Statistical depth functions provide from the "deepest" point a "center-outward ordering" of multidimensional data. In this sense, depth functions can measure the "extremeness" or "outlyingness" of a data point with respect to a given data set. Hence, they can detect outliers--observations that appear extreme relative to the rest of the observations. Of the various statistical depths, the spatial depth is especially appealing because of its computational efficiency and mathematical tractability. In this article, we propose a novel statistical depth, the kernelized spatial depth (KSD), which generalizes the spatial depth via positive definite kernels. By choosing a proper kernel, the KSD can capture the local structure of a data set while the spatial depth fails. We demonstrate this by the half-moon data and the ring-shaped data. Based on the KSD, we propose a novel outlier detection algorithm, by which an observation with a depth value less than a threshold is declared as an outlier. The proposed algorithm is simple in structure: the threshold is the only one parameter for a given kernel. It applies to a one-class learning setting, in which "normal" observations are given as the training data, as well as to a missing label scenario, where the training set consists of a mixture of normal observations and outliers with unknown labels. We give upper bounds on the false alarm probability of a depth-based detector. These upper bounds can be used to determine the threshold. We perform extensive experiments on synthetic data and data sets from real applications. The proposed outlier detector is compared with existing methods. The KSD outlier detector demonstrates a competitive performance.

It is estimated that less than 10 percent of the world's species have been described, yet species are being lost daily due to human destruction of natural habitats. The job of describing the earth's remaining species is exacerbated by the shrinking number of practicing taxonomists and the very slow pace of traditional taxonomic research. In this article, we tackle, from a novelty detection perspective, one of the most important and challenging research objectives in taxonomy ­ new species identification. We propose a unique and efficient novelty detection framework based on statistical depth functions. Statistical depth functions provide from the "deepest" point a "center-outward ordering" of multidimensional data. In this sense, they can detect observations that appear extreme relative to the rest of the observations, i.e., novelty. Of the various statistical depths, the spatial depth is especially appealing because of its computational efficiency and mathematical tractability. We propose a novel statistical depth, the kernelized spatial depth (KSD) that generalizes the spatial depth via positive definite kernels. By choosing a proper kernel, the KSD can capture the local structure of a data set while the spatial depth fails. Observations with depth values less than a threshold are declared as novel. The proposed algorithm is simple in structure: the threshold is the only one parameter for a given kernel. We give an upper bound on the false alarm probability of a depth-based detector, which can be used to determine the threshold. Experimental study demonstrates its excellent potential in new species discovery.

BACKGROUND: Mean-based clustering algorithms such as bisecting k-means generally lack robustness. Although componentwise median is a more robust alternative, it can be a poor center representative for high dimensional data. We need a new algorithm that is robust and works well in high dimensional data sets e.g. gene expression data. RESULTS: Here we propose a new robust divisive clustering algorithm, the bisecting k-spatialMedian, based on the statistical spatial depth. A new subcluster selection rule, Relative Average Depth, is also introduced. We demonstrate that the proposed clustering algorithm outperforms the componentwise-median-based bisecting k-median algorithm for high dimension and low sample size (HDLSS) data via applications of the algorithms on two real HDLSS gene expression data sets. When further applied on noisy real data sets, the proposed algorithm compares favorably in terms of robustness with the componentwise-median-based bisecting k-median algorithm. CONCLUSION: Statistical data depths provide an alternative way to find the "center" of multivariate data sets and are useful and robust for clustering.