Stellar parameter prediction and spectral simulation using machine learning - A systematic comparison of methods with HARPS observational data

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Volume 693 (January 2025)

A&A, 693 (2025) A256

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Table E.1

Notation.

term	explanation
s	Input spectrum.
l	Ground truth labels (spectra parameters).
$\hat{s}$ $\[\hat{\mathbf{s}}\]$	Spectrum prediction.
$\hat{l}$ $\[\hat{\mathbf{l}}\]$	Label prediction.
μ^k	The mean value of the kth label.
σ^k	The standard deviation of the kth label.
$\underline{l}$ $\[\underline{\mathbf{l}}\]$	Normalized labels according to Eq. 1.
l_scale	Scaled labels according to Eq. 1.
Δl^k	Intervention for kth label.

${\hat{l}}^{L I F}$ $\[\hat{\mathbf{l}}^{\mathrm{LIF}}\]$	Label-informed factors in the latent representation. They are nodes in the latent representation that are supervised with ground truth labels l during training. They are treated as label prediction $\hat{l}$ $\[\hat{\mathbf{l}}\]$ during inference.
u	Unsupervised factors in the latent representation. Associated nodes represent undetermined spectra parameters and statistically relevant features.
b	Latent representation is a concatenation of ${\hat{l}}^{L I F}$ $\[\hat{\mathbf{l}}^{\mathrm{LIF}}\]$ and u.
ℬ Latent space. M: b → s	Hidden generative process that maps latent representation to the spectrum.

D	Number of samples in the HARPS training dataset.
N	Number of pixels in the HARPS or ETC spectrum.
K	Number of supervised and injected labels.

q^c	Classic encoder – part of autoencoder (AE) – projects input spectrum into latent representation b.
q^p	Probabilistic encoder – part of variational autoencoder (VAE) – that projects input spectrum into distribution over latent representations.
μ	Mean values of latent representation b.
σ	Standard deviation of latent representation b.
p	Decoder – used by both VAE and AE – that projects latent representation b to output spectrum $\hat{s}$ $\[\hat{\mathbf{s}}\]$ .

L	Loss function.
L_rec	Reconstruction loss function.
L_lab	Label loss function.
$D_{K L}$ $\[D_{\mathbb{K} \mathbb{L}}\]$	Kullback-Leibler divergence (also known as information gain or relative entropy).
λ_lab	Label loss weight.
$λ_{K L}$ $\[\lambda_{\mathbb{K} \mathbb{L}}\]$	KL loss weight.

do(b, Δl^k, k)	do operator returns modified b′, where Δl^k is added to the kth element of b.
shift(s, v)	Doppler shifts of the spectrum s by radial velocity v.
∘	Function composition operator, g(f(x)) = (g ∘ f)(x).

$E$ $\[\mathbb{E}\]$	Expected value.

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