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New Sampling-Based Estimators for OLAP Queries

New Sampling-Based Estimators for OLAP Queries. Ruoming Jin , Kent State University Leo Glimcher , The Ohio State University Chris Jermaine , University of Florida Gagan Agrawal , The Ohio State University. Approximate Query Processing. AQP is an active area of DM research

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New Sampling-Based Estimators for OLAP Queries

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  1. New Sampling-Based Estimators for OLAP Queries Ruoming Jin, Kent State University Leo Glimcher, The Ohio State University Chris Jermaine, University of Florida Gagan Agrawal, The Ohio State University

  2. Approximate Query Processing • AQP is an active area of DM research • The goal is to provide accurate estimation of queries without access the entire databases • Especially useful and important for data warehouse and OLAP • Consider you have a total of 10,000 disks, each with 200GB (2PB) • Takes 1 hour to scan • Answering a single, simple aggregate query may need an hour • Unacceptable to analysts/end-users • If each disk cost $1000 year to maintain • One simple query can cost • $1572=10,000  $1000/ (365  24) • inhibitive cost

  3. OLAP Queries • Querying the Large Relational Tables composed of • Dimensional Attributes • Categorical data (Most) • Sex, Country, State, City, Product Code, Department, Color, … • Measure Attributes • Numerical data • Salary, Sales, Price, Number of Complaints, … • Aggregate Queries • Most AQP tailored to numerical data • Wavelets, kernels, histograms • Problematic for categorical data and high-dimensionality • Random Sampling • Well studied in statistical theory • Can handle high-dimension category data • Provide estimates of the query results as well as the estimate accuracy

  4. Confidence Interval • The measure for accuracy • COMPLAINTS(PROF, SEMESTER, NUM_COMPLAINTS) • SELECT SUM (NUM_COMPLAINTS) FROM COMPLAINS WHERE PROF = ‘Smith’ AND SEMESTER = ‘Fa03’ • A Confidence Bound: • With a probability of .95, Prof. Smith received 27 to 29 complaints in the Fall of 2003 Accuracy level Interval width =2

  5. How to estimate the confidence interval? • Uniform Sampling • Central limit theorem (CLT) • Delta Methods Assuming the distribution of an estimator ŷ of an aggregate query result y is approximately normal with mean E(ŷ), and variance V(ŷ) for a large sample, an approximate 95% confidence interval for the estimator is given by [ŷ-1.96SE(ŷ), ŷ+1.96SE(ŷ)] where 1.96 is the 0.975th percentile of the standard normal distribution, and SE(ŷ) is the standard error (the square root of the variance V(ŷ) ). Accuracy level Interval width = 3.92SE(ŷ)

  6. How to (cont’d) • Unequal Probability Sampling • Stratified Sampling • Separate Samples for Each Measure (Numerical) Attribute • Re-Sampling • Bootstrapping • Computational Intensive • Distribution-free • Chebyshev and Hoeffding bound • Loose bound

  7. Problem studied in this presentation • How to provide an accurate confidence interval together with an estimation? • Boosting the accuracy level • Reducing the interval width • Key idea: Ensemble Estimates • Find multiple (unbiased) estimators for each OLAP query • Linearly combine the individual estimators and derive the optimal coefficients to reduce the global variance • Handle the correlation among the individual estimators

  8. Example • Database describing student complaints

  9. Example • We sample the database…

  10. Example • And ask: How many complaints for Smith? Est: (21+7+8)/8×16=72; Answer: 121

  11. Why So Bad? • We missed two important records Oops!

  12. How we know something went wrong? • What if we know the total complaints of the entire table: SUM(NUM_COMPLAINTS) • Compare with the estimated total complaints of the entire table Est: (2+21+1+7+8+4+3+0)/8 × 16 = 92, Answer: 148 • One of the key ideas in the APA approach • Pre-aggregation of the low-dimensional aggregates • 0-dimensional fact: SUM(NUM_COMPLAINTS) =148 • 1-dimensional fact, for example, on SEMESTER SELECT SUM(NUM_COMPLAINTS) FROM COMPLAINTS GROUP-BY SEMESTER • Or higher, depending on the cost of such pre-aggregation • In our example, assuming only the 0-dimensional fact is know!

  13. How we can pull ourselves out? • APA use Maximal Likelihood Estimation (MLE) • Break data space based on relational selection predicates; 2m Quadrants • Compute aggregate for each quadrant • Characterize the error of the estimates using normal PDF (justification: CLT) • Pretend estimates are independent • Adjust the means to max likelihood • Subject to known facts about the data • Shows to be very accurate in various datasets, significantly better than plain sampling and stratified sampling • In our example, the New Estimation is 136.3 (answer was 121, the original estimation is 72) • However, loss of analytic guarantees on accuracy!

  14. Let us go back to the plain sampling • For the query: How many complaints for Smith? Est: (21+7+8)/8×16=72 (Answer: 121); The standard error (SE) is 68. 2 [ŷ-1.96SE(ŷ), ŷ+1.96SE(ŷ)]

  15. New Estimator: The Negative One • To answer the query: How many complaints for Smith? (Answer:121) • We first ask: How many complaints NOT for Smith? Est: (2+1+4+3+0)/8×16=20, The Negative Estimator: 148-20=128, Standard Error (SE) = 13.4

  16. How two is always better than one: The Ensemble Estimator • Linearly combining the direct (positive) estimator and the negative estimator • Estnew = αEstdirect + (1- α ) Estnegative (0 α 1) • Note since both the direct estimators and negative estimators are unbiased estimators, the ensemble estimator is also unbiased. • Choose the parameter α to minimize the variance the ensemble estimator • The ensemble estimator always is always more accurate • If the individual estimators are independent, the optimal value of the parameter α is V(Estdirect)/(V(Estdirect)+V(Estnegative )) • In our example, α=0.0373, Estnew=125.95, Standard Error (SE) = 13.1

  17. What if we have higher-dimensional facts? • Image we have the relational table EMPLOYEE(NAME, SEX,DEPARTMENT,JOB_TYPE, SALARY) • Query: SELECT SUM (SALARY) FROM EMPLOYEE WHERE SEX=‘M’ AND DEPARTMENT=‘ACCOUNT’ AND JOB_TYPE=‘SUPERVISOR’ • Pre-Aggregation • 1-dimesional facts

  18. More negative estimators SEX ‘M’ AND DEPARTMENT ‘ACCOUNT’ AND JOB_TYPE ‘SUPERVISOR’ b1^b2^b3, or b1^b2^b3 b1^b2^b3 b1^b2^b3 DEPARTMENT SEX SELECT SUM (SALARY) FROM EMPLOYEE WHERE SEX=‘M’ AND DEPARTMENT=‘ACCOUNT’ AND JOB_TYPE=‘SUPERVISOR’ b1 b2 b3 JOB_TYPE

  19. More negative estimators SEX ‘M’ AND DEPARTMENT ‘ACCOUNT’ AND JOB_TYPE ‘SUPERVISOR’ b1^b2^b3, or b1^b2^b3 b1^b2^b3 b1^b2^b3 b1^b2^b3 b1^b2^b3 DEPARTMENT SEX SELECT SUM (SALARY) FROM EMPLOYEE WHERE SEX=‘M’ AND DEPARTMENT=‘ACCOUNT’ AND JOB_TYPE=‘SUPERVISOR’ b1 b2 b3 JOB_TYPE

  20. More negative estimators (cont’d) SEX ‘M’ AND DEPARTMENT ‘ACCOUNT’ AND JOB_TYPE ‘SUPERVISOR’ b1^b2^b3, or b1^b2^b3 b1^b2^b3 b1^b2^b3 b1^b2^b3 b1^b2^b3 DEPARTMENT SEX SELECT SUM (SALARY) FROM EMPLOYEE WHERE SEX=‘M’ AND DEPARTMENT=‘ACCOUNT’ AND JOB_TYPE=‘SUPERVISOR’ b1 b2 b3 JOB_TYPE

  21. More negative estimators (cont’d) SEX ‘M’ AND DEPARTMENT ‘ACCOUNT’ AND JOB_TYPE ‘SUPERVISOR’ b1^b2^b3, or b1^b2^b3 b1^b2^b3 b1^b2^b3 b1^b2^b3 b1^b2^b3 DEPARTMENT SEX SELECT SUM (SALARY) FROM EMPLOYEE WHERE SEX=‘M’ AND DEPARTMENT=‘ACCOUNT’ AND JOB_TYPE=‘SUPERVISOR’ b1 b2 b3 JOB_TYPE

  22. Combining Positive and Negative Estimators in APA1+ • We will have multiple negative estimators • Estnew = α0 Estdirect + α1 Estnegative1 +α2 Estnegative2 +… 0 αi  1, α0+ α1+ α2+… = 1 • Decompose the negative estimators into the cell representations • Each cell in the cube correspond to a direct estimation • The variance of the cell can be estimated • We can use Lagrange multipliers to optimize all the parameters (αi) • We assume the direct estimations for each cell is independent • This procedure usually involve a linear solver for a linear equation

  23. Actually, the estimators are correlated • Fortunately, we are able to capture such correlation analytically • If each individual estimator is approximately normal, and they are independent, the combined estimator is also approximately normal • However, the correction effect results in a slightly different distribution • Analytically very close to the spherically symmetric distribution, of which normal distribution is a special case. • Empirically, it shows strong tendency to normal • We use normal distribution to derive the confidence interval

  24. Empirical Distribution of the Ensemble Estimators Empirical distribution of APA0+ Empirical distribution of APA1+

  25. Experimental Evaluation • Four datasets • Forest Cover data (from UCI KDD archive) • River Flow data • William Shakespeare data • Image Feature vector • Approximation techniques • Simple Random Sampling • Stratified Sampling • APA0+ • APA1+ • Queries • 2000 queries for each dataset

  26. Measure the estimated confidence interval • We generate 95% confidence intervals of all estimation techniques for each query • Accuracy level • What are the real chances the correct answers actually fall in the confidence intervals? • Interval width • How tight are the bounds of the confidence intervals?

  27. How good are the new estimators? • Accuracy of the confidence intervals (Expected: 95%) • APA1+ average around 90%, which was 23.2% higher than simple random sampling (the next best alternative in terms of accuracy) • The accuracy of APA0+, random sampling, and stratified sample are comparable, all less than 70% in average • Confidence interval width • The width of the confidence interval produced by APA1+ is only 1/2 the size of one from random sampling • Compared with stratified sampling, APA1+ is at least 20% smaller • The width of the confidence interval produced by APA0+ is around 15% smaller than random sampling

  28. Discussion • Overall, the new estimators work pretty well! • It’s very simple! • Significantly better than the random sampling • Significantly better than the stratified sampling • APA1+ is the only estimator which provides the confidence interval close to the theoretically expected accuracy and with much smaller width! • Suitable for both categorical, numerical data • APA0+, and APA1 unaffected by high dimensions! • Future work • How to apply this idea to work with more complicated aggregation functions?

  29. Thanks!!

  30. Roadmap • Approximate Query Processing and Confidence Interval • Motivating Example • Generalization and Handling Correlation • Experimental Results • Conclusions • Inspired by • Chris’s original APA approach (how to find multiple estimators) • Ensemble Classifiers in Statistical Learning

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