Cosmic Web Reconstruction through Density Ridges: Method and Algorithm

7 MB·English

by  Yen-Chi Chen

Mon.Not.R.Astron.Soc.000,1–18(2015) Printed31August2015 (MNLATEXstylefilev2.2)
Cosmic Web Reconstruction through Density Ridges:
Method and Algorithm
Yen-Chi Chen,1,3(cid:63) Shirley Ho,2,3 Peter E. Freeman,1,3
Christopher R. Genovese,1,3 Larry Wasserman1,3
1Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
2Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
5 3McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
1
0
2 31August2015

g
u
A ABSTRACT

8 Thedetectionandcharacterizationoffilamentarystructuresinthecosmicwebal-
2 lows cosmologists to constrain parameters that dictate the evolution of the Universe.
While many filament estimators have been proposed, they generally lack estimates
]
O of uncertainty, reducing their inferential power. In this paper, we demonstrate how
one may apply the Subspace Constrained Mean Shift (SCMS) algorithm (Ozertem
C
& Erdogmus 2011, Genovese et al. 2014) to uncover filamentary structure in galaxy
h. data. The SCMS algorithm is a gradient ascent method that models filaments as
p density ridges, one-dimensional smooth curves that trace high-density regions within
- the point cloud. We also demonstrate how augmenting the SCMS algorithm with
o
bootstrap-based methods of uncertainty estimation allows one to place uncertainty
r
t bands around putative filaments. We apply the SCMS first to the dataset generated
s
fromtheVoronoimodel.Thedensityridgesshowstrongagreementwiththefilaments
a
[ fromVoronoimethod.WethenapplytheSCMSmethoddatasetssampledfromaP3M
N-bodysimulation,withgalaxynumberdensitiesconsistentwithSDSSandWFIRST-

3 AFTA, and to LOWZ and CMASS data from the Baryon Oscillation Spectroscopic
v
Survey (BOSS). To further assess the efficacy of SCMS, we compare the relative loca-
3
tions of BOSS filaments with galaxy clusters in the redMaPPer catalog, and find that
0
redMaPPer clusters are significantly closer (with p-values <10−9) to SCMS-detected
3
filaments than to randomly selected galaxies.
5
0
Key words: cosmology: observations – large-scale structure of the Universe – meth-
.
1 ods: data analysis – methods: statistical
0
5
1
:
v
1 INTRODUCTION of the universe’s evolution (Bond et al. 1996; Zhang et al.
i
X 2009,2013).Filamentsalsoinfluencetheshape,angularmo-
Observations of the local universe made over the last four
mentum,andpeculiarvelocitiesofdarkmatterhaloes(Hahn
r decades show that on megaparsec scales, matter is dis-
a etal.2007b,a;Pazetal.2008;Hahnetal.2009;Zhangetal.
tributed in web-like structures−clusters, filaments, sheets,
2009; Jones et al. 2010; Zhang et al. 2013; Forero-Romero
andvoids−thatarisenaturallyfromthenon-linearevolution
et al. 2014), as well as the intrinsic alignments and lumi-
of initially small density fluctuations (Peebles 1980; Bond
nosities of nearby galaxies (Guo et al. 2015; Clampitt et al.
etal.1996;Jenkinsetal.1998;Colbergetal.2005;Springel
2014; Codis et al. 2014).
et al. 2005; Dolag et al. 2006). Of particular interest to us
are the filaments, one-dimensional structures that connect
As the review of Cautun et al. (2014) amply demon-
galaxy clusters and form at the boundaries of empty voids.
strates, the detection of filamentary structure is a non-
Filaments are of interest for several reasons. The detection
trivial problem for which many solutions have been pro-
andcharacterizationoffilamentsatarangeofredshiftspro-
posed. These solutions include methods that examine the
vides a means by which cosmologists can constrain theories
Hessianmatrixofthegalaxydensityfield,suchastheMul-
tiscaleMorphologyFilter(MMF;Arago´n-Calvoetal.2007,
2010a) and NEXUS and NEXUS+ (Cautun et al. 2013),
(cid:63) E-mail:[email protected] as well as segmentation-based methods, such as the Candy
(cid:13)c 2015RAS
2 Yen-Chi Chen et al.
model (Stoica et al. 2007; Stoica et al. 2005), the skeleton where
(Novikovetal.2006),theSpinemethod(Arago´n-Calvoetal.
G(x)=V(x)V(x)Tg(x) (2)
2010b), andDisPerSE models (Sousbie 2011), and the path
density method (Genovese et al. 2009). While all of these is the projected gradient. The fact that ridges have pro-
methods provide estimates of filamentary structure, none jected a gradient of 0 (and second eigenvalues being nega-
provide an assessment of estimator uncertainty. The fact tive)meansthatridgesarelocalmaximumsinthesubspace
thatfilamentestimatesarerandomsetspresentsasignificant spannedbyeigenvectorsv (x),··· ,v (x).Whenpissmooth
2 d
challenge to the construction of valid uncertainty measures and the eigengap
(Molchanov 2005).
β(x)=λ (x)−λ (x) (3)
In this paper, we introduce a new method for filament 1 2
detection based on the Subspace Constrained Mean Shift
is positive, the ridges have the properties of filaments,
(SCMS)algorithmofOzertem&Erdogmus(2011).Thesta-
i.e.smoothcurve-likestructureswithhighdensity(seeFig-
tistical properties of SCMS were studied in Genovese et al.
ure1).NotethatRwillincludemodesofthedensitypwhich,
(2014). The mathematical properties of density ridges and
in the context of cosmic filament detection, means that R
the statistical consistency of SCMS are discussed in Eberly
contains both filaments and galaxy clusters. Also note that
(1996); Genovese et al. (2014), and Chen et al. (2014a), re-
density ridges are more general objects than the skeleton
spectively, while Chen et al. (2014a) introduce an uncer-
models proposed in Novikov et al. (2006); Sousbie et al.
tainty measure to the ridge formalism that allows one to
(2008) and the Spine method (Arago´n-Calvo et al. 2010b).
quantitatively assess, in the context of the current paper,
Essentially, when d = 2,3, density ridges are the same as
putative cosmic filaments.
skeletons.
In§2,wedescribetheSCMSalgorithmandthemethods
Compared with other models, density ridges adapt in-
weusetoassesstheuncertaintyofitsfilamentestimates.In
formation from both gradient and Hessian matrix of den-
§3, we apply SCMS, first to a P3M N-body simulation out-
sity. In contrast, MMF (Arago´n-Calvo et al. 2007, 2010a),
put(Tracetal.2015),andthentolow-redshift(0.235(cid:54)z(cid:54)
NEXUS and NEXUS+ (Cautun et al. 2013) only use the
0.240) and high-redshift (0.530 (cid:54)z (cid:54) 0.535) data collected
information of second derivatives (they define filaments as
by the Baryon Oscillation Spectroscopic Survey (BOSS),
the regions with λ (x) < 0 and λ (x) ≈ λ (x) > λ (x)).
2 1 2 3
whichwasreleasedaspartofSDSSDataRelease11.Wealso
DisPerSE models (Sousbie 2011) define filaments as those
demonstrate the consistency between filaments detected by
gradient flows that start from saddle points and end up at
SCMSandgalaxyclusterslistedintheredMaPPercatalog.
local maximums, which utilize only the first derivatives.
In §4 we summarize our results and offer possible avenues
Anattractivefeatureforthedensityridgemodelisthat
for future methodological development. In Appendix A we
thestatisticaltheoryforconsistentlyestimatingthedensity
provide further detail on how to optimally select values for
ridge has been well-established (Genovese et al. 2014; Chen
thetuningparametersoftheSCMSalgorithm,whileinAp-
etal.2014a,b).WealsouseN-bodysimulationtoverifythe
pendix B we apply the algorithm to labeled simulated data
convergence of density ridges when we subsample different
generatedviatheVoronoimodelof(vandeWeygaert1994)
number of galaxies (see Section 3.2).
to show that it preferentially detects structures labeled as
filaments.Inasecondpaper,wewillprovideafullcatalogue
of filaments detected in SDSS data. 2.2 SCMS: Filament Detection
The algorithm consists of three steps described below and
listed in Algorithm 1. The first is to estimate the under-
lying density function p(x) given X ,··· ,X , the observed
1 n
2 SUBSPACE CONSTRAINED MEAN SHIFT:
locations of galaxies. We use the standard kernel density
ALGORITHM
estimator (see e.g. Wasserman 2006):
2.1 Density Ridge Formalism p(x)= 1 (cid:88)n K(cid:18)(cid:107)x−Xi(cid:107)(cid:19) , (4)
Assume that we observe n galaxies with locations (cid:98) nhd h
i=1
X ,··· ,X thatared−dimensionpoints;fordatafromtyp-
1 n
where K(·) is the smoothing kernel (e.g. a Gaussian), (cid:107)x−
ical astronomical surveys, d = 2 (if the galaxies are con-
X (cid:107) is the Euclidean distance between the point x and the
strainedtoaredshiftshell)ord=3.WemodelX ,··· ,X i
1 n
ith galaxy location X , and h is the smoothing bandwidth
asrandomvariablessampledfromanunknowndensityfunc- i
(the selection of which is discussed in Appendix A).
tion p.
In the second step, we denoise by applying a threshold
Formally, a density ridge (Eberly 1996; Ozertem & Er-
totheestimateddensityfunctionp(x)toeliminatetheeffect
dogmus 2011; Genovese et al. 2014; Chen et al. 2014a,b) (cid:98)
that galaxies in low-probability density regions, i.e. where
of p is defined as follows. Let g(x) = ∇p(x) and H(x)
p(x) < τ, would have on filament estimation. How one se-
be the gradient and Hessian, respectively, of p(x) and let (cid:98)
lects τ is also discussed in Appendix A. The denoising step
v (x),··· ,v (x) be the eigenvectors of the Hessian matrix,
1 d
with associated eigenvalues λ (x) (cid:62) λ (x) (cid:62) ··· (cid:62) λ (x). is not part of the original SCMS algorithm but is impor-
1 2 d
tant to increase its statistical power in low-density regions
We define V(x) to be the matrix of all eigenvectors orthog-
(see Figure 3. We note that a thresholding step is included
onal to the first, [v (x),··· ,v (x)], and the ridge set R as
2 d
in several filament-detection algorithms, including those of
e.g. Novikov et al. (2006) and Sousbie (2011).
R≡Ridge(p)={x:G(x)=0, λ (x)<0}, (1) Forthefinalstep,givenasetofgalaxiesinhigh-density
2
(cid:13)c 2015RAS,MNRAS000,1–18
Cosmic Web Reconstruction through Density Ridges 3
(a) (b)
Figure 1.Examplesofridges(bluecurves)inasmoothfunction.
lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 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l llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Figure 2. Pictoral overview of the SCMS algorithm (Step 4 in Algorithm 1). Each point in an initially uniform mesh (the blue dots
in the top-left panel) is moved to the closest density ridge (bottom right). The top-middle, top-right, bottom-left, bottom-middle, and
bottom-rightpanelsindicatethelocationsofthemeshpointsafter1,2,4,8,and16iterationsofthealgorithm,respectively.
regions,weapplytheoriginalversionoftheSCMS(Ozertem operator to p:
(cid:98)
& Erdogmus 2011) to detect filamentary structures. Given
a point x on a defined, uniform mesh, SCMS moves it ac- R(cid:98)=Ridge(p(cid:98)). (6)
cording to an“estimated projected gradient”given by
Essentially,R(cid:98)isverysimilartothefilamentsdefinedinSous-
G(cid:98)(x)=V(cid:98)(x)V(cid:98)(x)Tg(cid:98)(x), (5) bieetal.(2008);Bondetal.(2010);Choietal.(2010).Note
thataputativefilamentis,inthecontextofthisalgorithm,
where V(cid:98)(x),g(cid:98)(x) are estimates of the quantities V(x),g(x) asetofpointsandnotaone-dimensionalcurve.InStep4of
thatwedefineaboveinSection2.1.Onemayviewthispro- Algorithm 1, We further describe how we apply SCMS. In
cedure as estimating a ridge set R by applying the Ridge Figure 2, we illustrate the application of SCMS to uniform
(cid:13)c 2015RAS,MNRAS000,1–18
4 Yen-Chi Chen et al.
Without Thresholding With Thresholding
llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
(a) Withoutthresholding (b) Thresholding
Figure3.AnexampleofthecomparisonofSCMSwithandwithoutnoiseremoval.Thisisasimplesimulateddatasetwithclutternoise.
Ascanbeseeneasily,thresholdingthedensityremovesproblemsofclutternoise.
llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllll0llllllll lllllllll.lll0l0 llll0l.ll9llllllllll0lllllllll.l lllllllll0lllll8ll l llllll0ll0llll7lll llllllll.ll.lll lllll0lll0lllllllllllllll5ll6lllllllllll l lllllllllllllllllllllllllllllllllllllll lllllll0lllll lllllll.0lllll0ll.lll30llllllllll l4llllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllll0lll.l0llllllllll2lllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll0lllllll.lll0ll llllllll0l5lllllll.l ll0lllllllllll1llllllllll lllllllllllllllllll lllllll0llllllllllll.lllll0llllllllllllllll4lllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllll0lllll lllllllll.0llllll0llllllllllllllllll.llllllllll6ll0lllllllllllll l ll5lllllllllll0lll ll lllllll.lllll0llll l0llllllll0lllllll.llllllllll7llll0lllll.lllllllllll0lll ll8lllllllllllll9ll lll lllllllll0lllll l.lllllllllllll1lllll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllll0llllllll lllllllll.lll0l0 llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0lllllllllllll.llllllll9llllllllllllllllllllllllllllll0lllllllllllllllllllllllllllll.llllllll lllllllllllllllllllllllllllllllllll0lllllllllllllllllllllll8lll lllll lllllllllllllllllllllllllllllllllllllllllllllllll0lllll0llllllllllllllllllllllllllllllllll7llllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllll.lllll.lllllllllllllllll llllllllllllllll0lllll0lllllllllllllllllllllllllllllllllllllllllllllllllllll5llll6llllllllllllllllllllllllllllllllllllllllllll lllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllll0lllllllllllllllllllllll lllllllllllllllllllllll.ll0lllllllllllllllllllllll0llllllllllllllllll.llllllllllllllllllllllll30lllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll4llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllll0llllllllllllllllllllllllllllllll.lllllllllllllllllll0lllllllllllllllllllllllllllllllllllllllllllllllllllllll2lllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllll0lllllllllllllllllllllllllllll.llllllllllllllllllll0lllllllllllllll lllllllllllllllllllll0llll5llllllllllllllllllllllllllllllll.lll llllllllll0llllllllllllllllllllllllllllllllllllllllllll1lllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllll0llllllllllllllllllllllllllllllllllllllllllllllllllll.lllllllllllllllllllllllllllll0llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll4llllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll0llllllllllllllllllllllll llllllllllllllllllllllllllll.lll0llllllllllllllllllllllllll0lllllllllllllllllllllllllllllllllll.lllllllllllllllllllllllllllll6lllll0lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll lllllllllllll5lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0llllllllllllllllllllllllllllllllllllllllllllllllllllll lllll lllllllllllllllllllll.lllllllllll0llllllllllllll llll0lllllllllllllllllll0llllllllllllllllllllllll.llllllllllllllllllllll7llllllllll0llllllllllllllllllll.llllllllllllllllllllllllllllllllllllllllllllllll0llllllllllllllllllllllllll ll8lllllllllllll9ll lll lllllllll0lllll l.lllllllllllll1lllll llllllllllllllllllllllllllllllllllllllllllllllllllllll
(a) (b) (c)
Figure 4.AnexampleoftheapplicationofSCMS.(a)Theoriginaldata.(b)Contourplotshowingthekerneldensityestimateofthe
densityp.(c)Theridgeestimate(bluecurve).Notethatin(b),weremovepointswheretheestimateddensityislessthanathresholdτ.
mesh of points, and in Figure 3 we demonstrate the impor- asafunctionoftheunknowndensityfieldpandtheunknown
tanceofthethresholdingstep:theleftandrightpanelsshow filament set R, by utilizing bootstrap resampling.
putative filaments detected without and with thresholding, In this paper, we consider both the original version
respectively.Weobservethatthresholdinggreatlydecreases of bootstrap (Efron 1979) and the smooth bootstrap. The
the rate of false filament detection. smoothbootstrap(seee.g.Silverman&Young1987)isavari-
ant of the bootstrap that is useful in functional estimation
problems in which the bootstrap sample is drawn from the
2.3 SCMS: Filament Uncertainty Estimation
estimated density p instead of the original data. When the
(cid:98)
We quantify the uncertainty in the filament estimates pro- smoothing kernel is a bivariate Gaussian, we generate the
ducedbySCMSusingtheconceptoflocaluncertainty (Chen smooth bootstrap sample via the following two steps:
etal.2014a).Thelocaluncertaintyinanestimatedfilament
1. Generate the bootstrap sample.
R(cid:98) at a point x on the true filament R is the expected dis-
2. Add independent and identically distributed Gaus-
tance between x and the closest point to x on R(cid:98). This is sian noise with variance h2.
denoted by ρ(x) and is given by:
(cid:114) (cid:104) (cid:105) Unlike the bootstrap, the smooth bootstrap takes into ac-
 E d2(x) if x∈R countboththevarianceandthebiasoffilamentestimation,
ρ(x)= R(cid:98) , (9)
0 otherwise but with less precision in variance estimation with respect
to the bootstrap.
where d (x)=min{(cid:107)x−y(cid:107):y ∈R(cid:98)} and the notation E[·] Assume we generate B bootstrap samples, and each
denotesR(cid:98)the expected value operator. ρ(x) is the radius of a of them is denoted as {X1∗(b),··· ,Xn∗(b)}, b = 1,··· ,B.
local confidence ball that surrounds the point x: the more For each bootstrap sample, say X1∗(b),··· ,Xn∗(b), we com-
uncertain the true location of the estimated filament, the pute the density estimate p(cid:98)∗(b), the ridge estimate R(cid:98)∗(b) =
larger the value of ρ(x). We estimate ρ(x), which is defined Ridge(p∗(b)), and the confidence ball radii ρ (x) for all
(cid:98) (b)
(cid:13)c 2015RAS,MNRAS000,1–18
Cosmic Web Reconstruction through Density Ridges 5
Algorithm 1 SCMS (Subspace Constrained Mean Shift)
Input: Data {X ,··· ,X }. Smoothing bandwidth h. Threshold τ.
1 n
Step 1. Compute the density estimator p(x) via equation (4).
(cid:98)
Step 2. Select a mesh M of points. By default, we can take M={X ,··· ,X }.
1 n
Step 3. Thresholding: remove m∈M if p(m)<τ. Let the remaining mesh points be denoted M(cid:48).
(cid:98)
Step 4. For each x∈M(cid:48), perform the following subspace constrained mean shift until convergence:
Step 4-1. For i=1,··· ,n, compute
(cid:18) (cid:19)
x−X x−X
µ = i, c =K i
i h2 i h
Step 4-2. Compute the Hessian matrix
n (cid:18) (cid:19)
H(x)= 1 (cid:88)c µ µT − 1 I . (7)
n i i i h2
i=1
Step 4-3. PerformspectraldecompositiononH(x)andcomputeV(x)=(v (x),··· ,v (x)),theeigenvectorscorrespond-
2 d
ing to the smallest d−1 eigenvalues.
Step 4-4. Update x←−V(x)V(x)Tm(x)+x until convergence, where
(cid:80)n c X
m(x)= (cid:80)i=n1 ic i −x (8)
i=1 i
is called the mean shift vector.
Output: The collection of all remaining points.
(a) Localuncertainty. (b) Uncertaintyband.
Figure 5. An illustration of the uncertainty measure for SCMS. In (a), we display the uncertainty measures with different color (red:
highlyuncertain).TheunittothecoloristhesameasxandY axis.In(b),weshowtheuncertaintymeasuresbyagrayregionaround
thefilament(blue).NotethatthisshowsthattheSCMShasmoreuncertaintymeasuresaroundthehighlycurvedregionsandtheend
points.
x∈R(cid:98). We estimate ρ(x) by adding the B radius estimates taintymeasureisafeaturethatSCMSfilamentshave.This
in quadrature: measure has a geometric interpretation and can be consis-
tently estimated. See Chen et al. (2014a) for more involved
(cid:118)
(cid:117)(cid:117)1 (cid:88)B discussion. Note that other filament finders do have have
ρ(cid:98)(x)=(cid:116)B ρ2(b)(x), (10) such a statistically consistent error measurement.
b=1
InAlgorithm2weoutlinethecomputationalstepsthatone
2.4 SCMS: Boundary Bias
must follow to derive ρ(x).
(cid:98)
Note that calculating the uncertainty measure is not Whencomputedwithakerneldensityestimatorasinequa-
part to the SCMS algorithm–we can detect filaments with- tion(4),SCMSfilamentestimatessufferfromboundarybias
out using the uncertainty measure. However, this uncer- within ∼ two bandwidths of the edge of the observation
(cid:13)c 2015RAS,MNRAS000,1–18
6 Yen-Chi Chen et al.
Algorithm 2 Uncertainty Measure for SCMS
Input: Data {X ,··· ,X }. Smoothing bandwidth h. Threshold τ.
1 n
Step 1. Perform SCMS on {X1,··· ,Xn} to detect filaments; denote the estimated filaments by R(cid:98).
Step 2. Generate B bootstrap samples: X∗(b),··· ,X∗(b) for b=1,··· ,B.
1 n
Step 3. For each bootstrap sample, apply SCMS which yields R(cid:98)∗(b) for b=1,··· ,B.
Step 4. For each x∈R(cid:98), calculate ρ2 (x)=d2(x,R(cid:98)∗(b)), b=1,··· ,B.
(b)
Step 5. Compute ρ(x)=(cid:2)mean{ρ2(x),··· ,ρ2(x)}(cid:3)1/2.
(cid:98) 1 B
Output: ρ(x).
(cid:98)
region. This is a systematic deviation from the true fila- Combining the uncertainty measures and the coverage,
ment caused by the density estimator averaging over a re- wecanstudythepropertiesoftheuncertaintyband..Anun-
gion where no data can be observed, and it can degrade certaintybandforadetectedfilamentissimplytheunionof
the confidence band coverage probabilities near the bound- theconfidenceballscomputedforeachpointonthefilament,
ary. One remedy for boundary bias is to include additional i.e.
data immediately outside of the region of interest. Includ- (cid:91)
ing galaxies within 2h of the boundaries eliminates most U(cid:98)(k)=R(cid:98)⊕kρ(cid:98)≡ B(x,kρ(cid:98)(x)), (14)
of the boundary bias, since very little of the volume under x∈R(cid:98)
a bivariate Gaussian kernel lies beyond that point. If one where B(x,r) = {y : (cid:107)x−y(cid:107) (cid:54) r} represents the set of
cannot include additional data points outside the bound- pointswithinaballcenteredatxandwithradiusr.Denote
aries (for instance, due to overall survey limits), then one the region within the uncertainty band as A. The coverage
must be careful when interpreting filaments detected near for A is then
the boundaries. FCov(A)=Cov (A)
Rtrue
Number of points in (A∩R ) (15)
= true .
2.5 Filament Coverage Number of points in Rtrue
One can think of FCov(A) as the true positive coverage us-
Here we introduce some useful geometric concepts about
ingasetA.Forinstance,ifFCov(A)=0.8,thenonaverage,
coverage. Given two sets A and B. The coverage of B by
80% of the points on any given true filament lie within its
A is defined as
associated uncertainty band, and 20% lie outside the band.
Number of points in (A∩B)
Cov (A)= . (11) Thisinterpretationofcoveragediffersfromthestandardin-
B Number of points in B
terpretation of confidence band coverage, thus motivating
NotethatwhenAandBarecurves,theywillcontaininfinite our use of the term “uncertainty band” instead of “confi-
number of points. In this case, we will replace ‘number of dence band.”Figure 10 gives examples of the coverage for
points in’ by ‘the length of’. Similarly, we can define the uncertainty bands U(cid:98)(k) with k ∈ (0,3) and n = 250 and
coverage of A by B as Cov (B). 2500. As we observe in Figure 10, the coverage percentage
A
Given two collections of filaments R and R , since R depends sensitively on the sample size n; thus, we cannot
1 2 1
andR arecurvessothattheymaynotintersecteachother providesimplerulesforconvertingkσ uncertaintybandsto
2
ingeneralsothatthecoverageis0.Thus,insteadofdirectly coverage percentages.
compute their coverage, we consider a flatten version of R
1
(and R respectively). We define
2
3 SUBSPACE CONSTRAINED MEAN SHIFT:
R ⊕r={x:d(x,R )(cid:54)r} (12)
1 1 APPLICATIONS
asther-flattensetofR .ThenwedefinethecoverageofR
1 2 3.1 Voronoi Dataset
by R as a function of r as
1
Toshowtheeffectivenessofcapturingfilaments,wecompare
Number of points in (R ∩(R ⊕r))
Cov (r;R )= 2 1 . (13) the SCMS filaments (density ridges) to the filaments in the
R2 1 Number of points in R
2 Voronoimodel.TheVoronoimodel(vandeWeygaert1994)
Similarly, we can define Cov (r;R ). The two functions applies Voronoi tessellation to compute a density estimate
R1 2
Cov (r;R ) and Cov (r;R ) contain information about forgalaxiesaswellasthecurvatureofthatestimate.Givena
R1 2 R2 1
the similarity between R and R . curvatureestimate,theVoronoimethodassignsaclasslabel
1 2
In simulation, we are able to define true filaments, say to each galaxy, indicating the type of large-scale structure
Rtrue,andwewillhaveanestimatefilament,denotedasR(cid:98)n. to which to associate the galaxy. There are four possible
Then we call the quantity CovRtrue(r;R(cid:98)n) the true positive classes: cluster, filament, wall, and void.
coverage (ratiooftruefilamentsbeingcoveredbyestimated We use the SCMS algorithm to analyze a simulated
filaments)andwecall1−Cov (r;R )thefalse positive dataset (2563 galaxies, each with a class label, that span a
coverage (Cov (r;R ) is thR(cid:98)enratiotroufeestimated filament 100×100×100Mpc3box)generatedwiththeVoronoimodel
R(cid:98)n true
beingcoveredbytruthsothat1minusthisratioistheratio (M. A. Arago´n-Calvo, private communication). Figure 6
offalsepositive).SeeFigure9foranexampleoftruepositive showsacomparisonbetweenourdensityridges(bluecurves)
and false positive coverage. andgalaxieswithdifferentclasslabels(brown:cluster;red:
(cid:13)c 2015RAS,MNRAS000,1–18
Cosmic Web Reconstruction through Density Ridges 7
filament;green:wall;pink:void).Thetwomethodsgener- filamentcoverageis,asnotedabove,sensitivetothesample
ateremarkablysimilarresults:Voronoiclusters(i.e.galaxies sizenandthatthesmoothbootstrapprovidesconsiderably
labeled cluster) occur at the intersection points of density more conservative confidence bands, particularly for k (cid:46)2.
ridges; Voronoi filaments surround the density ridges; and The gray regions displayed in Figure 10(b) are the smooth
Voronoiwallsspansurfacesonwhichthedensityridgeslay. bootstrap confidence regions U(cid:98)(1), which we estimate con-
To further quantify the association between density tain 85% of the true filaments (cyan curves).
ridges and each Voronoi model class, we study their pro- Figure 11 illustrates the effect of boundary bias in the
jectiondistancesontoeachother.Notethatthedistribution n=2500subsamplebycomparingtheestimatesanduncer-
of projection distances is related to filament coverage; fur- tainties with padded and unpadded data near the bound-
ther discussion of this may be found in Chen et al. (2015). ary. Panels (a) and (b) show the boundary bias. Note that
Figure 7 displays the distributions of projection distances. the red curves are filaments estimated by using only points
In both panels, we see that the distribution for ridges ver- within the boundary (given by the orange rectangle). The
sus the Voronoi filaments peaks at distances (cid:46) 1 h−1Mpc bluecurvesarefilamentsdetectedbySCMSwithboundary
This indicates that the density ridges and the Voronoi fila- points (i.e. points outside the orange rectangle). As can be
ments are very similar. On the other hand, the projection seen,theestimationoffilamentswithoutboundarydata(red
distances from the density ridges increases as we consider curves)becomesmoreinaccurateasweapproachthebound-
clusters, walls, and voids; the distributions exhibit increas- ary. The boundary bias occurs for filaments with distances
ing positive skewness. lessthan10Mpc/h(2timessmoothingparameterh)tothe
boundaries. The uncertainty measures also show the influ-
ence of boundary bias. Figure 11(c) and 11(d) exhibit the
3.2 P3M N-body Simulation
uncertaintymeasuresforfilamentsestimatedwithandwith-
To further demonstrate the efficacy of SCMS, we apply it outboundarypoints.Asexpected,theuncertaintymeasures
to P3M N-body simulations from Trac et al. (2015), which in panel (d) increase as we move close to the boundary.
assume a ΛCDM cosmology with Ω =0.3, Ω =0.7, Ω =
m l b
0.045, h = 0.7, σ = 0.8 and n = 0.96. Each side of the
8 s
simulationboxisoflength1Gpc/h,andeachcontains20483
3.3 Sloan Digital Sky Survey
particles.
In Figure 8, we demonstrate that, as sample size in- 3.3.1 Data
creases,SCMSoutputsfilamentestimatesthatarecloserto
the true filaments (defined by the true density function); We further demonstrate the efficacy of SCMS by applying
the uncertainty measures capture SCMS errors due to the it to data from Data Release 12 (Alam et al. 2015) of the
sampling variability. We take a slice of the full simulation SloanDigitalSkySurvey(SDSS;Yorketal.2000).Together,
data(x,y∈[125,375]Mpc/handz∈[100,105]Mpc/h)and SDSSI,II(Abazajianetal.2009),andIII(Eisensteinetal.
smooth the data with smoothing bandwidth h=5 (recom- 2011)usedadrift-scanningmosaicCCDcamera(Gunnetal.
mendedbytheselectionruleinAppendixAwithA =0.4) 1998) to image over one third of the sky (14,555 square de-
0
to get the density function and the filaments (cyan curves). grees)infivephotometricbandpasses(Fukugitaetal.1996;
Figure 8(a) shows a contour plot for the density function. Smithetal.2002;Doietal.2010)toalimitingmagnitudeof
The original sliced data contains 88,406 points (gray dots). r(cid:39)22.5, using the dedicated 2.5-m Sloan Telescope (Gunn
Wedownsampletogetthreedifferentsubsamples;eachcon- et al. 2006) located at Apache Point Observatory in New
tains 250/2500/10000 particles. For each subsample (black Mexico. The imaging data were processed through a series
dots), we apply SCMS to detect filaments (blue curves). ofpipelinesthatperformastrometriccalibration(Pieretal.
Note that the convergence phenomena of Figure 8 are fur- 2003),photometricreduction(Luptonetal.2001),andpho-
therquantifiedbythetruepositiveandfalsepositivecover- tometric calibration (Padmanabhan et al. 2008). All of the
age plot in Figure 9. imaging was reprocessed as part of SDSS Data Release 8
Notethatthesparsestsubsamplen=250hasagalaxy (2011ApJS..193...29A; Aihara et al. 2011).
number density 5.56×10−4 Mpc−3 which is similar to the The Baryon Oscillation Spectroscopic Survey (BOSS)
numberdensityobservedinSDSSCMASSdata(∼4×10−4 has obtained spectra and redshifts for 1.35 million galaxies
Mpc−3).ThefuturesurveyWide-FieldInfraredSurveyTele- overafootprintcovering10,000squaredegrees.Thesegalax-
scope (WFIRST),1 a NASA mission with science objectives iesareselectedfromtheSDSS2011ApJS..193...29Aimaging
in exoplanet exploration, dark energy research and galac- and are being observed together with 160,000 quasars and
ticandextragalacticsurveys,willobserveanumberdensity approximately100,000ancillarytargets.Thetargetsareas-
similar to the n=2500 subsample (∼5.56×10−3 Mpc−3). signed to tiles of diameter 3◦ using a 2003AJ....125.2276B
We show the uncertainty measures and filament cover- algorithm that is adaptive to the density of targets on the
ageforn=2500inFigure10.Weplotfilamentcoveragefor sky (Blanton et al. 2003). Spectra are obtained using the
confidenceregionsU(cid:98)(k)fork∈(0,3)inFigure10(a),where double-armedBOSSspectrographs(Smeeetal.2013).Each
n = 250 and 2500, and where ρ is estimated by both the observation is performed in a series of 900-second expo-
(cid:98)
bootstrap(BT)andthesmoothbootstrap(SB).Thisrange sures, integrating until a minimum signal-to-noise ratio is
contains sample sizes that are in line with both CMASS achieved for the faint galaxy targets. This ensures a homo-
(n≈ 250) and WFIRST (n≈ 2500) data. We observe that geneous data set with a high redshift completeness of more
than97percentoverthefullsurveyfootprint.Redshiftsare
extracted from the spectra using the methods described in
1 http://wfirst.gsfc.nasa.gov/ Bolton et al. (2012). A summary of the survey design ap-
(cid:13)c 2015RAS,MNRAS000,1–18
8 Yen-Chi Chen et al.
(a) Allgalaxies (b) Galaxieswithlabel=clusters (c) Galaxieswithlabel=filaments
(d) Galaxieswithlabel=walls (e) Galaxieswithlabel=voids
Figure 6. AcomparisonbetweendensityridgesandVoronoimodel.Ineachpanel,thebluecurvesaredensityridgesusingallgalaxies.
Panel(b)-(e)displaythecomparisonofdensityridgestotheVoronoiclusters,filaments,wallsandvoids.Inpanel(c),weseearemarkable
similaritybetweendensityridgesandtheVoronoifilaments.
4 Clusters (Voronoi) cover Ridges 1.4 Ridges cover Clusters (Voronoi)
Filaments (Voronoi) cover Ridges Ridges cover Filaments (Voronoi)
l Walls (Voronoi) cover Ridges Ridges cover Walls (Voronoi)
l Voids (Voronoi) cover Ridges 1.2 l l Ridges cover Voids (Voronoi)
l
3 1.0 l
n n l l
Distributio012 llll lll lll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll llll Distributio0.00.20.40.60.8 llll llll lll lll lll lll lll lll llll llll llll llll llll llll llll llll llll llll llll llll
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
Distance (Mpc/h) Distance (Mpc/h)
Figure7. ThedistributionsforprojectiondistancesfromVoronoi-model-derivedstructuresontodensityridges(leftpanel)andvice-versa
(rightpanel).BothpanelsindicatethatdensityridgestracestructuresmostsimilartoVoronoifilaments.
pears in Eisenstein et al. (2011), and a full description is yieldsasamplewithamedianredshiftz=0.57andastellar
provided in Dawson et al. (2013). mass that peaks at log (M/M ) = 11.3 (Maraston et al.
10 (cid:12)
2013). Most CMASS targets are central galaxies residing in
BOSS selects two classes of galaxies to be targeted
dark matter haloes of mass ∼1013h−1M .
for spectroscopy using SDSS 2011ApJS..193...29A imaging: (cid:12)
WetestSCMSusingtwoslicesofdata:atlowandhigh
‘LOWZ’ and ‘CMASS’ (we refer the reader to Anderson
redshift. The low-z dataset comprises 1,158 galaxies in the
etal.(2014)forfurtherdescriptionoftheseclasses).Forthe
volume
LOWZ sample, the effective redshift is z = 0.32, slightly
eff
lowerthanthatoftheSDSS-IIluminousredgalaxies(LRGs)
135◦ (cid:54)RA(cid:54)175◦ 5◦ (cid:54)δ(cid:54)45◦ 0.235(cid:54)z(cid:54)0.240
as we place a redshift cut z < 0.43. The CMASS selection
(cid:13)c 2015RAS,MNRAS000,1–18
Cosmic Web Reconstruction through Density Ridges 9
(a) Densityfieldandtruefilaments (b) n=250(CMASS)
(c) n=2500(WFIRST) (d) n=10000
Figure 8. A simulated example to show the consistency of SCMS. This data is a slice of an N-body simulation in a box; the unit of
X and Y axes is Mpc/h. We take a slice with width 5 Mpc/h. The original sample contains 88,406 particles. The color contour is the
galaxy density field from the original sample with smoothing parameter h=5 and the true filaments (cyan) are density ridges of this
densityfield.Wesubsampleundervarioussizes.Thebluecurvesareestimatedfilamentsbasedonthesubsample(blackdots).Onecan
seeaclearpattern;assamplesize(forthesubsample)increases,theestimatedfilamentsareclosertothetruefilaments.SeeSection3.2
formoredetails.
while the high-z dataset lies in the volume second reason is that when z > 0.2, the number density is
very low, and performing a three-dimensional analysis will
135◦ (cid:54)RA(cid:54)175◦ 5◦ (cid:54)δ(cid:54)45◦ 0.530(cid:54)z(cid:54)0.535 produceresultswithlargestatisticalerrorsduetothesmall
samplesize.Lower-dimensionalanalysesresultindecreased
and contains 4,678 galaxies. Both samples have a very thin
statistical error; see e.g. Wasserman 2006.
redshiftrange∆z=0.005(thecorrespondingcomovingdis-
tanceisaround14−21Mpc)sothattheirconstituentgalax-
iesmaybeviewedaslyingonatwo-dimensionalsurfacewith
3.3.2 Results
coordinates (RA,δ).
There are two principal reasons for our choice to per- We apply SCMS to the low-z data with smoothing band-
formatwo-dimensionalanalysisoftheSDSSdata.Thefirst width h = 2.50◦ (41.8 Mpc) and threshold level τ =
is that there is too large a change in the number density of 1.02×10−3;wedisplayourresultsinFigure12.Forthehigh-
detected galaxies over the SDSS redshift range. The SCMS z data, h and τ are 2.03◦ (71.1 Mpc) and τ =7.52×10−4,
algorithm incorporates kernel density estimation to locate respectively; we display our results in Figure 13. Note that
densityridges,andKDErequiresafixedsmoothingparam- wehaveincludedadditionalgalaxieswithin5degreesofour
eterh.However,inlow-densityregions,hshouldbelargeto selected window to mitigate boundary bias.
obtain reliable results, while in high-density regions, h has As can be seen in Figures 12 and 13, SCMS filament
to be small so as to not oversmooth the point cloud. The estimates capture high density regions and they exhibit
(cid:13)c 2015RAS,MNRAS000,1–18
10 Yen-Chi Chen et al.
(a) Truepositivecoverage. (b) Falsepositivecoverage
Figure 9. True positive and false positive coverage. As can be seen, for all distances, the true positive coverage increases with sample
size,whereasthefalsepositivecoveragedecreases.
(a) Filamentcoverage. (b) N =2500,smoothbootstrap
Figure 10.Filamentcoveragebasedontheuncertaintymeasure.(a)ThefilamentcoverageFCov(U(cid:98)(k))asafunctionofk(x-axis).We
alsoprovidethecoverageforGaussiandistribution(probabilitybeingwithinkσtothecenterofGaussian)asareference.(b)Visualizing
theuncertaintybycolorandaconfidencesetforthesubsamplewithn=2,500withtheuncertaintymeasureestimatedviathesmooth
bootstrap.Thecyancurvesarethetruefilaments.NotethatthegrayregionsareU(cid:98)(k),k=1,equivalenttotheerrorbarfor1×σ,based
onthesmoothbootstrapestimate.Frompanel(a),weknowthatthegrayregionscontainabout85%truefilaments(cyancurves).The
unittothecolorinuncertaintybandisMpc,thesameasX andY axes.
one-dimensional, nearly connected structures. In addition, galaxies.Aswedemonstratebelow,theseputativefilaments
SCMSyieldssmoothfilaments;mostfilamentestimatorsdo have higher estimates of uncertainty.
not output such smooth structures (cf. Stoica et al. 2007;
We derive the uncertainties for the filament estimators
Stoicaetal.2005;Sousbie2011;Aanjaneyaetal.2012;Lecci
as described in Section 2.3 from the two test datasets; the
et al. 2013). We note that the filaments detected by SCMS
resultsforlow-z andhigh-z samplesaregiveninFigures14
will not actually connect with each other; points on merg-
and 15, respectively. We visualize local uncertainty using
ing filaments have eigengap β (equation 3) that asymptote
color, where red indicates locations where the filamentary
toward0,makingthedensityridgeill-definedsincethefirst
structure is highly uncertain. We also display uncertainty
andsecondeigenvaluesbecomeequal.Wenotethatinboth
regions as bands of varying width (shown in gray) centered
figurestherearepossiblyspuriousfilaments;forinstance,in
on the filaments. Our simulation study in Section 3.2 indi-
Figure12,at(RA,δ)=(165◦,40◦)and(165◦,20◦),weseefil-
catesthatthefilamentcoverageFCovfortheregionsinFig-
amentsthatareassociatedwitharelativelysmallnumberof
ures 14(a) and 15(a) is ≈ 45%, while that in Figures 14(b)
and 15(b), is ≈60%. We find that the overall structure for
(cid:13)c 2015RAS,MNRAS000,1–18