (sparse) |S|x|S| matrix M(o,t), called transition matrix

 M(o,t)[i,j] is the probability that object o will transition to state j at time t+1, given o is located at state i at time t

location space

time space

ta

Q

large probability small

probability

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Markov Chain Model for UST data

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 Incorporation of additional observations possible using Bayesian inference

 Each possible path can be associated with a probability

 Several probabilistic window queries possible

 Use sparse matrix operations for efficient implementation

 Details can be found in [1]

location space

time space

ta

Q

tb

large probability small

probability

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Indexing Uncertain Spatio-Temporal Data

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 Large number of objects and observations in a database

 Checking each pair of successive observations requires to much computation

 How to prune as many objects and observations as possible during query evaluation?

location space

time space

Q

“Which objects intersect Q with

a probability of at least τ?”

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Indexing Uncertain Spatio-Temporal Data

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 Large number of objects and observations in a database

 Checking each pair of successive observations requires to much computation

 How to prune as many objects and observations as possible during query evaluation?

 Temporal Pruning?

location space

time space

Q

“Which objects intersect Q with

a probability of at least τ?”

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Indexing Uncertain Spatio-Temporal Data

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 Index possible positions in

(location + time) in an R*-Tree

 Apply multistep filter

R-Tree Filter

location space

time space

Q

“Which objects intersect Q with

a probability of at least τ?”

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Indexing Uncertain Spatio-Temporal Data

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 Index possible positions in

(location + time) in an R*-Tree

 Apply multistep filter

R-Tree Filter

Diamond Filter

location space

time space

Q

“Which objects intersect Q with

a probability of at least τ?”

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Indexing Uncertain Spatio-Temporal Data

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 Index possible positions in

(location + time) in an R*-Tree

 Apply multistep filter

R-Tree Filter

Diamond Filter

True Hit Detection

location space

time space

Q

“Which objects intersect Q with

a probability of at least τ?”

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Indexing Uncertain Spatio-Temporal Data

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 Index possible positions in

(location + time) in an R*-Tree

 Apply multistep filter

R-Tree Filter

Diamond Filter

True Hit Detection

Probabilistic Pruning location space

time space

Q

“Which objects intersect Q with

a probability of at least τ?”

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Probabilistic Pruning

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 Lower bound the probability of the object (o) to move through the gray area only PLB(o in gray)

 Then PUB(o intersects Q) = 1- PLB(o in gray)

 If PUB(o intersects Q ) < τ

 o can be pruned

 How obtain PLB(o in gray) without computing it?

location space

time space

Q

“Which objects intersect Q with

a probability of at least τ?”

(20)

Probabilistic Pruning

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 Offline computation:

Precompute probability for o to stay in sub-diamonds

Simple variation of the window query for each sub-diamond

location space

time space

λ 0 1

λ

1

00 1

p

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Probabilistic Pruning

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 Offline computation:

Precompute probability for o to stay in sub-diamonds

Simple variation of the window query for each sub-diamond

location space

time space

λ 0 1

λ

1

00 1

p

(22)

Probabilistic Pruning

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 Offline computation:

Precompute probability for o to stay in sub-diamonds

Simple variation of the window query for each sub-diamond

location space

time space

λ 0 1

λ

1

00 1

p

(23)

Probabilistic Pruning

23

 Offline computation:

Precompute probability for o to stay in sub-diamonds

Simple variation of the window query for each sub-diamond

location space

time space

λ 0 1

λ

1

00 1

p

(24)

Probabilistic Pruning

24

 Offline computation:

Precompute probability for o to stay in sub-diamonds

Simple variation of the window query for each sub-diamond

location space

time space

λ 0 1

λ

1

00 1

p

(25)

Probabilistic Pruning

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 Offline computation:

Precompute probability for o to stay in sub-diamonds

Simple variation of the window query for each sub-diamond

location space

time space

λ 0 1

λ

1

00 1

p

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Probabilistic Pruning

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 Offline computation:

Storing all these probabilities is of course memory inefficient

Approximation with a linear

function using linear programming

location space

time space

λ 0 1

λ

1

00 1

p

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Probabilistic Pruning

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 During Query Processing:

Construct largest possible sub- diamond

Lookup of PLB(o in sub-diamond)

location space

time space

λ

1

00 1

p

Q

λ

0

1

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Experiments

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 Indexing UST data yields orders of magnitude over a scan based method (with temporal pruning)

 Using diamond and probabilistic filtering (UST-Tree) speeds up the R*-Tree by factor 3-4

Setting: 1000 objecs á 100

observations and 10000 possible locations in 2D

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Experiments

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 All Filters can be computeted efficiently in contrast to the verification step

(~500 ms in this example)

 Effectiveness of probabilistic filter is dependent on the threshold τ

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Experiments

30

 Probabilistic spatio-temporal window queries are usually CPU- bound

 I/O-cost of UST-Tree are higher in the filter step since more

information is stored in the leaves (=> higher tree)

 I/O-cost of UST-Tree are lower in the refinement step since less candidates have to be loaded from disk

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 Indexing based on the Markov-Chain Model for UST data

 Yields several orders of magnitudes over scan-based method

 Yields 3-4 times speedup over straightforward indexing

 Techniques for spatio-temporal and probabilistic pruning are applicable to other models for UST data

 Techniques are extendable to the multi-dimensional case

 More special cases are treated in the paper

 Other window queries

 Query window overlapping several diamonds of the same object

Summary

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[1] T. Emrich, H.-P. Kriegel, N. Mamoulis, M. Renz, and A. Züfle. Querying uncertain spatio-temporal data.

In Proceedings of the 28th International Conference on Data Engineering (ICDE), Washington, DC, 2012.

Related Work

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P

Index Entries at Leaf Level:

oid leaf level:

directory levels:

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location space

time space

Q Q

Referenzen

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