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In this paper, we propose a novel formulation for distance-based outliers that is based on the distance of a point from its k th nearest neighbor. We rank each point on the basis of its distance to its k th nearest neighbor and declare the top n points in this ranking to be outliers. In addition to developing relatively straightforward solutions to finding such outliers based on the classical nested-loop join and index join algorithms, we develop a highly efficient partition-based algorithm for mining outliers. This algorithm first partitions the input data set into disjoint subsets, and then prunes entire partitions as soon as it is determined that they cannot contain outliers. This results in substantial savings in computation. We present the results of an extensive experimental study on real-life and synthetic data sets. The results from a real-life NBA database highlight and reveal several expected and unexpected aspects of the database. The results from a study on synthetic data sets demonstrate that the partition-based algorithm scales well with respect to both data set size and data set dimensionality.

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AbstractÐDiscovering sequential patterns is an important problem in data mining with a host of application domains including medicine, telecommunications, and the World Wide Web. Conventional sequential pattern mining systems provide users with only a very restricted mechanism (based on minimum support) for specifying patterns of interest. As a consequence, the pattern mining process is typically characterized by lack of focus and users often end up paying inordinate computational costs just to be inundated with an overwhelming number of useless results. In this paper, we propose the use of Regular Expressions (REs) as a flexible constraint specification tool that enables user-controlled focus to be incorporated into the pattern mining process. We develop a family of novel algorithms (termed SPIRITÐSequential Pattern mIning with Regular expressIon consTraints) for mining frequent sequential patterns that also satisfy user-specified RE constraints. The main distinguishing factor among the proposed schemes is the degree to which the RE constraints are enforced to prune the search space of patterns during computation. Our solutions provide valuable insights into the trade-offs that arise when constraints that do not subscribe to nice properties (like antimonotonicity) are integrated into the mining process. A quantitative exploration of these trade-offs is conducted through an extensive experimental study on synthetic and real-life data sets. The experimental results clearly validate the effectiveness of our approach, showing that speedups of more than an order of magnitude are possible when RE constraints are pushed deep inside the mining process. Our experimentation with real-life data also illustrates the versatility of REs as a user-level tool for focusing on interesting patterns.