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Probabilistic Queries and Uncertain Data Sunil Prabhakar Department of Computer Sciences Purdue University Email: [email protected] http://www.cs.purdue.edu/homes/sunil Introduction The traditional database model expects data items to be modeled as sets (bags) of tuples consisting of precise attribute values. However, real-world data does not easily fit into this model if there is uncertainty in the information. Uncertainty comes from many sources: unreliable measurements and data sources, incomplete or missing information, irreconcilable facts, … This problem has been recognized for a long time (e.g. NULL values) and numerous models have been proposed. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 2 Introduction Long history of ideas for incorporating uncertain data in databases Many proposals for models Recent renewed interest in the area Some initial work on developing systems This tutorial provides a sampling of the area. More information at http://www.cs.purdue.edu/homes/sunil Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 3 Outline Motivating examples Proposed Models Implementation issues Efficiency Scalability Prototypes Open problems References Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 4 Application: Sensor databases sensor Database System queries Network Channel sensor External Environment e.g., temperature, moving objects, hazardous materials results sensor sensor user Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 5 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! Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 6 Answering a Minimum Query Recorded Temperature Current Temperature 30 x1 y0 20 10 Database: X Correct answer: Y x0 y1 0 oF x Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b y 7 Bounding Uncertainty with Dead-Reckoning Data values cannot change drastically The system negotiates a bound d with the sensor [v-d,v+d] (v, d) sensor System v Trade-off between data uncertainty and update frequency Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 8 Answering Minimum Query with Error-Bounded Readings Recorded Temperature Bound for Current Temperature 30 y0 20 x certainly gives the minimum temperature reading 10 x0 0 oF x y Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 9 Answering Minimum Query with Error-Bounded Readings Recorded Temperature Bound for Current Temperature uncertainty pdf 30 y0 20 How do we determine the answer to this query? Each sensor has some chance of given the minimum reading. Probabilistic Queries 10 x0 0 oF x Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b y 10 Probabilistic Queries As attribute values become uncertain (actually, imprecise), operators (e.g =, <,>) over these data need to be defined. These operators may no longer return Boolean results. Instead, given the probability distributions, they can return probabilistic answers Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 11 Answering Minimum Query with Error-Bounded Readings Recorded Temperature Bound for Current Temperature 30 y0 20 (X,0.7), (Y,0.3) Answers augmented with probabilistic guarantees 10 x0 0 oF x Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b y 12 Sensor Errors In the previous examples, uncertainty was introduced in order to avoid incorrect results Uncertainty may be inherent due to measurement errors, e.g. Most scientific instruments have well known errors GPS has a Gaussian distribution Micro-array data have a Lorentzian distribution Statistical results also have margins of error Similar to previous case Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 13 Data Privacy Uncertainty may sometimes be desirable in order to provide privacy for individuals. Instead of reporting an exact location to a Location-Based service provider, users can obfuscate their location to a small spatial region. This naturally results in ambiguity (uncertainty) in query results. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 14 Application: Protein Annotation Consider a protein database that records the functions of the proteins (annotations). Some function information is experimentally derived and has high confidence (certainty). More often, annotations are transferred based upon computational results HMMs Sequence similarity Rule bases Such annotations are inherently less reliable. As these annotations propagate, so do the errors. It is desirable to be able to capture the uncertainties in the annotations within the database. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 15 Application: Text Retrieval In text retrieval systems, answers to queries are typically inexact. For example, “Find documents on uncertain data management” Results are ranked in order of relevance to the query Thus, the answer can be viewed as having a probability of being part of the result relation When multiple conditions are tested -- how do we combine these rankings? Probabilistic modeling can help in this situation. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 16 Application: Data Integration & Cleaning When integrating multiple database, it is necessary to identify matches between tuples For many pairs, there is no clear Yes/No answer to the matching question Existing methods can provide a probability or degree of match which can be exploited in an application-specific manner. How should these uncertainties in the result of cleaning or integration be handled? Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 17 Unreliable Sources, Missing Data Consider the following cases: Information received from certain sources may not be entirely reliable (compromised sensors, poor quality of data, …). Information from multiple sources may be inconsistent, even contradictory. An attribute’s exact value may not be known, but it can be only one of few possibilities. Each of these cases are examples where the data is uncertain. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 18 Application Needs In summary, we see that there are numerous applications for which uncertainty in data is either inherent or desirable. Existing systems do not provide any support for uncertain data thereby compelling applications to morph their data to fit the model. There is a real need for the development of database systems that handle uncertain data. The characteristics of uncertainty are diverse and often application-dependent. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 19 Outline Motivating examples Proposed Models Implementation issues Efficiency Scalability Prototypes Open problems References Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 20 Uncertain Data Models There have been numerous proposal for models. Some distinguishing features include: Nature of uncertainty (probabilitic, …) Types of databases (Relational, XML,…) Complexity of uncertainty Granularity of uncertainty Handling correlations Handling missing data Types of uncertainty supported Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 21 Types of uncertainty models Qualitative models NULL values Definite, Indefinite, or Maybe [LS87,LS91] Quantitative models Probabilistic Dempster-Shafer (evidence-based) [LSS96, Lee92] Fuzzy sets (possibilities) [CUP06] Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 22 Probabilistic Models There are two main types of probabilistic data uncertainty addressed in recent work: Attribute uncertainty The value of an attribute of a tuple is not known precisely Modeled as a set or range of possible values with associated probabilities Tuple uncertainty The membership (presence) of an entire tuple within a relation is uncertain Maybe modeled as an probability attached to the tuple. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 23 Other Models Some systems consider both types ([GUP06]) Table uncertainty has also been proposed to handle coverage of a table (what percentage of tuples are present in the table) [Wid05]. Probabilistic database in semi-structured model XML data (Nierman & Jagadish) [NJ02] Acyclic data structure (Hung,Getoor & Subrahmanian) [HGS03] Fuzzy databases [GUP06] (possibility values) Uncertainty in Deductive Databases [LS97,LS01,LS03] Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 24 Tuple Uncertainty There has been a significant amount of work in this domain dating back (at least) to 1979. The basic idea is that the membership of a tuple in a relation is not certain. This uncertainty may reflect the degree of confidence that this tuple belongs to the relation or the degree of relevance of the tuple to the relation (a query answer). Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 25 Some Tuple Uncertainty Models Cavallo and Pittarelli [CP87] Fuhr and Roellke [RK97] Fuhr [Fuhr95] Dey and Sarkar [DS96] TRIO [Wid05] Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 26 Fuhr [FR97,Fuhr90,Fuhr95] Input relations are assumed to have attributes that have probabilistic events associated with them. These are assumed to be independent The evaluation of queries results in new tuples with complex events associated with them. These tuples may no longer be independent thus causing complications. Fuhr solves this problem using intensional semantics -- for each tuple, the complex event is derived. In the final step the probability value of this event is computed. This is very expensive and complicated. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 27 Dalvi & Suciu [DS04, DS05] Dalvi and Suciu explore extensional evaluations -the probability values of tuples after the application of operators are computed. However, this can lead to incorrect results in some cases. Notion of safe query plans. An algorithm to identify a safe extensional plan for a query is developed. May not always return a result. Heuristic plans and approximations are proposed for the case where the data complexity of the query is #P-complete. [DS05] addresses the case where input relation tuples are not independent. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 28 Information Source Tracking Fereidoon Sadri [FS91, FS95] Sources of data are assigned a reliability Query answers and derived data are also assigned a score that can be computed Each tuple is assigned a propositional formula that describes its certainty (in terms of the reliability of sources) -- vectors Sources are assumed to be independent Computing a query implies computing the vectors for each tuple and then computing the corresponding certainty -- requires certainty of sources Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 29 Information Source Tracking (Cont.) Possible worlds semantics: k sources, 2k possible relations Provided definitions of extended operators that guaranteed Soundness and completeness: I.e. the result of these operators over uncertain relations had the same set of possible words as applying regular relational operators over the possible worlds of the input relations Efficiency concerns due to large size of pwd. Algorithms for aggregations also developed, but mostly expensive or NP-Complete Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 30 Attribute Uncertainty The earliest example of work in this area is the notion of NULL values (Codd) The probabilistic data model (PDM) proposed in [BHP92] -- focus on discrete values ProbView [LLR+97] Continuous attribute case proposed for sensor data [CKP03] Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 31 Codd’s model for uncertainty NULL values are a means of capturing uncertainty with three-valued logic (T,F,M) A-mark and I-mark also introduced along with a four-valued logic (T, F, A, I) A-mark implies that the attribute value exists, but is not known. I-mark implies that the attribute value is undefined, or does not exist. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 32 Probabilistic Data Model Barbara, Garcia-Molina, Porter [BGP92] Discrete attribute uncertainty Key attributes are deterministic (precise) Notion of attribute groups (handles dependent data) Captures missing probability (no assumption) Probabilities may be user defined, statistically determined, due to staleness, etc. STUDENT Adam GPA 3.8 INTEREST ACC_EVAL 0.7[theory] 0.6[Y A] 0.3[*] 0.1[N A] 0.3[* *] Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 33 Probabilistic Data Model (cont.) Selects can refer to attributes or probabilities Selection conditions specify cutt-off probabilities Two flavors -- must and maybe (with or without the missing probability) SELECT APPLICANTS WHERE ACC_EVAL: V = [Y, *], P > 0.7 (Adam not in result -- Must semantics) SELECT APPLICANTS WHERE ACC_EVAL: v = [Y, *], p > 0.7 (Adam in result -- Maybe semantics) Natural joins allowed where join attribute must be key for one of the relations (not commutative) Project similarly defined for dropping attributes from groups Studied impact of missing probabilities on joins -- may lead to loss of information. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 34 Probabilistic Data Model (contd.) New operators: -SELECT, -Join: Based upon similarity of probability distributions STOCHASTIC: convert regular relation to probabilistic based upon given schema (freq gives probability) DISCRETE: convert probabilistic relation to a regular relation (based upon expected values) GROUP: merge two or more attribute groups into one Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 35 ProbView [LLR+97] Attribute values specified as alternative discrete values with probability intervals. Attribute uncertainty is converted to tuple uncertainty. Possible worlds are derived from this set with upper and lower bounds on probabilities. Annotated relations obtained by flattening probabilistic relations with path (expressions on worlds) Computing probabilities for queries is done via userspecified functions. Relational algebra operations are extended to handle the probability bounds and paths. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 36 Continuous Attribute Uncertainty fi(x) – uncertainty pdf [L uncertainty interval R] Cheng, Kalashnikov, Prabhakar [CKP03a, CKP04] Allow an attribute value to be a continuous range with an associated probability density function The cumulative probability over the interval should be 1 General continuous attribute uncertainty model Covers models used in various application domains, e.g., location uncertainty [WSCY99, PJ99] DNA microarray data error [BWW+02] Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 37 Probabilistic Nearest Neighbor Query At distance r, A is the nearest neighbor of Q if: A is at distance r from Q B,C,D are all located at distances > r from Q. The pdf pA(r) can be computed. C A r D B Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 38 Probabilistic Nearest Neighbor Query Compute pA(r) From the shortest distance of A to Q (nA) To the longest distance of A to Q (fA) C fA A nA Q D fA PA p A (r )dr nA B Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 39 Classification of Probabilistic Results Four classes of queries identified [CKP03b] 1. 2. Nature of result values Continuous: returns a single value e.g., Average query ([l,u], pdf) Discrete: returns a set of objects e.g., Range query ({(Ti,pi), pi>0}) Relationship between result values Independent: whether an object satisfies a query is independent of others e.g., Range query Interdependent: interplay between objects decides result e.g., Nearest-Neighbor query Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 40 Classification of Probabilistic Queries Continuous Discrete Independent Interdependent What is the temperature of sensor x? Which sensor has temp between 10oF and 30oF? What is the average temperature of the sensors? Which sensor gives the highest temperature? The notion of query answer quality was also introduced. For each class of queries, a metric for query quality was specified. Intuitively, this metric captures the degree of uncertainty in the answer (as compared to an answer derived over precise data). Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 41 Quality of Probabilistic Result Probabilistic queries: notion of result "quality" Example: range query (is Ti.z in range [l, u]?) regular range query "yes" or "no" probabilistic range query Score | pi 0.5 | 0.5 l u a) b) c) 1 | pi 0.5 | Score _ of _ an _ ERQ | R | iR 0.5 Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 42 Quality for Continuous-Interdependent Queries Query result: [l,u], {p(x) : x [l,u]} U[3,4] less ambiguous than U[1,100] Differential entropy u H ( X ) p( x ) log 2 p( x )dx l Measures uncertainty associated with r.v. X with pdf p max(H(X)) = log2(u-l) iff X~U[l,u] (most uncertain) Score _ of _ Value _ Aggr _ Query H ( X ) Metrics for other classes also proposed. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 43 Outline Motivating examples Proposed Models Implementation issues Efficiency Scalability Prototypes Open problems References Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 44 Implementation Challenges Many proposals have not addressed the issues of implementation Some models are known to be very expensive computationally, e.g. the model proposed in [FR97]. Is it possible to avoid enumeration of all possible worlds in order to compute queries? Notion of safe queries and extensional evaluation [DS04]. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 45 Extensional Semantics [DS04] Intensional evaluation is very expensive. Propose new extensional evaluation where probabilities are continuously maintained. Can lead to incorrect results -- develop the notion of safe extensional plans based upon PWD semantics. Extensional plans not always available. Some heuristics have been proposed. Can one do better? Work done in the context of queries with uncertain predicates (information retrieval). What about other domains? Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 46 Orion Query Evaluation [CKP03] Probabilistic Range Query example Recorded Temperature Uncertainty for Current Temperature {(T1,0.2),(T2,0.8)} 30 20 10 p1 12 25 f (z)dz p2 f 2 ( z)dz 1 15 10 0 oF T1 Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b T2 47 Probabilistic Threshold Range Query (PTRQ) Users are likely to be concerned with results that meet a given cutoff 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 How to exploit indexes for such queries? 1. 2. Use R-tree or interval index [AV96, KRVV96, MTT00] to find intervals intersecting [a,b] For each object retrieved, evaluate its probability of being within [a,b]. Return objects with probability ≥ p Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 48 Problem with Current Indexes Current Interval indexes do not consider probabilities during search Many irrelevant objects (probability < p) may be processed. New indexes for probabilistic data. Orion [CXP+04]: Probability Threshold Indexing (PTI) 1D interval R-tree with uncertainty Variance-based Clustering Transform intervals to 2D points and index based on variance Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 49 Pruning in a 1D R-Tree Q (p = 0.3) a b •Some intervals in the MBR may satisfy Q •Need to retrieve the contents of the MBR and evaluate Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 50 x-bounds in a PTI Node left-0.2-bound right-0.2-bound 0.2 left 0.2bound Li f i (y)dy 0.2 ≥ 0.8 Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 51 x-bounds in a PTI Node left-0.2-bound left-0.3-bound right-0.2-bound left-0-bound (MBR) left/right-0.5-boundright-0.3-bound right-0-bound (MBR) Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 52 Pruning with x-bounds left-0.2-bound right-0.2-bound Q (p = 0.3) a Q (p = 0.3) b a b An MBR is not retrieved if there exists an x-bound p>x a b on the right left ofofleft-x-bound right-x-bound Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 53 Drawback of PTI Extra overhead in storing x-bounds Small intervals near edges limit gains left-0.2-bound right-0.2-bound Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 54 Clustering 2D points cluster of large intervals y=Ri Points clustered When 2D points are in the same based means and clustered, intervals of vicinityon have similar variances (variancedifferent variances means and variances based clustering) are separated x=y (Li,Ri) variance of [Li,Ri] mean of [Li,Ri] cluster of smaller intervals x=Li Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 55 Answering PTRQ with 2D R-Tree Construct a R-tree over 2D points transformed from the intervals Convert PTRQ to a 2D-range query Query the 2D R-Tree Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 56 Querying Uniform pdf y = Ri Li Ri x=y b (Li,Ri) Q (p = 0.75) a a x(1-p)+yp b-a y(1-p)+xp a <x ≥ p(y-x) < y<≥bba Intervals containing ba Intervals Intervals Intervals containing containing in [a,b] [a,b] b a 1D View (Uniform pdf) b x =Li 2D View Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 57 Implemented Systems U. Washington Tuple-uncertainty Built as a layer over SQLServer 2000 Evaluation of similarity queries over certain data. Orion (Purdue) Attribute uncertainty Extension of PostgreSQL Defines new uncertain data types, and operators Boolean operations over uncertain data (thresholds) http://orion.cs.purdue.edu/ Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 58 Orion 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 Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 59 Example Queries Create a table with UNCERTAIN type CREATE table T( k INTEGER primary key, a UNCERTAIN); Insert Gaussian pdf (μ,σ) Insert into T values (1,‘(g,μ,σ)’); Display uncertain info. of a if a > 5 SELECT a FROM T where a > 5; Equality join of uncertain attributes (=% returns probability of equality) SELECT R.k, S.k, R.a =% S.a FROM R,S WHERE R.a = S.a; Entities with prob. giving min value of a (e.g., {(3,0.5), (5,0.3), (11,0.2)} SELECT Emin(T.a) from T; Min value of a for table T (UNCERTAIN) SELECT Vmin(T.a) from T; Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 60 Outline Motivating examples Proposed Models Implementation issues Efficiency Scalability Prototypes Open problems References Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 61 Models A large number of models have been proposed. Some are subsumed by others. Still unclear which is the best model (if any). What model should be used for what applications? What is the nature of uncertainty for important classes of applications? Which model(s) are applicable? Mapping model to user notions. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 62 Model issues Models What types of uncertainty does a model provide? Is the model complete? Closed? Query semantics for a given model How to handle missing data? Correlations? Models for specific domains? User interpretation and understandability. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 63 Implementation issues How should uncertainty be represented in the system? Efficient algorithms for query evaluation. Query optimization Operators over uncertain data. New types of queries. Index structures for uncertain data. Should we approximate? Threshold queries? How should probabilities (uncertainties) be attached to data? Query language extensions. User-interfaces -- how can users understand and control the impact of uncertainty? Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 64 References [AV96] L. Arge and J. S. Vitter. On dynamic interval management in external memory (extended abstract). In FOCS, p. 560-569, 1996. [BGP92] D. Barbara, H. Garcia-Molina and D. Porter. The management of probabilistic data. IEEE TKDE, 4(5):487-502, 1992. [BWW+02] J. Brody, B. Williams, B. Wold, and S. Quake Significance and statistical errors in the analysis of DNA microarray data. Proc. Of the National Academy of Sciences, U S A., 2002, 1;99(20). [CH89] C. Chatfield. The analysis of time series an introduction. Chapman and Hall, 1989. [CKP04] R. Cheng, D. V. Kalashnikov, and S. Prabhakar. Querying imprecise data in moving object environments. In IEEE TKDE, 2004. [CKP03b] R. Cheng, D. Kalashnikov, and S. Prabhakar. Evaluating probabilistic queries over imprecise data. In ACM SIGMOD 2003. [CPK03a] R. Cheng, S. Prabhakar, and D. V. Kalashnikov. Querying imprecise data in moving object environments. In IEEE ICDE 2003. [CP04] R. Cheng and S. Prabhakar. Using Uncertainty to Provide Privacy-Preserving and HighQuality Location-Based Services. In Workshop on Location Systems Privacy and Control, Mobile HCI’04. [CXP+04] 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. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 65 References [DGM+04] A. Deshpande, C. Guestrin, S. Madden, J. Hellerstein and W. Hong. Model-Driven Data Acquisition in Sensor Networks. In VLDB, 2004. [DGM05] A. Deshpande, C. Guestrin and S. Madden. Using Probabilistic Models for Data Management in Acquisitional Environments. In CIDR, 2005. [DS04] N. Dalvi and D. Suciu. Efficient Query Evaluation on Probabilistic Databases. In VLDB 2004. [DS05] N. Dalvi and D. Suciu. Answering Queries from Statistics and Probabilistic Views. In VLDB 2005. [FR97] N. Fuhr and T. Roelleke, A Probabilistic Relational Algebra for the Integration of Information Retrieval and Database Systems, ACM Transactoins on Information Systems, 15(1): 32-66, 1997. [Fuhr90] N. Fuhr. A Probabilistic Framework for Vague Queries and Imprecise Information in Databases. In VLDB, 1990. [Fuhr95] N. Fuhr. Probabilistic Datalog Logic for Powerful Retrieval Methods. In Proc. Of ACM SIGIR, 1995. [GUP06] J. Galindo, A. Urrutia, M. Piattini. Fuzzy Databases: Modeling, Design, and Implementation. Idea Group Publishing, ISBN: 1-59140-324-3 [HGS03] E. Hung, L. Getoor and V. S. Subrahmanian. PXML: A Probabilistic Semistructured Data Model and Algebra. In ICDE 2003. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 66 References [JSS94] S. Vrbsky and J.W.S. Liu. Producing approximate answers to set- and single-valued queries. The Journal of Systems and Software, 27(3),1994. [KRVV96] 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. [KT01] S. Khanna and W.C. Tan. On computing functions with uncertainty. In 20th ACM Symposium on Principles of Database Systems, 2001. [LCL+04] 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. [Lee92] S. K. Lee, An extensional relational database model for uncertain and imprecise information. In Proc. Of VLDB, 1992. [LLR+97] L. V. S. Lakshmanan, N. Leone, R. Ross, V. S. Subrahmanian: ProbView: A Flexible Probabilistic Database System. ACM Trans. Database Syst. 22(3): 419-469 (1997) [LS87] K. C. Liu and R. Sunderraman. An Extension to the Relational Model for Indefinite Databases, Proceedings of the ACM-IEEE Computer Society Fall Joint Computer Conference, Dallas, Texas, Pages 428--435, 1987 [LS91] K.C. Liu and R. Sunderraman, A Generalized Relational Model for Indefinite and Maybe Information, IEEE Transactions on Knowledge and Data Engineering, Vol. 3, No. 1, Pages 65--77, 1991 Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 67 References [LS97] L. V. S. Lakshmanan, F. Sadri: Uncertain Deductive Databases: A Hybrid Approach. Inf. Syst. 22(8): 483-508 (1997) [LS01] L. V. S. Lakshmanan, F. Sadri: On a theory of probabilistic deductive databases. TPLP 1(1): 5-42 (2001) [LS03] L. V. S. Lakshmanan, F. Sadri: On A Theory of Probabilistic Deductive Databases CoRR cs.DB/0312043: (2003) [LSS96] Lim, Srivastava, and Shekhar, An Evidential Reasoning Approach to Attribute Value Conflict Resolution in Database Integration, IEEE Transactions on Knowledge and Data Engineering, Vol. 8, No. 5, 1996 [MTT00] Y. Manolopoulos, Y. Theodoridis, and V. J. Tsotras. Chapter 4: Access methods for intervals. In Advanced Database Indexing, Kluwer, 2000. [NJ02] A. Nierman and H. V. Jagadish. ProTDB: Probabilistic Data in XML. In VLDB 2002.[PJ99] 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. [SWC+98] P. A. Sistla, O. Wolfson, S. Chamberlain, and S. Dao. Querying the uncertain position of moving objects. In Temporal Databases: Research and Practice. 1998. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 68 References [TWZ+02] G. Trajcevski, O. Wolfson, F. Zhang and S. Chamberlain. The Geometry of Uncertainty in Moving Objects Databases. In EDBT 2002. Springer LNCS 2287, pp. 233-250. [Wid05] J. Widom. Trio: A system for integrated management of data, accuracy and lineage. In CIDR, 2005. [WSCY99] O. Wolfson, P. Sistla, S. Chamberlain, and Y. Yesha. Updating and querying databases that track mobile units. Distributed and Parallel Databases, 7(3), 1999. Sunil Prabhakar, Probabilistic Queries and Uncertain Data, COMAD 2005b 69