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We bring the theory of rough paths to the study of non-parametric statisticson streamed data. We discuss the problem of regression where the input variableis a stream of information, and the dependent response is also (potentially) astream.  A certain graded feature set of a stream, known in the rough path literatureas the signature, has a universality that allows formally, linear regression tobe used to characterise the functional relationship between independentexplanatory variables and the conditional distribution of the dependentresponse.  This approach, via linear regression on the signature of the stream, isalmost totally general, and yet it still allows explicit computation. Thegrading allows truncation of the feature set and so leads to an efficient localdescription for streams (rough paths). In the statistical context this methodoffers potentially significant, even transformational dimension reduction.  By way of illustration, our approach is applied to stationary time seriesincluding the familiar AR model and ARCH model. In the numerical examples weexamined, our predictions achieve similar accuracy to the Gaussian Process (GP)approach with much lower computational cost especially when the sample size islarge.

Learning from the past, predicting the statistics for the future, learning an evolving system