A&A 561, A57 (2014)
DOI: 10.1051/0004-6361/201321143
c ESO 2013
Astronomy
&
Astrophysics
UPMASK: unsupervised photometric membership assignment
in stellar clusters
A. Krone-Martins and A. Moitinho
SIM – Faculdade de Ciências da Universidade de Lisboa, Ed. C8, Campo Grande, 1749-016 Lisboa, Portugal
e-mail: algol@sim.ul.pt
Received 21 January 2013 / Accepted 7 September 2013
ABSTRACT
Aims. We develop a method for membership assignment in stellar clusters using only photometry and positions. The method is aimed
to be unsupervised, data driven, model free, and to rely on as few assumptions as possible.
Methods. The approach followed in this work for membership assessment is based on an iterative process, principal component
analysis, clustering algorithm, and kernel density estimations. The method, UPMASK, is able to take into account arbitrary error
models. An implementation in R was tested on simulated clusters that covered a broad range of ages, masses, distances, reddenings,
and also on real data of cluster fields.
Results. Running UPMASK on simulations showed that the method effectively separates cluster and field populations. The overall
spatial structure and distribution of cluster member stars in the colour-magnitude diagram were recovered under a broad variety
of conditions. For a set of 360 simulations, the resulting true positive rates (a measurement of purity) and member recovery rates
(a measurement of completeness) at the 90% membership probability level reached high values for a range of open cluster ages
(107.1
−109.5
yr), initial masses (0.5−10 × 103
M ) and heliocentric distances (0.5−4.0 kpc). UPMASK was also tested on real data
from the fields of open cluster Haffner 16 and of the closely projected clusters Haffner 10 and Czernik 29. These tests showed that even
for moderate variable extinction and cluster superposition, the method yielded useful cluster membership probabilities and provided
some insight into their stellar contents. The UPMASK implementation will be available at the CRAN archive.
Key words. open clusters and associations: general – open clusters and associations: individual: Haner 10 – methods: data analysis –
methods: statistical – open clusters and associations: individual: Haner 16 – open clusters and associations: individual: Czernik 29
1. Introduction
Astronomical research using observations of star clusters often
meets a fundamental challenge: being able to recognise and
make use of the signature of cluster members in data sets that
are heavily contaminated by field stars. The signature can be of
different kinds. It can be an apparent over-density of stars in a
stellar field image or map, a cluster sequence in photometric diagrams,
clustering of stars with common kinematics in proper
motion, or radial velocity space, or similar parallaxes, among
others.
Ideally, given the constrained spatial origin of stars in a
cluster, membership would be determined through a threedimensional
selection of stars at similar distances, at least for
young clusters not too dynamically evolved or for the concentrated
members of older clusters. In practice, accurate parallactic
distances are not known except for the closest stars (Perryman
& ESA 1997) and indirect distance indicators (spectroscopic,
photometric, among others) will not be accurate enough to
distinguish members based on distances alone, except in very
favourable cases such as high latitude fields with little field
contamination.
After distances, proper motions are considered to be the
best member discriminator. The canonical Vasilevskis-Sanders
method (Vasilevskis et al. 1958; Sanders 1971) and variations
have been widely used in the literature for determinations of
cluster memberships and kinematics (Kharchenko et al. 2004;
Dias et al. 2006; Krone-Martins et al. 2010). These methods
assume bivariate Gaussian distributions for cluster and field
motions. It has been argued that assumptions on the distribution
might not be adequate in some situations, which has led other authors
to develop non-parametric approaches (e.g. Cabrera-Caño
& Alfaro 1990; Balaguer-Núñez et al. 2004; Javakhishvili et al.
2006, and references within). However, independently of the
details of each method, precise proper motions are required,
which are currently constrained to relatively short heliocentric
distances, since relatively bright stars (V ∼ 15–16) and measurable
angular motions are required. Moreover, in any case these
methods will not be able to distinguish cluster members in directions
dominated by the solar motion.
Spectroscopic radial velocities can also be used, alone or
combined with proper motions, for assessing cluster memberships.
Combined with photometry, spectroscopy provides a powerful
means for selecting cluster members since it allows one to
precisely pinpoint each spectral type in photometric diagrams,
determine individual reddenings and use reddening as a member
discriminator. But spectroscopy is a telescope-time-intensive approach,
which limits its use in characterising large numbers of
clusters. However, recent and planned large spectroscopic surveys
of the Milky Way such as the SDSS SEGUE and APOGEE
(Yanny et al. 2009; Majewski et al. 2007), RAVE (Steinmetz
et al. 2006), Gaia-ESO (Gilmore et al. 2012), and LAMOSTLEGUE
(Deng et al. 2012) will allow membership determinations
for some hundreds of clusters. Still, as in the case of proper
motions, stringent limits are imposed by the required brightness
of stars and solar motion.
Multi-band imaging (and the resulting photometric catalogues)
is by far the most wide-spread technique for studying
Article published by EDP Sciences A57, page 1 of 12
A&A 561, A57 (2014)
star clusters. Imaging allows obtaining data for large numbers of
stars over wide areas and reaches depths not accessible to other
techniques, or at least not accessible to other techniques without
a large investment of observing time. However, the rich numbers
of stars brought by imaging surveys also bring the problem of
strong contamination by field stars.
The extent to which cluster members need to be identified
is driven by the scientific question being addressed. In some
cases, a simple perception of the cluster presence can be enough.
These are the cases of determining cluster centres and radii from
star counts and classic isochrone eye-fitting of prominent cluster
sequences in photometric diagrams. Cluster luminosity and
mass functions can also be determined without knowledge of the
specific cluster members through statistical comparisons with
control fields (e.g. Moitinho et al. 1997, and many others).
However, under heavy field contamination visual techniques
break down since the eye no longer unambiguously identifies the
cluster signature. Determination of cluster reddenings, distances,
and ages by classical methods becomes almost a guess. Similarly
with simple statistical approaches: the cluster over-density (in
physical or magnitude space) becomes marginal when compared
with the fluctuations in the number of field stars, yielding uncertain,
or even useless, radii and luminosity and mass functions.
Additionally, recognising the sequence is not enough in
many studies and knowledge of individual memberships are required.
Examples range from hunting brown dwarfs and other
research on specific stellar types, to separating members from
clusters superimposed along their lines of sight, just to mention
a few. This is also the case of determining cluster parameters
through automated fitting of zero-age main-sequence (ZAMS)
or isochrone models to photometric sequences. It has long been
recognised that objective and precise model-fitting procedures
(as opposed to eye fitting) yielding repeatable results and rigorously
assessed uncertainties should be employed. However, only
recently have such methods began to appear more frequently
in the literature. The trigger has been the profusion of cluster
photometric data and the need to compare results from different
studies in a systematic way, together with common availability
of the necessary computing power. Examples of these approaches
are the τ2
method by Naylor & Jeffries (2006) and the
Bayesian inversion method by von Hippel et al. (2006), which
require samples of stars only composed of cluster members or
with very little contamination. Monteiro et al. (2010) adopted a
cross-entropy method for finding the best isochrone fit, but without
requiring a pure sample of cluster stars. Instead, they lessened
the effect of field contamination in the fits by adopting a
filtering and weighting scheme based on the distribution of stars
in coordinate space and multi-dimensional magnitude space –
this method was subsequently refined in Dias et al. (2012).
Another problem closely related to photometric cluster
membership determinations is the determination of individual
stellar parameters from photometry, including reddening and
stellar classification. Reddening can provide a criterion for membership
determinations. For clusters unaffected by variable reddening,
member stars appear reddened by the same amount,
which will in general be different for background and foreground
stars. This is specially applicable to the majority of open clusters,
which are located at low Galactic latitudes where integrated
extinction along the line of sight varies significantly with distance.
Individual stellar classification together with photometric
measurements allow a rough distance estimation and thus cleaning
the sample by selecting stars within a range of distances.
These stellar parameter determinations of cluster members are
in turn linked to the problem of determining cluster parameters
(reddening, distance, age, metallicity) by fitting isochrones to
photometric cluster sequences. If this can be done, the values for
all the member stars become determined, assuming they share
these properties. Both problems have a noticeable aspect in common:
the solutions are model dependent, relying on the adoption
of reference lines (isochrones, ZAMS) and reddening law. They
may also be sensitive to mismatches between the passbands used
in the observations and the ones assumed in the models.
The motivation of this paper is the identification of star cluster
members, buried in contaminated fields, in a purely datadriven
way without relying on models. The paper focuses on
the special case of membership determinations using exclusively
multi-band photometry and projected on-sky positions.
However, the method developed here – UPMASK – can be immediately
applied to other types of data, including spectrophotometry
(as produced by the forthcoming Gaia mission), radial
velocities, and proper motions. In principle, and as discussed in
more detail in Sect. 5, the method can be perfected in a way such
that it accommodates missing data and other measurement errors
distributions. This will be addressed in future work.
The remainder of this paper is organised as follows.
Section 2 describes the UPMASK method. Sections 3 and 4 report
on the results of UPMASK using simulated and real data,
respectively. Limitations and possible extensions of the method
are discussed in Sect. 5, followed by the conclusions of this work
in Sect. 61
.
2. The UPMASK method
2.1. Conceptual approach
Owing to their common origin and spatial confinement, members
of stellar clusters share common properties. Statistically,
this means that in most parameter spaces star cluster members
are expected to be clustered together. The clustering region,
however, can assume arbitrary formats such as tubular regions,
or even be disconnected. Field stars can also be clustered in certain
parameter spaces but are not expected to cluster in most (e.g.
the space defined by reddening, distance, and age).
In this paper, we define a stellar cluster as a spatial overdensity
of stars with a common origin2
. Conversely, field stars
are expected to be spatially scattered as they do not share a single
origin. According to this definition, random subsets of stellar
cluster members are spatially concentrated. This is not necessarily
the case with the measured photometric indexes. For example,
arbitrary subsets of cluster members are not necessarily
concentrated around a common colour index. In this view, the
positional space has a favoured place in the UPMASK method.
These are the main assumptions on which the UPMASK
method relies: cluster members will be clustered in most spaces,
including positional space. Field stars, even if clustered in some
spaces, are not expected to cluster in positional space.
Because of their position in the Galaxy, star clusters are affected
by tidal interactions that can reshape their spatial density
profiles, deviating from those of self-gravitating stellar systems,
such as the King profile. Also, although the morphologies of star
1
This article addresses star clusters as well as clusters of data (as
identified by clustering algorithms). For economy and readability, the
word cluster will be used to describe either whenever the context is
unambiguous.
2
Different schools have different definitions for star clusters. Another
popular and more restrictive definition requires clusters to be gravitationally
bound.
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A. Krone-Martins and A. Moitinho: UPMASK: unsupervised photometric membership assignment in stellar clusters
cluster colour−magnitude diagrams (CMDs) are fairly well described
by current stellar evolution models, not all evolutionary
stages, mass ranges, and even colour-magnitude combinations
are equally well described (e.g. Lyra et al. 2006; Soderblom
2010). In particular, there is still much to be improved for
younger clusters with pre-main sequence stars, especially for the
less massive stars. Finally, one should not exclude the possibility
that the data could show the unexpected, such as clusters with
more than one epoch of star formation, which could be particularly
relevant for globular clusters, merged clusters, or clusters
projected on the same line of sight.
Therefore, another central point in the design of the method
is that it should be data-driven. Apart from the spatial uniformity
of field members, UPMASK does not assume a priori parametrisations,
such as isochrones or King profiles, of any probability
distributions involved.
2.2. Dimensional transformation
Although UPMASK is a data-driven approach, it is not the data,
but the information they encode that drives the membership assignment.
However, several observable spaces, and particularly
multi-magnitude and multi-colour spaces, exhibit redundant information.
This is manifested through correlations between the
observables.
Ideally, membership assessment would be performed in
spaces of physically uncorrelated variables. In these spaces, the
distances between the observables representing these variables
are maximised. By eliminating correlations between observables,
and thus redundancy, these spaces have the smallest number
of parameters encoding (most of) the information content of
the original observables.
A common approach for building such spaces is provided by
the principal component analysis (PCA). This is a linear transformation
of the original data onto an orthogonal coordinate
system where projections of the data on the axes are ordered
by variance: the first principal component describes most of the
variance in the data, and so on. By finding an orthogonal coordinate
system, PCA transforms linearly correlated variables to
uncorrelated variables.
The examples in this paper use UBVRI photometric datasets.
In addition to using the original U, B, V, R and I magnitudes,
PCA was also performed using the U − B, B − V, V − I,
R − I colours and the (almost) reddening-free index3
Q as additional
observables. The reason is that PCA is designed for linear
correlations between the observables, but the interdependencies
between the UBVRI bands are non-linear (principal components
are expected to vary between different spectral types) and some
help had to be given by providing additional observables with a
higher capability of separating the intervening physical factors.
The resulting principal components were found to have most
of the variance in the first four components, with the fourth
providing a minor contribution. We interpret this as an indication
that these components contain information on the main four
physical quantities that intuitively are expected to contribute to
the variance in UBVRI photometric data: distance, reddening,
temperature, and, to a lesser degree, metallicity or surface gravity.
In the end, the justification for using linear PCA is the good
performance of UMASK reported in Sects. 3 and 4. In this
sense, linear PCA is to be taken as a useful heuristic with a physical
motivation.
3
Q = (U − B) − 0.72 × (B − V).
Although the exact number varies from data-set to data-set,
the first three principal components are responsible for 99% of
the variance in the data, with the fourth component contributing
to almost all of the remaining variance. For instance, in the
Haffner 16 data-set analysed in Sect. 4.1 the contributions from
the four most important PCs are 58.8%, 31.3%, 9.0%, and 0.9%.
We note that since the method is data-driven, it is also expected
to work on other kinds of parameter spaces. Naturally,
its success in segregating populations depends on the discriminating
capability of physical quantities encoded by the chosen
observables.
2.3. Implementation of the method
The concept described above was implemented in the R language
(R Development Core Team 2011). The implementation
relies on two iterative processes. The first process, called the
UPMASK kernel, implements the core of the concept. The second
process, which we call the outer loop, adds the capability of
taking measurement errors into account, as well as of relaxing
the dependency on the initial conditions chosen for the adopted
clustering algorithms.
2.3.1. Main iterative process: UPMASK kernel
The UPMASK kernel is the core of the UPMASK method. It is
an iterative loop composed of three main steps.
First, a PCA is performed on the selected observables, positions
excluded. Since the observables will in general have different
variances and span different ranges, they are first scaled to
unit variance. This scaling is applied to avoid that observables
with the highest variance dominate the principal components.
The PCA result will allow selecting the most significant principal
components. As discussed in Sect. 2.2, physical motivations
and information content of broadband UBVRI data have led us
to consider four principal components.
Then, a clustering analysis is conducted on the selected principal
components. Although the method concept is independent
of the adopted data-clustering algorithm, a choice has to be
made. In this paper we adopted one of the simplest algorithms,
k-means with a random initialisation, but others can be used as
well. The number of clusters, k, is not directly set by the user.
Instead, the user sets the number of stars per cluster. The number
of clusters will then be the total number of stars under analysis,
divided by the number of stars per cluster, rounded up to the
next integer. The tests performed in this work had best results
with values between 10 and 25 stars per clustering. Applications
of the method that adopted more complex algorithms might improve
the results or completely eliminate the need of user input.
Finally, the stars in each of the clusters output by the previous
step are tested for clustering in positional space (details in
the next subsection). If the data in a cluster are also clustered in
positional space, it is retained for the next iteration. Otherwise,
its stars are catalogued as field stars and are not considered in
subsequent iterations. At this point, some stars classified as field
stars may be members of stellar clusters. This will be addressed
by the re-sampling introduced in the outer loop described in
Sect. 2.3.3.
These steps are repeated until no more stars are added to the
list of field stars. A graphical representation of the steps composing
the UPMASK kernel can be found in Fig. 1.
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A&A 561, A57 (2014)
Start
Run PCA on the
photometry data
Select the 4 most
important PCs
Run the clustering algorithm
Is the object
list empty or is it the
same as in the last
iteration?
Select clusters that are not
compatible with random fields
No
Yes
Done
Assign the stars on list as
members of the cluster, and all
others as members of the field
Fig. 1. Flowchart of the UPMASK kernel.
2.3.2. Detection of spatial clustering
To assess whether a certain distribution of stars is clustered in
positional space, the UPMASK method employs kernel density
estimations. Density parameters determined for real data and
randomly generated uniform fields are compared. If the values
obtained for the real data and for the random fields are compatible,
the stars are classified as field members. The detailed
process is as follows:
First, a two-dimensional normal kernel density estimation is
performed on the positions of the stars within a k-means cluster,
adopting the bandwidth recommended by Venables & Ripley
(2002) for such kernels. The resulting density estimation function
Φ(x, y) is then sampled, resulting in a Φ set. From this set, a
distance between its maximum value (corresponding to the maximum
spatial density) and its mean value in units of standard
deviations is computed:
D(Φ) =
max(Φ) − Φ
σΦ
, (1)
where σΦ is the standard deviation of the set Φ.
Then, using the same number of stars and region size as in
the real data, a large set of realisations of random fields with
uniform position distribution is generated. For each of these random
realisations, a two-dimensional normal kernel density estimation
Ψi(x, y) is computed and also sampled, resulting in a Ψi
set. Afterwards, a set DΨ is constructed grouping the parameters
Di(Ψi) computed from each random realisation. Finally, the data
is considered as not compatible with a random realisation of a
uniform field if
D(Φ) ≥ DΨ + T × σDΨ
,
where DΨ is the mean of the DΨ set, σDΨ
is its standard deviation,
and T is the threshold level above σ.
The number of random realisations performed is 2000. This
value was empirically chosen. For larger numbers the computed
parameters vary by less than 0.5%. As this is a time-consuming
step, the current implementation of UPMASK uses a look-up
table. In this strategy the parameters are computed only the first
time they are needed in the analysis (per number of stars and
region area), and further requests of these values are retrieved
from the look-up table.
Finally, we adopted the thresholding level T = 1, but this
threshold level can be tuned for applications that require higher
purity in the final members.
2.3.3. Outer loop
Although the iterative UPMASK kernel described in Sect. 2.3.1
already performs a membership assignment, it does not implement
the whole UPMASK concept presented in Sect. 2.1. The
result is a binary classification, with stars listed as either belonging
to the field or to a stellar cluster. As mentioned in Sect. 2.3.1,
it is expected that the field catalogue will include cluster stars.
Conversely, the catalogue of star cluster members is expected to
include field stars.
A first goal of the outer loop is to break this binary classification
by assigning membership probabilities. A second goal
is to take the effects of observational errors into account in the
membership assignment process. It works as follows:
First, for each star in the data set, each original observable
(in our case the magnitudes) is replaced by a random draw
from a Gaussian distribution with the mean equal to the observable’s
value and σ equal to the measurement error. For observables
with manifestly non-Gaussian error distributions, other error
models can be easily plugged into the method.
Then, the UPMASK kernel described in Sect. 2.3.1 is run on
the newly generated data-set. If the kernel requires initialisation
of variables, it is done before processing the data-set. In the current
implementation, the k-means algorithm is initialised from
random means. Upon termination, the kernel will have classified
the stars in this set as cluster members or field stars.
The two steps described above are repeated until a userdefined
maximum number of iterations is reached. Then, the results
from all the runs are taken, and for each star the fraction
of runs is computed for which it was assigned as cluster member.
This fraction is an indicator of how certain the method is
in assigning a star as a member of a stellar cluster. In a frequentist
sense it is identified with a membership probability.
Since outer iterations run the UPMASK kernel many times, the
k-means clustering algorithm is also randomly initialised many
times. This reduces the dependency of the final results on the
choice of the clustering algorithm’s initialisation conditions. A
flowchart representing the outer iterations of UPMASK is shown
in Fig. 2.
3. Application to simulated data
UPMASK was validated by running the method on synthetic
data sets. In this section we describe the characteristics of these
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A. Krone-Martins and A. Moitinho: UPMASK: unsupervised photometric membership assignment in stellar clusters
Run the UPMASK kernel
Start
Draw random values from
the photometric data and
error models
Compute the membership
frequencies
Done
Max. iterations
reached?
No
Yes
Fig. 2. Flowchart of the outer loop.
data sets and present an assessment of the method’s classification
performance, caveats, and limitations.
3.1. Simulated data
Synthetic catalogues of UBVRI photometry of cluster fields were
generated to simulate observations of clusters spanning a range
of initial mass, age, and distance (see Table 1) along a line-ofsight
in the Milky Way. The field of view was set at 12 × 12 ,
the simulated CCD frames were defined to be 2048×2048 pixels
and the limiting magnitude at U, B, V, R, I ∼ 19.5, 20.5, 21.5, 22,
22 mag to match typical open cluster photometric tables found
in the literature.
The underlying Galactic field was created with the
TRILEGAL simulator4
(Girardi et al. 2005) for the direction
(l, b) = (120◦
, 0◦
) with an applied extinction of 1mag/kpc. The
photometric system of the simulations was set to UBVRIJHK
(Bessell 1990; Maíz Apellániz 2006). The other TRILEGAL parameters
(except for the field of view and magnitude limit) were
left at their defaults. Since TRILEGAL does not output individual
stellar coordinates, these were randomly generated with a
uniform distribution across the simulated CCD frame.
The open cluster photometry and CCD positions were created
using the MASSCLEAN5
package (Popescu & Hanson
2009). For each simulated cluster distance, the adopted visual
absorption, AV, was the value determined in TRILEGAL as
4
http://stev.oapd.inaf.it/cgi-bin/trilegal
5
http://www.physics.uc.edu/~bogdan/massclean.html
500 1000 5000 20000
020406080100
Contamination in original data (%)
TruepositiverateatP>90%
Fig. 3. True positive rate of the subsample with membership probabilities
higher than 90% (TPR90) against the field contamination in the
original data-set. Each dot corresponds to one of the simulations in the
grid defined in Table 1. The dots colours indicate the cluster masses: the
bluer, the more massive, the redder, the less massive.
Table 1. Parameter range covered by the simulations.
Parameter Values
Age [10n
yr] 7.1, 7.7, 8.0, 8.3, 8.7, 8.8, 9.0, 9.2, 9.5
Initial mass [103
M ] 0.5, 1, 2.5, 5, 10
Distance [kpc] 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0
corresponding to the same distance. This was done to ensure that
the location of the cluster sequence was consistent with the field
in the colour space.
MASSCLEAN generates error-free photometry, and thus,
magnitude dependent noise was added to the simulated photometry.
An exponential error model was used, with errors starting
at 0.01 mag for the brightest sources, reaching about 0.06 mag
at 1.5 mag before the limiting magnitude, and then steeply rising
to asymptotically reach infinity at the limiting magnitude.
3.2. Results on simulated data
3.2.1. Global analysis
One central aspect in the analysis of the simulations is the purity
level of the sample of identified cluster members. Since
UPMASK outputs individual membership probabilities for each
star, the purity analysis must be performed on the subset of the
data whose membership probability is above a certain cut-off.
As a measure of purity, we define the true positive rate (TPR)
as the ratio between the number of real cluster members in the
high-probability subset and its total size. A method capable of
creating the high-probability subset that consists purely of cluster
members would attain a TPR of 1 (or 100%) independently
of the degree of contamination in the global data-set.
In this work, we chose the cut-off level at a membership
probability of 90% and represent the corresponding TPR as
TPR90. Figure 3 depicts the TPR90 for all the simulated clusters
described in Table 1 against the contamination rate of the
corresponding original data-sets. The contamination rate is defined
as the ratio between the total number of stars in a field and
the number of simulated cluster stars.
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A&A 561, A57 (2014)
As seen in Fig. 3, even under heavy contamination the resulting
TPR90 is in general fairly high. As expected, the TPR is
depends on the contamination rate of the data-set. For a fixed
Galactic direction and limiting magnitude, the contamination
will increase with the cluster’s age, distance, and reddening and
decrease with increasing initial mass. Naturally, one does not
expect to obtain a simple relation between the TPR and the contamination
rate.
These results, obtained over a grid of 360 fields, spanning a
wide range of conditions, show that more than 63% of the simulated
clusters have a TPR90 ≥ 80%. This is a remarkable performance
for an unsupervised method that is only based on very
weak assumptions on the objects under analysis.
Figure 3 also shows that the TPR covers a broad range of values
depending on the contamination rate. Given that the Galactic
background is the same for all simulations, the contamination
rate depends on the initial mass of the cluster. In this figure,
the masses follow the contamination, with the lowest masses in
the rightmost part of the plot and the most massive clusters in the
left part. We note that even for the lowest initial masses (500 M
and 1000 M ) the TPR90 reaches high values.
By considering only membership probabilities above 90%,
it is expected that a number of cluster stars will not have been
included in our list of members. In other words, the list has a high
purity, but is not complete. Depending on the research goals,
one can increase completeness by relaxing the cut-off to lower
membership probability values such as 50%. The trade-off will
be a lower purity, or a lower TPR.
To measure the completeness of the sample of recovered
members, we defined the member recovery rate (MRR) as the
ratio between the number of members in the sample and the total
number of members. The MRR will be 1 (or 100%) when
all the members are classified as such and 0 when no members
are recovered. Considering a cut-off at 90% membership probability,
we obtained the MRR90 represented in Fig. 4 for clusters
over the simulated grid.
As expected, the MRR is higher for high-mass, and thus
richer, clusters (Fig. 4 lower right) than for lower mass objects
(Fig. 4 upper left) in the entire age-distance plane. However,
Fig. 4 also shows that even for the poorer clusters (500 M initial
mass), UPMASK has recovered with high purity a fraction
of the members significant enough to enable subsequent analyses
such as isochrone fitting, up to distances of more than 1 kpc
for most ages (photometric sequences of more distant and older
cluster are more embedded in the field distribution and are therefore
intrinsically harder to separate). For the high-mass clusters,
UPMASK can still yield good recovery rates (MRR90 ∼ 50%) at
distances of 3.5 kpc.
3.2.2. Case studies
In this section we examine more closely the simulation results.
Given the large number of simulated fields (360), we have chosen
two representative cases, that are however not the most favorable
ones.
The first case is a cluster with an initial mass of 103
M , has
an age of 109.5
yr (∼3.16 Gyr) and is located at 1.5 kpc. Figure 5
illustrates the results. Panels (a) and (b) show the distribution
of the initial cluster+field data in positional and V vs. (B − V)
spaces. In these plots, one cannot even guess at the existence
of a cluster. This particular example is heavily contaminated by
field stars: while there are 12249 field stars, there are only 101
cluster stars, which are represented in panels (c) and (d). The
7.5 8.0 8.5 9.0 9.5
0.51.52.53.5
log10(age) [yr]
Distance[kpc]
0
20
40
60
80
100
Memb. rec.(P>90%)/Memb. in data
7.5 8.0 8.5 9.0 9.5
0.51.52.53.5
log10(age) [yr]
Distance[kpc]
0
20
40
60
80
100
Memb. rec.(P>90%)/Memb. in data
7.5 8.0 8.5 9.0 9.5
0.51.52.53.5
log10(age) [yr]
Distance[kpc]
0
20
40
60
80
100
Memb. rec.(P>90%)/Memb. in data
7.5 8.0 8.5 9.0 9.5
0.51.52.53.5
log10(age) [yr]
Distance[kpc]
0
20
40
60
80
100
Memb. rec.(P>90%)/Memb. in data
Fig. 4. Dependency of member recovery rate (colour scale) for membership
probability above 90% on cluster age and distance for four initial
masses. Upper left: 500 M . Upper right: 1000 M . Lower left:
2500 M . Lower right: 5000 M . The 104
M simulations have an
MRR90 80 across most of the considered age-distance plane and
are not represented in the figure. The dots mark the regions of the plane
sampled by the simulations.
resulting membership assignment by UPMASK is represented
in panels (e) and (f).
Although from the global analysis presented in Sect. 3.2.1
one could expect this cluster to present poorer results, its members
were well recovered by the UPMASK method, mostly with
high ( 0.5) membership probabilities.
For some regions in the CMD minor voids are opened (ex.
between B − V = 1.1 and 1.2 mag). Gaps in the CMD of star
clusters can be real (e.g. de Bruijne et al. 2000), but this is not the
case of the present simulation. Here, the gaps are a consequence
of not requiring continuity between the different k-means clusters
classified as members/non-members. Nonetheless, they can
be filled by a non-parametric post-processing phase that would
determine whether stars lying between high-probability clusters
could belong to the star cluster. An alternative, but parametric,
approach could consist of first fitting an isochrone to the identified
members and then using an isochrone proximity criterion
for selecting stars in the gapped regions.
In any case, the comparison of the simulated cluster with
the recovered members shown in Fig. 5 shows that the general
features of the distribution of stars in the original cluster were
captured.
The second case represents a more massive cluster with
5 × 103
M and an age of 1 Gyr that is located at 4.0 kpc. The
UPMASK results and the initial data are represented in Fig. 6.
Because this cluster is more massive than that from the first
example, it shows up as a region with higher density in the position
space (Fig. 6a). It is, nonetheless, hardly distinguishable
in the CMD (Fig. 6b), except for a hint of a concentration of
red-clump stars.
Even in a highly contaminated data-set, the method was able
to separate the field and cluster populations. It was also able to
recover the cluster’s red-clump stars, as can be seen by comparing
the simulated members and the stars assigned as members
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A. Krone-Martins and A. Moitinho: UPMASK: unsupervised photometric membership assignment in stellar clusters
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(a)
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(b)
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(c)
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0.50
0.25
(f)
Fig. 5. Results of UPMASK on a simulated cluster with 103
M and
109.5
yr at 1.5 kpc. In a) and b) we plot the distribution of field+cluster
stars in positional space and in V vs. (B − V) space. In c) and d) we
show the distributions of cluster members only. In e) and f) we plot the
distributions of the cluster members identified by UPMASK. Dot size
and transparency encode membership probability.
(Fig. 6d and f). Although, as discussed in the previous example,
gaps may appear in the recovered sample of cluster members,
one of the strengths of UPMASK is precisely allowing for
such gaps. In the present case, it has allowed us to identify the
main-sequence and the red-clump stars, which are disconnected.
Finally, the overall morphological structure of the cluster in positional
space was also well recovered (Figs. 6c and e).
The effect of the number of stars per cluster when using the
k-means algorithm
The UPMASK method does not rely on the choice of a specific
algorithm for the clustering step in the UPMASK kernel (see
Sect. 2.3.1). Nonetheless, a choice must be made, and the current
implementation has used the k-means algorithm. The question
then arises of how to specify the number of k-means clusters, or
equivalently, as we have done, the number, n, of stars per cluster.
In Sect. 2.3.1 we mentioned that best results were achieved
with n between 10 and 25. Here we analyse of the effects of
varying n for both simulated clusters studied in this section.
Figure 7 exemplifies the dependency of purity and completeness
at the 50% and 90% membership levels for different
0 500 1000 1500 2000
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(a)
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(b)
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(d)
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Fig. 6. As Fig. 5, but for a simulated cluster with 5 × 103
M and 1 Gyr
at 4.0 kpc.
values of n. The TPR90 decreases continuously from n > 13
and the MMR90 reaches values around ∼25% for 20 > n > 30.
Meanwhile, the MMR50 quickly reaches ∼70% for n = 15, and
then slowly decreases. These figures indicate that adopting high
values of n may result in more contaminated assignments at a
fixed membership probability level. A noticeable feature however,
is that for a low value n = 7–10 not only TPR50 is very
high, but MMR50 is also high (within its range). This is an indication
that for less densely populated clusters the adoption of
lower values of n is recommended, as one could naively expect.
Although fine-tuning of n can always be performed for studies
requiring very high purity, the analysis in this section indicates
that values in the range of ∼10 to ∼25 can be adopted in
most applications. According to the cases studied here, a value
of n = 15 stars could be taken as an optimal default value.
Effect of the field size
It is known that for proper-motion-based membership studies,
unavoidable pre-selection of areas in proper motion space and/or
in coordinate space affects the derived membership probabilities
(e.g. Sánchez et al. 2010). Here we investigate how the the relative
areas of the field and cluster under analysis might influence
the output of UPMASK.
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A&A 561, A57 (2014)
5 10 15 20 25 30
020406080100
Stars per k-means cluster (n)
MRR50(blue)andTPR50(red)[%]
(a)
5 10 15 20 25 30
020406080100
Stars per k-means cluster (n)
MRR90(blue)andTPR90(red)[%]
(b)
5 10 15 20 25 30
020406080100
Stars per k-means cluster (n)
MRR50(blue)andTPR50(red)[%]
(c)
5 10 15 20 25 30
020406080100
Stars per k-means cluster (n)
MRR90(blue)andTPR90(red)[%]
(d)
Fig. 7. Dependency of the completeness and purity indicators at 50%
and 90% membership probability levels, with the adopted number of
stars per k-means cluster. In the top panels we show the cluster from
Fig. 5. In the bottom panels we show the cluster from Fig. 6.
The dependency of the true positive and the membership recovery
rates as a function of the ratio between the size of the
observed field of view and the cluster diameter is shown in
Fig. 8. Here, the cluster radius is considered as four times the
King’s profile core radius, since we found that this is roughly the
radius at which star counts become indistinguishable from those
of the field. The current implementation of UPMASK does not
perform the cluster/field disentangling on fields much smaller
than the cluster radius. We are investigating variations of the spatial
density estimation step in the UPMASK kernel for adapting
to smaller fields.
In Fig. 8 we present the results for the lower mass, nearby
cluster in the left plot and the more massive, farther removed
object in the right plot. In both cases, the TRP90 (in red) and
MMR90 (in blue) are shown for two extreme values of the number
n of stars per k-means cluster. The completeness, represented
by MMR90, remains fairly constant at the 10–20% level for the
lighter cluster over a wide range of field/cluster size ratios. For
the same cluster, we observe that the purity indicator, TRP90,
shows a stronger variation: while a lower value of n resulted in
highly pure sets, even reaching 100% for most size ratios, the results
for a higher n were poorer and degraded as the field of view
(FoV) grew. This test confirms that lower values of the number of
stars per k-means cluster should be adopted if the best solutions
are required for less massive objects, even though an acceptable
result is obtained with TRP90 above 50% for the worst case that
was analysed. For the farther removed, but more massive cluster
the results for purity are very stable at high levels above ∼85%.
The completeness, on the other hand, favours somewhat more
extended fields, which can increase the MMR90 up to ∼10%.
Strictly speaking, from the point of view of the method concept,
this increase is unexpected because there is no immediate reason
for such an increase beyond the cluster limits. Our current
explanation is that it could be due to the recipe for the spatial
clustering test. This will be analysed in future work.
1.0 1.2 1.4 1.6 1.8 2.0
020406080100
(field size) / (cluster diameter)
MRR90(blue)andTPR90(red)[%]
n=25
n=25
n=10
n=10
(a)
1.0 1.2 1.4 1.6 1.8 2.0
020406080100
(field size) / (cluster diameter)
MRR90(blue)andTPR90(red)[%]
n=25
n=25
n=10
n=10
(b)
Fig. 8. Dependency of the completeness and purity indicators at the
90% membership probability level with the field of view. In the left
panel we show the cluster from Fig. 5. In the right panel we show the
cluster from Fig. 6. Field size refers to the linear dimension of the simulated
square field of view.
Finally, as previously mentioned and according to Fig. 4,
the simulated clusters analysed here do not occupy particularly
favourable regions of the mass-distance space. Nevertheless, the
tests showed that even in these cases, useful purity and completeness
results are obtained. These examples show that the
UPMASK method is able to separate cluster/field populations
and moreover recovers the overall morphological structure and
features of the distribution of member stars on the CMD, as readily
seen in Figs. 5 and 6.
3.2.3. Fields with lower counts
Since UPMASK is based on statistical procedures, it is conceivable
that for fields with low stellar counts, small number
statistics could affect the results. To investigate this, we
analysed simulations of a low-mass cluster (500 M initial
mass, Myr, 81 stars) at three different Galactic coordinates:
(l, b)gal = {(120◦
, 0◦
), (180◦
, 15◦
), (180◦
, 25◦
)}, corresponding to
4956, 1975, and 916 field stars respectively.
Following the analysis in Sect. 3.2.2, for lower counts it is
advisable to adopt lower values for the number n of stars in each
k-means cluster. Adopting n = 7, the TPR90 obtained for the
three coordinates are ∼97.5%, ∼94.4%, and ∼100.0%, respectively.
Although high purity values were obtained, the completeness
in this last case (the least populated field) was low, with only
12% of the cluster stars recovered with a membership probability
> 90%. At the 50% membership probability level, however,
we obtained a purity of 90.5% and a completeness of 47%. In
the other cases, the completeness at 90% membership probability
was 48% and 42%, respectively.
Despite the evoked reasons for adopting a low n, we also
tested n = 25. The TRP90 obtained were ∼66.1%, ∼73.8%, and
∼85.5%. As expected, these are significantly lower than the purity
figures obtained for n = 7. Conversely, the completeness is
significantly higher, reaching 99%, 73%, and 58%, respectively.
These tests show that the choice of the number of stars per
k-means cluster does influence the best results in low count situations.
Nevertheless, the default value of n = 15 would produce
acceptable results in the analysed simulations.
4. Application to real data
Following the validation of UPMASK with simulated data presented
above, we tested the method on real data-sets. Here we
present results using UBVRI photometry of two cluster fields
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A. Krone-Martins and A. Moitinho: UPMASK: unsupervised photometric membership assignment in stellar clusters
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Fig. 9. Results of the UPMASK run on the UBVRI data of Haffner 16.
a) Distribution of all observed objects in positional space (X–Y CCD
coordinates). b) Distribution of all observed objects in the V vs. (B −
V) CMD. c) and d) Cluster membership probabilities are encoded as
dot sizes and transparencies in the CMD and positional spaces (less
probable = smaller and more transparent). e) and f) the same encoding,
but for field membership probabilities.
taken from Moitinho (2001), which have the same field of view
and limiting magnitude as those in the simulations. The first
field is centred on the open cluster Haffner 16. The second field
adds complexity by including two open clusters, Haffner 10 and
Czernik 29. Only stars with measurements in all five UBVRI
bands were considered.
4.1. Haffner 16
The original data for Haffner 16 and the results from the
UPMASK run are presented in Fig. 9. For positional data, we
used the CCD pixel coordinates. On-sky coordinates (right ascension
and declination, α and δ) could have been used instead.
This is expected to have no effect because UPMASK sets no particular
requirement on the shape of the spatial clustering.
Figure 9d shows that for V < 17 mag the cluster main sequence
emerges in the CMD from the high-probability members
alone. Immediately below, we find a gap extending almost to
V ∼ 18 mag, after which there is a new group of stars assigned
with high membership probabilities. At these fainter magnitudes
the U photometry becomes incomplete and has larger errors,
which could partially explain the scatter in the faint group.
Although it is beyond the scope of this paper to embark on a
detailed interpretation of the observations, we note that the faint
group appears as a blue bump in the overall distribution of stars.
This bump corresponds mostly to the stars with high membership
probability seen off the cluster centre in the upper left of
Fig. 9c. Together with the red clump at (B − V) ∼ 1.4 mag, this
might indicate the presence of a distant and older cluster in the
field. Another possible scenario is that we are seeing the signature
of the metal-poor thick disc reported in various fields in this
region of the Galaxy (Carraro et al. 2007, 2008, 2010).
Returning to the upper main sequence, we find some dispersion
in the high-probability members. This need not be due to
residual field contamination. The thickness of the main sequence
is well compatible with being due to unresolved binary members.
In fact, these stars are also clustered in positional space.
Another noticeable of feature UPMASK can be seen in
Figs. 9e and f. Here we see that the method avoided opening
holes in the CMD or positional spaces for the field distribution.
Holes in the distribution of the remaining field stars are a wellknown
artefact in parametric methods such as those based on
the Vasilevskis-Sanders approach. However, comparing Fig. 9e
and a, one can notice a residual concentration of stars in the centre
of the positional space in (e). Some of these stars are very
likely cluster members, but have been assigned low membership
probabilities and thus still appear as a concentration in the field.
As another test, we applied UPMASK only to the most central
part of the field, cutting 66% of the area. The results are
presented in Figs. 10c and d. Interestingly, the fainter highprobability
clump has disappeared and a hint of the cluster sequence
departing to the red below V ∼ 17 can be seen. This
red sequence is better distinguished in Fig. 11. An interesting
research possibility offered by applying UPMASK to these data
would be to verify whether this redder sequence might be composed
of pre-main sequence stars.
By comparing the results obtained from this reduced field
with those from the entire data-set (Fig. 9), we found that running
UPMASK on a fraction of the field influences the membership
probabilities. Dependency on the considered area is
also known to affect other membership-assessment algorithms
(Sánchez et al. 2010).
4.2. Haffner 10 and Czernik 29
The second test using real data was performed on the field of the
open clusters Haffner 10 and Czernik 29. These clusters were selected
because they are projected close to each other on the plane
of the sky and fit in the same CCD frame. As can be seen from
Fig. 12a, the angular separation of their centres is only ∼3.8 arcmin.
Another reason was that their sequences are not clearly
visible in the CMD of the original data shown in Fig. 12b, and
thus, these objects pose an additional challenge to fully automatic
methods.
Application of UPMASK to the full data-set resulted in
the cluster- and field-membership probabilities represented in
Figs. 12c, d and e, f, respectively. In panel (c), there are two
overlapping regions with a high concentration of objects with
high membership probabilities: one in the lower-left (hereafter
LL) and another in the upper-right corner (UR).
The objects defining these two regions occupy different loci
in the CMD shown in panel (d): those at the UR region are at
(B − V) < 0.9 mag; while the objects at the LL region are at
(B − V) > 0.9 mag, including those at (B − V) ∼ 1.5 mag.
We note that some of the UR objects can also be found at
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Fig. 10. UPMASK results for the central region of the Haffner 16 field.
Panels as in Fig. 9.
0.0 0.5 1.0 1.5
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1.00
0.75
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(a)
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2018161412
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V
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(b)
Fig. 11. V vs. (B−V) and V vs. (V−I) CMDs for the central region of the
Haffner 16 field. The dot size and transparency encode the square-root
of the membership probabilities.
(B − V) > 0.9 mag. We identify the UR region with Czernik 29
and the LL with Haffner 10. Figure 13 shows the V vs. (B − V)
CMDs of stars centred on these clusters.
Figure 13b reveals the probable members of Haffner 10 as
defining a main sequence with a broad colour spread and a red
clump. The spread in the main sequence could indicate some
field contamination among the stars classified as cluster members.
Given the faint magnitudes and in turn the larger measurement
errors, a certain broadening of the cluster sequence is
500 1000 1500
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Fig. 12. Same as Fig. 9, but for the field of Haffner 10 and Czernik 29.
also expected. However, as noted by Pietrukowicz et al. (2006),
the red clump constitutes a tilted branch parallel to the reddening/extinction
vector, which these authors argued to be evidence
for significant variable reddening. We note that the length of the
branch is similar to the spread in the main sequence, which could
thus be explained by variable reddening.
The other cluster projected on the same field, Czernik 29,
presents a more homogeneous result (Fig. 13a), although gaps
appear in the CMD, for instance, around (B − V) ∼ 0.65 mag, as
also occurred with simulated data-sets. This is a consequence of
not requiring continuity between the different sets classified as
member/non-member stars. However, not requiring continuity is
what allows the method to detect two (or more) different clusters
projected on the same line-of-sight (as well as the red giant
clump of Haffner 10).
With this case study, we have assessed the ability of
UPMASK to automatically find and assign membership probabilities
to two different overlapping clusters in the same field.
4.3. Remarks on the application of UPMASK to real data
The tests presented here pushed UPMASK close to the limits
where the simulations indicated that the method would start to
perform less well. Haffner 16 is at ∼3.7 kpc (Moitinho et al.
2006), while Haffner 10 and Czernik 29 are at ∼3.7 kpc and
∼3.0 kpc, respectively (Vázquez et al. 2010). These distances are
A57, page 10 of 12
A. Krone-Martins and A. Moitinho: UPMASK: unsupervised photometric membership assignment in stellar clusters
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Fig. 13. V vs. (B − V) CMD of stars centred on Czernik 29 (left) and
Haffner 10 (right). Membership is represented as in Fig. 9.
at the upper limits of those for which the simulations showed that
the method attained high completeness and purity (MRR90 and
TPR90). The Haffner 10 and Czernik 29 field added the challenge
of including two clusters in the same field-of-view. Moreover,
Haffner 10 has been reported to be affected by significant variable
extinction (Pietrukowicz et al. 2006), means that its members
are not tightly concentrated around a well-defined extinction
(one of the hypotheses for distinguishing cluster and field
stars).
Notwithstanding, the UPMASK method revealed interesting
characteristics of the studied objects in a completely automatic
analysis with few assumptions. The results presented in this section
indicate that UPMASK might be used with real data up to
greater distances limits, providing useful membership probabilities
and new insights into the nature of the studied clusters.
5. Present caveats and possible developments
The UPMASK method can take into account missing data during
the sampling step in the outer loop described in Sect. 2.3.3.
Currently, this is implemented in a very crude way, with a uniform
distribution over a specified range of the missing magnitude
(which is far from ideal), and therefore this feature was not
enabled during the tests presented here. Nonetheless, any distribution
can be plugged-in. A better way to address this problem,
would be to adopt distributions based on knowledge provided
by the available measurements of a given object. A possibility
would be to set constraints on the colours that use the missing
measurement, instead of allowing for any value. An example
of this approach could be to fill the missing values assuming
a (reddened) blackbody spectrum constrained by the available
photometric data. Treatment of missing data will be the subject
of future research.
The UPMASK method, although originally intended for
photometric and positional data alone, can also be promptly
adapted for use with other types of data, such as proper motions
and parallaxes. The parallaxes can be used together with positions
in the spatial clustering veto. Ideally, the spatial clustering
should be performed on the entire three-dimensional positional
space if reliable distance information is available.
The proper motions can be used in an intermediate step between
the spatial cluster veto and the photometric membership
or as variables additional to the principal components during the
k-means clustering step.
A possible future development could be to run the method
independently on cells of the photometric parameter space and
then combine the results a posteriori. The rationale behind this
idea is to enhance the clustering in principal component space.
As discussed in Sect. 2.2, the PCA produces a linear transformation,
the same for all the data. However, the problem is nonlinear
and different spectral types project onto different principal
components. Using cells means that when a cell has a high fraction
of star cluster members, there will be a dominant spectral
type in the cell and thus the principal components will be optimised
for that spectral type.
From the technical point of view, as expected from methods
based on Monte-Carlo-like heuristics, UPMASK is computeintensive.
Thus, in order to profit from multicore and sharedmemory
architectures, our R implementation parallelises the
outer loop via the multicore package (Urbanek 2011). This allows
an almost linear speed-up with the number of cores. As an
indication, at the time of the writing, running the UPMASK kernel
120 times on the field of Haffner (2096 stars with UBVRI
photometry) took about 7 min using 12 Intel Xeon E7-8837
cores (2.67 GHz).
6. Summary and conclusions
We have described the method UPMASK, which has been designed
to identify star cluster members in highly contaminated
fields using a minimal number of assumptions. Unlike field
stars, members of star clusters share common properties that
make them cluster in the spaces of observables related to those
properties.
A PCA step was introduced to minimise overweighting observables
and to maximise the separation of the different physical
parameters measured by the observables. For stellar UBVRI
photometry, these physical parameters are reddening, distance,
temperature, and to lesser degrees metallicity and surface gravity
for some spectral types. The PCA is specially important
for redundant information brought by different, but correlated,
colours.
The method was tested on UBVRI photometry and positions
of simulated and real star cluster fields. Results on simulations
showed that UPMASK is capable of assigning star cluster members
with good true positive rates and member recovery rates for
a broad range of open cluster ages, masses, distances from the
Sun, and contamination by field stars. Application to real data
showed that the method works and produces good results when
pushed to the limits indicated by the simulations.
Although the UPMASK implementation and tests presented
in this paper exclusively address photometry and positions, the
framework is more general and can be easily adapted to other
types of data. It might also be applied to clustering studies of
objects of a different nature. A possible use could be with extragalactic
data or any type of data where the following basic
premises apply:
– The member objects are located in the same region of positional
(or any physically favoured) space.
– The member objects share other common characteristics.
They should be clustered in some arbitrary parameter spaces.
– The non-member objects are not (significantly) clustered in
those parameter spaces.
Finally, we note that the source code of the implementation presented
in this paper will be distributed under an open-source licence.
As other R codes, it will be available through the CRAN
archive as the UPMASK package. The simulated data used in
this paper can be requested from the authors.
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A&A 561, A57 (2014)
Acknowledgements. This work was supported by the Portuguese Fundação
para Ciência e Tecnologia, FCT, project numbers SFRH/BPD/74697/2010 and
PTDC/CTE-SPA/118692/2010. This work has made use of the computing facilities
of the Laboratory of Astroinformatics (IAG/USP, NAT/Unicsul), whose purchase
was made possible by the Brazilian agency FAPESP (grant 2009/54006-4)
and the INCT-A. This work was written using the collaborative on-line ScribTeX
platform.
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