  We bring the theory of rough paths to the study of non-parametric statistics
on streamed data. We discuss the problem of regression where the input variable
is a stream of information, and the dependent response is also (potentially) a
stream.
  A certain graded feature set of a stream, known in the rough path literature
as the signature, has a universality that allows formally, linear regression to
be used to characterise the functional relationship between independent
explanatory variables and the conditional distribution of the dependent
response.
  This approach, via linear regression on the signature of the stream, is
almost totally general, and yet it still allows explicit computation. The
grading allows truncation of the feature set and so leads to an efficient local
description for streams (rough paths). In the statistical context this method
offers potentially significant, even transformational dimension reduction.
  By way of illustration, our approach is applied to stationary time series
including the familiar AR model and ARCH model. In the numerical examples we
examined, our predictions achieve similar accuracy to the Gaussian Process (GP)
approach with much lower computational cost especially when the sample size is
large.
