Christopher R. Genovese

We study nonparametric clustering of smooth random curves on the basis of the L2 gradient flow associated to a pseudo-density functional and we show that the clustering is well-defined both at the population and at the sample level. We provide an algorithm to mark significant local modes, which are associated to informative sample clusters, and we derive its consistency properties. Our theory is developed under weak assumptions, which essentially reduce to the integrability of the random curves, and does not require to project the random curves on a finite-dimensional subspace. However, if the underlying probability distribution is supported on a finite-dimensional subspace, we show that the pseudo-density and the expectation of a kernel density estimator induce the same gradient flow, and therefore the same clustering. Although our theory is developed for smooth curves that belong to an infinite-dimensional functional space, we also provide consistent procedures that can be used with real data (discretized and noisy observations).

Mode clustering is a nonparametric method for clustering that defines clusters using the basins of attraction of a density estimator's modes. We provide several enhancements to mode clustering: (i) a soft variant of cluster assignment, (ii) a measure of connectivity between clusters, (iii) a technique for choosing the bandwidth, (iv) a method for denoising small clusters, and (v) an approach to visualizing the clusters. Combining all these enhancements gives us a complete procedure for clustering in multivariate problems. We also compare mode clustering to other clustering methods in several examples

We introduce the concept of coverage risk as an error measure for density ridge estimation. The coverage risk generalizes the mean integrated square error to set estimation. We propose two risk estimators for the coverage risk and we show that we can select tuning parameters by minimizing the estimated risk. We study the rate of convergence for coverage risk and prove consistency of the risk estimators. We apply our method to three simulated datasets and to cosmology data. In all the examples, the proposed method successfully recover the underlying density structure.

We introduce the functional mean-shift algorithm, an iterative algorithm for estimating the local modes of a surrogate density from functional data. We show that the algorithm can be used for cluster analysis of functional data. We propose a test based on the bootstrap for the significance of the estimated local modes of the surrogate density. We present two applications of our methodology. In the first application, we demonstrate how the functional mean-shift algorithm can be used to perform spike sorting, i.e. cluster neural activity curves. In the second application, we use the functional mean-shift algorithm to distinguish between original and fake signatures.

The generalized density is a product of a density function and a weight function. For example, the average local brightness of an astronomical image is the probability of finding a galaxy times the mean brightness of the galaxy. We propose a method for studying the geometric structure of generalized densities. In particular, we show how to find the modes and ridges of a generalized density function using a modification of the mean shift algorithm and its variant, subspace constrained mean shift. Our method can be used to perform clustering and to calculate a measure of connectivity between clusters. We establish consistency and rates of convergence for our estimator and apply the methods to data from two astronomical problems.

The Lyman-alpha forest is a portion of the observed light spectrum of distant galactic nuclei which allows us to probe remote regions of the Universe that are otherwise inaccessible. The observed Lyman-alpha forest of a quasar light spectrum can be modeled as a noisy realization of a smooth curve that is affected by a `damping effect' which occurs whenever the light emitted by the quasar travels through regions of the Universe with higher matter concentration. To decode the information conveyed by the Lyman-alpha forest about the matter distribution, we must be able to separate the smooth `continuum' from the noise and the contribution of the damping effect in the quasar light spectra. To predict the continuum in the Lyman-alpha forest, we use a nonparametric functional regression model in which both the response and the predictor variable (the smooth part of the damping-free portion of the spectrum) are function-valued random variables. We demonstrate that the proposed method accurately predicts the unobservable continuum in the Lyman-alpha forest both on simulated spectra and real spectra. Also, we introduce distribution-free prediction bands for the nonparametric functional regression model that have finite sample guarantees. These prediction bands, together with bootstrap-based confidence bands for the projection of the mean continuum on a fixed number of principal components, allow us to assess the degree of uncertainty in the model predictions.

A filament is a high density, connected region in a point cloud. There are several methods for estimating filaments but these methods do not provide any measure of uncertainty. We give a definition for the uncertainty of estimated filaments and we study statistical properties of the estimated filaments. We show how to estimate the uncertainty measures and we construct confidence sets based on a bootstrapping technique. We apply our methods to astronomy data and earthquake data.

The next generation of telescopes, coming online in the next decade, will acquire terabytes of image data each night. Collectively, these large images will contain billions of interesting objects, which astronomers call sources. One critical task for astronomers is to construct from the image data a detailed source catalog that gives the sky coordinates and other properties of all detected sources. The source catalog is the primary data product produced by most telescopes and serves as an important input for studies that build and test new astrophysical theories. To construct an accurate catalog, the sources must first be detected in the image. A variety of effective source detection algorithms exist in the astronomical literature, but few, if any, provide rigorous statistical control of error rates. A variety of multiple testing procedures exist in the statistical literature that can provide rigorous error control over pixelwise errors, but these do not provide control over errors at the level of sources, which is what astronomers need. In this article, we propose a technique that is effective at source detection while providing rigorous control on sourcewise error rates. We demonstrate our approach with data from the Chandra X-ray Observatory Satellite. Our method is competitive with existing astronomical methods, even finding two new sources that were missed by previous studies, while providing stronger performance guarantees and without requiring costly follow up studies that are commonly required with current techniques.

The next generation of telescopes will acquire terabytes of image data on a nightly basis. Collectively, these large images will contain billions of interesting objects, which astronomers call sources. The astronomers' task is to construct a catalog detailing the coordinates and other properties of the sources. The source catalog is the primary data product for most telescopes and is an important input for testing new astrophysical theories, but to construct the catalog one must first detect the sources. Existing algorithms for catalog creation are effective at detecting sources, but do not have rigorous statistical error control. At the same time, there are several multiple testing procedures that provide rigorous error control, but they are not designed to detect sources that are aggregated over several pixels. In this paper, we propose a technique that does both, by providing rigorous statistical error control on the aggregate objects themselves rather than the pixels. We demonstrate the effectiveness of this approach on data from the Chandra X-ray Observatory Satellite. Our technique effectively controls the rate of false sources, yet still detects almost all of the sources detected by procedures that do not have such rigorous error control and have the advantage of additional data in the form of follow up observations, which will not be available for upcoming large telescopes. In fact, we even detect a new source that was missed by previous studies. The statistical methods developed in this paper can be extended to problems beyond Astronomy, as we will illustrate with an example from Neuroimaging.