11institutetext:  Centro Internacional Franco Argentino de Ciencias de la
Información y de Sistemas (CIFASIS, CONICET–UNR), Rosario, Santa Fé,
Argentina. 11email<EMAIL_ADDRESS>22institutetext: Instituto
De Astronomía Teórica y Experimental – Observatorio Astronómico Córdoba
(IATE–OAC–UNC–CONICET), Córdoba, Córdoba, Argentina 33institutetext: Western
Sydney University (WSU), Locked Bag 1797, Penrith South DC, NSW 2751,
Australia

# Automatic catalog of RR Lyrae from $\sim$14 million VVV light curves:
How far can we go with traditional machine-learning?

J. B. Cabral 11 2 2 F. Ramos 22 S. Gurovich 2233 P.M. Granitto 11

###### Abstract

Context. The creation of a 3D map of the bulge using RR Lyrae (RRL) is one of
the main goals of the VISTA Variables in the Via Lactea Survey (VVV) and
VVV(X) surveys. The overwhelming number of sources undergoing analysis
undoubtedly requires the use of automatic procedures. In this context,
previous studies have introduced the use of machine learning (ML) methods for
the task of variable star classification.

Aims. Our goal is to develop and test an entirely automatic ML-based procedure
for the identification of RRLs in the VVV Survey. This automatic procedure is
meant to be used to generate reliable catalogs integrated over several tiles
in the survey.

Methods. Following the reconstruction of light curves, we extracted a set of
period- and intensity-based features, which were already defined in previous
works. Also, for the first time, we put a new subset of useful color features
to use. We discuss in considerable detail all the appropriate steps needed to
define our fully automatic pipeline, namely: the selection of quality
measurements; sampling procedures; classifier setup, and model selection.

Results. As a result, we were able to construct an ensemble classifier with an
average recall of 0.48 and average precision of 0.86 over 15 tiles. We also
made all our processed datasets available and we published a catalog of
candidate RRLs.

Conclusions. Perhaps most interestingly, from a classification perspective
based on photometric broad-band data, our results indicate that color is an
informative feature type of the RRL objective class that should always be
considered in automatic classification methods via ML. We also argue that
recall and precision in both tables and curves are high-quality metrics with
regard to this highly imbalanced problem. Furthermore, we show for our VVV
data-set that to have good estimates, it is important to use the original
distribution more abundantly than reduced samples with an artificial balance.
Finally, we show that the use of ensemble classifiers helps resolve the
crucial model selection step and that most errors in the identification of
RRLs are related to low-quality observations of some sources or to the
increased difficulty in resolving the RRL-C type given the data.

###### Key Words.:

Methods: data analysis – Methods: statistical – Surveys – Catalogs – Stars:
variables: RR Lyrae – Galaxy: bulge

##  1 Introduction

The stars of RR Lyrae (RRL) were first imaged in the last decade of the 19th
century. Although it is uncertain as to who made the first observations, some
authors report that it was Williamina Fleming. Fleming led Pickering’s Harvard
Computer group in early studies of variable stars towards globular clusters
(Burnham, 1978; Silbermann & Smith, 1995). Towards the end of the 19th
century, J. C. Kapteyn and D. E. Packer independently published results of
variable stars which were also likely to be RRL stars (Smith, 2004). In the
decade following Fleming’s initial work, Bailey (1902) classified RRL into
three subtypes. Bailey also proposed that RRL be used as “standard candles” to
measure distances to clusters. Later works by Seares & Shapley (1914) found
that variability in RRL, or so-called “cluster variables,” was probably caused
by the radial pulsation of RRL stellar atmospheres. Several years later
Shapley (1918), used RRL to estimate the distance to globular clusters in the
Galaxy. Later, Baade (1946) used them to measure the distance to the galactic
center.

Now, over a century after Fleming’s initial discovery and after a plethora of
systematic studies, RRL have come to be known as a class of multi-mode
pulsating population II stars with spectral (color) type A or F that reside in
the so-called ”horizontal branch” of the C-M diagram. They are also considered
excellent standard candles with periodic light curves (LC) and characteristic
features that are sensitive to the chemical abundance of the primordial fossil
gas from which they formed Smith (2004). These short-period variables have
periods typically ranging between 0.2 and 1.2 days (Smith, 2004) and they are
old (aged $\sim$ 14 Gyr), with progenitor masses typically about 0.8 Solar
$M_{\odot}$ and absolute magnitudes consistent with core helium burning stars.
When it comes to how many RRL there are in the Galaxy, Smith (2004), estimated
some 85 000. We now know, thanks to a recent work by OGLE (Soszyński et al.,
2019) and Gaia (Gaia Collaboration et al., 2018) data releases, that there are
over 140000 RRL in The Galaxy. Interestingly from a galaxy formation and
evolution perspective, RRL chemical abundances could be used as evidence to
piece together the structural formation and evolution of the bulge, disk, and
halo components of the Galaxy, as well as in near-by ones. Lee (1992)
presented evidence that bulge RRL stars may even be older than the Galaxy Halo
stars, implying that galaxies may have formed inside-out. So RRL are also
important for testing the physics of pulsating stars and the stellar evolution
of low mas stars. RRLs have also been used extensively for studies of globular
clusters and nearby galaxies. They have been used successfully and
systematically as a rung in the extra-galactic distance ladder (de Grijs et
al., 2017) which, along with other methods, Cepheid distances, tip of the red
giant branch methods, etc. (Sakai & Madore, 2001), have provided primary
distance estimates to boot-strap distance measurements to external galaxies
(Clementini et al., 2001) and reddening determinations within our own.

The list of objectives of the VISTA Variables in the Via Lactea Survey (VVV,
Minniti et al. 2010), as designed, includes a $3(+1)$ dimensional mapping of
the bulge using RRLs. As shown by Dwek et al. (1995), most of the bulge light
is contained within ten degrees of the bulge center, so most of the mass,
assuming a constant mass-to-light ratio, should also be contained within this
radius.

Our study (and sample) was made after the OGLE-IV release but it precedes the
OGLE catalog presented in Soszyński et al. (2019) and Gaia Data release. We
use the VISTA data to extract characteristic RRL features based on NIR VVV
photometry. A star-by-star approach to the reduction and process analysis of
light curves (LC) is too costly, prone to external biases, and largely
inefficient due to the overwhelmingly high number of light curves collected by
modern surveys like VVV. The automatic methods developed for this study
statistically handle subtleties involved in normalizing the observational
parameter space to obtain a machine-eye view at the physical parameter space.
Our study, we hope, will serve to refine previous attempts at the use of
machine learning (ML) and data mining techniques to detect and tag RRL
candidates for studies of bulge dynamics of the Galaxy. We also attempt to
take into account the inhomogeneity of Galactic extinction and reddening, as
well as varying stellar density at different lines-of-sight, considering these
in a computationally statistical manner during the feature extraction and
classification steps.

While panchromatic studies from the UV through the near-IR (NIR) and beyond,
including spectroscopic studies, will undoubtedly better constrain the
observed and physical RRL parameter space, this is not yet possible, so for
this study we draw mostly on learning from visual data taken from the
literature and from NIR VVV photometry. Our objective class is the set of
known RRL stars from the literature. These stars are tagged based on OGLE-
III/OGLE-IV and other catalogs accessed through VizieR. They are listed in
Appendix A.

We wish to note that stellar population synthesis studies based on broadband
colors, for example: Bell & de Jong (2001) and Gurovich et al. (2010) appear
to suggest that the evolution of stellar mass in massive disk galaxies is
better constrained by NIR photometry than by broadband photometry at shorter
effective wavelengths. Although predicted effective temperatures for RRL with
given spectral types are more closely matched to bluer filters, we made a
compromise and chose to draw our characteristic feature set preferentially on
NIR light. Moreover, at NIR wavelengths, Galactic extinction effects (and
errors) due to differential reddening are also significantly reduced, so any
method of feature engineering for RRL classification using NIR data should
also prove to be insightful.

Previous works have attempted to develop ML methods for the identification of
variable stars on massive datasets. For example, Richards et al. (2011) and
Richards et al. (2012) introduced a series of more than 60 features measured
over folded light curves and used a dataset with 28 classes of variable stars.
Armstrong et al. (2015) discussed the use of an unsupervised method before a
classifier for K2 targets. Mackenzie et al. (2016) introduced the use of an
automatic method to extract features from a LC based on clustering of patterns
detected in the LC. Shin et al. (2009, 2012) introduced the use of an outlier
detection technique for the identification of variable stars and Moretti et
al. (2018) used another unsupervised technique, PCA, for the same task.
Pashchenko et al. (2018) considered the task of detecting variable stars over
a set of $\sim 30000$ OGLE-II sources with 18 features and several ML methods.
In the most relevant antecedent to our work, Elorrieta et al. (2016) described
the use of ML methods in the search of RRLs in the VVV. They consider the
complete development of the classifier, starting from the collected data (2265
RRLs and 15436 other variables), the extraction of 32 features and the
evaluation of several classifiers. As most previous works, they found that
methods based on ensembles of trees are the best performers for this task. In
this particular case they show an small edge in favor of boosting versions
over more traditional random forest (RF) (Breiman, 2001).

As always in this line of research, there are several ways in which we can
improve upon earlier results, particularly for the findings of Elorrieta et
al. (2016). First, they chose not to use color so as to avoid extinction
related difficulties and they do not use reddening-free indices. Second, as in
most previous works, they use a data sample that is clearly skewed in favor of
variable stars. This over-representation of RRLs produces, in general, over-
optimistic results, as we show later in this work. Lastly, they used
performance measures (ROC-AUC and F1 at a fixed threshold, see Section 3) that
are not appropriate for highly imbalanced problems and low performance.

Furthermore, modern surveys request for fully automatic methods, starting from
the extraction of a subset of stable features through the training and setup
of efficient classifiers up to the identification of candidate RRLs. In this
work, we attempt to develop a completely automatic method to detect RRLs in
the VVV survey. We show the effectiveness of our subset of features (including
color features), along with ways to automatically select the complete setup of
the classifier and to produce unbiased and useful estimates of its
performance.

Although we limit ourselves to classifying RRLs based on VVV aperture
magnitudes in this work, we note our method could be adapted for VVV point
spread function (PSF) magnitudes or used to classify other Variables such as
Cepheids, Transients, or even employed on other surveys such as the Vera Rubin
Observatory’s Legacy Survey of Space and Time (LSST) or Gaia (Brough et al.,
2020; Ivezić et al., 2019).

This paper is organized as follows. In Section 2, we outline the input data
based on the VVV survey: type, size, structure, and how we clean it. Section 3
describes the computational environment and method used in this work. Section
4 summarizes the steps taken to build and extract the features from the LC,
along with details on how we incorporate our positive class (variable stars)
into our dataset, and, finally, we present an evaluation of diverse feature
subsets. In Section 5, we describe the selection of an appropriate ML method
and in Section 6, we discuss the best way to sample the data in order to
obtain good performance and error estimations. Then we select the final
classifier (Section 7), analyze how some errors are produced (Section 8), and
discuss the data release (Section 9). We summarize our work in Section 10.

##  2 The VVV survey and data reduction

The VVV completed observations in 2015 after about 2000 hours of observations,
for which the Milky Way bulge and an adjacent section of the mid-plane (with
high star formation rates) were systematically scanned. Its scientific goals
were related to the structure, formation, and evolution of the Galaxy and
galaxies in general. The data for both VVV and VVV(X) is obtained by VIRCAM,
the VISTA infrared camera mounted on ESO’s VISTA survey telescope (Sutherland
et al., 2015), At the time of its commissioning, VIRCAM was the largest NIR
camera with 16 non-contiguous 2k x 2k detectors. All 16 detectors are exposed
simultaneously capturing a standard “pawprint” of a patch of sky. To fill the
spaces between the 16 detectors and for more uniform sampling, 6 pawprints are
observed consecutively at suitable chosen offsets of a large fraction of a
detector. When combined into a ”tile,” these six pawprints sample a
rectangular field of roughly 1.5 x 1.0 degrees, so that each pixel of sky is
observed in at least 2 pawprints. There are also two small strips in which
there is only one observation. These strips overlap another such strip on the
adjacent tile. Over the course of the survey, the same tile is re-observed for
variability studies. In the first year of VVV survey observations more
stricter conditions were required and for most tiles five astronomical pass-
bands where observed that included in increasing wavelength Z, Y, J, H, and
Ks, separated only by a couple hours. However, for a small number of tiles,
when the required survey conditions were not met (e.g., seeing), simultaneous
multi-band observations for that tile were made later, some, even, in
subsequent years. We would like to note that we use the Cambridge Astronomical
Survey Unit (CASU) search service to group the multi-epoch data for the
construction of our band-merge catalog for CASU version 1.3 release. Finally,
during the multi-epoch campaign mostly Ks observations were made, which we use
for our variability study for the same release. Typically, for each tile,
around 80 observation were recorded for the Ks band and only one, the first
tile epoch, for the other four bands. We also note that the input data
preprocessed by the CASU VISTA Data Flow System pipeline (Emerson et al.,
2004), include photometric and astrometric corrections, made for each pawprint
and tile (González-Fernández et al., 2018) image and catalog pre-processed
data produced by CASU as standard FITS files (Hanisch et al., 2001).

The work in this study is based mainly on the 3rd aperture photometry data (as
suggested by CASU for stellar photometry in less crowded fields) from the
single pawprint CASU FITS catalogs. The magnitudes are extrapolated or total
magnitudes. The FITS catalogs were converted to ascii catalogs using an
adapted CASU cat_fit_list.f program. Over the course of the survey, given the
high data rates ($\sim 200$ MB/pawprint), several hundred terabytes of
astronomical data were produced. Systematic methods are required to extract,
in a homogeneous way, astrophysical information across $\sim 1000$ tiles for a
wide range of scientific goals. The ”first” phase of the pre-reduction process
is done by CASU’s VISTA Data Flow System (VDFS), which produces, among other
results, the pawprint, and tile catalogs that we use as our inputs.

###  2.1 Data for this work

Based on the VDFS results, we create two kind of post-processed data sources:

Tile catalogs:

The $J$, $H$, and $Ks$ consolidated master source list were defined from the
first epoch of observations. These data were selected because source color
information is available and also because the ”first” epoch data have
constraints that include more stringent observational and prepossessing data
requirements (seeing, photometric conditional limits, etc.) than for any other
epoch of observation. The cross-matching proximity algorithm employed was
modified for multiple bands. It is based on a KD-tree implementation from the
SciPy Python library. It reduces the matching complexity of traditional
algorithms from $O(N^{2})$ to $O(LogN)$111The cross-matching assumes that
objects A and B from two different catalogs are the same if object A is the
closest to object B, and object B is the closest to object A, with a threshold
of $1/3\arcsec$ used to define which sources are the same across different
bands. The KD-tree implementation was developed in collaboration with Erik
Tollerud for the Astropysics library (the precursor of Astropy). Its
implementation is also based on the Scipy package (Tollerud, 2012; Astropy
Collaboration et al., 2013; Oliphant, 2007).

Pawprint Stacks:

Each pawprint stack is associated to a Tile Catalog and includes the six
pawprint observation of the detectors for a single Ks-band epoch.

##  3 Environment: software, hardware, methods.

We took VDFS data catalogs of CASUVER 1.3 to begin our processing. The
processing phase was made within a custom multiprocess environment pipeline
called Carpyncho (Cabral et al., 2016), developed on top of the Corral
Framework (Cabral et al., 2017), the Python 222https://www.python.org/
Scientific Stack Numpy (Van Der Walt et al., 2011), Scipy (Oliphant, 2007),
and Matplotlib (Hunter, 2007)). Additional astronomy routines employed were
provided by the Astropy (Astropy Collaboration et al., 2013) and PyAstronomy
(Czesla et al., 2019) libraries. The feature extraction from the light-curve
data was handled using feets (Cabral et al., 2018) packages, which, in turn,
had been previously reingineered from an existing package called FATS (Nun et
al., 2015) and later incorporated as an affiliated package to astropy. The
feature extraction was executed in a 50 cores CPU computer provided by the
IATE 333http://iate.oac.uncor.edu/. The Jupyter Notebooks (Ragan-Kelley et
al., 2014) (for interactive analysis), Scikit-Learn (Pedregosa et al., 2011)
(for ML), and Matplotlib (for the plotting routine) are our tools of choice in
the exploratory phase with the aim of selecting the most useful model and
creating the catalog.

We considered the use of four diverse classical ML methods as classifiers in
this work. We first selected the support vector machine (SVM) (Vapnik, 2013),
a method that finds a maximum-margin solution in the original feature space
(using the so–called linear kernel) or in a transformed feature space for
non–linear solutions; we selected the radial-base-function, or RBF kernels, as
a second classifier in this case. Third we selected the K-nearest neighbors
(KNN) method (Mitchell, 1997), a simple but efficient local solution. Last, we
selected RF, an ensemble-based method, which requires almost no tuning and
provides competitive performance in most datasets.

In order to estimate performance measures, we used two different strategies.
In some cases, we relied on an internal cross-validation procedure, usually
known as K-fold CV. In this case, the available data is divided in K separate
folds of approximately equal length and we repeat K times the procedure of
fitting our methods on (K-1) folds, using the last one as a test set. Finally,
the K measurements are averaged. In other cases, we use complete tiles to fit
the classifiers and other complete tiles as test sets.

We selected three performance measures that are appropriate for highly
imbalanced binary classification problems. The RRLs are considered the
positive class and all other sources as the negative class (Section 4.4). RRLs
correctly identified by a classifier are called true positives (TP) and those
not detected are called false negatives (FN). Other sources wrongly classified
as RRLs are called false positives (FP), and those correctly identified are
called true negatives (TN). Using the proportion of these four outcomes on a
given dataset, we can define two complementary measures. Precision is defined
as TP / (TP + FP). It measures the fraction of real RRLs detected over all
those retrieved by the classifier. Recall is defined as TP / (TP + FN). It
measures the proportion of the total of RRLs that are detected by the
classifier. Classifiers that output probabilities can change their decisions
simply by changing the threshold at which a case is considered as positive.
Using a very low threshold leads to high Recall and low Precision, as more
sources are classified as positive. High thresholds, on the other hand, lead
to the opposite result, high Precision and low Recall. Precision and Recall
should be always evaluated at the same time. One of the best ways to evaluate
a classifier in our context (highly imbalanced binary problem) is to consider
Precision-Recall curves, in which we plot a set of pairs of values
corresponding to different thresholds. In general, curves that approach the
top-right corner are considered as better classifiers.

A more traditional measure of performance for binary problems is the accuracy,
defined as (TP + TN) / (TP + FP + TN + FN). In relation to accuracy, it is
also customary to create ROC-curves and to measure the area under the curve
(ROC-AUC) as a global performance measure. We show in Section 5 that these two
measures do not provide any information in our case.

##  4 Feature extraction: from light-curves to features

To identify and typify a variable-periodic star of any type, it is necessary
to precisely measure its magnitude variation in a characteristic time period.
To this end, we need to match each source present in a Tile Catalog with all
the existing observations of the same source in all available Pawprint-Stacks
and then to reconstruct its corresponding time series.

All the features that we extracted for this work correspond to sources that
share two characteristics:

  * •

An average magnitude between $12$ and $16.5$, where the VVV photometry is
highly reliable (Gran et al., 2015).

  * •

At least 30 epochs in its light curve to make the features more reliable, each
source needs to have

###  4.1 Proximity cross-matching

We take measurements of positions and magnitudes for every source in each
Pawprint Stack catalog with known tile identification number, as well as the
mid-time of the observation. Since the input source catalogs do not have star
identification numbers, we use cross-matching to determine the correspondence
between observed sources in the Tile Catalog and the Pawprint Stack using our
matching algorithm.

###  4.2 Date–time correction

The dates in the Pawprint Stack are recorded as the average date across all
the ”pawprints” involved in the ”stack” in Modified Julian Days (MJD) in UTC.
The recorded MJDs are subject to the light travel time delays caused by the
Earth’s orbital motion around the Sun. These delays are relevant for short-
period variable stars. In consequence, we transform our dates to Heliocentric
Julian Days (HJD), which modifies the MJD using differences in the position of
the Earth with respect to the Sun and the source (Eastman et al., 2010).

###  4.3 Period

Once the observation instances for each source have been identified and the
HJD calculated, the next step consists of calculating periods for our sources.
The VVV is an irregular time-sampled survey, so for sampling consistency
across the survey, we make the assumption that the frequency of observational
data is random. Figure 1 shows the observed magnitude of an RRL as a function
of HDJ time of measurement. No periodic signal is evident since the cadence
sampling is on the order of tens of days, which far exceeds the expected
period of RRL stars. We know RRL in the NIR have approximately sinusoidal
light curves with variations between $\sim 0.1$ and $\sim 0.5$ mags as opposed
to $\sim 0.2$ and $\sim 2.0$ mags in bluer broad-band filters, and if we
assume that therein may lie hidden useful features, to increase sensitivity in
the feature analysis that follows, we automatically analyse the Periodogram,
extract a period, and proceed by folding the RRL time series by this period.
We extract what we expect may be many useful features from the folded LC in
the following section.

Figure 1: Lightcurve of an RRL AB variable star from Gran et al. (2015)
identified with the ID VVV J2703536.01-412829.4. The horizontal axis is the
measurement date in heliocentric-corrected days (HJD) and the vertical one is
the source magnitude.

We note that to recover the period of the RRL source, we use the family of
Fast Lomb-Scargle methods (Lomb, 1976; Scargle, 1982; VanderPlas, 2018). This
method finds the least squares error fit of a sinusoid to the sampled data.
Folding the LC by this period puts the LC in-phase, as shown in Fig. 2,
allowing for the extraction of interesting features, for example the first
Fourier Components that have been found in other work to be sensitive to
[Fe/H] (e.g., Kovács & Walker, 2001).

Figure 2: Folded light curve for the same star as Fig. 1, using a period of
$\sim 0.757$ days. The horizontal axis is the phase, and the vertical shows
the source magnitude. For visual ease, two periods are shown.

###  4.4 Tagging: positive and negative samples

We selected our positive samples using the following procedure. First, we
identified all the variable stars in our Tile Catalogs (Section 2.1) by
proximity cross-matching (using a threshold of $1/3$ arc sec) mainly from the
variable-star catalogs of OGLE-III (Udalski, 2003; Soszyński et al., 2011),
OGLE-IV (Udalski et al., 2015; Soszyński et al., 2014) and from other catalogs
accessed through VizieR (Ochsenbein et al., 2000), listed in Appendix A we
give all literature RRL with common spatial fields to the VVV. From those
variable sources, we selected only the RRLs as our positive class and
discarded the rest. Then, any other source present in our Tile Catalogs were
considered as members of the negative class.

It is worth mentioning that our positive and negative classes cannot be
considered as completely accurate as they were not validated with a manual
(visual or spectroscopic) inspection of each source. In this context, it is
possible that a reduced number of already known RRLs were missed in our cross-
matching process and were, as a consequence, incorrectly labeled as negative.
These missed RRLs will most probably appear as FP to our classifiers,
increasing the estimated error levels of our methods slightly over the levels
that would be estimated with a verified dataset.

The selected sample of RRLs, Unknown (negative class), and VVV tile-ids are
shown in Table 1 and Fig. 3.

Table 1: Number of epochs in Ks band, RRL, and other stars by tile.

Tile | Epochs | Size | RRL | Unknown | %
---|---|---|---|---|---
b206 | 73 | 157825 | 47 | 157778 | 0.030%
b214 | 74 | 149557 | 34 | 149523 | 0.023%
b216 | 73 | 168996 | 43 | 168953 | 0.025%
b220 | 73 | 209798 | 65 | 209733 | 0.031%
b228 | 73 | 199853 | 28 | 199825 | 0.014%
b234 | 73 | 293013 | 126 | 292887 | 0.043%
b247 | 73 | 406386 | 192 | 406194 | 0.047%
b248 | 74 | 417839 | 218 | 417621 | 0.052%
b261 | 74 | 555693 | 252 | 555441 | 0.045%
b262 | 74 | 573873 | 314 | 573559 | 0.055%
b263 | 94 | 568110 | 317 | 567793 | 0.056%
b264 | 94 | 595234 | 307 | 594927 | 0.052%
b277 | 73 | 718567 | 429 | 718138 | 0.060%
b278 | 74 | 742153 | 436 | 741717 | 0.059%
b360 | 74 | 939110 | 669 | 938441 | 0.071%
b396 | 73 | 486639 | 15 | 486624 | 0.003%
Total | 1216 | 7182646 | 3492 | 7179154 | 0.049%

Note: Last column shows the proportion of RRLs over other sources, in
percentage.

Figure 3:  Map of the bulge tiles of the VVV survey. In blue, we mark the
tiles used for this paper.

###  4.5 Features extraction

The feature extraction process (including the period as previously described)
was carried out mostly using feets (Cabral et al., 2018). We obtained a set of
sixty-two features, which are summarized in Table 2. A brief description of
these features (accompanied by references) can be found in Appendix B. Also,
in the following paragraph, we include a more detailed explanation of a
particular new set of features considered in this work.

Table 2: Features selected for the creation of catalogs of RRL stars on tiles
of VVV

Features |  |  |
---|---|---|---
Amplitude | Autocor_length | Beyond1Std | Con
Eta_e | FluxPercentileRatioMid20 | FluxPercentileRatioMid35 | FluxPercentileRatioMid50
FluxPercentileRatioMid65 | FluxPercentileRatioMid80 | Freq1_harmonics_amplitude_0 | Freq1_harmonics_amplitude_1
Freq1_harmonics_amplitude_2 | Freq1_harmonics_amplitude_3 | Freq1_harmonics_rel_phase_1 | Freq1_harmonics_rel_phase_2
Freq1_harmonics_rel_phase_3 | Freq2_harmonics_amplitude_0 | Freq2_harmonics_amplitude_1 | Freq2_harmonics_amplitude_2
Freq2_harmonics_amplitude_3 | Freq2_harmonics_rel_phase_1 | Freq2_harmonics_rel_phase_2 | Freq2_harmonics_rel_phase_3
Freq3_harmonics_amplitude_0 | Freq3_harmonics_amplitude_1 | Freq3_harmonics_amplitude_2 | Freq3_harmonics_amplitude_3
Freq3_harmonics_rel_phase_1 | Freq3_harmonics_rel_phase_2 | Freq3_harmonics_rel_phase_3 | Gskew
LinearTrend | MaxSlope | Mean | MedianAbsDev
MedianBRP | PairSlopeTrend | PercentAmplitude | PercentDifferenceFluxPercentile
PeriodLS | Period_fit | Psi_CS | Psi_eta
Q31 | Rcs | Skew | SmallKurtosis
Std | c89_c3 | c89_hk_color | c89_jh_color
c89_jk_color | c89_m2 | c89_m4 | n09_c3
n09_hk_color | n09_jh_color | n09_jk_color |
n09_m2 | n09_m4 | ppmb |

Aside from the period, magnitude variation, and LC morphology, variable stars
may also be characterized by their temperatures or intrinsic color(s) as a
function of time. A color may be calculated as the difference of two standard
broad-band magnitudes with one important caveat: colors are sensitive to LOS
or external foreground reddening. In fact, its effect increases towards the
mid-plane of the Galaxy. At visual wavelengths that are within our range of
tiles, this is a serious problem because an absorption of several magnitudes
in the $V-band$ is observed near the galactic mid-plane towards the bulge of
the Galaxy. Extinction depends on dust grain type, size, spacing, and density,
but generally affects shorter wavelengths preferentially, hence causing
reddening . We correct the VVV data using different extinction laws and maps
obtained from the literature. We also include reddening-free indices that are
photometric indices chosen to be relatively insensitive to the effect of
extinction. For further information, see Catelan et al. (2011).

This work was planned and partly executed prior to the Schultheis et al.
(2014) 3D maps and the incorporation into the updated version of the BEAM-II
calculator that provides 3D distances. Thus, to maintain a homogeneous
analysis across all fields, we chose to correct for reddening by using the 2D
dust-maps from BEAM-I and dust laws employing the work of Gonzalez et al.
(2011) and Gonzalez et al. (2012) for which we estimate the absorption
coefficients of our sources with the BEAM-I - A VVV and 2MASS Bulge Extinction
And Metallicity Calculator 444http://mill.astro.puc.cl/BEAM/calculator.php.
Unfortunately the extinction maps of the VVV(X) zones were not available when
we developed this work, so in this paper, we limit ourselves to the VVV tiles.

The BEAM mappings have pixel resolutions of 2 arcmin x 2 arcmin or 6 arcmin x
6 arcmin, depending on the density of the red-clump bulge stars in the field.
In some cases, BEAM fails to calculate the extinction to a source. In those
situations, we replace it with the average of the extinction of the hundred
nearest neighbors, weighed by angular proximity.

Thus, for each source we obtain the $A_{v}$ absorption magnitude and use it to
calculate the reddening and convert it from the photometry of the 2MASS survey
(Skrutskie et al., 2006) to the photometry of VVV (González-Fernández et al.,
2018), given the two laws of extinction, Cardelli et al. (1989) and Nishiyama
et al. (2009). These laws describe the amount of extinction as a function of
wavelength.

Finally, the last stumbling block in calculating color is that, unfortunately,
the VVV multi-band observations were only made in the first epoch
systematically across all fields. Obtaining a good color-index is not possible
because, first, the classic calculation of color index (implemented in feets)
cannot be used, since it requires several observations in both bands to
subtract; and second, because the first epoch of two light curves may have
been observed in different phases, and the relations between the subtraction
of two points of different phases in two bands, even for the same star, is not
constant.

Under these conditions, the following strategy was chosen: The colors were
calculated using only the first observation epoch and both extinction laws; in
addition, a pseudo phase multi band was calculated, which is related to where
in the phase is the first epoch of the source, also including the number of
epochs; finally, the pseudo-magnitudes and pseudo-colors (with both extinction
laws) proposed in the work of Catelan et al. (2011) were calculated. These
values are reddening free indices.

The following color features were obtained. The reddening free indices are
shown with subscript “m2”, “m4”, and “c3”:

c89_c3 -

$C3$ Pseudo-color using the Cardelli et al. (1989) extinction law (Catelan et
al., 2011).

c89_ab_color -

Extinction-corrected color from the first epoch data between the band $a$ and
the band $b$ using the Cardelli et al. (1989) extinction law, where $a$ and
$b$ can be the bands $H$, $J,$ and $K_{s}$.

c89_m2, c89_m4 -

$m2$ (defined as $H-1.13(J-K_{s})$) and $m4$ (defined as $K_{s}-1.22(J-H)$)
pseudo-magnitudes using the Cardelli et al. (1989) extinction law (Catelan et
al., 2011).

n09_c3 -

$C3$ Pseudo-color using the Nishiyama et al. (2009) extinction law o (Catelan
et al., 2011).

n09_ab_color -

Extinction corrected color from the first epoch data between the band $a$ and
the band $b$ using the Nishiyama et al. (2009) extinction law, where $a$ and
$b$ can be the bands $H$, $J$ and $K_{s}$.

n09_m2, n09_m4

$m2$ and $m4$ pseudo-magnitudes using the Nishiyama et al. (2009) extinction
law (Catelan et al., 2011).

ppmb -

“Pseudo-Phase Multi-Band”. This index sets the first time in phase with
respect to the average time in all bands, using the period calculated by
feets.

$PPMB=frac(\frac{|mean(HJD_{H},HJD_{J},HJD_{K_{s}})-T_{0}|}{P}),$

where $HJD_{H}$, $HJD_{J}$ and $HJD_{K_{s}}$ are the time of observations in
the band $H$, $J$ and $K_{s}$; $T_{0}$ is the time of observation of maximum
magnitude in the $K_{s}$ band; $mean$ calculates the mean of the three times,
$frac$ returns only the decimal part of the number, and $P$ is the extracted
period.

###  4.6 Evaluation of feature subsets

We consider three subsets of features based on the root of feature type. We
bundle all color-derived features into the ”color feature” sub-type. Features
that are related to the extracted period are placed into the ”period based”
sub-type, including, for example, PeriodLS, Period_fit, Psi_eta, Psi_CS, ppmb
and all the Fourier components (See Appendix B). This subset depends on the
correct determination of the period as a first step, which makes them less
reliable than features that are measured directly from the observations. Then,
in the third and final subset, we include all classical features with this
last property that we call ”magnitude-based” features.

We designed a first experiment and established a handle on the predictive
power of our selected feature sets and restricted subsets. We selected four
tiles in the VVV Bulge footprint; $b234$, $b261$, $b278,$ and $b360$, as a
compromise between RRL number density and coverage. For each tile, we created
a reduced dataset with all the RRL stars and 5,000 unknown sources. We use RF
as classifier, with the setup explained in the next section (we will also
support this decision in that section). For every tile, we train four
classifiers: one with the full set of features, another with only the period +
magnitude subsets, a third with the color + magnitude subsets and a fourth
with the period + color feature subsets. Every classifier was then tested
using all other tiles as test sets (obviously using the same feature subsets).

Figure 4: Recall vs. precision curves of trained RF classifiers for different
feature subsets. On the left, center, and right panels, the test tiles are
$b234$, $b261$, and $b360$, respectively. All models were trained on tile
$b278$.  Figure 5: Precision for different subsets of features for the tiles
$b234$, $b261$, $b278,$ and $b360$ for a fixed recall ($\sim 0.9$). From left
to right: the first panel includes the total set of features; in the second,
only the magnitude- and period-based features is used; in the third, the
magnitude- and extinction-based ones; and, finally, the extinction- and
period-based features are used. In all cases, the rows indicate which tile was
used for train the RF classifier and the columns indicate the one used for the
test.

First, we evaluate the results using precision–recall curves. While all the
validation curves can be found in Appendix C, we present the curves trained
only with the tile $b278$ in Fig. 4. We selected $b278$ not only because it is
the one with the most RRL, but also for historical reasons; Baade’s window is
centred therein. It can be seen in Fig. 4 that the Period subset of features
contains most of the information since its removal causes the greatest loss in
performance for all test cases (the green curve corresponding to magnitude +
color is always the lowest and furthest from the 1-1 upper-right corner
point). However the Color subset of features is also significantly relevant
since its removal produces considerable loss to performance (but to a smaller
degree than does the removal of the Period subset). Finally, the Magnitude
subset is the least informative of all subsets. Its removal produces marginal
performance loss, which is somewhat expected if we consider RRL as standard
candles that are isotropically distributed about the bulge, the typical RRL’s
apparent magnitude should be weakly correlated to their class type. The full
set of curves shows the same behaviour (see Appendix C).

In order to make a quantitative comparison, we selected a fixed value of
Recall ($\sim 0.9$) and produced tables with the corresponding precision
values for a RF classifier trained with each subset of features. Figure 5
shows that in all cases, the results are consistent with the previous
qualitative analysis as far as the relative importance of our RRL feature
subtype.

##  5 Model selection

In a second experiment, we compared the performance of the four classifiers
considered in this work: SVM with linear kernel, SVM with RBF kernel, KNN and
RF.

Most of the classifiers have hyperparameters that need to be set to optimal
values. We carried out a grid search considering all possible combinations of
values for each hyperparameter over a fixed list. We used a 10 k-folds setup
on tile $b278$, considering the precision for a fixed recall of $\sim 0.9$ as
the performance measure.

With this setup, we selected the following hyperparameters values:

SVM-Linear:

$C=50$.

SVM-RBF:

$C=10$ and $\gamma=0.003$.

KNN:

$K=5$ with a $manhattan$ metric; also, the importance of the neighbor class
wasn’t weighted by distance.

RF:

We created $500$ decision trees with Information-Gain as metric, the maximum
number of random selected features for each tree is the $log_{2}$ of the total
number of features, and the minimum number of observations in each leaf is
$2$.

We trained the four classifiers using the same datasets and general setup as
in the previous experiments. Again, we selected a threshold for each
classifier that results in a fixed Recall of $\sim 0.9$. Figure 6 presents the
results of the experiment. In this case, we show (in addition to the
Precision), the ROC-AUC and the accuracy (for the same threshold as
precision).

Figure 6: Accuracy, ROC-AUC, and precision for different classifiers. All
details are the same as in Fig. 5. Figure 7: Recall vs. recision curves for
different classification methods. In the left, center, and right panel, the
test tiles are $b234$, $b261,$ and $b360$, respectively. All models were
trained on tile $b278$.

In the first place, all classifiers have a similar (and quite high) accuracy
and ROC-AUC for all test and train cases. Both measurements give the same
weight to positive and negative classes and, as a consequence, both are
dominated by the overwhelming majority of negative class. The comparison with
Precision values (with very different values for different cases) shows why
both measurements are not informative in the context of this work. When
analysing, then, only the precision results, the best method is clearly RF,
followed by KNN, SVM-RBF, and, finally, SVM-Linear.

In Fig. 7, we show the complete precision vs. recall curves for tile $b278$
(the complete set of curves for all tiles can be found in the Appendix D). All
figures show that in any situation RF is the best method (or comparable to the
best with few exceptions). Given these results, we proceed to selecting RF as
the choice classifier for the remainder of this work.

##  6 Sampling size and class imbalance

Figure 8: Recall vs. precision curves for different imbalance levels. All
models were trained on tile $b278$. The left, center, and right panels are for
the test tiles: $b234$, $b261,$ and $b360$, respectively. Test and
corresponding training sets have the same imbalance level.

The treatment of highly imbalanced datasets posses several problems to
learning, as is discussed, for example, in a recent work by Hosenie et al.
(2020). In particular, there is a problem that is usually not considered by
practitioners which considers whether learning be improved by changing the
balance between classes. Or whether, by changing the imbalance, we can
validate a real performance boost.

Among the various strategies in the literature, Japkowicz & Stephen (2002)
provides a scheme that is appropriate to our problem, achieved by subsampling
on the negative class, as we already implemented in the first two experiments.
Aside from a windfall in gained sensitivity towards the objective class,
reducing the sample size is desirable as it always leads to a considerable
decrease in the computational burden for any ML solution. This scheme is
implicitly used in most previous works; only a small sample of the negative
class is considered (Pashchenko et al., 2018; Mackenzie et al., 2016).

In our third experiment, we evaluate the relationship between the size of the
negative class and the performance of the classifiers. We use RF as
classifier, with the same setup as the previous section. We also use the same
four tiles and evaluation method (we train RF on one tile and test it on the
other three, using precision–recall curves).

We considered five different sizes for the negative class. First, we used all
sources available (”full sample”). Then we considered three fixed size
samples: 20000 sources (”large sample”), 5000 sources (”mid sample”), and 2500
sources (”small sample”). Finally, we considered a completely balanced sample,
equal to the number of RRLs in the tile (”one–to–one sample”). Each sample was
taken from the precedent one so for example, the small sample for tile $b278$
was taken from the mid sample of the same tile. In all cases, we used all the
RRLs in the corresponding tile.

Figure 8 shows the results of the experiment for tile $b278$ (results for all
datasets in all the experiments in this section can be found in Appendix E).
The figure shows that the one-to-one sample is clearly superior in performance
than are the imbalanced samples: there is a clear advantage in using balanced
datasets over highly imbalanced ones and not only in performance, but also in
running time.

There is a hidden flaw in our last results. We compared curves taken from
datasets with different imbalances. If we apply a one–to–one model to predict
a new tile, it will have to classify all the unknown sources, and not only a
sample taken from them. In that situation, the number of FP will certainly
increase from what the model estimated using a reduced sample. A fairer
comparison involves the use of a complete tile to estimate our performance
metrics, or at least the use of a corrected estimate of the performance that
takes into account the proportions in the evaluated sample and on the full
sample.

If we assume that the training sample is a fair sample (even though the RRL
number densities may vary from tile to tile) within a specific tile, say b278,
we could argue that local densities will be conserved and, as a consequence,
for each FP in the sample, there should be a higher number of incorrect
sources in the full dataset, with a proportion inverse to the sampling
proportion. This reasoning leads to the equation:

$FP^{*}=FP\times\frac{S_{F}}{S_{R}},$

where FP* is the estimate of the real number of FP, FP is the number measured
on the reduced sample, SF is the size of the Full sample, and SR is the size
of the reduced sample. Using this value, we can produce a corrected estimate
of the Precision on the sample:

$P^{*}=\frac{TP}{TP+FP^{*}}.$ (1)

where P* is the estimation of the corrected precision and TP is the value
estimated in the reduced sample. On the other side, recall values only use the
positive class for its estimation and do not need a sample size correction.

In Fig. 9, we show ”corrected precision” recall curves, using Equation 1, for
the same classifiers as in the previous figure. The correction seems to be a
good approximation only in the low Recall regions, where the number of FP and
TP is always bigger. For higher thresholds, when TP and FP become smaller,
there is a clear ”discrete” effect in the correction and it becomes useless.

Figure 9: Recall vs. corrected precision (P*) curves for different imbalance
levels. In the left, center, and right panels, the test tiles are $b234$,
$b261,$ and $b360$, respectively. All models were trained on tile $b278$. Test
sets have the same imbalance level than the corresponding training set.

Therefore, the only valid method to compare the different samplings is to
estimate performances using the complete tiles, with the associated
computational burden. Figure 10 shows the corresponding results. The
performances of the diverse sampling strategies are mostly similar, as the
only full sample results that are consistently better than the rest. Our
results show that when there are enough computational resources, the best
modeling strategy is to use all data available.

Figure 10: Recall vs Precision curves for different imbalance levels, using
always test sets of full size. In the left, center and right panel the test
tiles are $b234$, $b261$ and $b360$. All models were trained on tile $b278$.

##  7 Setting the final classifier

Throughout the previous sections, we show that the best modeling strategy is
to use a RF classifier with all the features available and training and
testing on complete tiles. We applied this recipe to all the 16 tiles included
in our study. Fig. 11 shows the corresponding results. There are very diverse
behaviours, depending on the training and test tiles. Overall, it seems that
we can select a recall value of 0.5 and find precision values over 0.5 in
almost all cases. It means that we can expect, in a completely automatic way,
to recover at least half of the RRLs in a tile paying the price of manually
checking a maximum of a false alarm for each correct new finding.

There is a last problem to be solved. In a real situation, we do not have the
correct labels and we cannot fix the threshold directly to secure the desired
recall level. We need to set the expected recall value using the training
data, and assume it will be similar to the recall that is actually observed in
the test data. We used a 10-fold CV procedure to estimate the performance of
our classifiers on training data only and selected the corresponding
thresholds that leads to a recall of $\sim 0.5$. In Fig. 12, we show the full
results of our final individual classifiers. Each panel shows the prediction
of a different tile using all the classifiers trained on the other tiles. We
added a red point to each curve showing the observed recall value on each test
set for the thresholds set by 10 folds CV. We find that in most situations,
our procedure leads to real recall values in the 0.3 to 0.7 zone.

Figure 11: Tables of precision and recall, training and testing with complete
data of the 16 tiles. The last (separated) row shows the estimated precision
(left) achieved by selecting a threshold value that sets the recall to $\sim
0.5$ (right), training and testing on each tile with a 10 K-Folds procedure.
The upper rows shows the recall and precision values observed on the test set
(columns) when training on diverse tiles (rows), using the thresholds selected
by 10 K-Folds (used on the last row).

On the other hand, another clear result from Figs. 11 and 12 is that
classifiers trained over different tiles produce very diverse results on a
given tile and, of course, we cannot know in advance which classifier will
work better for each tile. A practical solution to this problem is the use of
ensemble classifiers (Rokach, 2010; Granitto et al., 2005). An ensemble
classifier is a new type of classifier formed by the composition of a set of
(already trained) individual classifiers. In order to predict the class of a
new source, the ensemble combines the prediction of all the individual
classifiers that form it. The ensemble is in fact using information from all
tiles at the same time with no additional computational cost. This additional
information can produce improved performances in some cases and, more
importantly, it usually produces more uniform results over diverse situations
(diverse tiles). In our case, we combined all classifiers trained on the
diverse tiles in order to make predictions of a new tile. In this evaluation
phase, we exclude the classifier trained on the tile that is being tested. Our
combination method is to average over all classifiers (with equal weight) the
estimated probabilities of being an RRL. The threshold for this ensemble
classifiers is taken as the average of individual thresholds.

Figure 12 also shows the results of the ensemble for all tiles in the paper as
a bold black line. It is clear from the figure that the ensemble is equal or
better, on average, than any of the individual classifiers. In Fig. 13, we
show the specific recall and accuracy values for each tile using the ensemble
classifier with the previous setup. We also include the average of each column
of Fig. 11 as a comparison (the average of all classifiers). It is clear from
the figure that our automatic procedure produce a classifier that works well
in all tiles and produce the expected result of recall and precision (all but
one recall values are bigger than 0.3 and all but one precision values are
bigger than 0.5).

Figure 12: Precision vs. recall curves for the final classifiers on each test
tile. In each panel, the colored curves represent a classifier trained with a
tile, while the black curve is the ensemble classifier. Red dots correspond to
a precision and recall fixed at a threshold equivalent to the recall $\sim
0.5$ when training and testing on the same tile with 10 k-folds. The blue
stars represent the working point selected by averaging the thresholds
corresponding to the red points. Figure 13: Precision (left) and recall
(right) values of the ensemble classifier (see text). For comparison, we
include in the first row the average of the values or each individual
classifier in the ensemble, as shown in Fig. 11.

##  8 Analysis of misidentifications

Figure 14: Light curves in time (blue on the left) and in two phases (red on
the right) for the five FPs with the highest probability of being an RRL. The
titles of each left panel shows probabilities, tile, and identifier of the
source. The titles on the right panel shows the estimated period, right
ascension ($RA_{Ks}$), and declination ($Dec_{Ks}$). Figure 15:  Same as Fig.
14 for the five FNs with the lowest probability of being an RRL

There are two classes of possible errors (misidentifications) for our ensemble
classifier. First, there are FPs. These are sources that are identified as
RRLs by our method but are not registered as so in the catalogs (variable
stars discarded in another survey or present in new catalogs that were not
taken into account in our work). In Fig. 14, we show the light curves
corresponding to the five FPs identified with the highest probabilities. It is
clear that all five are potential RRLs, with clear periodicity. Most FPs are
similar to these and based on this evidence, we consider them as RRL
candidates more than errors.

Second, there are FNs. These are already known RRLs that our pipeline assigned
to the unknown-star class. In Fig. 15 we show the light curves corresponding
to the five FNs identified with the lowest probabilities. These low-
probability FNs are stars for which available observations on VVV were not
adequate as to find the periodicity in the light curves for a correct features
extraction. Typically, these errors are more related to the limitations on the
feature extraction process than to limitations of the classifier itself.

We also found another distinct group of FNs with the help of an innovative use
of ML methods. We devised a new experiment looking for explanations about why
some RRLs are misidentified. Our hypothesis is that if there are some patterns
that clearly distinguishes FNs and TPs (RRLs that are correctly and
incorrectly identified), a good classifier will be able to learn the patterns.

We started by creating a dataset to learn. To achieve this we trained a RF
with the complete data of all available tiles. RF has a simple and unbiased
way to estimate the classification probabilities of training samples that is
equivalent to a K-fold procedure, called Out-of-Bag (OOB) estimation (Breiman,
2001). We set the threshold of our RF to obtain an OOB Recall $\sim 0.9$.
Next, we used this RF to OOB-classify all RRLs in our dataset and split them
in two classes: TPs or FNs. At this point, we create a reduced dataset of only
RRLs, with a new positive class of those correctly identified in the last
step, and a new negative class with the rest of the RRLs. Finally, we train a
new RF on this reduced dataset, trying to differentiate between these two kind
of RRLs. We find that this RF has an OOB error level well below random
guessing, showing that there are patterns that discriminate these two classes.

The RF method is also capable of estimating the relative importance of each
feature in the dataset for the classification task (again, see Breiman 2001
for a detailed explanation of the procedure). Figure 16 presents in a boxplot
the relative importance of all our features for the problem of splitting
between the FN and TP RRLs. We can see that there are six features that are
outliers from the entire set, which means that these six features are highly
related to the learned patterns. The six features are
freq1_harmonic_amplitude_0 (the first amplitude of the Fourier component),
Psi_CS (folded range cumulative sum), ppmb (pseudo-Phase Multi-Band), PeriodLS
(the period extracted with the Lomb-Scargle method), Psi_eta (phase folded
dependency of the observations respect or the previous one), and Period_fit
(the period false alarm probability). It is interesting to note that all are
period-based.

Figure 16:  Distribution of the relative feature importance of a RF classifier
trained with all the RRLs and the objective class of TP and FN; see text for
more.  Figure 17:  Distributions of the TP and FN classes for the RRL stars in
all tiles according to the three most important features to differentiate
them. In the top row all the stars are compared, in the mid row only the RRL-
AB type, and in the low row the RRL-C type. Left column corresponds to
period_fit, mid column to Psi_eta and right column to PeriodLS. All features
were converted to z-scores and are dimensionless.

For a more detailed analysis, we compared how the FN and TP classes are
distributed in our sample of RRLs according to these six features. The
complete results can be found in Appendix F, but we show in Fig. 17 the
results for the three more relevant features. In the figure, we also divided
the sample into RRLs-AB and RRLs-C types because we found with these plots
that the shoulders in the distribution are related to RRL subtypes. In the
lower panel of the figure, there is an evident relationship between subtypes
and FNs. In fact, we find that from the 155 FN in our dataset, 98 are RRL-C.
The complete analysis shows that we have two main sources of FN; first some
sources are poorly observed in VVV, leading to poor estimates of periods and
all derived features; second, RRL-C stars are much more difficult to identify
than AB type RRL.

##  9 Data release

All the data produced in this work, light curves, features for each source and
our catalog of RRL candidates are available to the community via the Carpyncho
tool-set facility 555https://carpyncho.github.io/ (Cabral et al., 2020). At
this time, we share $\sim 14.3$ million $K_{s}$ band light curves and $\sim
11.2$ million features, all for curves that have at least 30 epochs.

We also publish a list of $242$ RRL candidates distributed in 11 tiles as
shown in Table 3 (The full list of candidates can be found in Appendix F). We
find using our final classifier a large sample of 117 candidate RRL in tile
b396 and it is our estimate that more than half of these candidates are bona
fide RRL stars. This number is between one and two orders of magnitude higher
than the RRL that were previously known, in part because this region lies
mostly outside of the OGLE-IV sensitivity footprint for RRL and our study was
concluded prior to the Gaia data Release 2 and RRL catalogs of Clementini et
al. (2019). Nevertheless a comparison of our RRL candidates and those of
Clementini et al. (2019) can be used to refine our method (as we hope to do
after more data is released onto the Carpyncho facility). Given the importance
of RRL as dynamical test particles for the formation and evolution modeling of
the bulge in the Galaxy, we promptly present our first data release of RRL, as
described above, to the community.

Table 3: Number of RRL candidates per tile. Tile | Candidates
---|---
b206 | 3
b214 | 17
b216 | 21
b220 | 15
b228 | 18
b234 | 23
b247 | 2
b248 | 1
b263 | 1
b360 | 24
b396 | 117

##  10 Summary, conclusions, and future work

In this work, we derive a method for the automatic classification of RRL
stars. We begin by discussing the context of RRL as keystones for stellar
evolution and pulsation astrophysics and their importance as rungs on the
intra- and extragalactic distance scale ladder, as well as for galaxy
formation models. We base our models on RRL that have previously been
classified in the literature prior to Gaia DR2. We match VVV data to those
stars, and extract features using the feets package affiliated to astropy,
presented in Cabral et al. (2018). We explore the difficulty inherent in
existing semi-automatic methods as found in the literature and set out to test
some of these pitfalls to learn from them to build a more robust classifier of
RRL for the VVV survey based on a newly crafted ML tool.

We start by considering a set of traditional period- and intensity-based
features, plus a subset of new pseudo-color and pseudo-magnitudes-based
features. We show that period-based features are the most relevant for RRLs
identification and that the incorporation of color features also add
information that improves performance. We next discuss the choice of
classifier and find that as in previous studies, classifiers based on multiple
classification trees are the best for this particular task. We used RF in this
work because of its reliability and internal OOB estimations, but boosting is
a valid option, as shown by Elorrieta et al. (2016).

We discuss extensively the need for appropriate quality metrics for these
highly imbalanced problems and the issues related to the use of reduced
samples. We show that precision-recall tables and curves are well suited to
these problems and that the best strategy for obtaining good estimations of
performance is to work on the complete, highly imbalanced dataset.

The last step of the pipeline is the model selection process. We show that the
threshold of our classifiers can be correctly selected using the internal
K-fold method and that the use of an ensemble classifier can overcome the
problem of selecting of the appropriate tile for modeling.

In the last sections of this work, we discuss how this analysis led us to a
reliable and completely automatic selection of RRL candidates, with a
precision$\geq 0.6$ and a recall$\geq 0.3$ in almost all tiles. Averaging over
all tiles except b396 we have a recall of 0.48 and a precision of 0.86, or
what is the same, we expect to recover the $48\%$ of RRLs candidate stars in
any new tile, and more than 8 out of 10 of them will be confirmed as RRLs. We
left tile b396 aside because there were previously no classified RRL in this
region.

We also discuss the most common errors of the automatic procedure, related to
the low quality of some observations and the more difficult identification of
the RRL-C type. All data produced in this work, including light curves,
features, and the catalog of candidates has been released and made public
trough the Carpyncho tool.

In terms of future work, we propose to extend the data collection contained in
Carpyncho to more tiles in the VVV, along with the complete publication of the
internal database (Tile-Catalogs and Pawprint-Stacks). We wish to cross-check
our candidate RRL with those of Gaia-2 and determine metallicities for our RRL
based on our derived light curves to make spatial metallicity maps for our RRL
and independent dust extinction maps. We hope to tune our classifier for other
variables. In particular, to delta-Scuti type variables that have historically
been misclassified as RRL, and to determine confusion thresholds for our model
RRL candidates. Although the color features used for this work are based on
the first epoch VVV data we hope to be able to incorporate more color epochs
from a subsample of tiles for which that information exists in later epochs.
This will allow for the introduction of color features that are not fixed in
time. We also wish to tune our classifier to other populations of RRL, for
example, the disk population and the Sag dSph population of RRL that both fall
within the foot-print of the VVV and VVV(X) data. We could hope to tune our
classifier for halo RRL, given that some halo RRL are also within the VVV and
VVV(X) data. We also hope to explore the change of the photometric base of our
VVV RRL data, from one that is based on aperture magnitudes to one that is
based on PSF magnitudes, as derived from the work of Alonso-García et al.
(2018). This will permit a higher level of precision in feature observables
for future studies of RRL.

###### Acknowledgements.

The authors would like to thank to their families and friends, and also IATE
astronomers for useful comments and suggestions. This work was partially
supported by the Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET, Argentina) and the Secretaría de Ciencia y Tecnología de la
Universidad Nacional de Córdoba (SeCyT-UNC, Argentina). F.R and J.B.C are
supported by fellowships from CONICET. P.M.G. acknowledge partial support from
PIP0066-CONICET. Some processing was achieved with Argentine VO (NOVA)
infrastructure, for which the authors express their gratitude. SG would like
to thank Ken Freeman for infecting him with the RRL Bug and Marcio Catelan,
Dante Minniti, Roberto Saito, Javier Alonso and other members of the VVV
scientific team for their genuine intellectual generosity. We gratefully
acknowledge data from the ESO Public Survey program ID 179.B-2002 taken with
the VISTA telescope and products from the Cambridge Astronomical Survey Unit
(CASU). J.B.C. thanks to Jim Emerson for extend explanations about the VISTA
instruments, and finally Bruno Sánchez, Laura Baravalle and Martín Beroiz for
the continuous support and friendship. This research has made use of the
http://adsabs.harvard.edu/, Cornell University xxx.arxiv.org repository,
adstex (https://github.com/yymao/adstex), astropy and the Python programming
language.

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##  Appendix A List of catalogs accessed in this work

  1. 1.

The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable
Stars. XI. RR Lyrae Stars in the Galactic Bulge (Soszyński et al. 2011).

  2. 2.

Gaia Data Release 2. Summary of the contents and survey properties (Gaia
Collaboration et al. 2018).

  3. 3.

Metallicity Analysis of MACHO Galactic Bulge RR0 Lyrae Stars from their Light
Curves (Kunder & Chaboyer 2008).

  4. 4.

VizieR Online Data Catalog: Updated catalog of variable stars in globular
clusters (Clement+ 2017) (Clement 2017).

  5. 5.

The International Variable Star Index (VSX) (Watson et al. 2006).

  6. 6.

The Extinction Toward the Galactic Bulge from RR Lyrae Stars (Kunder et al.
2008).

  7. 7.

A mid-infrared study of RR Lyrae stars with the Wide-field Infrared Survey
Explorer all-sky data release (Gavrilchenko et al. 2014).

  8. 8.

Blazhko effect in the Galactic bulge fundamental mode RR Lyrae stars - I.
Incidence rate and differences between modulated and non-modulated stars
(Prudil & Skarka 2017).

  9. 9.

The Gaia mission (Gaia Collaboration et al. 2016).

  10. 10.

Mapping the outer bulge with RRab stars from the VVV Survey (Gran et al.
2016).

  11. 11.

Establishing the Galactic Centre distance using VVV Bulge RR Lyrae variables
(Majaess et al. 2018).

  12. 12.

Over 38000 RR Lyrae Stars in the OGLE Galactic Bulge Fields (Soszyński et al.
2014).

  13. 13.

General catalogue of variable stars: Version GCVS 5.1 (Samus’ et al. 2017).

  14. 14.

Machine-learned Identification of RR Lyrae Stars from Sparse, Multi-band Data:
The PS1 Sample (Sesar et al. 2017).

  15. 15.

Catalogue of variable stars in open cluster fields (Zejda et al. 2012).

  16. 16.

Cyclic variations in O-C diagrams of field RR Lyrae stars as a result of LiTE
(Liška et al. 2016).

  17. 17.

Catalog of Fundamental-Mode RR Lyrae Stars in the Galactic Bulge from the
Optical Gravitational Lensing Experiment (Collinge et al. 2006).

  18. 18.

The Optical Gravitational Lensing Experiment. The Catalog of Periodic Variable
Stars in the Galactic Bulge. I. Periodic Variables in the Center of the
Baade’s Window (Udalski et al. 1994).

  19. 19.

The OGLE Collection of Variable Stars. Classical, Type II, and Anomalous
Cepheids toward the Galactic Center (Soszyński et al. 2017).

##  Appendix B Features related to feets

Most of these descriptions are adapted from the feets documentation
(https://feets.readthedocs.io).

Amplitude ($A_{K_{s}}$)

the half of difference between the median of the $5\%$ upper values and the
median of the $5\%$ of the lower values of the $K_{s}$ band (Richards et al.
2011).

Autocor_length

Auto-Correlation Length is the cross correlation of the signal with itself. Is
useful to find patterns like noise hidden periodic signal (Kim et al. 2011).

Beyond1Std

Percentage of points beyond one standard deviation from the weighted mean. For
a normal distribution, it should take a value close to $0.32$ (Richards et al.
2011).

Con

“Consecutive points” . number of three consecutive points greater or lesser
than $2\sigma$ (normalized by $N-200$). This index was created by the OGLE
survey to find variable stars (Wozniak 2000; Kim et al. 2011).

Eta_e ($\eta^{e}$)

Verify the dependency of the observations respect or the previous ones (Kim et
al. 2014).

FluxPercentileRatioMid

The “Middle flux percentiles Ratio” characterizes the distributions of sorted
magnitudes based on their fluxes percentiles666Flux or luminosity is the
amount of energy given by some star in one unit of time. If $F_{i,j}$ is the
difference between the flux percentil $j$ and the flux percentile $i$ we can
calculate:

  * •

$FluxPercentileRatioMid20=F_{40,60}/F_{5,95}$

  * •

$FluxPercentileRatioMid35=F_{32.5,67.5}/F_{5,95}$

  * •

$FluxPercentileRatioMid50=F_{25,75}/F_{5,95}$

  * •

$FluxPercentileRatioMid65=F_{17.5,82.5}/F_{5,95}$

  * •

$FluxPercentileRatioMid80=F_{10,90}/F_{5,95}$

Fourier components

The first three Fourier components for the first three period candidates.
Freqk_harmonics_amplitude_i y Freqk_harmonics_rel_phase_j represents the i-nth
amplitude ant the j-nth phase of the signal for the k-nth period. (Richards et
al. 2011).

Gskew

Median-of-magnitudes based measure of the skew

$Gskew=m_{q3}+m_{q97}-2m$

Where: $m_{q3}$ is the median of magnitudes lesser or equal than the quantile
3, $m_{q97}$ is the median of magnitudes greater or equal than the quantile
97; and $m$ is the median of magnitudes (Bowley 1901).

LinearTrend

Linear tendency. the slope of the linear regression of the data (Richards et
al. 2011).

MaxSlope

Maximum absolute slope between two consecutive observations(Richards et al.
2011).

Mean

Mean of magnitudes (Kim et al. 2014).

MedianAbsDev

The Median of the absolute deviations, is defined as the median of the
difference of every observed magnitude with the median of the entire time-
serie. In simbols $MedianAbsoluteDeviation=median(|mag-median(mag)|)$
(Richards et al. 2011).

MedianBRP

\- The “Median Buffer Range Percentage” can be interpreted as the percentage
closest to the median. Proportion of the magnitudes between the 10th part of
the total range around the median (Richards et al. 2011).

PairSlopeTrend

Considering the last 30 (time-sorted) measurements of source magnitude, the
fraction of increasing first differences minus the fraction of decreasing
first differences (Richards et al. 2011).

PercentAmplitude

Largest percentage difference between either the max or min magnitude and the
median. (Richards et al. 2011).

PercentDifferenceFluxPercentile

Ratio of $F_{5,95}$ (converted to magnitude) over the median magnitude
(Richards et al. 2011).

PeriodLS

Lomb-Scargle Period. Is the first period extracted with the Lomb-Scargle
method (Kim et al. 2011, 2014; VanderPlas 2018).

Period_fit

“Period Fitness”. Measure of the reliability of the PeriodLS value. The
reliability falls as it approaches the value of $1$ (Kim et al. 2011, 2014;
VanderPlas 2018).

Psi_CS ($\Psi_{CS}$)

$R_{C}S$ applied to the phase-folded light curve (generated using the period
as estimated from the Lomb-Scargle method) (Kim et al. 2011, 2014).

Psi_eta ($\Psi_{\eta})$

$\eta^{e}$ index calculated from the folded light curve (generated using the
period estimate from the Lomb-Scargle method) (Kim et al. 2011, 2014).

Q31 ($Q_{3-1}$)

Is the difference between the third quartile, $Q_{3}$ , and the first
quartile, $Q_{1}$ of magnitudes (Kim et al. 2014).

Rcs ($R_{CS}$)

“Range Cumulative Sum” is literally the range of the cumulative sum of
magnitudes normalized by $1/{N\sigma}$, where $N$ is the number of
observations and $\sigma$ is the standard deviation of magnitudes.(Kim et al.
2011).

Skew

The skewness of magnitudes (Kim et al. 2011).

SmallKurtosis

Small sample kurtosis of the magnitudes. For a normal distribution this values
is $\sim 0$ (Richards et al. 2011).

Std

The standard deviation of magnitudes (Richards et al. 2011).

##  Appendix C Feature analysis

Figure 18: Precision vs. recall curves for different features sub-sets for
different tile combinations in training and testing. Every plot is one
combination of train and test and every curve is one feature subset.

##  Appendix D Model selection

Figure 19: Precision vs. recall curves for different models for different tile
combinations in training and testing. Every plot is one combination of
training and testing and every curve is one model.

##  Appendix E Unbalance analysis

Figure 20:  Precision vs. recall curves training and testing on the same
sample sizes (one-to-one, small, mid, large, and full) for different tile
combinations in training and testing. Every plot is one combination of
training and testing and every curve is one sample size.  Figure 21:
Precision∗ vs. recall curves training on different sample sizes (one-to-one,
small, mid, large, and full) and testing on the full sample for different tile
combinations in training and testing. Every plot is one combination of train
and test and every curve is one sample size  Figure 22:  Precision vs. recall
curves training on different sample sizes (one-to-one, small, mid, large, and
full) and testing on the full sample for different tile combinations in
training and testing. Every plot is one combination of train and test and
every curve is one sample size

##  Appendix F Analysis of misidentifications

Figure 23:  Distributions of the TP and FN classes for the RRL stars in all
tiles according to the six most important features to differentiate them. In
the top row, all the stars are compared, in the mid row, only the RRL-AB type,
and in the low row, the RRL-C type. From right to left, the columns
corresponds to: period_fit, Psi_eta, PeriodLS, ppmb, Psi_CS, and
Freq1_harmonics_amplitude_0. All features were converted to z-scores and are
dimensionless.

##  Appendix G Catalog

Table 4: Candidates 1–61 candidates to RRL sorted by probability of being an
RRL.

| ID | Tile | $RA_{K_{s}}$ | $Dec_{K_{s}}$ | Period | Mean Mag. | Amplitude | Prob.
---|---|---|---|---|---|---|---|---
|  |  | (J2000, Deg.) | (J2000, Deg.) | (Days) |  |  |
1 | VVV J175011.98-184159.6 | b396 | 267.54992 | -18.69989 | 0.62997 | 14.306558 | 0.17800 | 0.852
2 | VVV J175228.32-174548.8 | b396 | 268.11802 | -17.76356 | 0.61577 | 14.396361 | 0.16400 | 0.849
3 | VVV J175008.72-180538.0 | b396 | 267.53635 | -18.09389 | 0.51236 | 14.313710 | 0.17350 | 0.838
4 | VVV J175033.09-184405.2 | b396 | 267.63789 | -18.73477 | 0.56379 | 14.307435 | 0.15550 | 0.837
5 | VVV J174957.06-185056.2 | b396 | 267.48776 | -18.84894 | 0.63820 | 14.029275 | 0.16350 | 0.828
6 | VVV J175125.96-181823.4 | b396 | 267.85817 | -18.30650 | 0.56330 | 13.940565 | 0.15800 | 0.827
7 | VVV J175110.05-181645.5 | b396 | 267.79188 | -18.27930 | 0.55656 | 14.007191 | 0.14900 | 0.825
8 | VVV J175137.35-182130.6 | b396 | 267.90562 | -18.35851 | 0.55214 | 14.010037 | 0.17300 | 0.825
9 | VVV J175252.40-183041.0 | b396 | 268.21834 | -18.51140 | 0.55839 | 14.610723 | 0.15575 | 0.825
10 | VVV J180447.32-401217.9 | b216 | 271.19715 | -40.20497 | 0.60829 | 14.435616 | 0.16500 | 0.812
11 | VVV J173339.15-285501.0 | b360 | 263.41313 | -28.91696 | 0.55995 | 14.503172 | 0.16450 | 0.808
12 | VVV J174806.73-183056.2 | b396 | 267.02806 | -18.51561 | 0.49765 | 13.864885 | 0.17500 | 0.804
13 | VVV J175153.98-175623.5 | b396 | 267.97493 | -17.93985 | 0.64240 | 13.658129 | 0.14800 | 0.804
14 | VVV J175244.49-181004.1 | b396 | 268.18538 | -18.16781 | 0.64917 | 13.571275 | 0.16850 | 0.801
15 | VVV J174910.42-182413.2 | b396 | 267.29342 | -18.40367 | 0.79211 | 13.726371 | 0.16125 | 0.796
16 | VVV J175222.29-181747.2 | b396 | 268.09285 | -18.29646 | 0.56563 | 13.610202 | 0.15900 | 0.791
17 | VVV J174739.05-182535.8 | b396 | 266.91271 | -18.42661 | 0.57735 | 14.112968 | 0.16725 | 0.791
18 | VVV J181233.16-345705.0 | b234 | 273.13815 | -34.95137 | 0.56021 | 14.337519 | 0.16450 | 0.787
19 | VVV J175017.49-180054.8 | b396 | 267.57288 | -18.01521 | 0.46292 | 14.436433 | 0.16850 | 0.786
20 | VVV J174901.70-180909.8 | b396 | 267.25710 | -18.15271 | 0.54296 | 13.406229 | 0.16800 | 0.785
21 | VVV J175055.31-183812.4 | b396 | 267.73045 | -18.63678 | 0.48131 | 14.234667 | 0.16350 | 0.785
22 | VVV J175221.02-174952.7 | b396 | 268.08760 | -17.83132 | 0.69004 | 14.610855 | 0.15650 | 0.776
23 | VVV J175115.83-174827.0 | b396 | 267.81597 | -17.80751 | 0.70771 | 13.750252 | 0.13600 | 0.775
24 | VVV J180201.49-395209.4 | b216 | 270.50620 | -39.86927 | 0.73553 | 14.096023 | 0.15350 | 0.766
25 | VVV J175001.34-181438.8 | b396 | 267.50558 | -18.24411 | 0.62780 | 14.558194 | 0.17975 | 0.764
26 | VVV J175010.14-183116.9 | b396 | 267.54227 | -18.52135 | 0.47182 | 14.711811 | 0.17300 | 0.762
27 | VVV J174924.53-180746.9 | b396 | 267.35220 | -18.12969 | 0.43561 | 14.542198 | 0.16450 | 0.761
28 | VVV J175123.43-185840.2 | b396 | 267.84763 | -18.97782 | 0.46822 | 13.648694 | 0.17150 | 0.759
29 | VVV J175305.84-175213.9 | b396 | 268.27435 | -17.87053 | 0.70312 | 14.216122 | 0.12600 | 0.757
30 | VVV J184352.31-245246.4 | b214 | 280.96797 | -24.87954 | 0.74086 | 14.267529 | 0.15825 | 0.755
31 | VVV J175038.85-174859.9 | b396 | 267.66187 | -17.81664 | 0.48545 | 14.521467 | 0.19125 | 0.755
32 | VVV J184022.72-235648.3 | b228 | 280.09466 | -23.94674 | 0.46551 | 14.685971 | 0.16550 | 0.752
33 | VVV J175339.18-175938.2 | b396 | 268.41323 | -17.99396 | 0.47261 | 14.356423 | 0.19000 | 0.751
34 | VVV J175129.85-175305.1 | b396 | 267.87437 | -17.88475 | 0.55569 | 14.347469 | 0.16850 | 0.746
35 | VVV J175039.31-183449.6 | b396 | 267.66379 | -18.58044 | 0.53109 | 14.455916 | 0.14850 | 0.744
36 | VVV J180212.50-394734.6 | b216 | 270.55208 | -39.79296 | 0.48912 | 13.581348 | 0.12550 | 0.740
37 | VVV J175134.20-181323.8 | b396 | 267.89250 | -18.22328 | 0.45851 | 14.495703 | 0.16800 | 0.733
38 | VVV J180301.33-402052.2 | b216 | 270.75555 | -40.34783 | 0.59245 | 14.108386 | 0.12500 | 0.732
39 | VVV J184645.80-243125.1 | b214 | 281.69085 | -24.52364 | 0.45093 | 14.274632 | 0.16700 | 0.731
40 | VVV J174942.51-173958.2 | b396 | 267.42712 | -17.66616 | 0.57721 | 14.137341 | 0.16500 | 0.727
41 | VVV J180136.63-395020.7 | b216 | 270.40262 | -39.83907 | 0.65193 | 14.284261 | 0.13950 | 0.727
42 | VVV J174941.08-183449.9 | b396 | 267.42116 | -18.58052 | 0.50584 | 14.818811 | 0.18050 | 0.723
43 | VVV J181058.84-335034.6 | b234 | 272.74518 | -33.84294 | 0.52978 | 14.666947 | 0.17350 | 0.721
44 | VVV J184441.70-241120.2 | b214 | 281.17373 | -24.18896 | 0.50501 | 14.437030 | 0.16300 | 0.721
45 | VVV J175205.44-183754.7 | b396 | 268.02266 | -18.63185 | 0.46989 | 14.627549 | 0.19850 | 0.719
46 | VVV J184400.27-252157.8 | b214 | 281.00111 | -25.36606 | 0.60092 | 14.390530 | 0.14450 | 0.717
47 | VVV J180649.57-315928.9 | b263 | 271.70656 | -31.99136 | 0.52976 | 14.088983 | 0.16700 | 0.717
48 | VVV J184056.31-244819.8 | b228 | 280.23464 | -24.80550 | 0.51275 | 14.201942 | 0.14650 | 0.716
49 | VVV J181338.52-341911.1 | b234 | 273.41051 | -34.31974 | 0.58462 | 14.159077 | 0.12650 | 0.714
50 | VVV J174756.74-182743.2 | b396 | 266.98641 | -18.46201 | 0.41117 | 14.609176 | 0.17600 | 0.709
51 | VVV J175042.93-181324.5 | b396 | 267.67887 | -18.22347 | 0.48632 | 14.535879 | 0.20100 | 0.708
52 | VVV J174923.39-180950.5 | b396 | 267.34745 | -18.16404 | 0.42305 | 14.362825 | 0.16550 | 0.705
53 | VVV J181059.73-345617.5 | b234 | 272.74887 | -34.93819 | 0.42151 | 14.415356 | 0.14900 | 0.704
54 | VVV J172934.42-293248.5 | b360 | 262.39341 | -29.54682 | 0.60060 | 14.289349 | 0.16000 | 0.700
55 | VVV J173346.21-290936.8 | b360 | 263.44255 | -29.16023 | 0.74245 | 14.218582 | 0.15050 | 0.698
56 | VVV J174803.95-181602.0 | b396 | 267.01645 | -18.26724 | 0.58704 | 14.803940 | 0.15400 | 0.692
57 | VVV J181301.46-342837.9 | b234 | 273.25609 | -34.47721 | 0.58798 | 14.180629 | 0.14250 | 0.683
58 | VVV J181748.88-345945.4 | b220 | 274.45365 | -34.99594 | 0.43784 | 14.434854 | 0.16200 | 0.680
59 | VVV J183750.44-241844.8 | b228 | 279.46016 | -24.31244 | 0.45987 | 14.459764 | 0.17775 | 0.679
60 | VVV J184520.08-243150.0 | b214 | 281.33366 | -24.53054 | 0.48765 | 13.647779 | 0.12800 | 0.679
61 | VVV J175144.83-185532.0 | b396 | 267.93678 | -18.92556 | 0.67278 | 14.511625 | 0.14775 | 0.679

Note: Column (1) is the order; column (2) the tile where the candidate is
located; columns (3) and (4), RA and Dec in J2000 coordinates; column (5) is
the calculated Period; column (6) the mean of magnitudes, column (7) is the
amplitude; and finally column (5) is the probability to be a RRL. Th Appendix
B describes in detail how Amplitude ($A_{KS}$) and Period ($PeriodLS$) are
calculated.

Table 5: Candidates 62–122 to RRL sorted by probability to be an RRL.

| ID | Tile | $RA_{K_{s}}$ | $Dec_{K_{s}}$ | Period | Mean Mag. | Amplitude | Prob.
---|---|---|---|---|---|---|---|---
|  |  | (J2000, Deg.) | (J2000, Deg.) | (Days) |  |  |
62 | VVV J175147.56-181433.8 | b396 | 267.94816 | -18.24273 | 0.52625 | 14.224069 | 0.15100 | 0.679
63 | VVV J180146.84-401017.7 | b216 | 270.44515 | -40.17158 | 0.52174 | 14.466260 | 0.15850 | 0.677
64 | VVV J175227.50-183530.5 | b396 | 268.11458 | -18.59182 | 0.54910 | 14.802849 | 0.17700 | 0.676
65 | VVV J180443.52-400343.2 | b216 | 271.18132 | -40.06201 | 0.72308 | 13.555667 | 0.14750 | 0.676
66 | VVV J175055.99-175233.4 | b396 | 267.73330 | -17.87594 | 0.46532 | 14.872555 | 0.17500 | 0.674
67 | VVV J184422.22-241105.0 | b214 | 281.09258 | -24.18472 | 0.60786 | 14.318809 | 0.11650 | 0.673
68 | VVV J175113.40-180449.9 | b396 | 267.80582 | -18.08053 | 0.60792 | 14.606886 | 0.12600 | 0.673
69 | VVV J180418.77-394938.4 | b216 | 271.07822 | -39.82732 | 0.47581 | 14.746172 | 0.17100 | 0.666
70 | VVV J175014.44-181511.2 | b396 | 267.56017 | -18.25312 | 0.61917 | 14.003946 | 0.11600 | 0.665
71 | VVV J174948.48-181052.8 | b396 | 267.45200 | -18.18134 | 0.60773 | 13.766404 | 0.13100 | 0.665
72 | VVV J175140.73-175300.8 | b396 | 267.91971 | -17.88357 | 0.50270 | 14.285242 | 0.16900 | 0.663
73 | VVV J184136.04-235324.2 | b228 | 280.40016 | -23.89006 | 0.41126 | 14.430500 | 0.15275 | 0.661
74 | VVV J175251.58-174457.5 | b396 | 268.21493 | -17.74931 | 0.53431 | 14.240718 | 0.14200 | 0.661
75 | VVV J181227.59-344031.0 | b234 | 273.11497 | -34.67529 | 0.68363 | 14.420188 | 0.12550 | 0.660
76 | VVV J184714.69-245037.1 | b214 | 281.81120 | -24.84363 | 0.48505 | 14.464294 | 0.13850 | 0.657
77 | VVV J175339.54-180642.2 | b396 | 268.41476 | -18.11172 | 0.55641 | 14.699515 | 0.13850 | 0.651
78 | VVV J173152.91-300840.5 | b360 | 262.97044 | -30.14458 | 0.61330 | 14.432890 | 0.18150 | 0.649
79 | VVV J174933.60-182711.6 | b396 | 267.38999 | -18.45321 | 0.44022 | 14.449957 | 0.17450 | 0.648
80 | VVV J180935.02-335151.8 | b248 | 272.39593 | -33.86439 | 0.52426 | 14.511369 | 0.16475 | 0.647
81 | VVV J173331.56-293241.3 | b360 | 263.38151 | -29.54482 | 0.63007 | 14.545403 | 0.20550 | 0.642
82 | VVV J175232.88-174312.6 | b396 | 268.13700 | -17.72017 | 0.39186 | 14.471931 | 0.13750 | 0.641
83 | VVV J180614.11-351239.5 | b247 | 271.55880 | -35.21098 | 0.54152 | 14.269311 | 0.15500 | 0.640
84 | VVV J175106.12-182433.3 | b396 | 267.77550 | -18.40924 | 0.46868 | 14.784901 | 0.15750 | 0.634
85 | VVV J173141.82-293139.8 | b360 | 262.92425 | -29.52772 | 0.44879 | 14.908584 | 0.18700 | 0.634
86 | VVV J181129.86-345913.9 | b234 | 272.87441 | -34.98718 | 0.54909 | 14.575766 | 0.14350 | 0.632
87 | VVV J174928.62-183945.2 | b396 | 267.36927 | -18.66256 | 0.54239 | 13.419081 | 0.17200 | 0.631
88 | VVV J175152.88-183749.0 | b396 | 267.97035 | -18.63028 | 0.49317 | 14.854412 | 0.18050 | 0.631
89 | VVV J175052.39-175426.6 | b396 | 267.71830 | -17.90740 | 0.77854 | 14.433668 | 0.12800 | 0.630
90 | VVV J180034.35-402401.8 | b216 | 270.14313 | -40.40049 | 0.44718 | 14.312008 | 0.20100 | 0.629
91 | VVV J175031.50-182011.6 | b396 | 267.63123 | -18.33656 | 0.61785 | 14.412606 | 0.11700 | 0.629
92 | VVV J174838.44-181758.4 | b396 | 267.16017 | -18.29957 | 0.57399 | 14.631496 | 0.12800 | 0.627
93 | VVV J180445.81-401231.1 | b216 | 271.19087 | -40.20865 | 0.55307 | 14.232366 | 0.11900 | 0.626
94 | VVV J175058.63-183908.0 | b396 | 267.74430 | -18.65222 | 0.61298 | 14.228781 | 0.15500 | 0.625
95 | VVV J175225.27-180705.0 | b396 | 268.10530 | -18.11804 | 0.75172 | 13.901447 | 0.15350 | 0.622
96 | VVV J181436.10-343718.5 | b220 | 273.65043 | -34.62180 | 0.45274 | 14.497057 | 0.12100 | 0.622
97 | VVV J181417.48-342508.0 | b234 | 273.57282 | -34.41888 | 0.46647 | 14.367053 | 0.18750 | 0.621
98 | VVV J175153.50-175251.9 | b396 | 267.97290 | -17.88109 | 0.32565 | 14.150061 | 0.14500 | 0.621
99 | VVV J182201.32-351345.9 | b206 | 275.50550 | -35.22942 | 0.50528 | 13.985881 | 0.18275 | 0.615
100 | VVV J173148.72-290052.4 | b360 | 262.95300 | -29.01456 | 1.19559 | 14.766105 | 0.19850 | 0.611
101 | VVV J173109.12-285519.9 | b360 | 262.78799 | -28.92219 | 0.39245 | 13.379934 | 0.15350 | 0.610
102 | VVV J173121.82-300816.1 | b360 | 262.84090 | -30.13782 | 0.54021 | 14.708926 | 0.21350 | 0.610
103 | VVV J174943.68-183005.5 | b396 | 267.43200 | -18.50152 | 0.33120 | 13.979621 | 0.14050 | 0.608
104 | VVV J175216.33-184401.9 | b396 | 268.06803 | -18.73387 | 0.35511 | 14.137985 | 0.13850 | 0.608
105 | VVV J180123.10-351918.6 | b247 | 270.34624 | -35.32184 | 0.49902 | 14.580500 | 0.12550 | 0.607
106 | VVV J180606.03-393827.8 | b216 | 271.52513 | -39.64106 | 0.70484 | 14.824082 | 0.19150 | 0.607
107 | VVV J181825.02-341206.7 | b220 | 274.60423 | -34.20187 | 0.50807 | 14.838024 | 0.18050 | 0.605
108 | VVV J175232.89-181925.8 | b396 | 268.13705 | -18.32383 | 0.45452 | 14.444980 | 0.18550 | 0.605
109 | VVV J175051.64-190024.2 | b396 | 267.71517 | -19.00673 | 0.70033 | 14.043564 | 0.13525 | 0.596
110 | VVV J173132.57-293253.4 | b360 | 262.88571 | -29.54816 | 0.65042 | 14.655571 | 0.18800 | 0.593
111 | VVV J175033.71-182745.1 | b396 | 267.64046 | -18.46252 | 0.56814 | 13.757696 | 0.15200 | 0.591
112 | VVV J184302.38-242841.9 | b214 | 280.75990 | -24.47829 | 0.50483 | 14.915676 | 0.17425 | 0.591
113 | VVV J174837.69-182243.3 | b396 | 267.15703 | -18.37871 | 0.33390 | 14.533334 | 0.14800 | 0.591
114 | VVV J175109.70-185006.0 | b396 | 267.79043 | -18.83501 | 0.61546 | 14.330465 | 0.11000 | 0.589
115 | VVV J175005.18-180035.5 | b396 | 267.52157 | -18.00987 | 0.36925 | 14.595371 | 0.17600 | 0.585
116 | VVV J180305.11-402803.8 | b216 | 270.77130 | -40.46773 | 0.63171 | 14.711038 | 0.16000 | 0.584
117 | VVV J184422.12-243004.0 | b214 | 281.09219 | -24.50112 | 0.64824 | 14.122191 | 0.09525 | 0.581
118 | VVV J175135.60-185156.8 | b396 | 267.89834 | -18.86577 | 0.44414 | 13.379947 | 0.19450 | 0.580
119 | VVV J173138.32-293219.7 | b360 | 262.90968 | -29.53882 | 0.53319 | 14.716767 | 0.20625 | 0.577
120 | VVV J175256.82-174505.0 | b396 | 268.23677 | -17.75140 | 0.39167 | 14.240672 | 0.10350 | 0.577
121 | VVV J181017.83-335048.2 | b234 | 272.57428 | -33.84673 | 0.49756 | 14.588391 | 0.13550 | 0.577
122 | VVV J184236.63-241216.9 | b228 | 280.65264 | -24.20471 | 0.35425 | 14.707500 | 0.15325 | 0.576

Note: All details are the same as in Table 4.

Table 6: Candidates 123–182 to RRL sorted by probability to be an RRL.

| ID | Tile | $RA_{K_{s}}$ | $Dec_{K_{s}}$ | Period | Mean Mag. | Amplitude | Prob.
---|---|---|---|---|---|---|---|---
|  |  | (J2000, Deg.) | (J2000, Deg.) | (Days) |  |  |
123 | VVV J184402.60-243818.9 | b214 | 281.01084 | -24.63858 | 0.42983 | 14.758015 | 0.12200 | 0.574
124 | VVV J175029.05-182011.5 | b396 | 267.62105 | -18.33653 | 0.37973 | 14.448916 | 0.12900 | 0.573
125 | VVV J175223.79-175532.0 | b396 | 268.09910 | -17.92557 | 0.52771 | 14.435710 | 0.11800 | 0.572
126 | VVV J181254.17-342725.3 | b234 | 273.22572 | -34.45702 | 0.44624 | 14.966774 | 0.19450 | 0.572
127 | VVV J175959.30-402059.7 | b216 | 269.99709 | -40.34992 | 0.64158 | 14.301632 | 0.12250 | 0.570
128 | VVV J175151.29-185422.0 | b396 | 267.96372 | -18.90612 | 0.39547 | 13.960515 | 0.09925 | 0.569
129 | VVV J181514.67-341758.0 | b220 | 273.81113 | -34.29944 | 0.60604 | 14.253613 | 0.11000 | 0.567
130 | VVV J173104.93-293408.7 | b360 | 262.77054 | -29.56908 | 0.45803 | 14.737644 | 0.20200 | 0.567
131 | VVV J184353.62-251759.5 | b214 | 280.97344 | -25.29986 | 0.56023 | 14.644209 | 0.16525 | 0.567
132 | VVV J183959.38-244738.9 | b228 | 279.99743 | -24.79414 | 0.53939 | 13.037118 | 0.10975 | 0.565
133 | VVV J173514.52-291508.9 | b360 | 263.81048 | -29.25249 | 0.60347 | 14.150329 | 0.19300 | 0.565
134 | VVV J184134.85-250632.3 | b228 | 280.39522 | -25.10898 | 0.59658 | 14.105015 | 0.12575 | 0.565
135 | VVV J180038.10-402217.3 | b216 | 270.15877 | -40.37148 | 0.63669 | 14.317283 | 0.10650 | 0.564
136 | VVV J181612.09-351210.7 | b220 | 274.05039 | -35.20296 | 0.63611 | 13.779581 | 0.11650 | 0.563
137 | VVV J182050.39-350506.0 | b206 | 275.20996 | -35.08501 | 0.54185 | 14.600896 | 0.15525 | 0.562
138 | VVV J173121.98-291332.8 | b360 | 262.84160 | -29.22578 | 0.34461 | 14.369184 | 0.17950 | 0.554
139 | VVV J181022.76-342819.9 | b234 | 272.59482 | -34.47220 | 0.44854 | 14.647595 | 0.15500 | 0.554
140 | VVV J184149.94-235645.8 | b228 | 280.45808 | -23.94604 | 0.62714 | 14.046736 | 0.08225 | 0.553
141 | VVV J175116.73-181532.6 | b396 | 267.81971 | -18.25906 | 1.20681 | 14.490817 | 0.18550 | 0.551
142 | VVV J175147.25-183024.2 | b396 | 267.94687 | -18.50673 | 0.56745 | 14.233916 | 0.12000 | 0.551
143 | VVV J175234.11-183947.4 | b396 | 268.14212 | -18.66317 | 0.98240 | 14.363797 | 0.19450 | 0.549
144 | VVV J180330.66-404605.5 | b216 | 270.87773 | -40.76819 | 0.70014 | 13.499436 | 0.09650 | 0.549
145 | VVV J174822.20-182630.9 | b396 | 267.09251 | -18.44192 | 0.72387 | 14.495015 | 0.11400 | 0.548
146 | VVV J173334.49-293202.7 | b360 | 263.39370 | -29.53409 | 0.45166 | 14.863846 | 0.20450 | 0.547
147 | VVV J180122.05-394751.2 | b216 | 270.34186 | -39.79756 | 0.49847 | 14.106507 | 0.15600 | 0.545
148 | VVV J175010.71-185558.2 | b396 | 267.54461 | -18.93284 | 0.49126 | 14.976538 | 0.19000 | 0.542
149 | VVV J183827.60-243516.4 | b228 | 279.61501 | -24.58788 | 0.35679 | 14.186794 | 0.13650 | 0.541
150 | VVV J174851.35-183557.1 | b396 | 267.21395 | -18.59920 | 0.47823 | 14.828702 | 0.22250 | 0.540
151 | VVV J181357.00-343414.0 | b220 | 273.48750 | -34.57056 | 0.55161 | 14.387049 | 0.10675 | 0.540
152 | VVV J174818.47-184240.3 | b396 | 267.07695 | -18.71118 | 0.65717 | 14.957061 | 0.20900 | 0.538
153 | VVV J175055.40-174744.1 | b396 | 267.73082 | -17.79559 | 0.47883 | 14.617106 | 0.21250 | 0.537
154 | VVV J174729.91-183118.1 | b396 | 266.87463 | -18.52169 | 0.54615 | 14.372712 | 0.10175 | 0.536
155 | VVV J184302.38-242841.9 | b228 | 280.75993 | -24.47831 | 0.50485 | 14.931221 | 0.18625 | 0.534
156 | VVV J175335.32-181114.5 | b396 | 268.39718 | -18.18737 | 0.25673 | 12.874030 | 0.16425 | 0.534
157 | VVV J174858.77-182134.1 | b396 | 267.24489 | -18.35946 | 0.63239 | 14.226136 | 0.21800 | 0.533
158 | VVV J175311.35-175231.5 | b396 | 268.29730 | -17.87541 | 0.60246 | 14.833838 | 0.14625 | 0.532
159 | VVV J183725.74-243751.1 | b228 | 279.35726 | -24.63087 | 0.31666 | 13.241353 | 0.12375 | 0.532
160 | VVV J181254.45-341459.0 | b234 | 273.22689 | -34.24972 | 0.48941 | 14.722361 | 0.16850 | 0.532
161 | VVV J181159.60-343500.1 | b234 | 272.99835 | -34.58335 | 0.63124 | 13.644068 | 0.10050 | 0.532
162 | VVV J175007.04-184545.3 | b396 | 267.52931 | -18.76258 | 0.33112 | 14.772765 | 0.15050 | 0.530
163 | VVV J175044.68-183310.8 | b396 | 267.68615 | -18.55301 | 0.38966 | 14.333492 | 0.09600 | 0.527
164 | VVV J181522.12-350716.6 | b220 | 273.84219 | -35.12128 | 0.28828 | 13.992228 | 0.12800 | 0.526
165 | VVV J175012.91-184052.0 | b396 | 267.55378 | -18.68112 | 0.29193 | 14.171443 | 0.19950 | 0.524
166 | VVV J181759.92-345805.5 | b220 | 274.49967 | -34.96821 | 0.65434 | 14.148250 | 0.14200 | 0.523
167 | VVV J175208.24-175314.8 | b396 | 268.03433 | -17.88744 | 0.42989 | 15.173374 | 0.19400 | 0.522
168 | VVV J181246.66-343549.4 | b234 | 273.19443 | -34.59704 | 0.54813 | 14.803782 | 0.12500 | 0.522
169 | VVV J174936.98-180337.6 | b396 | 267.40408 | -18.06046 | 0.41483 | 14.933797 | 0.16425 | 0.521
170 | VVV J175106.82-181234.8 | b396 | 267.77842 | -18.20966 | 0.43832 | 14.792894 | 0.16100 | 0.521
171 | VVV J180506.41-401414.1 | b216 | 271.27669 | -40.23724 | 0.64023 | 13.677761 | 0.11450 | 0.518
172 | VVV J181646.84-352508.5 | b220 | 274.19518 | -35.41902 | 0.50942 | 14.424503 | 0.19825 | 0.518
173 | VVV J174914.37-183922.0 | b396 | 267.30988 | -18.65612 | 0.41291 | 15.073359 | 0.17400 | 0.516
174 | VVV J173212.93-285453.9 | b360 | 263.05386 | -28.91498 | 0.53633 | 14.733316 | 0.20700 | 0.514
175 | VVV J181320.86-342628.8 | b234 | 273.33691 | -34.44133 | 0.45602 | 14.793616 | 0.19550 | 0.514
176 | VVV J175251.71-174540.4 | b396 | 268.21545 | -17.76123 | 0.47730 | 13.572054 | 0.07950 | 0.513
177 | VVV J181826.64-344401.2 | b220 | 274.61100 | -34.73367 | 0.52195 | 14.418227 | 0.11250 | 0.513
178 | VVV J173230.95-285727.1 | b360 | 263.12895 | -28.95752 | 0.68201 | 14.479752 | 0.18100 | 0.511
179 | VVV J181935.38-344801.2 | b220 | 274.89743 | -34.80032 | 0.32298 | 14.337214 | 0.10075 | 0.507
180 | VVV J174816.23-182243.9 | b396 | 267.06761 | -18.37887 | 0.48778 | 14.747733 | 0.12450 | 0.506
181 | VVV J184124.57-242949.1 | b228 | 280.35238 | -24.49697 | 0.49971 | 14.895250 | 0.18825 | 0.506
182 | VVV J180126.41-401920.6 | b216 | 270.36006 | -40.32240 | 0.64301 | 13.891366 | 0.10400 | 0.506

Note: All details are the same as in Table 4.

Table 7: Candidates 183–242 to RRL sorted by probability to be an RRL.

| ID | Tile | $RA_{K_{s}}$ | $Dec_{K_{s}}$ | Period | Mean Mag. | Amplitude | Prob.
---|---|---|---|---|---|---|---|---
|  |  | (J2000, Deg.) | (J2000, Deg.) | (Days) |  |  |
183 | VVV J175013.79-183031.1 | b396 | 267.55744 | -18.50862 | 0.34924 | 14.675773 | 0.11650 | 0.505
184 | VVV J175059.00-183434.7 | b396 | 267.74583 | -18.57631 | 0.44657 | 12.109485 | 0.14300 | 0.505
185 | VVV J174748.65-182609.1 | b396 | 266.95272 | -18.43586 | 0.59155 | 13.216329 | 0.08250 | 0.504
186 | VVV J180333.03-394105.2 | b216 | 270.88765 | -39.68478 | 0.37988 | 14.838353 | 0.15650 | 0.504
187 | VVV J180951.71-342819.1 | b234 | 272.46545 | -34.47197 | 0.49563 | 14.171205 | 0.20600 | 0.503
188 | VVV J182113.23-344729.2 | b206 | 275.30511 | -34.79144 | 0.42821 | 14.373597 | 0.16075 | 0.501
189 | VVV J175115.23-183311.3 | b396 | 267.81348 | -18.55315 | 0.37318 | 15.065302 | 0.18700 | 0.500
190 | VVV J175113.82-183107.5 | b396 | 267.80758 | -18.51876 | 0.44981 | 14.559584 | 0.11400 | 0.500
191 | VVV J181342.66-342803.1 | b234 | 273.42773 | -34.46752 | 0.46203 | 14.800813 | 0.15300 | 0.499
192 | VVV J173244.85-285440.1 | b360 | 263.18688 | -28.91113 | 0.59835 | 14.608091 | 0.17000 | 0.499
193 | VVV J175213.79-180827.9 | b396 | 268.05744 | -18.14107 | 0.65586 | 14.255504 | 0.14250 | 0.498
194 | VVV J175112.95-190345.5 | b396 | 267.80394 | -19.06263 | 0.27137 | 13.806954 | 0.11950 | 0.497
195 | VVV J183751.54-242902.7 | b228 | 279.46475 | -24.48409 | 0.64905 | 15.044247 | 0.17200 | 0.497
196 | VVV J174819.09-183741.8 | b396 | 267.07953 | -18.62827 | 0.45167 | 14.658939 | 0.11150 | 0.496
197 | VVV J175404.34-175811.2 | b396 | 268.51808 | -17.96977 | 0.35266 | 12.790197 | 0.20000 | 0.496
198 | VVV J175008.70-180601.2 | b396 | 267.53625 | -18.10034 | 0.46554 | 13.881969 | 0.07250 | 0.493
199 | VVV J184000.98-244740.2 | b228 | 280.00407 | -24.79450 | 0.38897 | 14.727768 | 0.14550 | 0.489
200 | VVV J173148.03-295554.0 | b360 | 262.95015 | -29.93167 | 0.28216 | 14.507204 | 0.23100 | 0.489
201 | VVV J184124.06-245403.1 | b228 | 280.35026 | -24.90086 | 0.31884 | 14.484309 | 0.15000 | 0.488
202 | VVV J175246.17-183139.2 | b396 | 268.19237 | -18.52755 | 0.58697 | 14.545273 | 0.13425 | 0.487
203 | VVV J175027.88-180122.6 | b396 | 267.61615 | -18.02295 | 0.54782 | 14.288711 | 0.18500 | 0.487
204 | VVV J173148.40-291123.4 | b360 | 262.95166 | -29.18984 | 0.42390 | 13.387236 | 0.16650 | 0.487
205 | VVV J173316.61-285514.8 | b360 | 263.31919 | -28.92079 | 0.34464 | 13.989234 | 0.21200 | 0.487
206 | VVV J181102.10-345941.2 | b234 | 272.75876 | -34.99478 | 0.41287 | 13.929947 | 0.11550 | 0.486
207 | VVV J183940.52-242911.6 | b228 | 279.91882 | -24.48655 | 0.48238 | 12.878464 | 0.12350 | 0.486
208 | VVV J175210.36-181822.1 | b396 | 268.04315 | -18.30613 | 0.30762 | 13.991246 | 0.11900 | 0.486
209 | VVV J181347.03-350812.9 | b220 | 273.44596 | -35.13691 | 0.26189 | 14.301592 | 0.16825 | 0.486
210 | VVV J175151.82-181311.0 | b396 | 267.96592 | -18.21973 | 0.49593 | 13.722261 | 0.18550 | 0.485
211 | VVV J174858.70-184406.4 | b396 | 267.24456 | -18.73512 | 0.17722 | 14.191008 | 0.19150 | 0.485
212 | VVV J184708.08-242541.1 | b214 | 281.78366 | -24.42808 | 0.61992 | 13.572956 | 0.10350 | 0.483
213 | VVV J175234.27-174823.7 | b396 | 268.14279 | -17.80658 | 0.28366 | 14.614862 | 0.08400 | 0.482
214 | VVV J181117.24-344747.3 | b234 | 272.82184 | -34.79648 | 0.34690 | 12.191316 | 0.15100 | 0.481
215 | VVV J182028.27-342613.1 | b220 | 275.11780 | -34.43698 | 0.51191 | 14.473590 | 0.17750 | 0.480
216 | VVV J181357.03-343414.2 | b234 | 273.48763 | -34.57061 | 0.55163 | 14.394436 | 0.12000 | 0.480
217 | VVV J175107.70-185907.0 | b396 | 267.78210 | -18.98529 | 0.63272 | 13.664555 | 0.12500 | 0.480
218 | VVV J180415.22-394522.6 | b216 | 271.06343 | -39.75627 | 0.51927 | 14.897313 | 0.17375 | 0.480
219 | VVV J173022.12-291526.9 | b360 | 262.59217 | -29.25747 | 0.15626 | 14.303645 | 0.15150 | 0.477
220 | VVV J175219.02-180707.9 | b396 | 268.07925 | -18.11886 | 0.32261 | 14.891803 | 0.17300 | 0.477
221 | VVV J181323.20-343115.3 | b234 | 273.34668 | -34.52092 | 0.62302 | 14.166402 | 0.09300 | 0.477
222 | VVV J174905.20-180158.6 | b396 | 267.27168 | -18.03295 | 0.33799 | 14.726076 | 0.09450 | 0.476
223 | VVV J184536.67-241612.7 | b214 | 281.40280 | -24.27018 | 0.84289 | 14.059838 | 0.08800 | 0.475
224 | VVV J175236.25-174554.4 | b396 | 268.15106 | -17.76510 | 0.43096 | 12.355095 | 0.21250 | 0.475
225 | VVV J184712.36-244841.3 | b214 | 281.80150 | -24.81147 | 0.47995 | 13.991441 | 0.13450 | 0.475
226 | VVV J184439.95-245216.2 | b214 | 281.16644 | -24.87117 | 0.34216 | 13.976015 | 0.06650 | 0.474
227 | VVV J184630.34-251453.7 | b214 | 281.62643 | -25.24826 | 0.42034 | 13.330500 | 0.10200 | 0.473
228 | VVV J181945.49-341939.4 | b220 | 274.93955 | -34.32761 | 0.94775 | 13.411339 | 0.17300 | 0.472
229 | VVV J175151.57-180235.9 | b396 | 267.96486 | -18.04329 | 0.33626 | 14.480251 | 0.08650 | 0.471
230 | VVV J174951.30-182038.2 | b396 | 267.46375 | -18.34395 | 0.41009 | 14.992596 | 0.15350 | 0.469
231 | VVV J180531.65-400424.3 | b216 | 271.38188 | -40.07343 | 0.92920 | 12.674581 | 0.15250 | 0.469
232 | VVV J181241.96-350513.8 | b234 | 273.17484 | -35.08717 | 0.66180 | 14.033288 | 0.11700 | 0.468
233 | VVV J184238.41-244132.2 | b214 | 280.66006 | -24.69229 | 0.34091 | 14.220500 | 0.09525 | 0.468
234 | VVV J181509.42-341210.4 | b234 | 273.78926 | -34.20289 | 0.29957 | 13.641538 | 0.12900 | 0.468
235 | VVV J175120.85-173644.3 | b396 | 267.83687 | -17.61230 | 1.27136 | 14.019130 | 0.09850 | 0.466
236 | VVV J184201.71-242900.8 | b228 | 280.50713 | -24.48357 | 0.32113 | 13.079305 | 0.16625 | 0.466
237 | VVV J184100.22-245141.5 | b228 | 280.25091 | -24.86152 | 0.78273 | 13.253191 | 0.08775 | 0.465
238 | VVV J173432.50-291813.8 | b360 | 263.63542 | -29.30383 | 0.26678 | 13.976676 | 0.10700 | 0.463
239 | VVV J175006.12-181502.0 | b396 | 267.52550 | -18.25055 | 0.39397 | 14.903939 | 0.11800 | 0.462
240 | VVV J174957.74-182458.3 | b396 | 267.49059 | -18.41621 | 0.62848 | 13.894237 | 0.09000 | 0.461
241 | VVV J181854.12-341347.6 | b220 | 274.72552 | -34.22988 | 0.35558 | 13.953846 | 0.07000 | 0.461
242 | VVV J173353.59-285439.9 | b360 | 263.47329 | -28.91109 | 0.56931 | 14.740961 | 0.13800 | 0.461

Note: All details are the same as in Table 4.

Figure 24:  Folded light curve for the candidates 1 to 27 to RRL sorted by
probability to be an RRL. In the title of every plot are displayed number of
the source, the VVV identification and the probability to be a RRL. The
horizontal axis is the phase, and the vertical shows the source magnitude. For
visual ease, two periods are shown.  Figure 25:  Folded light curve for the
candidates 28 to 54 to RRL sorted by probability of being an RRL. All details
are the same as in Fig. 24. Figure 26:  Folded light curve for the candidates
55 to 81 to RRL sorted by probability of being an RRL. All details are the
same as in Fig. 24. Figure 27:  Folded light curve for the candidates 82 to
108 to RRL sorted by probability of being an RRL. All details are the same as
in Fig. 24. Figure 28:  Folded lightcurve for the candidates 109 to 135 to RRL
sorted by probability of being an RRL. All details are the same as in Fig. 24.
Figure 29:  Folded light curve for the candidates 136 to 162 to RRL sorted by
probability of being an RRL. All details are the same as in Fig. 24. Figure
30:  Folded light curve for the candidates 163 to 189 to RRL sorted by
probability of being an RRL. All details are the same as in Fig. 24. Figure
31:  Folded light curve for the candidates 190 to 216 to RRL sorted by
probability of being an RRL. All details are the same as in Fig. 24. Figure
32:  Folded light curve for the candidates 217 to 242 to RRL sorted by
probability of being an RRL. All details are the same as in Fig. 24.