For many KDD applications finding the outliers, i.e. the rare events, is more interesting and useful than finding the common cases, e.g. detecting criminal activities in E-commerce. Being an outlier, however, is not just a binary property. Instead, it is a property that applies to a certain degree to each object in a data set, depending on how 'isolated' this object is, with respect to the surrounding clustering structure. In this paper, we formally introduce a new notion of outliers which bases outlier detection on the same theoretical foundation as density-based cluster analysis. Our notion of an outlier is 'local' in the sense that the outlier-degree of an object is determined by taking into account the clustering structure in a bounded neighborhood of the object. We demonstrate that this notion of an outlier is more appropriate for detecting different types of outliers than previous approaches, and we also present an algorithm for finding them. Furthermore, we show that by combining the outlier detection with a density-based method to analyze the clustering structure, we can get the outliers almost for free if we already want to perform a cluster analysis on a data set.

Cluster analysis is a primary method for database mining. It is either used as a stand-alone tool to get insight into the distribution of a data set, e.g. to focus further analysis and data processing, or as a preprocessing step for other algorithms operating on the detected clusters. Almost all of the well-known clustering algorithms require input parameters which are hard to determine but have a significant influence on the clustering result. Furthermore, for many real-data sets there does not even exist a global parameter setting for which the result of the clustering algorithm describes the intrinsic clustering structure accurately. We introduce a new algorithm for the purpose of cluster analysis which does not produce a clustering of a data set explicitly; but instead creates an augmented ordering of the database representing its density-based clustering structure. This cluster-ordering contains information which is equivalent to the density-based clusterings corresponding to a broad range of parameter settings. It is a versatile basis for both automatic and interactive cluster analysis. We show how to automatically and efficiently extract not only 'traditional' clustering information (e.g. representative points, arbitrary shaped clusters), but also the intrinsic clustering structure. For medium sized data sets, the cluster-ordering can be represented graphically and for very large data sets, we introduce an appropriate visualization technique. Both are suitable for interactive exploration of the intrinsic clustering structure offering additional insights into the distribution and correlation of the data.

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For many KDD applications, such as detecting criminal activities in E-commerce, finding the rare instances or the outliers, can be more interesting than finding the common patterns. Existing work in outlier detection regards being an outlier as a binary property. In this paper, we contend that for many scenarios, it is more meaningful to assign to each object a degree of being an outlier. This degree is called the local outlier factor (LOF) of an object. It is local in that the degree depends on how isolated the object is with respect to the surrounding neighborhood. We give a detailed formal analysis showing that LOF enjoys many desirable properties. Using real-world datasets, we demonstrate that LOF can be used to find outliers which appear to be meaningful, but can otherwise not be identified with existing approaches. Finally, a careful performance evaluation of our algorithm confirms we show that our approach of finding local outliers can be practical.

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