Importancia de la Extensión

In a subsequent work by Branimir Sesar (Sesar, Hernitschek et al. 2016), features extracted by the author (see Table 5.7) are used together with template fitting in order to determine periods of RR Lyrae candidates and thus enhance the sample purity even more. The method is outlined here, as the cleaner sample together with precise distance estimates is used in the analysis presented in Chapter 6.

Table 5.4. The Catalog of Variable Sources in PS1 3π

Column FITS Format Code Description

1 E right ascension in degrees 2 E declination in degrees

3 E scalar variability quantity ˆχ2_{, Equ. (5.1)}

4 E best fit structure function parameter ωr(r band variability amplitude) on log-spaced grid

5 E best fit structure function parameter τ (time scale) on log-spaced grid 6 E error-weighted mean gP1band magnitude hgP1i

7 E error-weighted mean rP1band magnitude hrP1i

8 E error-weighted mean iP1band magnitude hiP1i

9 E error-weighted mean zP1band magnitude hzP1i

10 E error-weighted mean yP1band magnitude hyP1i

11 E W1-W2 color from WISE

12 E pQSO

13 E pRRLyrae

Note. — Structure of the Catalog of Variable Sources in PS1 3π. The Catalog of Variable Sources is available for PV2 in its entirety in machine-readable format in the supplementary material to Hernitschek et al. (2016). A table structure is shown here for guidance regarding its form and content.

5.6.1 Template Fitting

For period finding, a template based method was applied. The paper by Sesar, Hernitschek et al. (2016) states that previous tests with the multi-band periodogram of VanderPlas and Ivezi´c (2015) (for a description see Sec. 4.1.3) had shown that it will fail with the data at hand for nearly half of the S82 RR Lyrae with PS1 3π photometry. The reason is that the model by VanderPlas and Ivezi´c (2015) is a mathematical multi-band light curve, thus for sparse data, the optimal estiamated light-curve shape (the best-fit model) will not necessarily be physical.

Sesar, Hernitschek et al. (2016) adopt a set of 482 of griz models, consisting of the lightcurve templates derived in the work by Sesar et al. (2010) from SDSS S82 RR Lyrae (with SDSS photometry). In total, the template set consists of 379 type ab multi-band templates and 104 type c multi-band templates for RRab and RRc stars, respectively.

In the subsequent analysis, Sesar, Hernitschek et al. (2016) use all yPS1 and zPS1 observations as

if they would be from the same band (zPS1), as they had found from phased PS1 3π light curves

of RR Lyrae that both are identical within photometric uncertainties.

To find the best-fit values of these parameters, the phase of each light curve given an assumed period P is

φ(t|P, φ0) = (t− 2400000) moduloP

In this equation, t is given in heliocentric Julian days, and the phase offset _{−0.5 < φ}0 < 0.5 is

used to enforce the maximum of the light curve occurring at φ = 0. Then, a χ2-like statistic is minimized,

χ2_k= X m=gP1,rP1,iP1,zP1,yP1 Nobs X n=1  mn− mk(φ(tn|P, φ0)|F, r0) σmn 2 . (5.8)

In this equation, σmn is the photometric uncertainty for the n-th observation in the m =

gP1, rP1, iP1, zP1, yP1 band. Sesar, Hernitschek et al. (2016) use the Differential Evolution al-

gorithm of Storn and Price (1997) in order to minimize. this algorithm is very fast compared to others tested, and is already implemented in scipy (Millman and Aivazis 2011).

During the template fitting, to a given PS1 3π light curve first every template is fitted. While doing so, the minimization is constrained to periods from 0.4–0.9 days for type ab templates, and to 0.2–0.5 days for type c templates, as being typical for these classes. After that, depending on the type of the best-fit template, only type ab or c are fitted, now setting a more restrictive prior on the permissible periods, begin in the range of 2 min around the previous best-fit template. For further inference, only the best-fit outcome from this method is used.

As a test set, the S82 periods from SDSS were used by Sesar, Hernitschek et al. (2016). The method is capable of recovering the period for 85% out of 440 RR Lyrae. Even with a precision of 1 sec, the method can recover the period for 73% from PS1 3π photometry.

Thus, period estimation from template fitting is a powerful tool in feature extraction for subse- quent classification.

5.6.2 A Cleaner Sample

The method is then applied by Sesar, Hernitschek et al. (2016) to full PS1 3π, where a more rigid cut than the one described in Section 5.3.2 is needed. For being processed, they require that the sources meet the following conditions after outlier cleaning took place:

(i) at least two epochs in each gP1, rP1, iP1 bands, and at least a total of two epochs in zP1,

yP1

(ii) a total of at least 23 epochs

(iii) an uncertainty-weighted mean magnitude of 15 <hmi < 21.5 in at least one of the gP1, rP1,

iP1 bands.

Sesar, Hernitschek et al. (2016) use the estimated period _{P, a set of 10 features comparable to} the ones used within this work (see Table 5.7), as well as ∼20 other features. The complete set of features is described in Sesar, Hernitschek et al. (2016) and are used to train a supervised classifier comparable to the approach described in Section 5.4.3.

(i) the training set is similar to the S82-part of the training set used in this work (see Section 5.4.3), with the difference of sources must have at minimum 23 (instead of 10) epochs (ii) the feature set of Table 5.7 is adopted, but replaced the feature stellar_locus (see Equ.

(5.3)) used by Hernitschek et al. (2016) with gP1− iP1

(iii) instead of a RFC, a gradient boosting (as described in Section 4.2.2, implemented as XGBoost (Chen and Guestrin 2016) is used.

The approach uses three subsequently more detailed classifiers to overcome the huge computa- tional effort of template fitting. As one template fitting takes ∼30 min CPU time per source, it will not be feasible to carry it out for the majority of the sources.

The three subsequent classifiers, as outlined in Sesar, Hernitschek et al. (2016) use (i) optical/NIR colors and variability features

This analysis uses the feature set of Table 5.7, but replace the feature stellar_locus by Her- nitschek et al. (2016) by gP1− iP1. The analysis is in purity and completeness comparable to the

results presented by the author in Sec. 5.5.2. The classifier by Sesar, Hernitschek et al. (2016) also selects samples that are, for example, _{∼80% pure and ∼80% complete. A cut is set on the} outputted classification score to reduce the sample by more than three orders of magnitude, while losing only 2% of the true RR Lyrae.

(ii) multi-band periodogram

Multi-band periodograms are calculated for the sources passing step (i). The best 20 periods, as well as their power (i.e. height of the periodogram at the given period) are extracted. The classifier now uses the features of (i) plus the 20 best periods and their powers, resulting in 50 features.

(iii) template fitting

To sources with a high classifier score from (ii), template fitting is applied.

This strategy avoids wasting CPU time for estimating periods or fitting templates to sources that are likely not RR Lyrae.

Comparing the purity and completeness of this classifier to the one by the author as described in Sec. 5.5.2, after the final step (iii) samples that are, for example, 90% pure and 90% complete can be selected.

Fig. 5.20 compares the purity-completeness curves from the classifier developed in this work to the purity-completeness curves after step (iii) from Sesar, Hernitschek et al. (2016) in order to show the remarkably high purity and completeness that can be reached using multi-band template fitting.

For exploring the Galactic halo, precise distances are important. To estimate distances, methods as Equ. 5.5 rely on knowledge on the metallicity, that is mostly not available.

Having the period at hand due to the analysis described before, it can be used for a more precise distance determination. In detail, this period can be used in combination with a period-absolute

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 purity completeness 14.5 < r_P1 < 18.5 (~40 kpc) this work Sesar+(2016) (a) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 purity completeness 19.7 < r_P1 < 20.7 (~80 kpc) this work Sesar+(2016) (b)

Figure 5.20 Trade-off between purity and completeness for the classifier within this work and the classifier of Sesar, Hernitschek et al. (2016).

Comparison of both classifiers for the (a) bright, (b) faint end of the sample. For both magnitudes ranges, the classifier by Sesar, Hernitschek et al. (2016) using period fitting provides a significant improvement, enabling the selection of samples being almost 100% pure at a completeness of ∼80% (faint end) or even > 90% (bright end). magnitude-metallicity (PLZ) relation to measure the distances of RR Lyrae with a precision of 3% without knowledge on the metallicity (Sesar, Hernitschek et al. 2016).

The distance estimation by Sesar, Hernitschek et al. (2016) relies on empirical studies by Catelan et al. (2004), Sollima et al. (2006), Marconi et al. (2015) and other. They have shown that the absolute magnitude of a RR Lyrae can be modeled as

M = α log₁₀(P/Pref) + β([Fe/H]− [Fe/H]ref) + Mref +  (5.9)

where Mref is the absolute magnitude at a reference period Pref and metallicity [Fe/H]ref. The

variables α and β give the dependence of the absolute magnitude on period and metallicity, respectively. The intrinsic scatter in the absolute magnitude is modeled by , being a standard normal variable with mean 0 and standard deviation σM.

The details of how to constrain distances using this method are outlined in detail in Sesar, Hernitschek et al. (2016).

In the end, distance moduli of PS1 3π RR Lyrae candidates are computed from the flux-averaged mean iP1 magnitude as

with an uncertainty in MiP 1 of 0.06 mag, i.e. a∼3% uncertainty in distance.

These distances of a sample selected by the method described in Section 5.6.2 (having a purity of 0.9, completeness of 0.8 at 80 kpc) will be used in Chapter 6 in order to precisely map the geometry of Sagittarius stream.