Draft:Weighted overlap coefficient

• Comment: some secondary sources would be great imo Astropedian (talk) 17:50, 2 June 2026 (UTC)

---

The Weighted Overlap Coefficient (WOC) is a statistical method, introduced in 2022, for quantifying the morphological similarity between two two- or three-dimensional scalar fields. It was developed in the context of observational cosmology to compare the spatial distributions of dark matter, member galaxies and intracluster light in simulated galaxy clusters, but the underlying construction is general and can in principle be applied to any pair of co-registered scalar maps.

The method was proposed as an alternative to the Modified Hausdorff Distance (MHD) and to cross-correlation techniques, both of which had previously been used in the astrophysical literature to compare dark-matter and intracluster-light distributions. A reference Python implementation, pywoc, is distributed under the GPL-3.0 license.

Background

A recurring problem in extragalactic astrophysics is to decide, in a quantitative way, whether two scalar fields defined on the same spatial domain share the same underlying morphology. For example, comparing the similarity of the spatial distributions of the X-ray and Optical components of a galaxy cluster. Earlier attempts to address this problem in the cluster context fall into three main families:

• Radial profile comparison. Azimuthally averaged one-dimensional profiles can be compared directly, but this discards all azimuthal information about asymmetries, substructure and merging activity.
• Modified Hausdorff Distance. The MHD measures the distance between point sets extracted from contours of the two fields and has been used to compare intracluster light with dark matter.
• Cross-correlation. Two-point cross-correlation between weak-lensing mass maps and galaxy density fields has also been employed.

Outside astrophysics, similar problems are addressed by the niche-overlap measures of theoretical ecology and by the Earth mover's distance, the latter of which has become popular as a loss function in some machine learning applications.

The authors of the WOC method argue that when one wishes to compare the shape of two distributions without making any prior assumption about the relation between their absolute amplitudes, the methods above can be biased or insensitive to substructure, motivating a new approach.

Definition

Schematic of the Weighted Overlap Coefficient. The reference distribution (solid contours) and the comparison distribution (dashed contours) are sliced at three nested density-threshold levels; at each level the overlap area is evaluated, and the levels are combined with weights based on the contour area and the signal at each level.

The Weighted Overlap Coefficient compares two co-registered scalar maps, denoted $I_{1}$ (the reference) and $I_{2}$, that share the same pixel or voxel grid. The construction proceeds in six steps:

1. Both maps are smoothed on a common spatial scale.
2. A signal centre is chosen in the reference map (for example, the peak of $I_{1}$), the one-dimensional radial profile of $I_{1}$ about that centre is computed, and the profile is evaluated at $n$ equally spaced radii to give $n$ density thresholds.
3. For each threshold the enclosed region of $I_{1}$ (the set of pixels above that threshold) is identified, and its total area is recorded; if the level set is disconnected, the areas of the separate components are summed.
4. In the comparison map $I_{2}$, density thresholds are then found that enclose the same total area as the corresponding levels in $I_{1}$. These are determined iteratively, starting from the peak of $I_{2}$ and lowering the threshold in small steps until the area constraint is met. Matching areas rather than thresholds is what allows the comparison to be made independent of the absolute amplitudes of the two fields.
5. At each level, the area of the intersection of the two regions is evaluated and converted into an overlap fraction.
6. The level-by-level overlaps are combined into a single similarity score via a weighted sum.

The result is given by

$\mathrm {WOC} (I_{1},I_{2})\;=\;{\frac {\displaystyle \sum _{i=1}^{n}a_{i}\,{\bigl (}w_{A,i}+w_{\rho _{1},i}+w_{\rho _{2},i}{\bigr )}}{\displaystyle \sum _{i=1}^{n}{\bigl (}w_{A,i}+w_{\rho _{1},i}+w_{\rho _{2},i}{\bigr )}}},$ 1

with the weights normalised so that

$\sum _{i=1}^{n}w_{A,i}\;=\;\sum _{i=1}^{n}w_{\rho _{1},i}\;=\;\sum _{i=1}^{n}w_{\rho _{2},i}\;=\;1.$ 2

Here $a_{i}\in [0,1]$ is the fractional overlap area at the $i$-th level, $w_{A,i}$ is the area enclosed at level $i$ relative to the area of the outermost ($n$-th) contour, and $w_{\rho _{1},i}$ and $w_{\rho _{2},i}$ are the (normalised) density thresholds at level $i$ in $I_{1}$ and $I_{2}$ respectively. The construction guarantees $\mathrm {WOC} \in [0,1]$, with unity reached when the two fields share identical morphology and zero indicating no spatial coincidence at any level.

Because the comparison is built from the geometry of nested contours of equal enclosed area rather than from a pixel-by-pixel intensity match, the WOC is, by construction, sensitive to morphology rather than to overall amplitude. This is intended for cases in which the relation between the signal strengths of the two fields is either unknown or is not the quantity of interest. The estimator can be applied to either two-dimensional images or three-dimensional cubes and supports the use of a binary mask for excluding bad pixels or survey footprints.

Comparison with the Modified Hausdorff Distance

In the original presentation, the WOC was benchmarked against the MHD on idealised test cases and on simulated galaxy clusters. The authors report that the WOC remains stable in the presence of multiple substructures, whereas the MHD can be dominated by a small number of outlying contour points, and that the WOC discriminates more clearly between dynamically relaxed and dynamically disturbed cluster configurations.

Application to galaxy clusters

The method was applied to six simulated galaxy clusters drawn from the Galaxy Replacement Technique (GRT) simulation, a high-resolution N-body simulation performed with the Gadget3 code in which dark-matter halos that fall into a cluster are replaced by higher-resolution halos containing exponential stellar disks. The six clusters span a range of dynamical states, from systems whose last major merger occurred at high redshift to a system currently undergoing a major merger.

The two-dimensional dark-matter map of each projected cluster was compared, using the WOC, with maps of the satellite galaxies, of all stellar particles, and of the brightest cluster galaxy plus intracluster light (BCG+ICL). The authors report two main findings:

• Of the visible components considered, the BCG+ICL component most faithfully traces the projected dark-matter distribution.
• Among the simulated clusters, dynamically relaxed systems show stronger BCG+ICL–dark matter similarity than dynamically young systems, suggesting that the WOC measured between BCG+ICL and a mass map could in principle be used as a dynamical-stage estimator. The same trend is weaker when measured with the MHD.

These results extend earlier work that had argued, on different statistical grounds, for intracluster light as a luminous tracer of dark matter in clusters.

Implementation

The reference implementation is the open-source Python package pywoc, maintained on GitHub. The package exposes a single high-level function, woc(map1, map2, radii, ...), that accepts two-dimensional or three-dimensional NumPy arrays of identical shape, an optional boolean mask, an explicit pixel size, and parameters controlling the radial profile (number of bins, maximum radius, choice of centre). Optional acceleration via Numba, radial-profile vectorisation and a regression test suite were added in subsequent releases.

The package is installable from the Python Package Index as pywoc, with optional extras for plotting and for the tutorial notebooks distributed with the source repository.

See also

• Intracluster light
• Dark matter halo
• Hausdorff distance
• Wasserstein metric
• N-body simulation
• Brightest cluster galaxy
• Overlap coefficient

External links

• WOC source code on GitHub
• Yoo et al. (2022) on arXiv
• pywoc on the Python Package Index

Category:Computational astronomy Category:Galaxy clusters Category:Dark matter Category:Free statistical software