  The cosmological 21 cm signal is one of the most promising avenues to study
the Epoch of Reionization. One class of experiments aiming to detect this
signal is global signal experiments measuring the sky-averaged 21 cm brightness
temperature as a function of frequency. A crucial step in the interpretation
and analysis of such measurements is separating foreground contributions from
the remainder of the signal, requiring accurate models for both components.
Current models for the signal (non-foreground) component, which may contain
cosmological and systematic contributions, are incomplete and unable to capture
the full signal. We propose two new methods for extracting this component from
the data: Firstly, we employ a foreground-orthogonal Gaussian Process to
extract the part of the signal that cannot be explained by the foregrounds.
Secondly, we use a FlexKnot parameterization to model the full signal component
in a free-form manner, not assuming any particular shape or functional form.
This method uses Bayesian model selection to find the simplest signal that can
explain the data. We test our methods on both, synthetic data and publicly
available EDGES low-band data. We find that the Gaussian Process can clearly
capture the foreground-orthogonal signal component of both data sets. The
FlexKnot method correctly recovers the full shape of the input signal used in
the synthetic data and yields a multi-modal distribution of different signal
shapes that can explain the EDGES observations.
