Department of Statistics Unitmark
Dietrich College of Humanities and Social Sciences

CMU TopStat

The CMU Topological Statistics group is a research group at Carnegie Mellon University. The emphasis of our research is on statistical problems related to topological inference.

Cosmic web reconstruction through density ridges: catalogue

Y.-C. Chen, S. Ho, J. Brinkmann, P. E. Freeman, C. R. Genovese, D. P. Schneider, & L. Wasserman
Publication Date:
October, 2016

We construct a catalogue for filaments using a novel approach called SCMS (subspace constrained mean shift). SCMS is a gradient-based method that detects filaments through density ridges (smooth curves tracing high-density regions). A great advantage of SCMS is its uncertainty measure, which allows an evaluation of the errors for the detected filaments. To detect filaments, we use data from the Sloan Digital Sky Survey, which consist of three galaxy samples: the NYU main galaxy sample (MGS), the LOWZ sample and the CMASS sample. Each of the three data set covers different redshift regions so that the combined sample allows detection of filaments up to z = 0.7. Our filament catalogue consists of a sequence of two-dimensional filament maps at different redshifts that provide several useful statistics on the evolution cosmic web. To construct the maps, we select spectroscopically confirmed galaxies within 0.050 < z < 0.700 and partition them into 130 bins. For each bin, we ignore the redshift, treating the galaxy observations as a 2-D data and detect filaments using SCMS. The filament catalogue consists of 130 individual 2-D filament maps, and each map comprises points on the detected filaments that describe the filamentary structures at a particular redshift. We also apply our filament catalogue to investigate galaxy luminosity and its relation with distance to filament. Using a volume-limited sample, we find strong evidence (6.1σ-12.3σ) that galaxies close to filaments are generally brighter than those at significant distance from filaments.

Cosmic web reconstruction through density ridges: method and algorithm

Y.-C. Chen, S. Ho, P. E. Freeman, C. R. Genovese, & L. Wasserman
Publication Date:
November, 2015

The detection and characterization of filamentary structures in the cosmic web allows cosmologists to constrain parameters that dictate the evolution of the Universe. While many filament estimators have been proposed, they generally lack estimates of uncertainty, reducing their inferential power. In this paper, we demonstrate how one may apply the subspace constrained mean shift (SCMS) algorithm (Ozertem & Erdogmus 2011; Genovese et al. 2014) to uncover filamentary structure in galaxy data. The SCMS algorithm is a gradient ascent method that models filaments as density ridges, one-dimensional smooth curves that trace high-density regions within the point cloud. We also demonstrate how augmenting the SCMS algorithm with bootstrap-based methods of uncertainty estimation allows one to place uncertainty bands around putative filaments. We apply the SCMS first to the data set generated from the Voronoi model. The density ridges show strong agreement with the filaments from Voronoi method. We then apply the SCMS method data sets sampled from a P3M N-body simulation, with galaxy number densities consistent with SDSS and WFIRST-AFTA, and to LOWZ and CMASS data from the Baryon Oscillation Spectroscopic Survey (BOSS). To further assess the efficacy of SCMS, we compare the relative locations of BOSS filaments with galaxy clusters in the redMaPPer catalogue, and find that redMaPPer clusters are significantly closer (with p-values <10-9) to SCMS-detected filaments than to randomly selected galaxies.

Investigating galaxy-filament alignments in hydrodynamic simulations using density ridges

Y.-C. Chen, S. Ho, A. Tenneti, R. Mandelbaum, R. Croft, T. DiMatteo, P. E. Freeman, C. R. Genovese, & L. Wasserman
Publication Date:
December, 2015

In this paper, we study the filamentary structures and the galaxy alignment along filaments at redshift z = 0.06 in the MassiveBlack-II simulation, a state-of-the-art, high-resolution hydrodynamical cosmological simulation which includes stellar and AGN feedback in a volume of (100 Mpc h-1)3. The filaments are constructed using the subspace constrained mean shift (SCMS; Ozertem & Erdogmus; Chen et al.). First, we show that reconstructed filaments using galaxies and reconstructed filaments using dark matter particles are similar to each other; over 50 per cent of the points on the galaxy filaments have a corresponding point on the dark matter filaments within distance 0.13 Mpc h-1 (and vice versa) and this distance is even smaller at high-density regions. Second, we observe the alignment of the major principal axis of a galaxy with respect to the orientation of its nearest filament and detect a 2.5 Mpc h-1 critical radius for filament's influence on the alignment when the subhalo mass of this galaxy is between 109 M h-1 and 1012 M h-1. Moreover, we find the alignment signal to increase significantly with the subhalo mass. Third, when a galaxy is close to filaments (less than 0.25 Mpc h-1), the galaxy alignment towards the nearest galaxy group is positively correlated with the galaxy subhalo mass. Finally, we find that galaxies close to filaments or groups tend to be rounder than those away from filaments or groups.

Non-parametric 3D map of the intergalactic medium using the Lyman-alpha forest

J. Cisewski, R. A. C. Croft, P. E. Freeman, C. R. Genovese, N. Khandai, M. Ozbek, & L. Wasserman
Publication Date:
May, 2014

Visualizing the high-redshift Universe is difficult due to the dearth of available data; however, the Lyman-alpha forest provides a means to map the intergalactic medium at redshifts not accessible to large galaxy surveys. Large-scale structure surveys, such as the Baryon Oscillation Spectroscopic Survey (BOSS), have collected quasar (QSO) spectra that enable the reconstruction of H I density fluctuations. The data fall on a collection of lines defined by the lines of sight (LOS) of the QSO, and a major issue with producing a 3D reconstruction is determining how to model the regions between the LOS. We present a method that produces a 3D map of this relatively uncharted portion of the Universe by employing local polynomial smoothing, a non-parametric methodology. The performance of the method is analysed on simulated data that mimics the varying number of LOS expected in real data, and then is applied to a sample region selected from BOSS. Evaluation of the reconstruction is assessed by considering various features of the predicted 3D maps including visual comparison of slices, probability density functions (PDFs), counts of local minima and maxima, and standardized correlation functions. This 3D reconstruction allows for an initial investigation of the topology of this portion of the Universe using persistent homology.