The ilium forward modelling algorithm for multivariate parameter estimation and its application to derive stellar parameters from Gaia spectrophotometry

I introduce an algorithm for estimating parameters from multidimensional data based on forward modelling. It performs an iterative local search to effectively achieve a non-linear interpolation of a template grid. In contrast to many machine-learning approaches, it avoids fitting an inverse model and the problems associated with this. The algorithm makes explicit use of the sensitivities of the data to the parameters, with the goal of better treating parameters which only have a weak impact on the data. The forward modelling approach provides uncertainty (full covariance) estimates in the predicted parameters as well as a goodness-of-fit for observations, thus providing a simple means of identifying outliers. I demonstrate the algorithm, ilium, with the estimation of stellar astrophysical parameters (APs) from simulations of the low-resolution spectrophotometry to be obtained by Gaia. The AP accuracy is competitive with that obtained by a support vector machine. For zero extinction stars covering a wide range of metallicity, surface gravity and temperature, ilium can estimate Teff to an accuracy of 0.3 per cent at G= 15 and to 4 per cent for (lower signal-to-noise ratio) spectra at G= 20, the Gaia limiting magnitude (mean absolute errors are quoted). [Fe/H] and log g can be estimated to accuracies of 0.1–0.4 dex for stars with G≤ 18.5, depending on the magnitude and what priors we can place on the APs. If extinction varies a priori over a wide range (0–10 mag) – which will be the case with Gaia because it is an all-sky survey – then log g and [Fe/H] can still be estimated to 0.3 and 0.5 dex, respectively, at G= 15, but much poorer at G= 18.5. Teff and AV can be estimated quite accurately (3–4 per cent and 0.1–0.2 mag, respectively, at G= 15), but there is a strong and ubiquitous degeneracy in these parameters which limits our ability to estimate either accurately at faint magnitudes. Using the forward model, we can map these degeneracies (in advance) and thus provide a complete probability distribution over solutions. Additional information from the Gaia parallaxes, other surveys or suitable priors should help reduce these degeneracies.