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Abstract Discoveries of gaps in data have been important in astrophysics. For example, there are kinematic gaps opened by resonances in dynamical systems, or exoplanets of a certain radius that are empirically rare. A gap in a data set is a kind of anomaly, but in an unusual sense: instead of being a single outlier data point, situated far from other data points, it is a region of the space, or a set of points, that is anomalous compared to its surroundings. Gaps are both interesting and hard to find and characterize, especially when they have nontrivial shapes. We present in this paper a statistic that can be used to estimate the (local) “gappiness” of a point in the data space. It uses the gradient and Hessian of the density estimate (and thus requires a twice-differentiable density estimator). This statistic can be computed at (almost) any point in the space and does not rely on optimization; it allows us to highlight underdense regions of any dimensionality and shape in a general and efficient way. We illustrate our method on the velocity distribution of nearby stars in the Milky Way disk plane, which exhibits gaps that could originate from different processes. Identifying and characterizing those gaps could help determine their origins. We provide in an appendix implementation notes and additional considerations for finding underdensities in data, using critical points and the properties of the Hessian of the density. 7 7 A Python implementation of t methods presented here is available at https://github.com/contardog/FindTheGap . 

The latticework structure known as the cosmic web provides a valuable insight into the assembly history of large-scale structures. Despite the variety of methods to identify the cosmic web structures, they mostly rely on the assumption that galaxies are embedded in a Euclidean geometric space. Here, we present a novel cosmic web identifier called sconce (Spherical and CONic Cosmic wEb finder) that inherently considers the 2D (RA, DEC) spherical or the 3D (RA, DEC, z) conic geometry. The proposed algorithms in sconce generalize the well-known subspace constrained mean shift (scms) method and primarily address the predominant filament detection problem. They are intrinsic to the spherical/conic geometry and invariant to data rotations. We further test the efficacy of our method with an artificial cross-shaped filament example and apply it to the SDSS galaxy catalogue, revealing that the 2D spherical version of our algorithms is robust even in regions of high declination. Finally, using N-body simulations from Illustris, we show that the 3D conic version of our algorithms is more robust in detecting filaments than the standard scms method under the redshift distortions caused by the peculiar velocities of haloes. Our cosmic web finder is packaged in python as sconce-scms and has been made publicly available.

 

This paper studies the linear convergence of the subspace constrained mean shift (SCMS) algorithm, a well-known algorithm for identifying a density ridge defined by a kernel density estimator. By arguing that the SCMS algorithm is a special variant of a subspace constrained gradient ascent (SCGA) algorithm with an adaptive step size, we derive the linear convergence of such SCGA algorithm. While the existing research focuses mainly on density ridges in the Euclidean space, we generalize density ridges and the SCMS algorithm to directional data. In particular, we establish the stability theorem of density ridges with directional data and prove the linear convergence of our proposed directional SCMS algorithm.

 

In this paper, we propose a new clustering method inspired by mode-clustering that not only finds clusters, but also assigns each cluster with an attribute label. Clusters obtained from our method show connectivity of the underlying distribution. We also design a local two-sample test based on the clustering result that has more power than a conventional method. We apply our method to the Astronomy and GvHD data and show that our method finds meaningful clusters. We also derive the statistical and computational theory of our method. 

Directional data consist of observations distributed on a (hyper)sphere, and appear in many applied fields, such as astronomy, ecology, and environmental science. This paper studies both statistical and computational problems of kernel smoothing for directional data. We generalize the classical mean shift algorithm to directional data, which allows us to identify local modes of the directional kernel density estimator (KDE). The statistical convergence rates of the directional KDE and its derivatives are derived, and the problem of mode estimation is examined. We also prove the ascending property of the directional mean shift algorithm and investigate a general problem of gradient ascent on the unit hypersphere. To demonstrate the applicability of the algorithm, we evaluate it as a mode clustering method on both simulated and real-world data sets.