International audienceThe compelling and intrinsic network of galaxies builds up a complex structure for the Universe. In this work positive association between long bridging structures named galaxy filaments and a photometric redshift galaxy dataset is under investigation. Possible positive association is studied by the use of a bivariate J-function.Le réseau de filaments formé par la position des galaxies est une des structures les plus fascinantes dans notre Univers. Dans cet article, nous étudions des possibles associations entre le réseau de filaments déjà détecté et des nouvelles observations. Ces observations très récentes sont obtenues à partir des mesures du décalage vers le rouge photométrique. L'utilisation de la fonction J-bivariée tend à indiquer une association positive entre les filaments existants et les nouvelles observations

Kipper, Rain

Kruuse, Maarja

Stoica, Radu

Tempel, Elmo

English

Archive Ouverte en Sciences de l'Information et de la Communication

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The bivariate J-function to analyse positive association
between galaxies and galaxy filaments
Maarja Kruuse, Elmo Tempel, Rain Kipper, Radu Stoica
To cite this version:
Maarja Kruuse, Elmo Tempel, Rain Kipper, Radu Stoica. The bivariate J-function to analyse positive
association between galaxies and galaxy filaments. 51es Journées de Statistique 2019, Jun 2019, Nancy,
France. ￿hal-02305994￿
The bivariate J-function to analyse positive
association between galaxies and galaxy
filaments
Maarja Kruuse 1, Elmo Tempel 2, Rain Kipper 3 & Radu Stoica 4
1 Tartu Observatory, University of Tartu, Observatooriumi 1, 61602 To˜ravere, Estonia, maarja@to.ee
2 Tartu Observatory, University of Tartu, Observatooriumi 1, 61602 To˜ravere, Estonia,
elmo.tempel@to.ee
3 Tartu Observatory, University of Tartu, Observatooriumi 1, 61602 To˜ravere, Estonia,
rain.kipper@to.ee
4 Universite´ de Lorraine, CNRS, IECL, F-54000 Nancy, France, radu-stefan.stoica@univ-lorraine.fr
Abstract. The compelling and intrinsic network of galaxies builds up a complex struc-
ture for the Universe. In this work positive association between long bridging structures
named galaxy filaments and a photometric redshift galaxy dataset is under investigation.
Possible positive association is studied by the use of a bivariate J-function.
Keywords. Methods: statistical – galaxies: statistics – large-scale structure of Uni-
verse
1 Introduction
Large scale galaxy surveys show the intrinsic structure of the Universe: bridging vast
filamentary patterns, galaxy clusters, sheets, superclusters and immense regions almost
devoid of galaxies named voids (Jo˜eveer et al. (1978)). These maps are dominated by
galaxy filaments, which connect the structures into a web (Pimbblet et al. (2005)). A
mathematical framework named the Bisous model (Stoica et al. (2010)) has been applied
on observed large scale galaxy datasets in the paper Tempel et al. (2014). In the latter
paper they estimated the filamentary pattern from Sloan Digital Sky Survey (York et
al. (2010)) (SDSS) spectroscopic galaxy data (these galaxies have precise distances in
redshift1 space).
In this study we analyse whether these galaxy filaments (Tempel et al. (2014)) are pos-
itively associated with the Sloan Digital Sky Survey photometric redshift galaxies dataset
(these galaxies have rather uncertain distance estimations in redshift space) compiled in
(Beck et al. (2016)). The galaxies can be viewed as points in 3-dimensional Euclidean
space (Martinez & Saar (2001)). The Cartesian coordinates of these galaxies are point
locations in 3−dimensional Euclidean space and other properties can be considered as
marks. Thus the galaxies situated in a survey region can be seen as a realization of a
marked point process in a compact set.
1Characteristic that estimates the measure of distance in cosmology.
1
− 50 − 45 − 40
15
20
25
30
λ(deg)
η(d
eg
)
Figure 1: Visualization of the two datasets in spherical sky-coordinates: kernel smoothed
2-dimensional density from the distribution of photometric galaxies on the viewed region
of the sphere (grey background density) and the filamentary spines (orange spines). All
the objects locate in the distance range of 200− 400 Mpc and the darker grey denotes a
higher density of photometric galaxies.
2 Data
In this section the analysed photometric redshift galaxies dataset and the filamentary
pattern catalogue are characterized. Limitations will be set on all of the observed objects
dependent on their distance to the observer.
Figure 1 depicts the two studied datasets in spherical sky coordinates η and λ. From
visual inspection one can notice that the higher density of filamentary spines seems to fol-
low the higher density of photometric redshift galaxies. Further on this possible clustering
effect is analysed with the use of the bivariate J−function.
2.1 Filamentary network
The filamentary spines catalogue holds information about 46 403 objects. They form a
random set of objects Yfil in the 3−dimensional Euclidean space. The mapping of a point
2
in R3 to the spherical coordinates latitude η and longitude λ is done using
η = 2 arctan
(
y√
x2 + y2
− x
)
,
λ = arcsin
(
z√
(x2 + y2 + z2)
)
. (1)
In paper (Kruuse et al. submitted) the mean distance used to characterise spine Y
distance to the observer O is defined as:
d¯(O, Y ) =
1
l(Y )
∫
Y
d(O, y)dy, (2)
where l(Y ) is the associated spine length and d(O, y) is the distance from the observer O
to the point y of the spine Y . The integral in (2) is computed along the considered spine.
These mean distances will be used to limit the filamentary spines into the distance range
of 200− 400 Mpc. Filamentary spines, which satisfy this condition, are further analysed.
The dataset Yfil will be of 20 367 filamentary spines. The chosen distance range carries
the bulk of the filamentary spines.
2.2 Photometric galaxies
The rather uncertain redshift zphoto estimations (δzphoto ≤ 0.05) in the catalogue (Beck
(2005)) are used to limit the galaxies into the distance range of 200 − 400 Mpc. Which
leaves us with the dataset of 236 850 photometric galaxies. Their locations are defined on
the sphere by their radian latitude η and longitude λ. They are viewed as a configuration
of points Xphoto in a sphere S
2.
3 The bivariate J−function
The datasets studied are fit for being analysed with spatial statistics tools (Baddeley et
al. (2015); Martinez & Saar (2001)). One can consider the photometric galaxies set like a
point processes realization and the filamentary set as a realization of a random set. The
bivariate J−function can be used to study possible positive association between these
objects.
Thorough mathematical description of the theory of point processes and probabilistic
models therein are written in van Lieshout (2000), Møller & Waagepetersen (2004), Illian
et al. (2008), Chiu et al. (2013) and Baddeley et al. (2015). Here we will present the
border corrected estimations for the empty space function, nearest-neighbour function
and the bivariate J−function as they are described in van Lieshout & Baddeley (1996),
3
Foxall & Baddeley (2002) and Baddeley et al. (2015). All the described summary statis-
tics have exact formulas for the homogeneous Poisson point process, which represents a
completely random pattern. A rigorous representation of the Poisson point process is
written Baddeley et al. (2015).
3.1 Empty space function
First of all the shortest distances from a point w ∈ W to a subset A ⊂ W is noted by the
following form
d(w,A) = infa∈A‖w − a‖.
The border corrected estimator for the empty space function from any arbitrary point
wi to the random set of Y is of the following form
F̂Y (r) =
∑
i 1{d(wi,W c) ≥ r}1{d(wi, Y ) ≤ r}∑
i 1{d(wi,W c) ≥ r}
(3)
with W c the border of W and {wi, i = 1, 2, . . .} a finite family of arbitrary points in
W . The estimation F̂ is a cumulative distribution function of smallest distances from
arbitrary points to the closest object in the random set.
3.2 Nearest-neighbour function
The estimator of the nearest neighbour distribution from any point in X to the random
set Y is of the following form
ĜX,Y (r) =
∑
i 1{d(xi,W c) ≥ r}1{d(xi, Y ) ≤ r}∑
i 1{d(xi,W c) ≥ r}
, (4)
where {xi, i = 1, . . .} is the observed finite point configuration of X. The nearest-
neighbour function is a cumulative distribution function of smallest distances from a
point of the point process X to the nearest object in the random set Y . The estimation
of Ĝ measures association between the objects of type X and type Y . If the point process
X and the random set of objects Y are independent, then ĜX,Y ≡ F̂X .
3.3 Bivariate J−function
The bivariate J function is obtained by the nearest-neighbour function GX,Y (r) and empty
space function FY (r) in the following form:
ĴX,Y (r) =
1− ĜX,Y (r)
1− F̂Y (r)
. (5)
4
Distance r (radians)
cu
m
u
la
tiv
e 
di
str
ib
u
tio
n 
fu
nc
tio
n 
va
lu
e
0 0.005 0.01 0.015
0
0.
2
0.
4
0.
6
0.
8
1
G(Xphoto, Yfil)
F(Yfil)
Distance r (radians)
J(r
)
0 0.005 0.01 0.015
0
0.
2
0.
4
0.
6
0.
8
1
J(Xphoto, Yfil)
Theoretical case of independence
Figure 2: On the left panel: the results of the empty space function F̂Yfil (red dotted line)
and nearest-neighbour function ĜXphoto,Yfil(r) (black continuous line). On the right panel:
the results of the bivariate J−function ĴXphoto,Yfil with the 0.95−confidence interval. The
theoretical case of independence is noted by the dashed dark line.
The J−function measures association between X and Y (Foxall & Baddeley 2002). If X
and Y are independent, then JX,Y = 1. Values of JX,Y close to 1 suggest independence,
values higher than 1 suggest negative association and values lower than 1 suggest positive
association.
4 Applying the bivariate J-function
In this section we will present the results obtained to analyse possible positive association
between the photometric galaxies Xphoto and filamentary spines Yfil, which locate at the
distance range of 200 − 400 Mpc from the observer. The border corrected empty space
function F̂Yfil , border corrected nearest-neighbour function ĜXphoto,Yfil(r) and the bivariate
J− function ĴXphoto,Yfil are presented.
Figure 2 left-hand panel represents the results for the border corrected estimations of
the G and F statistics. The values for the empty space function are below of the values
of the nearest neighbour function for small values of distance r. To investigate whether
positive association between the photometric galaxies and filamentary spines exists the
bivariate J−function is presented in the Fig. 2 right-hand panel.
Figure 2 right-hand panel draws the result for the bivariate J−function (eq. (5)). The
decreasing ĴXphoto,Yfil values below the theoretical reference case (which would indicate
5
independence between observed sets) represent positive association between the photo-
metric galaxies and all filamentary spines. Indicating that the photometric galaxies carry
information about the filamentary network, which has been detected from the spatial
distribution of spectroscopic galaxies.
5 Conclusion
The bivariate J−function is well fit to study possible positive association between com-
plex filamentary structures and galaxies, which can be described as a realisation of a
point process. The statistical signal of positive association indicates that the photometric
galaxies hold information about the location of the filamentary network. The result ob-
tained from the bivariate J−function encourages to use the photometric galaxies. They
would increase the number density of galaxies per volume of space, which will contribute
to a more detailed analysis of the cosmic web. A thorough analysis with the bivariate
J−function to study eventual correlation between galaxies and galaxy filaments is done
in Kruuse et al. (submitted to A & A).
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The bivariate J-function to analyse positive association between galaxies and galaxy filaments

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