Get the Most out of Your Sample: Optimal Unbiased Estimators using Partial Information

Random sampling is an essential tool in the processing and transmission of data. It is used to summarize data too large to store or manipulate and meet resource constraints on bandwidth or battery power. Estimators that are applied to the sample facilitate fast approximate processing of queries posed over the original data and the value of the sample hinges on the quality of these estimators. Our work targets data sets such as request and traffic logs and sensor measurements, where data is repeatedly collected over multiple {\em instances}: time periods, locations, or snapshots. We are interested in queries that span multiple instances, such as distinct counts and distance measures over selected records. These queries are used for applications ranging from planning to anomaly and change detection. Unbiased low-variance estimators are particularly effective as the relative error decreases with the number of selected record keys. The Horvitz-Thompson estimator, known to minimize variance for sampling with "all or nothing" outcomes (which reveals exacts value or no information on estimated quantity), is not optimal for multi-instance operations for which an outcome may provide partial information. We present a general principled methodology for the derivation of (Pareto) optimal unbiased estimators over sampled instances and aim to understand its potential. We demonstrate significant improvement in estimate accuracy of fundamental queries for common sampling schemes.Comment: This is a full version of a PODS 2011 pape

What you can do with Coordinated Samples

Sample coordination, where similar instances have similar samples, was proposed by statisticians four decades ago as a way to maximize overlap in repeated surveys. Coordinated sampling had been since used for summarizing massive data sets. The usefulness of a sampling scheme hinges on the scope and accuracy within which queries posed over the original data can be answered from the sample. We aim here to gain a fundamental understanding of the limits and potential of coordination. Our main result is a precise characterization, in terms of simple properties of the estimated function, of queries for which estimators with desirable properties exist. We consider unbiasedness, nonnegativity, finite variance, and bounded estimates. Since generally a single estimator can not be optimal (minimize variance simultaneously) for all data, we propose {\em variance competitiveness}, which means that the expectation of the square on any data is not too far from the minimum one possible for the data. Surprisingly perhaps, we show how to construct, for any function for which an unbiased nonnegative estimator exists, a variance competitive estimator.Comment: 4 figures, 21 pages, Extended Abstract appeared in RANDOM 201