Parallel coordinates are an established technique to visualize high-dimensional data, in particular for data mining purposes. A major challenge is the ordering of axes, as any axis can have at most two neighbors when placed in parallel on a 2D plane. By extending this concept to a 3D visualization space we can place several axes next to each other. However, finding a good arrangement often does not necessarily become easier, as still not all axes can be arranged pairwise adjacently to each other. Here, we provide a tool to explore complex data sets using 3D-parallel-coordinate-trees, along with a number of approaches to arrange the axes.

When comparing clustering results, any evaluation metric breaks down the available information to a single number. However, a lot of evaluation metrics are around, that are not always concordant nor easily interpretable in judging the agreement of a pair of clusterings. Here, we provide a tool to visually support the assessment of clustering results in comparing multiple clusterings. Along the way, the suitability of a couple of clustering comparison measures can be judged in different scenarios.

In this paper, we propose an original solution for the general reverse k-nearest neighbor (RkNN) search problem. Compared to the limitations of existing methods for the RkNN search, our approach works on top of any hierarchically organized tree-like index structure and, thus, is applicable to any type of data as long as a metric distance function is defined on the data objects. We will exemplarily show how our approach works on top of the most prevalent index structures for Euclidean and metric data, the R-Tree and the M-Tree, respectively. Our solution is applicable for arbitrary values of k and can also be applied in dynamic environments where updates of the database frequently occur. Although being the most general solution for the RkNN problem, our solution outperforms existing methods in terms of query execution times because it exploits different strategies for pruning false drops and identifying true hits as soon as possible.

In this paper, we propose an efficient and effective method to find arbitrarily oriented subspace clusters by mapping the data space to a parameter space defining the set of possible arbitrarily oriented subspaces. The objective of a clustering algorithm based on this principle is to find those among all the possible subspaces, that accommodate many database objects. In contrast to existing approaches, our method can find subspace clusters of different dimensionality even if they are sparse or are intersected by other clusters within a noisy environment. A broad experimental evaluation demonstrates the robustness, efficiency and effectivity of our method.

Correlation clustering aims at the detection of data points that appear as hyperplanes in the data space and, thus, exhibit common correlations between different subsets of features. Recently proposed methods for correlation clustering usually suffer from several severe drawbacks including poor robustness against noise or parameter settings, incomplete results (i.e. missed clusters), poor usability due to complex input parameters, and poor scalability. In this paper, we propose the novel correlation clustering algorithm COPAC (COrrelation PArtition Clustering) that aims at improved robustness, completeness, usability, and efficiency. Our experimental evaluation empirically shows that COPAC is superior over existing state-of-the-art correlation clustering methods in terms of runtime, accuracy, and completeness of the results.

In high dimensional data, clusters often only exist in arbitrarily oriented subspaces of the feature space. In addition, these so-called correlation clusters may have complex relationships between each other. For example, a correlation cluster in a 1-D subspace (forming a line) may be enclosed within one or even several correlation clusters in 2-D superspaces (forming planes). In general, such relationships can be seen as a complex hierarchy that allows multiple inclusions, i.e. clusters may be embedded in several super-clusters rather than only in one. Obviously, uncovering the hierarchical relationships between the detected correlation clusters is an important information gain. Since existing approaches cannot detect such complex hierarchical relationships among correlation clusters, we propose the algorithm ERiC to tackle this problem and to visualize the result by means of a graph-based representation. In our experimental evaluation, we show that ERiC finds more information than state-of-the-art correlation clustering methods and outperforms existing competitors in terms of efficiency.

In this paper, we propose an approach for efficient approximative RkNN search in arbitrary metric spaces where the value of k is specified at query time. Our method uses an approximation of the nearest-neighbor-distances in order to prune the search space. In several experiments, our solution scales significantly better than existing non-approximative approaches while producing an approximation of the true query result with a high recall.

Correlation clustering aims at grouping the data set into correlation clusters such that the objects in the same cluster exhibit a certain density and are all associated to a common arbitrarily oriented hyperplane of arbitrary dimensionality. Several algorithms for this task have been proposed recently. However, all algorithms only compute the partitioning of the data into clusters. This is only a first step in the pipeline of advanced data analysis and system modelling. The second (post-clustering) step of deriving a quantitative model for each correlation cluster has not been addressed so far. In this paper, we describe an original approach to handle this second step. We introduce a general method that can extract quantitative information on the linear dependencies within a correlation clustering. Our concepts are independent of the clustering model and can thus be applied as a post-processing step to any correlation clustering algorithm. Furthermore, we show how these quantitative models can be used to predict the probability distribution that an object is created by these models. Our broad experimental evaluation demonstrates the beneficial impact of our method on several applications of significant practical importance.

The reverse k-nearest neighbor (RkNN) problem, i.e. finding all objects in a data set the k-nearest neighbors of which include a specified query object, is a generalization of the reverse 1-nearest neighbor problem which has received increasing attention recently. Many industrial and scientific applications call for solutions of the RkNN problem in arbitrary metric spaces where the data objects are not Euclidean and only a metric distance function is given for specifying object similarity. Usually, these applications need a solution for the generalized problem where the value of k is not known in advance and may change from query to query. However, existing approaches, except one, are designed for the specific R1NN problem. In addition - to the best of our knowledge - all previously proposed methods, especially the one for generalized RkNN search, are only applicable to Euclidean vector data but not for general metric objects. In this paper, we propose the first approach for efficient RkNN search in arbitrary metric spaces where the value of k is specified at query time. Our approach uses the advantages of existing metric index structures but proposes to use conservative and progressive distance approximations in order to filter out true drops and true hits. In particular, we approximate the k-nearest neighbor distance for each data object by upper and lower bounds using two functions of only two parameters each. Thus, our method does not generate any considerable storage overhead. We show in a broad experimental evaluation on real-world data the scalability and the usability of our novel approach.

The detection of correlations between different features in high dimensional data sets is a very important data mining task. These correlations can be arbitrarily complex: one or more features might be correlated with several other features, and both noise features as well as the actual dependencies may be different for different clusters. Therefore, each cluster contains points that are located on a common hyperplane of arbitrary dimensionality in the data space and thus generates a separate, arbitrarily oriented subspace of the original data space. The few recently proposed algorithms designed to uncover these correlation clusters have several disadvantages. In particular, these methods cannot detect correlation clusters of different dimensionality which are nested into each other. The complete hierarchical structure of correlation clusters of varying dimensionality can only be detected by a hierarchical clustering approach. Therefore, we propose the algorithm HiCO (hierarchical correlation ordering), the first hierarchical approach to correlation clustering. The algorithm determines the cluster hierarchy, and visualizes it using correlation diagrams. Several comparative experiments using synthetic and real data sets show the performance and the effectivity of HiCO

Many important industrial applications rely on data mining methods to uncover patterns and trends in large data warehouse environments. Since a data warehouse is typically updated periodically in a batch mode, the mined patterns have to be updated as well. This requires not only accuracy from data mining methods but also fast availability of up-to-date knowledge, particularly in the presence of a heavy update load. To cope with this problem, we propose the use of online data mining algorithms which permanently store the discovered knowledge in suitable data structures and enable an efficient adaptation of these structures after insertions and deletions on the raw data. In this paper, we demonstrate how hierarchical clustering methods can be reformulated as online algorithms based on the hierarchical clustering method OPTICS, using a density estimator for data grouping. We also discuss how this algorithmic schema can be specialized for efficient online single-link clustering. A broad experimental evaluation demonstrates that the efficiency is superior with significant speed-up factors even for large bulk insertions and deletions.