Functional Regression for Quasar Spectra
  The Lyman-alpha forest is a portion of the observed light spectrum of distant
galactic nuclei which allows us to probe remote regions of the Universe that
are otherwise inaccessible. The observed Lyman-alpha forest of a quasar light
spectrum can be modeled as a noisy realization of a smooth curve that is
affected by a `damping effect' which occurs whenever the light emitted by the
quasar travels through regions of the Universe with higher matter
concentration. To decode the information conveyed by the Lyman-alpha forest
about the matter distribution, we must be able to separate the smooth
`continuum' from the noise and the contribution of the damping effect in the
quasar light spectra. To predict the continuum in the Lyman-alpha forest, we
use a nonparametric functional regression model in which both the response and
the predictor variable (the smooth part of the damping-free portion of the
spectrum) are function-valued random variables. We demonstrate that the
proposed method accurately predicts the unobservable continuum in the
Lyman-alpha forest both on simulated spectra and real spectra. Also, we
introduce distribution-free prediction bands for the nonparametric functional
regression model that have finite sample guarantees. These prediction bands,
together with bootstrap-based confidence bands for the projection of the mean
continuum on a fixed number of principal components, allow us to assess the
degree of uncertainty in the model predictions.
