Managing Uncertainty in ConstantlyEvolving Environments

1
Managing Uncertainty inConstantly-Evolving
Environments

• Reynold Cheng (ckcheng_at_cs.purdue.edu)
• http//www.cs.purdue.edu/homes/ckcheng
• Department of Computer Science
• PhD Oral Defense
• Major Advisor Prof. Sunil Prabhakar
• March 31st, 2005

2
Sensor Databases
Goal data retrieval in a correct, efficient and
scalable manner
3
Data Uncertainty

• Due to limited network bandwidth and battery
  power, readings are sampled
• The value of the entity being monitored (e.g.,
  temperature, location) is changing
• Most of the time the database stores old values
• Query results can be incorrect!

4
Answering Minimum Query with Database Readings
Recorded Temperature
30
Current Temperature
x1
y0
20

• Database X
• Correct answer Y

10
x0
y1
0
oF
x
y
5
Bounding Uncertainty with Dead-Reckoning

• Data values cannot change drastically
• The system negotiates a bound d with the sensor

v-d,vd
System
(v, d)
sensor
v

• Trade-off between data uncertainty and update
  frequency

6
Answering Minimum Query with Error-Bounded
Readings
Recorded Temperature
30
Bound for Current Temperature
y0
20

• x certainly gives the minimum temperature reading

10
x0
0
oF
x
y
7
Answering Minimum Query with Error-Bounded
Readings
Recorded Temperature
uncertainty pdf
30
Bound for Current Temperature
y0
20

• (X,0.7), (Y,0.3)
• Answers augmented with probabilistic guarantees
• Measurement error also a source of uncertainty

10
x0
0
oF
x
y
8
Probabilistic Queries

• We represent the imprecision in the value of the
  data as an interval with associated pdf
• Query answers are augmented with probabilities
• Probabilistic queries give us a correct (possibly
  less precise) answer, instead of a potentially
  incorrect answer

9
Related Work

• Wolfson et al. DPD99 and Pfoser et al. ISSD99
  discussed probabilistic range queries for moving
  objects
• Olston and Widom SIGMOD02 discussed tradeoff
  between precision and performance of querying
  replicated data
• Probabilistic queries ICDE03,SIGMOD03,TKDE04,VLDB
  04c
• Deshpande et al. presented probabilistic
  prediction for sensor values VLDB04b
• Recent proposals CIDR05a,CIDR05b for
  probabilistic modeling for uncertain data

10
Probabilistic Databases

• Probabilistic Database each tuple is augmented
  with a probability value (Barbara,Garcia-Molina
  Porter) TKDE92.
• Dalvi Suciu VLDB04a studied efficient query
  evaluation with ranked results.
• Probabilistic database in semi-structured model
• XML data (Nierman Jagadish) VLDB02
• Acyclic data structure (Hung,Getoor
  Subrahmanian) ICDE03

11
My Dissertation

• Characterization of uncertainty in sensor and
  mobile databases
• Classification, evaluation and quality of
  probabilistic queries
• Efficient techniques for supporting uncertainty
  management in databases

12
Outline

• Introduction
• Related Work
• Uncertain Data Probabilistic Queries
• Indexing for Probabilistic Range Queries
• Probabilistic Join Processing
• Experimental Results
• Prototype

13
Outline

• Introduction
• Related Work
• Uncertain Data Probabilistic Queries
• Indexing for Probabilistic Range Queries
• Probabilistic Join Processing
• Experimental Results
• Prototype

14
Uncertainty Model
fi(x) uncertainty pdf
Ti.z
Li
Ri
uncertainty interval

• Ti.z dynamic attribute (e.g., temperature,
  locations)
• Used in various application domains, e.g.,
• location uncertainty DPD99, ISSD99
• DNA microarray data error NAS02

15
Classification of Probabilistic Queries

• Nature of answer
• Value-based returns a single value
• e.g., Average query (l,u, pdf)
• Entity-based returns a set of objects
• e.g., Range query ((Ti,pi), pigt0)
• Aggregation
• Aggregate interplay between objects decides
  result e.g., Nearest-Neighbor query
• Non-aggregate whether an object satisfies a
  query is independent of others
• e.g., Range query

16
Classification of Probabilistic Queries

• Only probabilistic range query(entity-based
  non-aggregate class) studied before WS99,ISSD99

17
Quality of Probabilistic Result

• Probabilistic queries notion of result "quality"
• Motivating example range query (is Ti.z in range
  l, u?)
• regular range query
• "yes" or "no"
• probabilistic range query
• Propose metrics for each class of probabilistic
  queries

18
Quality for Value- Aggregate Queries

• Query result l,u, p(x) x ? l,u
• U3,4 less ambiguous than U1,100
• Differential entropy
• Measures uncertainty associated with r.v. X with
  pdf p
• max(H(X)) log2(u-l) iff XUl,u (most
  uncertain)

19
Outline

• Introduction
• Related Work
• Uncertain Data Probabilistic Queries
• Indexing for Probabilistic Range Queries
• Probabilistic Join Processing
• Experimental Results
• Prototype

20
Probabilistic Range Query
Recorded Temperature
Uncertainty for Current Temperature
30
20

• (T1,0.2),(T2,0.8)

10
0
oF
T1
T2
21
Probabilistic Threshold Range Query (PTRQ)

• Users are likely to be concerned with results
  with a high probability
• Retrieve sensor ids with readings between 10oF to
  25oF with probability 0.7
• PTRQ Given a,b and p, return Ti where
  Prob(value of Ti is inside a,b) p

22
Solving PTRQ with Interval Indexes

• Use R-tree or interval index FOCS96, JCSS96,
  ADI00 to find intervals intersecting a,b
• For each object retrieved, evaluate its
  probability of being within a,b
• Return objects with probability p

23
The Problem of Current Indexes

• Current Interval indexes do not consider
  probabilities during search
• Many irrelevant objects (probability lt p) may be
  processed.
• Our approach
• Probability Threshold Indexing (PTI)
• 1D interval R-tree with uncertainty
• Variance-based Clustering
• Transform intervals to 2D points and index based
  on variance

24
Pruning in a 1D R-Tree

• Some intervals in the MBR may satisfy Q
• Need to retrieve the contents of the MBR and
  evaluate

25
x-bounds in a PTI Node
left-0.2-bound
right-0.2-bound
? 0.2
0.8
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x-bounds in a PTI Node
left-0-bound (MBR)
right-0-bound (MBR)
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Pruning with x-bounds
left-0.2-bound
right-0.2-bound

• An MBR is not retrieved if there exists an
  x-bound
• p gt x
• b on the left of left-x-bound

• An MBR is not retrieved if there exists an
  x-bound
• p gt x
• a on the right of right-x-bound

28
Implementation of PTI
29
Drawback of PTI

• Extra overhead in storing x-bounds
• Small intervals near edges limit gains

right-0.2-bound
left-0.2-bound
30
Clustering 2D points
cluster of large intervals
yRi

• When 2D points are clustered, intervals of
  different variances are separated

• Points clustered based on means and variances
  (variance-based clustering)

xy
(Li,Ri)
cluster of smaller intervals
xLi
31
Answering PTRQ with 2D R-Tree

• Construct a 2D R-tree over uncertain data by
  indexing (meani,variancei)
• Query the 2D R-Tree
• For uniform pdf, a PTRQ can be converted to a
  2D-range query

32
Querying Uniform pdf
y Ri
Li
Ri
xy
Q (p 0.75)
b
a
y(1-p)xp a Intervals containing a
a ltx lt y lt b Intervals in a,b
x(1-p)yp ? b Intervals containing b
b-a p(y-x) Intervals containing a,b
a
b
x Li
a
b
1D View (Uniform pdf)
2D View
33
Outline

• Introduction
• Related Work
• Uncertain Data Probabilistic Queries
• Indexing for Probabilistic Range Queries
• Probabilistic Join Processing
• Experimental Results
• Prototype

34
Table Join

• Join is an important database problem
• Join operator , ?, gt, lt

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Join over Uncertainty

• How do we define comparison operators for
  uncertain data?

36
Semantics of Comparison Operators

• We studied the meaning of ,?, gt,lt
• Imprecision about two values satisfying a
  comparison is expressed by a probability value

37
Equality Join

• In continuous domain, 2 real values are equal at
  a point with zero probability
• Resolution c a is equal to b if they are within
  c of each other.

38
Efficient Join Processing

• Computing joins is costly
• Probability Threshold p can help
• 3 pruning techniques between
• 2 Items,
• 2 Pages, and
• 2 Indices
• Example Equality join with resolution c

39
Item-Level Pruning

• Assume cumulative distribution function Fi(x)
• Let la,b,c be max(La- c, Lb - c) ,
• Let ua,b,c be min(Ra c, Rb c)
• P(a c b) is at most
• min(Fa(ua,b,c) - Fa(la,b,c), Fb(ua,b,c) -
  Fb(la,b,c)) (?)
• Equation (?) is easy to compute
• If Equation (?) lt p, then a and b can be pruned

40
Page-Level Pruning

• Goal Prune pages R and S without examining
  individual items
• Solution Place x-bounds on R and S, and perform
  4 tests with x-bounds

41
Page-Level Pruning

• Goal Prune pages R and S without examining
  individual items
• Solution Place x-bounds on R and S, and perform
  4 tests with x-bounds

R.left-x-bound
R.right-x-bound
Page R
42
BNLJ and U-BNLJ

• BNLJ
• Block-Nested-Loop Join
• Joins 2 lists of unordered pages
• U-BNLJ
• Place x-bounds on every page
• Use page-level pruning techniques

43
Index-Level Pruning

• Extension of page-level pruning
• Construct PTI on the relation
• Perform page-level pruning on the page storing
  the node (consists of x-bounds for children nodes)

44
INLJ and U-INLJ

• INLJ
• Index-Nested-Loop Join
• Construct a R-tree or interval index for inner
  relation
• U-INLJ
• Construct PTI for inner relation

45
Outline

• Introduction
• Related Work
• Uncertain Data Probabilistic Queries
• Indexing for Probabilistic Range Queries
• Probabilistic Join Processing
• Experimental Results
• Prototype

46
Experiment 1 Threshold Range Query

• Compare number of I/Os between
• 1D R-tree on intervals only
• PTI (1D R-tree with probability thresholds)
• 2D variance-based clustering (called Extensive)

47
Simulation Model

• 100K uncertain data, with length uniformly
  distributed in 0,10000 and uniform uncertainty
  pdf
• 10K PTRQs with length of a,b normally
  distributed and p ? 0.1,1
• Each PTI node contains five x-bounds, where x
  ?0.1,0.3,0.5,0.7,0.9

48
Scalability of Indexes

• Both PTI and Extensive outperform R-tree
• Answering PTRQ with R-tree requires more
  computation
• Extensive needs about 50 less I/Os than PTI

49
Query Probability Threshold

• R-tree does not benefit from the increasing value
  of p
• When p is 0.5, Extensive is 4 times better than
  PTI

50
Experiment 2 Threshold Join

• 2 tables of uncertain data are generated
• Compare interval joins and x-bound-enhanced
  equality joins
• BNLJ vs. U-BNLJ
• INLJ vs. U-INLJ
• Measure no. of candidate pairs

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BNLJ vs. U-BNLJ
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INLJ vs. U-INLJ
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Effect of Resolution c
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Outline

• Introduction
• Related Work
• Uncertain Data Probabilistic Queries
• Indexing for Probabilistic Range Queries
• Probabilistic Join Processing
• Experimental Results
• Prototype

55
U-DBMS Prototype

• A system for handling uncertain data
• Meta-queries for specifying data uncertainty
  (e.g., uncertainty interval, type of uncertainty
  pdf,)
• Extension of SQL operators to support different
  probabilistic query classes
• Measurement of probabilistic answer quality
• Allows easy addition of new uncertain data types
  (e.g., uncertain pdf) and query operators

56
Architecture of U-DBMS

• UNCERTAIN class data structures (e.g.,
  histogram) and access methods (e.g., find
  variance of uncertainty)
• New data types and operators do not interfere
  with the original DBMS
• Implemented with PL/SQL and external C code

57
Example Queries
58
Summary

• Data staleness in sensor and mobile databases can
  render incorrect query answers
• We propose uncertainty interval as a trade-off
  between data imprecision and update frequency
• We perform an extensive study of uncertainty
  management classification, quality and efficient
  evaluation of probabilistic queries, indexing,
  joins
• Prototype implementation using PostgreSQL

59
Related Publications

• TKDE04 R. Cheng, D. V. Kalashnikov, and S.
  Prabhakar. Querying imprecise data in moving
  object environments. In IEEE TKDE, 2004.
• VLDB04c R. Cheng, Y. Xia, S. Prabhakar, R.
  Shah, and J. S. Vitter. Efficient indexing
  methods for probabilistic threshold queries over
  uncertain data. In VLDB 2004.
• SIGMOD03 R. Cheng, D. Kalashnikov, and S.
  Prabhakar. Evaluating probabilistic queries over
  imprecise data. In ACM SIGMOD 2003.
• ICDE03 R. Cheng, S. Prabhakar, and D. V.
  Kalashnikov. Querying imprecise data in moving
  object environments. In IEEE ICDE 2003.
• HCI04 R. Cheng and S. Prabhakar. Using
  Uncertainty to Provide Privacy-Preserving and
  High-Quality Location-Based Services. In Workshop
  on Location Systems Privacy and Control, Mobile
  HCI04.
• VSSN04 K.Y. Lam, R. Cheng, B. Liang and J.
  Chau. Sensor Node Selection for Execution of
  Continuous Probabilistic Threshold Queries in
  Wireless Sensor Networks. In VSSN, ACM Multimedia
  2004.

60
References

• FOCS96 L. Arge and J. S. Vitter. On dynamic
  interval management in external memory (extended
  abstract). In FOCS, p. 560-569, 1996.
• TKDE92 D. Barbara, H. Garcia-Molina and D.
  Porter. The management of probabilistic data.
  IEEE TKDE, 4(5)487-502, 1992.
• NAS02 Significance and statistical errors in
  the analysis of DNA microarray data J. Brody, 
  B. Williams,? B. Wold,? and S. Quake. Proc. Natl.
  Acad. Sci., U S A., 2002, 199(20).
• CH89 C. Chatfield. The analysis of time series
  an introduction. Chapman and Hall, 1989.
• VLDB04a N. Dalvi and D. Suciu. Efficient Query
  Evaluation on Probabilistic Databases. In VLDB
  2004.
• VLDB04b A. Deshpande, C. Guestrin, S. Madden,
  J. Hellerstein and W. Hong. Model-Driven Data
  Acquisition in Sensor Networks. In VLDB, 2004.
• CIDR05a A. Deshpande, C. Guestrin and S.
  Madden. Using Probabilistic Models for Data
  Management in Acquisitional Environments. In
  CIDR, 2005.
• ICDE03 E. Hung, L. Getoor and V. S.
  Subrahmanian. PXML A Probabilistic
  Semistructured Data Model and Algebra. In ICDE
  2003.

61
References

• JCSS96 P. C. Kanellakis, S. Ramaswamy, D.
  Vengroff, and J. S. Vitter. Indexing for data
  models with constraints and classes. In J. Comp.
  Syst. Sci, 52(3)589-612, 1996.
• ADI00 Y. Manolopoulos, Y. Theodoridis, and V.
  J. Tsotras. Chapter 4 Access methods for
  intervals. In Advanced Database Indexing, Kluwer,
  2000.
• VLDB02 A. Nierman and H. V. Jagadish. ProTDB
  Probabilistic Data in XML. In VLDB 2002.
• DPD99 O. Wolfson, P. Sistla, S. Chamberlain,
  and Y. Yesha. Updating and querying databases
  that track mobile units. Distributed and Parallel
  Databases, 7(3), 1999.
• SIGMOD02 C. Olston and J. Widom. Best-effort
  cache synchronization with source cooperation.
  In Proc. Of the ACM SIGMOD 2002.
• ISSD99 D. Pfoser and C. S. Jensen. Capturing
  the Uncertainty of Moving-Object Representations,
  in Proc. of the Sixth International Symposium on
  Spatio Databases, Hong Kong, July 20-23, 1999,
  pp. 111-132.
• CIDR05b J. Widom. Trio A system for integrated
  management of data, accuracy and lineage. In
  CIDR, 2005.

62
Thank You

• Imprecise data is prevalent in increasing number
  of applications. I believe uncertainty management
  is an emerging and important area.

Life is Uncertain... Eat Dessert First! - S.
Gordon and H. Brecher
63
How to define uncertainty pdf?

• The form of uncertainty pdf depends on the
  application e.g., Gaussian distribution models
  measurement error.
• If no information about pdf is known, a simple
  way is to assume uniform pdf a pessimistic
  estimation
• Can also use more sophisticated techniques, based
  on time-series analysis on past data for pdf
  derivation CH89

64
Classical Decomposition

• For a discrete time series, let Xt be a random
  variable at time t
• Xt mt st Yt
• mt trend, a slowly-moving function
• moving-average filter, exponential smoothing,
  curve fitting/regression
• st seasonal component periodic function
• Yt noise component
• Example mt2t1,stsin(t),YtN(0,1)
• pdf(100) N(201sin(100),1)