Predicting Quasar Continua Near Lyman-$\alpha$ with Principal Component
  Analysis
  Measuring the proximity effect and the damping wing of intergalactic neutral
hydrogen in quasar spectra during the epoch of reionization requires an
estimate of the intrinsic continuum at rest-frame wavelengths $\lambda_{\rm
rest}\sim1200$-$1260$ {\AA}. In contrast to previous works which used composite
spectra with matched spectral properties or explored correlations between
parameters of broad emission lines, we opted for a non-parametric predictive
approach based on principal component analysis (PCA) to predict the intrinsic
spectrum from the spectral properties at redder (i.e. unabsorbed) wavelengths.
We decomposed a sample of $12764$ spectra of $z\sim2$-$2.5$ quasars from
SDSS/BOSS into 10 red-side ($1280$ {\AA} $<\lambda_{\rm rest}<2900$ {\AA}) and
6 blue-side ($1180$ {\AA} $<\lambda_{\rm rest}<1280$ {\AA}) PCA basis spectra,
and constructed a projection matrix to predict the blue-side coefficients from
a fit to the red-side spectrum. We found that our method predicts the blue-side
continuum with $\sim6$-$12\%$ precision and $\lesssim1\%$ bias by testing on
the full training set sample. We then computed predictions for the blue-side
continua of the two quasars currently known at $z>7$: ULAS J1120+0641
($z=7.09$) and ULAS J1342+0928 ($z=7.54$). Both of these quasars are known to
exhibit extreme emission line properties, so we individually calibrated the
precision of the continuum predictions from similar quasars in the training
set. We find that both $z>7$ quasars, and in particular ULAS J1342+0928, show
signs of damping wing-like absorption at wavelengths redward of Ly$\alpha$.
