Neural Networks 16 (2003) 405–410
www.elsevier.com/locate/neunet
2003 Special Issue
Using self-organizing maps to identify potential halo white dwarfs
Enrique Garcı́a-Berroa,*,1, Santiago Torresa, Jordi Isernb,1
a
Departament de Fı́sica Aplicada, Universitat Politècnica de Catalunya, Jordi Girona Salgado S/N, Mòdul B-4,
Campus Nord, 08034 Barcelona, Spain
b
Institut de Ciències de l’Espai, C.S.I.C., Edifici Nexus, Gran Capità 2-4, 08034 Barcelona, Spain
Abstract
We present the results of an unsupervised classification of the disk and halo white dwarf populations in the solar neighborhood. The
classification is done by merging the results of detailed Monte Carlo (MC) simulations, which reproduce very well the characteristics of the
white dwarf populations in the solar neighborhood, with a catalogue of real stars. The resulting composite catalogue is analyzed using a
competitive learning algorithm. In particular we have used the so-called self-organized map. The MC simulated stars are used as tracers and
help in identifying the resulting clusters. The results of such an strategy turn out to be quite satisfactory, suggesting that this approach can
provide an useful framework for analyzing large databases of white dwarfs with well determined kinematical, spatial and photometric
properties once they become available in the next decade. Moreover, the results are of astrophysical interest as well, since a straightforward
interpretation of several recent astronomical observations, like the detected microlensing events in the direction of the Magellanic Clouds,
the possible detection of high proper motion white dwarfs in the Hubble Deep Field and the discovery of high velocity white dwarfs in the
solar neighborhood, suggests that a fraction of the baryonic dark matter component of our galaxy could be in the form of old and dim halo
white dwarfs.
q 2003 Elsevier Science Ltd. All rights reserved.
PACS: 95.35. þ d; 95.75.Pq; 95.80. þ p; 97.10.Yp; 97.20.Rp; 98.35.Gi; 98.35.Ln
Keywords: Stars; White dwarfs; Dark matter; Mathematical procedures and computer techniques; Catalogues
1. Introduction
White dwarfs are the most common end-point of stellar
evolution. Moreover, white dwarfs are well-studied objects.
In fact, the relative simplicity of their constitutive physics
allows us to obtain very detailed evolutionary models
(Salaris, Garcı́a-Berro, Hernanz, Isern, & Saumon, 2000,
and references therein). Although these evolutionary
models can be extremely sophisticated, it can be said that
their evolution is essentially a cooling process—see, for
instance, the recent review of Fontaine, Brassard, and
Bergeron (2001)—during which the degenerate and almost
isothermal core releases gravothermal energy which is
evacuated through the partially degenerate atmosphere,
whereas the hydrostatic equilibrium is achieved mostly by
the pressure of the nearly degenerate electrons. This
atmosphere, in turn, controls the rate at which the energy
is radiated away. Additionally, white dwarfs have very long
* Corresponding author. Tel.: þ 34-93-401-6898; fax: þ34-93-401-6090.
E-mail addresses: [email protected] (E. Garcı́a-Berro), [email protected]
(S. Torres), [email protected] (J. Isern).
1
Institut d’Estudis Espacials de Catalunya, Spain.
evolutionary time scales, which are comparable to the age of
our galaxy. Due to these facts, white dwarfs provide us with
an invaluable tracer of the early evolution of our galaxy and,
consequently, allow us to explore how our galaxy, and other
galaxies, formed and evolved (both chemically and
kinematically).
Given their intrinsic faintness, white dwarfs are difficult
to detect at large distances and, thus, the currently available
surveys reach modest distances, at most 300 pc. In fact, a
large fraction of white dwarfs has been found in proper
motion surveys. Thus, the vast majority of known white
dwarfs belong to the solar neighborhood. Whether these
white dwarfs are members of the known galactic disk
populations (namely the thin and the thick disk) or are halo
members is a crucial issue. There is now a widespread
consensus that the distribution of faint white dwarfs in the
solar neighborhood is in good agreement with the
expectations of the standard old thick disk population,
being their space density of about 0.005 pc23, with possibly
a small fraction of halo white dwarfs present in the sample.
The observational situation has improved dramatically in
the last few years with the advent of the Hubble Space
0893-6080/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0893-6080(03)00010-8
406
E. Garcı́a-Berro et al. / Neural Networks 16 (2003) 405–410
Telescope and large ground based telescopes. For instance,
faint white dwarfs have been already detected in several
open and globular galactic clusters, and there are some
evidences that the galactic halo white dwarf population has
been already detected, although this particular topic is still
the subject of large controversies. To be more specific, halo
white dwarfs have received a continuous interest during
almost one decade from both the theoretical (Isern,
Garcı́a-Berro, Hernanz, Mochkovitch, & Torres, 1998;
Mochkovitch, Garcı́a-Berro, Hernanz, Isern, & Panis,
1990; Tamanaha, Silk, Wood, & Winget, 1990) and the
observational points of view. From this last point of view it
is important to stress the big effort of Liebert, Dahn, and
Monet (1989) who studied a high proper motion sample and
from it derived the very first (although severely incomplete)
halo white dwarf luminosity function. Later Flynn, Gould,
and Bahcall (1996) and Méndez, Minnitti, De Marchi,
Baker, and Couch (1996) studied the white dwarf content of
the Hubble deep field. Moreover, the MACHO team
reported the discovery of microlenses towards the large
Magellanic cloud and claimed that about 20% of the dark
matter in the Galaxy could be in the form of white dwarfs
(Alcock et al., 1997). However, recent analyses (Alcock,
2000) have shown that the fraction of dark matter in the
form of white dwarfs is smaller, of the order of # 10%, in
good agreement with the theoretical expectations of Isern
et al. (1998). More recently, Ibata, Irwin, Bienaymé, Scholz,
and Guibert (2000) have reported the discovery of two
extremely cool white dwarfs in the solar neighborhood with
very high proper motion, making them very likely
observational counterparts of a putative ancient halo white
dwarf population. Other faint white dwarfs with extremely
large proper motions have been also discovered recently
(Hambly, Smartt, & Hodgkin, 1997; Hambly et al., 1999;
Hodgkin et al., 2000) making use of stacked photographic
plates. Increasing attention has been paid to this topic since
the very recent discovery (Oppenheimer, Hambly, Digby,
Hodgkin, & Saumon, 2001) of 38 new, nearby and old white
dwarfs with large space velocities. Whether these white
dwarfs belong to the thick disk or to the halo is still the
subject of a strong debate. In summary, although there are
evidences of a possible detection of the halo white dwarf
population, given the scarce number of halo white dwarfs it
is difficult to ascertain whether a small sample of these
objects remains hidden in the current catalogs and how this
putative population could be identified. In this paper we
describe how to identify the population of halo white dwarfs
in the existing white dwarf catalogues.
2. Method and results
With the advent of large astronomical databases the need
of efficient techniques to improve automatic classification
strategies has lead to a considerable amount of new
developments in the field. Among these techniques
the most promising ones are based in artificial intelligence
algorithms. Neural networks have been used successfully in
several fields such as pattern recognition, financial analysis,
biology – see Kohonen (1990) for an excellent review – and
in astronomy. For instance, Bazell and Peng (1998) used
these techniques to automatically discriminate stars from
galaxies, Naim, Lahav, Sodre, and Storrie-Lombardi (1995)
used them to classify galaxies according to their morphology, Serra-Ricart, Aparicio, Garrido, and Gaitan (1996)
found the fraction of binaries in stars clusters, and
Hernández-Pajares and Floris (1994) used such techniques
to classify populations in the Hipparcos Input Catalogue.
The common characteristic of all the existing neural
network classification techniques is the existence of a
learning process very much in the same manner as human
experts manually classify. Generally speaking there are two
different approaches: the supervised and the unsupervised
learning methods. The main advantage of the last class of
methods is that they require minimum manipulation of the
input data and, thus, the results are supposedly more
reliable. Their leading exponent is the so-called Kohonen
self-organizing map (SOM). Although we refer the reader to
the specific literature (Kohonen, 1997) we will summarize
here the basic features of the Kohonen SOM. The basic
principle of this technique is to map a multi-dimensional
input space ðSÞ into a bi-dimensional space ðLÞ: In fact, the
SOM is the result of a vector quantization algorithm that
places a number of reference vectors of a high-dimensional
input space into a bi-dimensional lattice in an ordered
fashion. When local order relationships are defined between
the reference vectors, the relative values of the latter are
made to depend on each other as if their neighboring values
would lie along an ‘elastic surface’. This surface is defined
as a non-linear regression of the reference vectors through
the data points. A mapping from a high-dimensional space
onto a two-dimensional lattice of points is thereby also
defined. Such a mapping can be used to visualize metric
ordering relationships of the input samples. Thus, neighbor
groups in L have similar properties. The mapping is
obtained as an unsupervised learning process. This process
may be used to find clusters in the input data, and to classify
individual objects within these clusters.
The catalog of McCook and Sion (1999) is a compilation
of the observational data of 2249 spectroscopically
identified white dwarfs. In order to classify the stellar
populations presumably present in this catalog a set of
variables describing their properties should be adopted. It
should be noted that the larger the set of variables adopted,
the smaller the number of objects that will have determinations for all the variables. Conversely, if the number of
variables in the set is small we could be disregarding
valuable information. We have adopted a minimal set in
order to be able to analyze the largest possible number of
objects in the catalog. The variables adopted in this study
are: the absolute visual magnitude MV ; the proper motion m;
the galactic coordinates ðl; bÞ; the parallax p; and a color
E. Garcı́a-Berro et al. / Neural Networks 16 (2003) 405–410
index, B 2 V: This reduces considerably the number of
objects with all the determinations, but allows a secure
classification. We have found very convenient to use the
reduced proper motion defined as H ¼ MV 2 5 log p þ 5
log m; instead of m itself because the resulting groups are
easier to visualize.
The first step to be done, previous to any attempt to
classify the above mentioned catalog is to determine if there
exists any linear relationship between the set of variables
that we have chosen. To this regard we have performed a
principal component analysis of the set of white dwarfs
which have observational determinations of all the necessary data. We have not found any zero eigenvalue, meaning
that all the chosen variables are independent. We have also
found that all the eigenvalues are significant and, hence,
none of these variables can be disregarded.
The statistical classification of an observational database
usually ends up with the detection of groups in the input
space that require an ‘a posteriori’ analysis. Since we are
interested in detecting different stellar populations, simultaneously with the clustering process we mix in the input
data a synthetic population of tracer stars that will allow us
to label the groups detected by the classification algorithm
as halo, disk or intermediate population. The results of the
classification procedure are not sensitive to the fine details
of these synthetic populations. These synthetic tracer stars
have been produced using a Monte Carlo (MC) simulator.
The description of the MC simulation of the disk population
can be found in Garcı́a-Berro, Torres, Isern, and Burkert
(1999). A comprehensive discussion of the results of the
MC simulation of the halo population will be published
elsewhere. However, and for the sake of completeness, a
brief summary of the inputs is given here. We have adopted
a standard, Salpeter-like, initial mass function (Salpeter,
1961). The halo was supposed to be formed 14 Gyr ago in an
intense burst of star formation of 1 Gyr of duration. The
sensitivity of our results to the exact value of these two
parameters is very small. The stars are randomly distributed
in a sphere of radius 200 pc centered in the sun according to
a density profile given by the expression
rðrÞ /
ða2 þ R2( Þ
;
ða2 þ r 2 Þ
r being the galactocentric radius, a < 5 kpc and
R( ¼ 8:5 kpc. The velocities of the tracer stars were
randomly drawn according to normal distributions for
both the radial and the tangential components, with velocity
dispersions as given in Marković and Sommer-Larsen
(1997); the adopted rotation velocity Vc is 220 km s21 :
The values of the velocity dispersions depend on the
galactocentric coordinates, but inside the above
mentioned sphere of 200 pc of radiuspffiffitheir values are
roughly the same sr . st . Vc = 2 , 155 km s 21 :
The adopted density of halo white dwarfs was logðnÞ .
25:39 pc23 M21
bol at logðL=L( Þ . 24; in accordance with
407
the results of Liebert et al. (1989). The remaining inputs
were the same adopted in Garcı́a-Berro et al. (1999). In
order to reproduce accurately the properties of the real
catalog both MC simulations were required to meet the
additional set of criteria: d $ 08; 8:5 # MV # 16:5; m #
4:1 in: yr21 and 0:006 # p # 0:376 in:; which are derived
from the subset of 232 white dwarfs which have all the
determinations. An added value of the above-described
procedure of mixing tracer and real stars is that in this way
we can check the accuracy of the classification algorithm
and the quality of the MC simulations.
The simulated samples mimic fairly well the observational sample as can be seen in Fig. 1, where the results of
the MC simulations for the disk and the halo are compared
with the observational sample in the reduced proper motioncolor diagram. As can be seen in this diagram the two
simulated samples are easily visualized. Similar diagrams
can be produced for each pair of variables and the results of
the MC simulations compare equally well with the real data.
After that, we have run the public domain neural network
software SOM_PAK (available at http://www.cis.hut.fi/
nnrc/som_pak/) with a catalog constructed as described.
We have used a Gaussian kernel. We have used a linearly
decreasing learning rate aðtÞ ¼ Að1 2 t=BÞ; where A and B
are constants. In a first step the learning rate was chosen to
be high ðA ¼ 0:5; B ¼ 103 Þ; whereas in subsequent runs we
used a slow learning rate ðA ¼ 0:02; B ¼ 106 Þ: On its hand,
the width of the kernel was chosen to be three nodes for the
first run and in subsequent runs we actualized only one
neighbor node. Finally, in the first run 103 iterations were
used whereas in subsequent runs 106 iterations were
required. The number of nodes was chosen in such a way
that the error function was minimized. Nevertheless it is
worth mentioning here that a compromise between the
number of nodes and the number of white dwarfs in the
observational catalog had to be reached. If the number of
nodes was too large there were too few white dwarfs in the
resulting groups and, hence, the interpretation turned out to
be difficult. On the contrary if the number of nodes was too
small the confusion error became too large. However, the
halo white dwarfs identified using the neural network were
the same for reasonable choices of the grid and, thus, the
identifications can be considered as safe. We also checked
different geometries of the grid, and the results turned out to
be independent of the considered geometry.
The SOM of the input catalog, after three passes over the
entire sample and with a grid of 5 £ 5 nodes is shown in
Fig. 2. The groups have been assigned either to halo (‘H’) or
disk (‘D’) populations if the percentage of tracer stars of one
of the populations was larger than 70%. In the groups
labeled as ‘I’ (intermediate population) neither the halo nor
the disk tracers were in excess of this recognition
percentage. As can be seen in Fig. 2 all the halo groups
are close neighbors and, furthermore, the intermediate
population groups are surrounded by halo and disk groups.
A good measure of the overall quality of the classification
408
E. Garcı́a-Berro et al. / Neural Networks 16 (2003) 405–410
Fig. 1. Reduced proper motion-color diagram for the MC simulations of the disk (solid dots) and the halo (solid squares) and of the observational sub-sample
(open triangles).
scheme can be obtained by checking how many of the
synthetic stars are misclassified. This results in the following
confusion matrix
!
0:98 0:03
C¼
0:02 0:97
where the matrix element C11 indicates the percentage of
disk tracers classified in disk groups, C21 is the percentage
of disk tracers missclassified in halo groups, and so on.
This matrix is very close to unity, and thus the
classification seems to be secure. More confidence in
this classification comes from the fact that the vast
majority of old disk white dwarfs in the sample of
Liebert, Dahn, and Monet (1988) are in the groups (0,2)
and (2,1) which are labeled as intermediate population.
Moreover, LHS 56, LHS 147 and LHS 291 belong to the
group (1,0) which clearly is a halo group, and LHS 2984
belongs to the group (0,0), all these objects were already
identified as halo members by Liebert et al. (1989), and
were used to build their halo white dwarf luminosity
function. The only object of the sample of Liebert et al.
(1989) misclassified is LHS 282 which is classified in the
group (0,2), which is intermediate population. All this
evidence points in the same direction: the classification is
correct. The percentages of halo tracers in the groups
labeled as halo can be found for each of the halo groups
in Fig. 2. Since all of them are larger than 80% all these
groups can be securely labeled as halo.The so-called
Sammon map of our groups is shown in Fig. 3. As it can
be seen there, the distances between the disk nodes
(labeled as ‘D’) are very small and, thus, these groups
have similar characteristics. The same occurs for the halo
groups (labeled as ‘H’) and thus this set of groups is
homogeneous as well. However, the distances between
Fig. 2. SOM of the sample of white dwarfs, see text for details. The group
(0,0) is located in the lower left corner of the diagram and the group (4,4) is
located in the upper right corner. As a rule of thumb H increases from right
to left and MV decreases downwards in the diagram.
E. Garcı́a-Berro et al. / Neural Networks 16 (2003) 405–410
409
Fig. 3. Sammon map of the SOM of Fig. 2.
this last set of groups and the former set are much larger
than distances inside both sets of groups. Moreover, the
groups of intermediate population lie also at considerable
distances from both sets of groups. All these facts allow
us to conclude that the classification scheme is well
defined. However, and for the sake of reliability we have
only identified as halo candidates those white dwarfs
belonging to groups which do not have a disk neighbor,
namely (0,0), (0,1), (1,0) and (2,0). One interesting
property of these white dwarfs is that all of them have
MV # 14 and only four have proper motions in excess of
1:0 00 yr21 ; being the average kml ¼ 0:87 00 yr21 : However,
most of them have p # 0:03 00 ; and are clustered around
p , 0:01 in:; leading to tangential velocities in excess of
200 km s21 for 11 of our candidates. Only one candidate
has a tangential velocity smaller than 100 km s21 : Therefore the detected population is intrinsically bright and
distant. The halo white dwarf candidates detected here can
be found in Table 1.
Table 1
Halo white dwarf candidates identified using the neural network algorithm,
along with their corresponding group and properties
We have shown that an artificial intelligence algorithm
is able to classify the catalog of spectroscopically
identified white dwarfs and ultimately detect several
potential halo white dwarfs. Some of these white dwarfs
were already proposed as halo objects by Liebert et al.
(1989). We have found as well that our halo candidates
are bright and distant, and that most of them have large
tangential velocities. The final answer to whether or not
old white dwarfs are significant contributors to the
baryonic dark matter will come from surveys which
hopefully will be able to identify several high velocity and
very cool white dwarfs. Such searches –the best example
being the Sloan Digital Sky Survey (Harris et al., 2001)–
are now underway. Most of these surveys are based on
proper motion and color selection criteria, and will
provide us with a fairly large amount of observational
data which should be studied carefully. Moreover, future
astrometric missions, like GAIA, will yield a huge amount
of white dwarfs (Figueras et al., 1999). Processing all the
data in order to identify the halo white dwarf population
will, undoubtedly, require the use of complex classification techniques. Among these, Neural Networks, and in
particular the SOM method, seem to provide excellent
results and, hence, have a very promising future.
Name
Group
MV
m (in. yr21)
p (in.)
B2V
Sp. type
LHS 2984p
LHS 3007
G 028-027
G 098-018
G 138-056
G 184-012
LP 640-069
LHS 56p
LHS 147p
LHS 151
LHS 291p
LHS 529
LHS 1927
G 038-004
LHS 3146
G 021-015
G 035-026
G 128-072
G 271-106
GR 363
(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(0,1)
(0,1)
(0,1)
(0,1)
(0,1)
(1,0)
(1,0)
(2,0)
(2,0)
(2,0)
(2,0)
(2,0)
(2,0)
11.62
13.06
12.41
11.81
13.34
13.18
12.75
13.51
13.64
13.46
13.39
13.94
11.41
12.31
11.88
11.54
11.12
12.53
11.77
11.39
0.930
0.636
0.281
0.426
0.692
0.427
0.284
3.599
2.474
1.142
1.765
1.281
0.661
0.428
0.579
0.390
0.335
0.457
0.396
0.133
0.015
0.028
0.003
0.003
0.006
0.017
0.009
0.069
0.016
0.053
0.012
0.046
0.009
0.010
0.024
0.015
0.007
0.025
0.014
0.003
0.03
0.29
0.03
0.38
0.37
0.26
0.29
0.36
0.40
0.33
0.11
0.64
0.11
0.17
0.17
0.05
20.14
0.21
0.18
20.03
DA
DA
DQ
DA
DA
DC
DA
DA
DC
DA
DQ
DA
DA
DA
DA
DA
DA
DA
DA
DA
The stars already identified in Liebert et al. (1989) are marked with an
asterisk.
3. Summary and conclusions
410
E. Garcı́a-Berro et al. / Neural Networks 16 (2003) 405–410
Acknowledgements
Part of this work was supported by the Spanish DGES
project PB98-1183-C03-02, the MCYT grants ESP98-1348,
AYA2000-1785 and HA2000-0038, and by the CIRIT.
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