On biased reservoir sampling in the presence of stream evolution

The method of reservoir based sampling is often used to pick an unbiased sample from a data stream. A large portion of the unbiased sample may become less relevant over time because of evolution. An analytical or mining task (eg. query estimation) which is specific to only the sample points from a recent time-horizon may provide a very inaccurate result. This is because the size of the relevant sample reduces with the horizon itself. On the other hand, this is precisely the most important case for data stream algorithms, since recent history is frequently analyzed. In such cases, we show that an effective solution is to bias the sample with the use of temporal bias functions. The maintenance of such a sample is non-trivial, since it needs to be dynamically maintained, without knowing the total number of points in advance. We prove some interesting theoretical properties of a large class of memory-less bias functions, which allow for an efficient implementation of the sampling algorithm. We also show that the inclusion of bias in the sampling process introduces a maximum requirement on the reservoir size. This is a nice property since it shows that it may often be possible to maintain the maximum relevant sample with limited storage requirements. We not only illustrate the advantages of the method for the problem of query estimation, but also show that the approach has applicability to broader data mining problems such as evolution analysis and classification.