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.

Abstract. Both, the number and the size of spatial databases, such as geographic or medical databases, are rapidly growing because of the large amount of data obtained from satellite images, computer tomography or other scientific equipment. Knowledge discovery in databases (KDD) is the process of discovering valid, novel and potentially useful patterns from large databases. Typical tasks for knowledge discovery in spatial databases include clustering, characterization and trend detection. The major difference between knowledge discovery in relational databases and in spatial databases is that attributes of the neighbors of some object of interest may have an influence on the object itself. Therefore, spatial knowledge discovery algorithms heavily depend on the efficient processing of neighborhood relations since the neighbors of many objects have to be investigated in a single run of a typical algorithm. Thus, providing general concepts for neighborhood relations as well as an efficient implementation of these concepts will allow a tight integeration of spatial knowledge discovery algorithms with a spatial database management system. This will speed-up both, the development and the execution of spatial KDD algorithms. For this purpose, we define a small set of database primitives, and we demonstrate that typical spatial KDD algorithms are well supported by the proposed database primitives. By implementing the database primitives on top of a commercial database management system, we show the effectiveness and efficiency of our approach, experimentally as well as analytically. The paper concludes by outlining some interesting issues for future research in the emerging field of knowledge discovery in spatial databases.

Due to the computerization and the advances in scientific data collection we are faced with a large and continuously growing amount of data which makes it impossible to interpret all this data manually. Therefore, the development of new techniques and tools that support the human in transforming data into useful knowledge has been the focus of the relatively new and interdisciplinary research area "knowledge discovery in databases". Knowledge discovery in databases (KDD) has been defined as the non-trivial process of discovering valid, novel, potentially useful and ultimately understandable patterns from data, a pattern is an expression in some language describing a subset of the data or a model applicable to that subset (Fayyad et al., 1996). The process of KDD is interactive and iterative, involving several steps such as data selection, data reduction, data mining, and the evaluation of the data mining results. The heart of the process, however, is the data mining step which consists of the application of data analysis and discovery algorithms that, under acceptable computational efficiency limitations, produce a particular enumeration of patterns over the data (Fayyad et al., 1996). While a lot of research has been conducted on knowledge discovery and data mining in relational databases (see e.g. (Chen et al., 1996) or (Fayyad, 1997) for an overview), only a few works deal with knowledge discovery in spatial databases (see (Gueting, 1994) for an introduction to spatial databases, (Koperski et al., 1996) for an overview of spatial data mining). Finding implicit regularities, rules or patterns hidden in spatial databases is an important task, e.g. for geo-marketing, traffic control or environmental studies. A spatial database contains objects which are characterized by a spatial location and/or extension as well as by several non-spatial attributes. Figure 7.1 illustrates a spatial database on Bavaria as an example. Depicted is the relation Communities containing polygons which represent communities in a geographic information system. This spatial database on Bavaria-referred to as the BA-VARIA database-is used in some of the following sections as test database for our algorithms. The database contains the ATKIS 500 data and the Bavarian part of the statistical data obtained by the

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.

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.