Sperical and Conic Cosmic Web Finder in Python

SCONCE-SCMS (Spherical and CONic Cosmic wEb finder with the extended SCMS algorithms [1]) is a Python library for detecting the cosmic web structures (primarily cosmic filaments and the associated cosmic nodes) from a collection of discrete observations with the extended subspace constrained mean shift (SCMS) algorithms on the 2D (RA,DEC) celestial sphere \(\mathbb{S}^2\) or the 3D (RA,DEC,redshift) light cone \(\mathbb{S}^2\times\mathbb{R}\).

(Notes: RA – Right Ascension, i.e., the celestial longitude; DEC – Declination, i.e., the celestial latitude.)

A quick introduction to the methodology

The subspace constrained mean shift (SCMS) algorithm [2] is a gradient ascent typed method dealing with the estimation of local principal curves, more widely known as density ridges in statistics [3]. The one-dimensional density ridge traces over the curves where observational data are highly concentrated and thus serves as a natural model for cosmic filaments in our Universe [4]. One advantage of modeling cosmic filaments as density ridges is that they can be efficiently estimated by the kernel density estimator (KDE) and the subsequent SCMS algorithm in a statistically consistent way.

Whereas the standard SCMS algorithm [2] is well-suited for identifying the density ridges in any “flat” Euclidean space \(\mathbb{R}^D\), it exhibits large bias in estimating the density ridges on the data space with a non-linear curvature; see examples in [1] and [5]. In astronomy, however, one often encounters observations from the 2D (RA,DEC) celestial sphere \(\mathbb{S}^2\) or the 3D (RA,DEC,redshift) light cone \(\mathbb{S}^2\times\mathbb{R}\). To resolve the estimation bias of the standard SCMS algorithm on these two data spaces, we propose our extended SCMS algorithms (DirSCMS [5] and DirLinSCMS [6] methods) that are adaptive to the spherical and conic geometries, respectively. At a high level, we utilize the directional or directional-linear KDEs to estimate the underlying density function and carefully design the iteration formulae for our extended SCMS algorithms.

More details can be found in Methodology and the reference paper [1].

Note

This project is under active development.

Contents:

• Installation guide
  • Dependencies
  • Quick Start
• Methodology
  • Standard SCMS Algorithm on the Euclidean Space \(\mathbb{R}^D\)
  • Directional SCMS (DirSCMS) Algorithm on the Unit Sphere \(\mathbb{S}^q\)
  • Directional-linear SCMS (DirLinSCMS) Algorithm on the 3D Light Cone \(\mathbb{S}^2\times \mathbb{R}\)
  • References
• Example Codes for Using the Mean Shift Algorithms in SCONCE-SCMS
  • Example 1: Mode-seeking on a 2D Gaussian mixture density
  • Example 2: Mode-seeking on a von Mises-Fisher (vMF) mixture density on \(\mathbb{S}^2\)
  • Example 3: Mode-seeking on a cylinder (\(\mathbb{S}^1\times \mathbb{R}\) data)
• Example Codes for Using the SCMS Algorithms in SCONCE-SCMS
  • Example 1: A circle crossing the North and South poles on the unit sphere \(\mathbb{S}^2\)
  • Example 2: 3D Spiral Curve
• API References
  • Standard KDE, mean shift, and SCMS algorithms in a flat Euclidean space \(\mathbb{R}^d\)
  • Directional KDE, mean shift, and SCMS algorithms on the unit (hyper-)sphere \(\mathbb{S}^q\)
  • Directional-linear KDE, mean shift, and SCMS algorithms on the directional-linear product space \(\mathbb{S}^q\times\mathbb{R}\)
  • Knot detection on a set of filamentary points
  • Utilities for SCONCE-SCMS

How to Cite SCONCE-SCMS

If you use sconce-scms in your research, please cite the following papers:

• “Y. Zhang, R. S. de Souza, and Y.-C. Chen (2022). SCONCE: A cosmic web finder for spherical and conic geometries. Monthly Notices of the Royal Astronomical Society, 517 (1): 1197–1217.”

• “Y.-C. Chen, S. Ho, P. E. Freeman, C. R. Genovese, and L. Wasserman (2015). Cosmic web reconstruction through density ridges: method and algorithm. Monthly Notices of the Royal Astronomical Society, 454 (1), 1140-1156.”

References

[1] (1,2,3)

Zhang, Y., de Souza, R. S., and Chen, Y.-C. (2022). SCONCE: A cosmic web finder for spherical and conic geometries. Monthly Notices of the Royal Astronomical Society, 517 (1): 1197–1217.

[2] (1,2)

Ozertem, U. and Erdogmus, D. (2011). Locally defined principal curves and surfaces. Journal of Machine Learning Research, 12, 1249-1286.

[3]

Genovese, C.R., Perone-Pacifico, M., Verdinelli, I. and Wasserman, L. (2014). Nonparametric ridge estimation. The Annals of Statistics, 42 (4), 1511-1545.

[4]

Chen, Y.-C., Ho, S., Freeman, P.E., Genovese, C.R. and Wasserman, L. (2015). Cosmic web reconstruction through density ridges: method and algorithm. Monthly Notices of the Royal Astronomical Society, 454 (1), 1140-1156.

[5] (1,2)

Zhang, Y. and Chen, Y.-C. (2022). Linear convergence of the subspace constrained mean shift algorithm: from Euclidean to directional data. Information and Inference: A Journal of the IMA, iaac005, https://doi.org/10.1093/imaiai/iaac005.

[6]

Zhang, Y. and Chen, Y.-C. (2021). Mode and ridge estimation in euclidean and directional product spaces: A mean shift approach. arXiv preprint arXiv:2110.08505, https://arxiv.org/abs/2110.08505.