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

Regression analysis aims to use observational data from multiple observations to develop a functional relationship relating explanatory variables to response variables, which is important for much of modern statistics, and econometrics, and also the field of machine learning. In this paper, we consider the special case where the explanatory variable is a stream of information, and the response is also potentially a stream. We provide an approach based on identifying carefully chosen features of the stream which allows linear regression to be used to characterise the functional relationship between explanatory variables and the conditional distribution of the response; the methods used to develop and justify this approach, such as the signature of a stream and the shuffle product of tensors, are standard tools in the theory of rough paths and seem appropriate in this context of regression as well and provide a surprisingly unified and non-parametric approach. We believe that the insight provided by this paper will provide additional tool in the toolbox for studying sequential data. Our reduction of this regression problem for streams to a linear problem is clean, systematic, and efficient in minimizing the effective dimensionality. The clear gradation of finite dimensional approximations increases its usefulness. In examples we considered, we use the autoregressive calibration (AR approach) and Gaussian processes regression (GP approach) as two benchmarks, our approach presents itself in a more robust and flexible restricted form compared with the AR approach, while as a non-parametric approach, it achieves similar accuracy to the GP approach with much lower computational cost especially when the sample size is large. Popular techniques in time series analysis such as AR, ARCH and GARCH can be incorporated to our model.