# On the Quality of Deep Representations for Kepler Light Curves Using Variational Auto-Encoders
## 2 Materials and Methods
### Light Curve Representation Methods
#### 2.1.1 Model-Based Representations  
The Mandel-Agol simulation [3] models the transit of a spherical planet around a spherical star assuming a non-linear limb darkening model on the light source. It does this by modeling the opacity observed on the light intensity according to the planet's position with a function over time \(\ell c(t)\). When the planet is in front of the star, the opacity is at maximum (\(\ell c<1\)). On the other hand, when the planet orbits without blocking the star light, the opacity is minimum (\(\ell c=1\)). When the planet is close to being in front of the star, the intensity \(\ell c(t)\) is modeled as a polynomial based on the limb darkening of the star. This requires knowing the distance from the center of the planet to the center of the parent star, as well as the radius of each one of the bodies, the transit period, the inclination, and the limb darkening models (coefficients). There are techniques that find the model parameters through least squares (LS) or Markov Chain Monte Carlo (MCMC).  
However, light curves can be observed not only by a transit, but several inherent and exogenous processes might be involved in the variability of the observed intensity of a star. Therefore, several machine learning techniques have been used to classify variable stars, typically by manually extracting specialized features from the light curve as representation and then applying classic pattern recognition methods to them. For example, [PERSON] et al. [12] presents a catalog of variable stars where 53 specialized features are extracted from light curves through statistics such as kurtosis, skewness, standard deviation, and Stetson, plus other features based on the period and frequency analysis of a Lomb-Scargle [13] fitted model. [PERSON] et al. [17] also worked on classifying variable stars on the Catalina Real-Time Transient Survey (CRTS) and the Kepler Mission, extracting similar features from the light curves. In the work of [PERSON] et al. [18], statistical descriptors are used as inputs for a random forest algorithm that can detect anomalous light curves based on probabilistic learning models. These outliers are removed from the training set used in variable stars' classification.  
For transit vetting, we found the so-called Autovetter [5] which uses the random forest model over the features derived from the statistics pipeline on the Kepler mission for vetting candidate objects on the Kepler Threshold-Crossing Event (TCE) data. Anotherapproach is to represent the light curve as phase-aligned sections called \"folds\" to address the problem of irregular sampling in transits. This folded light curve centers the transit and stacks all the times on which it occurs, as shown in Figure 1. In this sense, the most common is to bin the folded light curve based on a window proportional to the estimated transit period. For example, as a dimensionality reduction method (embedding representations) of folded light curves, [PERSON] et al. [11] proposed a locality preserving projection to filter out light curves with non-transit-like shapes, while [PERSON] et al. [19] proposes a self-organization map on the classification of true planets.  
Neural networks and deep learning [16] algorithms have become very popular in solving problems where feature extraction from data is non-trivial. These algorithms have been successfully applied for transit vetting in recent years by using the binned folded light curve representation. For example, [PERSON] [6] used a one-dimensional convolutional neural network (1D CNN) model on vetting candidate exoplanets on the Kepler TCE data with the global and local representation of the folded light curve. [PERSON] et al. [7] used a similar approach to [6] for detecting transit shape objects trained on simulated data and evaluated using the Kepler mission dataset. [PERSON] et al. [8] also used a 1D CNN model for distinguishing and vetting candidate exoplanets among variable stars (four classes) on the Wide Angle Search for Planets dataset.