  Two nonparametric methods are presented for forecasting functional time
series (FTS). The FTS we observe is a curve at a discrete-time point. We
address both one-step-ahead forecasting and dynamic updating. Dynamic updating
is a forward prediction of the unobserved segment of the most recent curve.
Among the two proposed methods, the first one is a straightforward adaptation
to FTS of the $k$-nearest neighbors methods for univariate time series
forecasting. The second one is based on a selection of curves, termed \emph{the
curve envelope}, that aims to be representative in shape and magnitude of the
most recent functional observation, either a whole curve or the observed part
of a partially observed curve. In a similar fashion to $k$-nearest neighbors
and other projection methods successfully used for time series forecasting, we
``project'' the $k$-nearest neighbors and the curves in the envelope for
forecasting. In doing so, we keep track of the next period evolution of the
curves. The methods are applied to simulated data, daily electricity demand,
and NOx emissions and provide competitive results with and often superior to
several benchmark predictions. The approach offers a model-free alternative to
statistical methods based on FTS modeling to study the cyclic or seasonal
behavior of many FTS.
