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OLAP Over Uncertain and Imprecise Data Adapted from a talk by T.S. Jayram (IBM Almaden) with Doug Burdick (Wisconsin), Prasad Deshpande (IBM), Raghu Ramakrishnan (Wisconsin), Shivakumar Vaithyanathan (IBM) Adapted by S. Sudarshan Dimensions in OLAP Automobile All Truck Sedan Civic Camry F150 Sierra Location All East West CA TX NY MA Measures, Facts, and Queries Automobile Auto = F150 Loc = NY Repair = $200 ALL Truck Sedan Civic Camry F150 MA NY East p2 p1 Auto = Truck Loc = East SUM(Repair) = ? Sierra ALL TX West Location p7 p6 p8 p4 p5 CA p3 Cell Restriction on Imprecision We restrict the sets of values in an imprecise fact to either: 1. A singleton set consisting of a leaf level member of the hierarchy, or, 2. The set of all the leaf level members under some non-leaf level member of the hierarchy. Cells and Regions A region is a vector of attribute values from an imprecise domains of each dimension of the cube. A cell is a region in which all values are leaf level members. Let reg(R) represent the set of cells in a region R. Queries on precise data A query Q = (R, M, A) refers to a region R, a measure M, and an aggregate function A. Eg : (<Ambassador, Location>, Repairs, Sum) The result of the query in a precise database is obtained by applying A on the measure M of all cells in R. For the example above, the result is (P1 + P2) Extend the OLAP model to handle data ambiguity Imprecision Uncertainty Imprecision Auto = F150 Loc = East Repair = $200 Automobile ALL Truck Sedan Civic Camry F150 Sierra p2 MA East p9 p11 NY p1 ALL TX West Location p7 p6 p8 p10 p5 CA p3 p4 Representing Imprecision using Dimension Hierarchies Dimension hierarchies lead to a natural space of “partially specified” objects Sources of imprecision: incomplete data, multiple sources of data Motivating Example Query: COUNT Truck F150 MA p3 Sierra p4 East We propose desiderata that enable p5 appropriate definition of query semantics for imprecise data NY p1 p2 Queries on imprecise data Consider the query region <Pune, Model> in the figure. It overlaps two imprecise facts P4 and P5. Three (naive) options for including fact in query: Contains: consider only if contained in query Overlaps: consider if overlapping query None: ignore all imprecise facts Desideratum I: Consistency Truck F150 Sierra MA p3 p4 p5 East NY p1 p2 Consistency specifies the relationship between answers to related queries on a fixed data set Notions of Consistency Generic idea: if query region is partitioned, and aggregate applied on each partition, then aggregate q on whole region must be consistent in some ways with aggregates qi on partitions General idea: alpha consistency for property alpha Specific forms of consistency discussed in detail in paper Sum consistency (for count/sum) Boundedness consistency (for average) Contains option : Consistency Intuitively, consistency means that the answer to a query should be consistent with the aggregates from individual partitions of the query. Using the Contains option could give rise to inconsistent results. For example, consider the sum aggregate of the query above and that of its individual cells. With the Contains option, will the individual results add up to be the same as the collective? Desideratum II: Faithfulness Data Set 1 F150 Data Set 2 Sierra p4 p5 p3 p4 p1 p2 F150 NY p2 NY NY p1 Sierra MA p3 MA MA p5 F150 Data Set 3 Sierra p5 p4 p3 p1 p2 Faithfulness specifies the relationship between answers to a fixed query on related data sets Notion of result quality relative to the quality of the data input to the query. – For example, the answer computed for Q=F150,MA should be of higher quality if p3 were precisely known. Formal definitions of both Consistency and Faithfulness depend on the underlying aggregation operator Can we define query semantics that satisfy these desiderata? F150 Query Semantics p5 MA p4 p3 NY F150 Sierra p1 p2 F150 Sierra p5 w1 p4 w2 p2 F150 p1 Sierra p4 p2 F150 NY p3 w3 MA p5 NY [Kripke63,…] MA Possible Worlds p3 p5 p4 w4 NY NY p1 MA MA p3 Sierra Sierra p5 p4 p3 p1 p2 p1 p2 Possible Worlds Query Semantics Given all possible worlds together with their probabilities, queries are easily answered (using expected values) But number of possible worlds is exponential! Allocation Allocation gives facts weighted assignments to possible completions, leading to an extended version of the data Size increase is linear in number of (completions of) imprecise facts Queries operate over this extended version Key contributions: Appropriate characterization of the large space of allocation policies Designing efficient allocation policies that take into account the correlations in the data Storing Allocations using Extended Data Model Truck F150 Sierra MA NY East ID FactID Auto Loc Repair Weight 1 1 F150 NY 100 1.0 2 2 Sierra NY 500 1.0 5 4 Sierra MA 200 1.0 p5 p4 p3 p1 p2 Advantages of EDM No extra infrastructure required for representing imprecision Efficient algorithms for aggregate queries : SUM and COUNT : linear time algo. AVERAGE : slightly complicated algorithm running in O(m + n3) for m precise facts and n imprecise facts. Aggregating Uncertain Measures Opinion pooling: provide a consensus opinion from a set of opinions Θ. The opinions in Θ as well as the consensus opinion are represented as pdfs over a discrete domain O linear operator LinOp(Θ) produces a consensus pdf P that is a weighted linear combination of the pdfs in Θ, Allocation Policies For every region r in the database, we want to assign an allocation pc, r to each cell c in Reg(r), such that ∑c Reg(r) pc, r = 1 Three ways of doing so: 1. Uniform : Assign each cell c in a region r an equal probability. pc, r = 1 / |Reg(r)| Allocation Policies For every region r in the database, we want to assign an allocation pc, r to each cell c in Reg(r), such that ∑c Reg(r) pc, r = 1 However, we can do better. Some cells may be naturally inclined to have more probability than others. Eg : Mumbai will clearly have more repairs than Bhopal. We can do this automatically by giving more probability to cells with higher number of precise facts. 2. Count based : where Nc is the number of precise facts in cell c Allocation Policies For every region r in the database, we want to assign an allocation pc, r to each cell c in Reg(r), such that ∑c Reg(r) pc, r = 1 Again, we can arguably get a better result by looking at not just the count, but rather than the actual value of the measure in question. 3. Measure based : next slide. Measure Based Allocation Assumes the following model : The given database D with imprecise facts has been generated by randomly injecting imprecision in a precise database D'. D' assigns value o to a cell c according to some unknown pdf P(o, c). If we could determine this pdf, the allocation is simply pc, r = P(c) / ∑ c' in Reg(r) P(c') Classifying Allocation Policies Measure Correlation Ignored Used Ignored Used Dimension Correlation Uniform Count EM Results on Query Semantics Evaluating queries over extended version of data yields expected value of the aggregation operator over all possible worlds intuitively, the correct value to compute Efficient query evaluation algorithms for SUM, COUNT consistency and faithfulness for SUM, COUNT are satisfied under appropriate conditions Dynamic programming algorithm for AVERAGE Unfortunately, consistency does not hold for AVERAGE Alternative Semantics for AVERAGE APPROXIMATE AVERAGE E[SUM] / E[COUNT] instead of E[SUM/COUNT] simpler and more efficient satisfies consistency extends to aggregation operators for uncertain measures Maximum Likelihood Principle A reasonable estimate for this function P can be that which maximises the probability of generating the given imprecise data set D. Example : Suppose the pdf depends only on the cells and is independent of the measure values. Thus, the pdf is a mapping : C where C is the set of cells. This pdf can be found by maximising the likelihood function : ( ) = r D ∑c Reg(r) (c) EM Algorithm The Expectation Maximization algorithm provides a standard way of maximizing the likelihood, when we have some unknown variables in the observation set. Expectation step (compute data): Calculate the expected value of the unknown variables, given the current estimate of variables. Maximization step (compute generator): Calculate the distribution that maximizes the probability of the current estimated data set. EM Algorithm : Example Initialization Step: Data: [4, 10, ?, ?] Initial mean value: 0 New Data: [4, 10, 0, 0] Step 1: New Mean: 3.5 New Data:[4, 10, 3.5, 3.5] Step 4: New Mean: 6.5625 New Data: [4, 10, 6.5625, 6.5625] Step 2: New Mean: 5.25 New Data: [4, 10, 5.25, 5.25] Step 5: New Mean: 6.7825 New Data: [4, 10, 6.7825, 6.7825] Step 3: New Mean: 6.125 New Data: [4, 10, 6.125, 6.125] Result: New Mean: 6.890625 EM Algorithm : Application Experiments : Allocation run time Experiments : Query run time Experiments : Query run time Experiments : Accuracy Uncertainty Measure value is modeled as a probability distribution function over some base domain e.g., measure Brake is a pdf over values {Yes,No} sources of uncertainty: measures extracted from text using classifiers Adapt well-known concepts from statistics to derive appropriate aggregation operators Our framework and solutions for dealing with imprecision also extend to uncertain measures Summary Consistency and faithfulness desiderata for designing query semantics for imprecise data Allocation is the key to our framework Efficient algorithms for aggregation operators with appropriate guarantees of consistency and faithfulness Iterative algorithms for allocation policies Correlation-based Allocation Involves defining an objective function to capture some underlying correlation structure a more stringent requirement on the allocations solving the resulting optimization problem yields the allocations EM-based iterative allocation policy interesting highlight: allocations are re-scaled iteratively by computing appropriate aggregations

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