Multivariate Nonparametric Methodology Studies

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DSM9616187 Scott Nonparametric methodology is widely used in one and two dimensions, but not in high dimensions. This research focuses on the mid-range dimensions and provides a deeper understanding of the implications of the curse of dimensionality and related problems associated with massive data sets. Particular emphasis has been given to multivariate regression and density estimation problems, and closely related applications such as clustering and ridges. Anecdotal evidence has suggested a gap between the apparent successes of nonparametric methodology in practice and the poor performance predicted by theory. We have examined new points of view, especially related to locally adaptive estimation. Higher quality estimation has often required use of negative kernels, but our results have shown that equivalent gains are possible in regions where the Hessian is indefinite, often in the tails which dominate in higher dimensions. In addition, we have developed a class of locally adaptive but not higher order algorithms that work better in practical problems and avoid problems of negativity. We have addressed problems arising from high dimensionality in several ways. We have created algorithms for finding interesting subspaces from the density estimation point of view. Such subspaces are defined by maximal bias content, sequentially peeling off low bias subspaces. We have examined semiparametric models for density estimation that can work better than ordinary nonparametric algorithms, extending feasibility by several extra dimensions. Visualization is especially important when dealing with medium dimensional data and the growing body of massive data sets. One example of a new visualization tool is provided by the density grand tour, which performs an ordinary grand tour but displays a real-time view of a derived density estimate, the averaged shifted histogram. We have found that traversing ridges and contours is useful to control or constrain viewing. We h ave extended our density visualization capabilities to regression surfaces and related problems in visual clustering and visual discrimination applications. Visualization is also important for organizing complicated multiple testing problems is clustering, such as our results in mode estimation and testing based on the mode tree. We have investigated a local testing algorithm for collapsing modes as the basis for an improved clustering algorithm. A natural extension has been demonstrated for multiprocessor and parallel architectures for massive data sets. A great challenge in mathematical sciences is provided by massive data sets. At a recent National Research Council workshop, numerous scientists identified critical statistical needs in their work: alternatives to principal components, specialized visualization tools for exploring massive data, better clustering algorithms, and techniques for handling nonstationary data. Results from our research directly impact three of these four critical opportunities. This program represents a comprehensive and long-term attack on a host of important data analytic problems in multivariate estimation. %%% Statistical techniques that do not require formulae to be written down explicitly are called nonparametric methods and include the well-known histogram as a simple example. Such techniques are widely used with data in one and two dimensions, but not in higher dimensions where most of the grand challenge problems are to be found. This research focuses on the mid-range dimensions where many serious theoreticians have expressed concern that nonparametric methods may not work. However, it is well-known that many practicing scientists and engineers have been successfully using nonparametric methods with data from signal processing, image understanding, data mining, among a wide array of real problems. This research is providing a deeper understanding of the implications of the so-called curse of dimensionality and particular p roblems associated with massive data sets. Particular emphasis is given to problems in multivariate regression and density estimation, as well as closely related applications such as clustering and ridges. We have obtained a new understanding of how locally adaptive estimation should work in overcoming the usual limitations of nonparametric methodology in several dimensions. For higher dimensional data, we have developed algorithms for finding maximally interesting subspaces from the density estimation point of view. Such subspaces are defined by maximal bias content and are constructed sequentially, peeling off low bias subspaces. Beyond two dimensions, visualization is a critical task, especially as related to the growing body of massive data sets. One example of a success is provided by our new density grand tour, which provides a new way of looking at high dimensional data in real-time. We have extended our density estimation visualization capabilities to regression surfaces. Visualization is also very useful for examining data to detect the presence of clusters. Such clusters are critical for determining the usefulness of data collected for proposes such as character recognition, remote sensing crop identification, ground water pollution, as well as many more specialized engineering and scientific applications. Multiprocessor and parallel architectures versions of these algorithms are particularly relevant in the massive data set situation. A great challenge in mathematical sciences is provided by handling massive data sets. At a recent National Research Council workshop, numerous scientists identified critical statistical needs in their work: alternatives to principal components, specialized visualization tools for exploring massive data, better clustering algorithms, and techniques for handling nonstationary data. Results from our research directly impact three of these four critical opportunities. This program represents a comprehensive and long-term attack on a host of important data analytic problems in multivariate estimation. Nonparametric methodology seems to work well in the hands of experts, and this research is designed to not only aid the expert but to facilitate the use of the methodology by a wider audience. ***