| Content | 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 talk, we consider the special case where the explanatory variable is a data 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 provide a unified and non-parametric approach with potential significant dimension reduction. To further improve the efficiency of the signature method, we can combine the non-linear regression method (e.g. neural network) with the signature feature set. Numerical examples are provided to show the superior performance of the proposed method. Lastly I will show that the signature based method have achieved the state-of-the-art results in online handwritten text recognition and action recognition. |