Introduction

A filament is a one-dimensional, smooth, connected structure embedded in a multi-dimensional space that characterizes the high density regions.

A statistical model for the filaments is the density ridge.

The below are some examples for density ridges on smooth density functions.

We use kernel density estimation to reconstruct the density function from data and identify the density ridges as filaments.

The density ridges have strong agreement to the result from the Voronoi model:

We thank Miguel Aragon for providing the Voronoi dataset.

Each black dot in the left panel is a galaxy. The Voronoi method assign a label to each galaxy to denote if this galaxy belongs to a cluster, a filament, a wall or in a void. We apply our method (density ridges) using all galaxies, ignoring the labels. Our filaments (density ridges) are the blue curves in the middle and right panel. In the middle panel, the brown dots are galaxies with Voronoi label "cluster". Remarkably, these galaxies tend to occur at around the intersection of density ridges. In the right panel, the red dots are galaxies with Voronoi label "filament"--all of these galaxies are just points around density ridges! Therefore, the density ridges have strong agreement with the Voronoi model.

After we estimate the filaments, we can evaluate the uncertainty for filament using the bootstrap.

The statistical consistency for this uncertainty measurement has been proved in one of our recent work.

We display local uncertainty by color (red:high) and confidence sets (wide regions:high).

We apply our method to Sloan Digit Sky Survey (SDSS), Data Release 12 (including main galaxy sample from NYU catalogue).

The following is a slice of our detected filaments along with the uncertainty measurement.

Each black point is a galaxy at certain position. We use our method to estimate filaments and display the uncertainty measures along filaments.

We compare our result to the galaxy clusters given by the redMaPPer catalogues (bold red points are the clusters):

The first slice is at redshifts z=0.105-0.110 and the second slice is at redshifts z=0.470-0.475. As can be seen, most clusters are located on the ridges--this shows that our filaments are consistent with the cluster catalogue.

An introduction slide - This is an introduction slide for this project from the talk at ASIAA, Taiwan.

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