Before revision:  We provide an approach based on identifying carefully chosen features of the streamwhich 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 streamand the shuffle product of tensors, are standard tools in the theory of rough pathsand seem appropriate in this context of regression as well and provide a surprisingly unified and non-parametric approach.
After revision:  We provide an approach based on identifying carefully chosen features of the streamwhich allows linear regression to be used to characterise the functional relationship between explanatory variables and the conditional distribution of the dependent response. This approach, via linear regression on the signature of a streamand the shuffle product of tensors, are standard tools in the theory of rough pathsand seem appropriate in this context of regression as well and provide a surprisingly unified and non-parametric approach.