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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

Abstract

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.

Suggested Citation

  • Daniel Levin & Terry Lyons & Hao Ni, 2013. "Learning from the past, predicting the statistics for the future, learning an evolving system," Papers 1309.0260, arXiv.org, revised Mar 2016.
  • Handle: RePEc:arx:papers:1309.0260
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    File URL: http://arxiv.org/pdf/1309.0260
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    References listed on IDEAS

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    1. Tim Bollerslev, 2008. "Glossary to ARCH (GARCH)," CREATES Research Papers 2008-49, Department of Economics and Business Economics, Aarhus University.
    2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
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    1. repec:bla:jorssa:v:180:y:2017:i:2:p:353-407 is not listed on IDEAS
    2. 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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