On the Quality of Deep Representations for Kepler Light Curves Using Variational AutoEncoders
Abstract
:1. Introduction
2. Materials and Methods
2.1. Light Curve Representation Methods
2.1.1. ModelBased Representations
2.1.2. SelfGenerated Representations
2.1.3. Variational AutoEncoders
2.2. Extending VRAE to Include Temporal Information
2.2.1. Problem Setup
2.2.2. VRAE Including Delta Times
2.2.3. VRAE with Embedded ReScaling
 Rescale data: The first layer of the encoder ${q}_{\varphi}(\xb7)$ rescales the data by dividing on the Fstd. This step is performed in order to use the standardized version of the data, as the literature recommends.
 Encode: The encoder ${q}_{\varphi}(\xb7)$ adds the Fstd as an input pattern to the coding task by ${q}_{\varphi}\left({\overrightarrow{z}}^{\left(i\right)}\right{\overrightarrow{x}}^{\left(i\right)},{\overrightarrow{\delta}}^{\left(i\right)},{s}^{\left(i\right)})=\mathcal{N}({\overrightarrow{\mu}}^{\left(i\right)},\mathrm{diag}\left({\overrightarrow{\sigma}}^{\left(i\right)}\right))$ in order to extract the information on it.
 Sample: The sampled latent variable is given by: ${\widehat{\overrightarrow{z}}}^{\left(i\right)}={\overrightarrow{\mu}}^{\left(i\right)}+{\overrightarrow{\sigma}}^{\left(i\right)}\odot \overrightarrow{\u03f5}$, with $\overrightarrow{\u03f5}\sim \mathcal{N}(\overrightarrow{0},\mathbf{I})$.
 Reconstruct: The decoder ${p}_{\theta}\left({\overrightarrow{x}}^{\left(i\right)}\right{\overrightarrow{z}}^{\left(i\right)},{\overrightarrow{\delta}}^{\left(i\right)})$ adds the Fstd to the reconstruction task in order to estimate the original Fstd by ${p}_{\theta}\left({s}^{\left(i\right)}\right{\overrightarrow{z}}^{\left(i\right)})$.
 Rescale reconstruction: The last layer of the decoder rescales the data, by returning the reconstructed Fstd (multiplied by it). This final step is performed in order to obtain a reconstructed time series on the unscaled values’ representation.
Algorithm 1 Forward pass VRAE${}_{t}$. 
Input: ${\overrightarrow{x}}_{s}^{\left(i\right)}$— scaled measurements of the time series ${\overrightarrow{\delta}}^{i)}$— delta times of the time series Output: ${\widehat{\overrightarrow{x}}}_{s}^{\left(i\right)}$— reconstructed scaled time series

Algorithm 2 Forward pass SVRAE${}_{t}$. 
Input: ${\overrightarrow{x}}^{\left(i\right)}$— measurements of the time series ${\overrightarrow{\delta}}^{(\ell )}$— delta times of the time series Output: ${\widehat{\overrightarrow{x}}}^{\left(i\right)}$— reconstructed time series

2.2.4. Loss Function
3. Experimental Setup and Results
3.1. Dataset
3.1.1. Data Representation
3.1.2. Data Selection and Augmentation
 Check for Kepler flags (in the metadata) and remove objects with “secondary event” or “not transitlike” flags;
 Remove objects with a “transit score” (in the Kepler metadata) less than $0.55$;
 Perform a Mandel–Agol fit and remove objects with (SMSE) residual greater than 1.
3.2. Model Assessment and Implementation
3.2.1. Reconstruction Validation
3.2.2. Disentanglement Validation
3.2.3. Classification Validation
3.2.4. Model Implementation
3.3. Results
3.3.1. Is the Time Needed?
3.3.2. Quality Evaluation
3.3.3. Application of the Learned Representation
4. Discussion
5. Conclusions
Future Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
AE  AutoEncoder 
AutoC  AutoCorrelation 
BKJD  Barycentric Kepler Julian Day 
BJD  Barycentric Julian Day 
CRTS  Catalina RealTime Transient Survey 
CNN  Convolutional Neural Network 
DiffM  Mean of the Differences 
FPCA  Fourier plus PCA 
GRU  Gated Recurrent Unit 
KOI  Kepler Objects of Interest 
LS  Least Squares 
LSST  Legacy Survey of Space and Time 
MA  Mandel–Agol 
MAST  Mikulski Archive for Space Telescopes 
MAE  Mean Absolute Error 
MCMC  Markov Chain Monte Carlo 
MI  Mutual Information 
MLP  MultiLayer Perceptron 
MSE  Mean Squared Error 
MSLE  Mean Squared Logarithm Error 
NASA  National Aeronautics and Space Administration 
NMI  Normalized MI 
PCA  Principal Component Analysis 
Pcorr  Pearson Correlation 
PcorrA  Pcorr in Absolute Values 
RNN  Recurrent Neural Network 
RAE  Recurrent AutoEncoder 
RAE${}_{t}$  RAE plus Time Information 
SVRAE${}_{t}$  VRAE${}_{t}$ with ReScaling 
SMSE  ReScaled Mean Squared Error 
RMSE  Root Mean Squared Error 
SpectralH  Spectral Entropy 
TESS  Transiting Exoplanet Survey Satellite 
TCE  ThresholdCrossing Event 
VAE  Variational AutoEncoder 
VRAE  Variational Recurrent AutoEncoder 
VRAE${}_{t}$  VRAE plus Time Information 
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Reconstruction  Denoising  Residual Noise  

Method  Time  RMSE ↓  MAE ↓  AutoC ↑  DiffM ↓  SpectralH ↑ 
RAE${}_{t}$  ×  0.630  0.448  0.429  0.206  0.889 
✓  0.680  0.475  0.502  0.125  0.895  
VRAE${}_{t}$  ×  0.689  0.480  0.559  0.074  0.900 
✓  0.688  0.484  0.594  0.068  0.901 
Reconstruction  Denoising  Residual Noise  

Method  Config.  RMSE ↓  MAE ↓  AutoC ↑  DiffM ↓  SpectralH ↑ 
Light Curve      0.273  0.784  0.840  
Passband  1–500  1.081  0.624  0.968  0.047  0.824 
50–1500  1.041  0.655  0.831  0.200  0.842  
50–2500  0.959  0.640  0.670  0.363  0.846  
Moving avg  3  0.719  0.461  0.704  0.274  0.876 
5  0.841  0.513  0.784  0.170  0.860  
10  0.937  0.553  0.843  0.089  0.843  
MA fit  ktransit  0.799  0.514  0.693  0.028  0.873 
RAE${}_{t}$  16  0.680  0.475  0.502  0.125  0.895 
VRAE${}_{t}$  16  0.688  0.484  0.594  0.068  0.901 
SVRAE${}_{t}$  16  0.724  0.489  0.611  0.064  0.898 
Representation  Pcorr  PcorrA  MI  NMI 

Metadata  $0.064$  $0.162$  $0.275$  $0.044$ 
(Raw) FPCA  $0.000$  $0.000$  $0.210$  $0.027$ 
(Fold) FPCA  $0.000$  $0.000$  $0.061$  $0.008$ 
RAE${}_{t}$  $0.057$  $0.277$  $0.255$  $0.033$ 
VRAE${}_{t}$  $0.012$  $0.168$  $0.122$  $0.016$ 
SVRAE${}_{t}$  $0.003$  $0.138$  $0.072$  $0.009$ 
Representation  Input  NonExoplanet  Exoplanet  Global  

Dim  P  R  ${\mathit{F}}_{1}$  P  R  ${\mathit{F}}_{1}$  ${\mathit{F}}_{1}$Ma  
Metadata  10  $94.62$  $86.05$  $90.13$  $77.81$  $90.91$  $83.85$  $87.00$ 
GlobalFolded  T  $83.55$  $83.48$  $83.51$  $69.35$  $69.45$  $69.36$  $76.45$ 
Unsupervised Methods  
(Raw) FPCA  16  $77.24$  $75.55$  $76.38$  $59.15$  $61.40$  $60.26$  $68.32$ 
(Fold) FPCA  16  $81.82$  $87.27$  $84.46$  $75.04$  $66.37$  $70.44$  $77.45$ 
RAE${}_{t}$  16  $85.35$  $82.55$  $83.93$  $71.37$  $75.44$  $73.35$  $78.64$ 
VRAE${}_{t}$  16  $87.37$  $85.75$  $86.55$  $76.06$  $78.51$  $77.27$  $81.91$ 
SVRAE${}_{t}$  16  $88.88$  $86.93$  $87.89$  $78.17$  $81.14$  $79.63$  $83.76$ 
Supervised Methods  
RNN${}_{t}$  T  $88.88$  $88.45$  $88.67$  $78.73$  $79.43$  $79.09$  $83.87$ 
1D CNN  T  $91.57$  $87.09$  $89.27$  $78.01$  $85.10$  $81.40$  $85.33$ 
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Mena, F.; Olivares, P.; Bugueño, M.; Molina, G.; Araya, M. On the Quality of Deep Representations for Kepler Light Curves Using Variational AutoEncoders. Signals 2021, 2, 706728. https://doi.org/10.3390/signals2040042
Mena F, Olivares P, Bugueño M, Molina G, Araya M. On the Quality of Deep Representations for Kepler Light Curves Using Variational AutoEncoders. Signals. 2021; 2(4):706728. https://doi.org/10.3390/signals2040042
Chicago/Turabian StyleMena, Francisco, Patricio Olivares, Margarita Bugueño, Gabriel Molina, and Mauricio Araya. 2021. "On the Quality of Deep Representations for Kepler Light Curves Using Variational AutoEncoders" Signals 2, no. 4: 706728. https://doi.org/10.3390/signals2040042