Mathematical framework

The mathematical approach of breads is presented in Agrawal et al. 2023. We reiterate here, adapted from section 3.2 of that paper:

breads, or the Broad Repository for Exoplanet Analysis, Detection, and Spectroscopy, is a flexible framework that allows forward modeling of data from moderate to high-resolution spectrographs. The philosophy of breads is to have the users choose a data class, a forward model function, and a fitting strategy.

Instrument Data Classes normalize the data format, simplifying reduction across different spectrographs while allowing for specific behaviors of each instrument to also be coded into their own specific class.

The forward model (FM) aims to reproduce the data d as d = FM + n, where n is the noise. The FM is a function not only of relevant astrophysical parameters of the planet and the host star but also some nuisance parameters. For a general FM within breads, nuisance parameters do not contain physical information about the planet but are needed to model the data accurately. For example, for the specific FM used in Agrawal et al. 2023, the linear parameters that model the spurious contribution of the host star, the contribution from telluric-only component, and the contribution from the residual principal components are all nuisance parameters. Meanwhile, planetary characteristics (which are needed to model its spectrum) such as effective temperature, surface gravity, and radial velocity or its position relative to the star are normal astrophysical parameters and not nuisance parameters.

We distinguish between linear and nonlinear parameters in any forward model function used within the breads framework because breads performs an analytical marginalization of all of its linear parameters, as described in Ruffio et al. (2019), to improve the tractability of the problem. For the specific FM used in Agrawal et al. 2023, the contribution from each FM component is a linear parameter. Indeed, the posteriors for these linear parameters can be calculated analytically without a sampling algorithm such as Markov Chain Monte Carlo (MCMC), allowing for increased speed, and higher-dimensional or complex models (Ruffio et al. 2019, 2021). The definition of a data structure and a forward model leads to the definition of a likelihood assuming Gaussian white noise, which can then be used to either optimize the parameters through a maximum likelihood or derive their posteriors.

Examples of Fitting Methods include a simple grid search optimization, more general optimizers (e.g., Nelder-Mead), or even posterior sampling algorithms such as MCMC. The grid search can, for example, be used to compute detection maps or cross-correlation functions by varying, respectively, the position of the planet or its RV.