Ly\alpha\ emission line reconstruction for high-$z$ QSOs
  We introduce an intrinsic Ly\alpha\ emission line profile reconstruction
method for high-$z$ quasars (QSOs). This approach utilises a covariance matrix
of emission line properties obtained from a large, moderate-$z$ ($2 \leq z \leq
2.5$), high signal to noise (S/N > 15) sample of BOSS QSOs. For each QSO, we
complete a Monte Carlo Markov Chain fitting of the continuum and emission line
properties and perform a visual quality assessment to construct a large
database of robustly fit spectra. With this dataset, we construct a covariance
matrix to describe the correlations between the high ionisation emission lines
Ly\alpha, C IV, Si IV + O IV] and C III], and find it to be well approximated
by an $N$-dimensional Gaussian distribution. This covariance matrix
characterises the correlations between the line width, peak height and velocity
offset from systemic while also allowing for the existence of broad and narrow
line components for Ly\alpha\ and C IV. We illustrate how this covariance
matrix allows us to statistically characterise the intrinsic Ly\alpha\ line
solely from the observed spectrum redward of 1275\AA. This procedure can be
used to reconstruct the intrinsic Ly\alpha\ line emission profile in cases
where Ly\alpha\ may otherwise be obscured. Applying this reconstruction method
to our sample of QSOs, we recovered the Ly\alpha\ line flux to within 15 per
cent of the measured flux at 1205\AA\ (1220\AA) ~85 (90) per cent of the time.
