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 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. We believe that the insight provided by this paper will provide additional tool in the toolbox for studying sequential data. Our reductionof 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 .
We bring the theory of rough paths to the study of non-parametric statistics on streamed data. We discuss the problem of regression where the input variable is a stream of information, and the dependent response is also (potentially) a stream. A certain graded feature set of a stream, known in the rough path literature as the signature, has a universality that allows formally, linear regression to be used to characterise the functional relationship between independent explanatory variables and the conditional distribution of the dependent response. This approach, via linear regression on the signature of the stream, is almost totally general, and yet it still allows explicit computation. The grading allows truncation of the feature set and so leads to an efficient local description for streams (rough paths). In the statistical context this method offers potentially significant, even transformational dimension reduction. By way of illustration, our approach is applied to stationary time series including the familiar AR model and ARCH model. In the numerical examples we examined, our predictions achieve similar accuracy to the Gaussian Process (GP) approach with much lower computational cost especially when the sample size is large .
R
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
We bring the theory of rough paths to the study of non-parametric statistics on streamed data. We discuss the problem of regression where the input
A
dependent
R
potentially
(potentially)
R
We provide an approach based on identifying carefully chosen features of the streamwhich allows
A certain graded feature set of a stream, known in the rough path literature as the signature, has a universality that allows formally,
A
independent
R
response; the methods used to develop and justify this approach, such as
dependent response. This approach, via linear regression on
R
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. We believe that the insight provided by this paper will provide additional tool in the toolbox for studying sequential data. Our reductionof 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
the stream, is almost totally general, and yet it still allows explicit computation. The grading allows truncation of the feature set and so leads to an efficient local description for streams (rough paths). In the statistical context this method offers potentially significant, even transformational dimension reduction. By way of illustration, our approach is applied to stationary time series including the familiar AR model and ARCH model. In the numerical examples we examined, our predictions achieve
R
GP
Gaussian Process (GP)
D
. Popular techniques in time series analysis such as AR, ARCH and GARCH can be incorporated to our model