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

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  • Daniel Levin
  • Terry Lyons
  • Hao Ni
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    Abstract

    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. To illustrate the approach we consider the problem of using data to predict the conditional distribution of the near future of a stationary, ergodic time series and compare it with probabilistic approaches based on first fitting a model. We believe 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. Although the approach is non-parametric, it presents itself in computationally tractable and flexible restricted forms in examples we considered. Popular techniques in time series analysis such as AR, ARCH and GARCH can be seen to be special cases of our approach, but it is not clear if they are always the best or most informative choices.

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    File URL: http://arxiv.org/pdf/1309.0260
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1309.0260.

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    Date of creation: Sep 2013
    Date of revision: Sep 2013
    Handle: RePEc:arx:papers:1309.0260

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    Web page: http://arxiv.org/

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    1. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    2. Tim Bollerslev, 2008. "Glossary to ARCH (GARCH)," CREATES Research Papers 2008-49, School of Economics and Management, University of Aarhus.
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    Cited by:
    1. Lajos Gergely Gyurk\'o & Terry Lyons & Mark Kontkowski & Jonathan Field, 2013. "Extracting information from the signature of a financial data stream," Papers 1307.7244, arXiv.org, revised Jul 2014.

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