A comparison of quasar emission reconstruction techniques for $z\geq5.0$ Lyman-$\alpha$ and Lyman-$\beta$ transmission

Reconstruction techniques for intrinsic quasar continua are crucial for the precision study of Lyman-$\alpha$ (Ly-$\alpha$) and Lyman-$\beta$ (Ly-$\beta$) transmission at $z>5.0$, where the $\lambda<1215 A$ emission of quasars is nearly completely absorbed. While the number and quality of spectroscopic observations has become theoretically sufficient to quantify Ly-$\alpha$ transmission at $5.0<z<6.0$ to better than $1\%$, the biases and uncertainties arising from predicting the unabsorbed continuum are not known to the same level. In this paper, we systematically evaluate eight reconstruction techniques on a unified testing sample of $2.7<z<3.5$ quasars drawn from eBOSS. The methods include power-law extrapolation, stacking of neighbours, and six variants of Principal Component Analysis (PCA) using direct projection, fitting of components, or neural networks to perform weight mapping. We find that power-law reconstructions and the PCA with fewest components and smallest training sample display the largest biases in the Ly-$\alpha$ forest ($-9.58\%/+8.22\%$ respectively). Power-law extrapolations have larger scatters than previously assumed of $+13.1\%/-13.2\%$ over Ly-$\alpha$ and $+19.9\%/-20.1\%$ over Ly-$\beta$. We present two new PCAs which achieve the best current accuracies of $9\%$ for Ly-$\alpha$ and $17\%$ for Ly-$\beta$. We apply the eight techniques after accounting for wavelength-dependent biases and scatter to a sample $19$ quasars at $z>5.7$ with IR X-Shooter spectroscopy, obtaining well-characterised measurements for the mean flux transmission at $4.7<z<6.3$. Our results demonstrate the importance of testing and, when relevant, training, continuum reconstruction techniques in a systematic way.