Mark Correlations: Relating Physical Properties to Spatial Distributions

Abstract

Abstract. Mark correlations provide a systematic approach to look at objects both distributed in space and bearing intrinsic information, for instance on physical properties. The interplay of the objects' properties (marks) with the spatial clustering is of vivid interest for many applications; are, e.g., galaxies with high luminosities more strongly clustered than dim ones? Do neighbored pores in a sandstone have similar sizes? How does the shape of impact craters on a planet depend on the geological surface properties? In this article, we give an introduction into the appropriate mathematical framework to deal with such questions, i.e. the theory of marked point processes. After having clarified the notion of segregation effects, we define universal test quantities applicable to realizations of a marked point processes. We show their power using concrete data sets in analyzing the luminosity-dependence of the galaxy clustering, the alignment of dark matter halos in gravitational N -body simulations, the morphologyand diameter-dependence of the Martian crater distribution and the size correlations of pores in sandstone. In order to understand our data in more detail, we discuss the Boolean depletion model, the random field model and the Cox random field model. The first model describes depletion effects in the distribution of Martian craters and pores in sandstone, whereas the last one accounts at least qualitatively for the observed luminosity-dependence of the galaxy clustering.

Correlations in the orientations of galaxy clusters

“…Gottlöber et al (2002) discussed the correlations of galaxy sized halos depending on the merging history with MCFs (see also Gottlöber et al 2001). In a recent review Beisbart et al (2002) show other physical applications of the MCFs and also how the MCFs can be calculated from stochastic models. To make this article self contained, we repeat briefly the basics of MCFs and then extend the formalism to allow for vector marks: Consider the set of N points…”

“…Beisbart et al (2002) provide some further explanations and discuss in which sense A(r), F (r) provide a complete characterization of the vector correlations.…”

SELFI: an object-based, Bayesian method for faint emission line source detection in MUSE deep field data cubes

“…This can be used to model the spatial distribution of many physical phenomena, including astronomical systems (see Neyman & Scott 1952;Scargle & Babu 2003;and Beisbart et al 2002, for detailed examples). In our case, the spatial distribution of the galaxies in the two spatial dimensions of the MUSE data cube can be represented by a configuration of points, i.e., it can be interpreted as a realization of a point process.…”

Measuring the Cosmic Web

“…Beisbart, Kerscher & Mecke 2002). This statistical estimator is defined as the average of the inner galaxy properties mhere taken as color index U − R or R magnitude -as a function of separation r and can be written as (⟨m⟩ is the average over the mark on the whole sample)…”