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 arXiv preprint arXiv:2207.07001.”

• “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 arXiv preprint arXiv:2207.07001

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.