In this talk, we consider the setting: a random realization of an evolving dynamical system, and explain how, using notions common in the theory of rough paths, such as the signature, and shuffle product, one can provide a new united approach to the fundamental problem of predicting the conditional distribution of the near future given the past. We will explain how the problem can be reduced to a linear regression and least squaresanalysis. The approach is clean and systematic and provides a clear gradation of finite dimensional approximations. The approach is also non-parametric and very general but still presents itself in computationally tractable and flexible restricted forms for concrete problems. Popular techniques in time series analysis such as GARCH can be seen to be restricted special cases of our approach but it is not clear they are always the best or most informative choices. Some numerical examples will be shown in order to compare our approach and standard time series models.
