Mining Top-K Frequent Itemsets Through Progressive Sampling

We study the use of sampling for efficiently mining the top-K frequent itemsets of cardinality at most w. To this purpose, we define an approximation to the top-K frequent itemsets to be a family of itemsets which includes (resp., excludes) all very frequent (resp., very infrequent) itemsets, together with an estimate of these itemsets' frequencies with a bounded error. Our first result is an upper bound on the sample size which guarantees that the top-K frequent itemsets mined from a random sample of that size approximate the actual top-K frequent itemsets, with probability larger than a specified value. We show that the upper bound is asymptotically tight when w is constant. Our main algorithmic contribution is a progressive sampling approach, combined with suitable stopping conditions, which on appropriate inputs is able to extract approximate top-K frequent itemsets from samples whose sizes are smaller than the general upper bound. In order to test the stopping conditions, this approach maintains the frequency of all itemsets encountered, which is practical only for small w. However, we show how this problem can be mitigated by using a variation of Bloom filters. A number of experiments conducted on both synthetic and real bench- mark datasets show that using samples substantially smaller than the original dataset (i.e., of size defined by the upper bound or reached through the progressive sampling approach) enable to approximate the actual top-K frequent itemsets with accuracy much higher than what analytically proved.Comment: 16 pages, 2 figures, accepted for presentation at ECML PKDD 2010 and publication in the ECML PKDD 2010 special issue of the Data Mining and Knowledge Discovery journa

Processing count queries over event streams at multiple time granularities

Management and analysis of streaming data has become crucial with its applications in web, sensor data, network tra c data, and stock market. Data streams consist of mostly numeric data but what is more interesting is the events derived from the numerical data that need to be monitored. The events obtained from streaming data form event streams. Event streams have similar properties to data streams, i.e., they are seen only once in a fixed order as a continuous stream. Events appearing in the event stream have time stamps associated with them in a certain time granularity, such as second, minute, or hour. One type of frequently asked queries over event streams is count queries, i.e., the frequency of an event occurrence over time. Count queries can be answered over event streams easily, however, users may ask queries over di erent time granularities as well. For example, a broker may ask how many times a stock increased in the same time frame, where the time frames specified could be hour, day, or both. This is crucial especially in the case of event streams where only a window of an event stream is available at a certain time instead of the whole stream. In this paper, we propose a technique for predicting the frequencies of event occurrences in event streams at multiple time granularities. The proposed approximation method e ciently estimates the count of events with a high accuracy in an event stream at any time granularity by examining the distance distributions of event occurrences. The proposed method has been implemented and tested on di erent real data sets and the results obtained are presented to show its e ectiveness