The Annals of Statistics

Aggregation of predictors for nonstationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes

Christophe Giraud, François Roueff, and Andres Sanchez-Perez

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Abstract

In this work, we study the problem of aggregating a finite number of predictors for nonstationary sub-linear processes. We provide oracle inequalities relying essentially on three ingredients: (1) a uniform bound of the $\ell^{1}$ norm of the time varying sub-linear coefficients, (2) a Lipschitz assumption on the predictors and (3) moment conditions on the noise appearing in the linear representation. Two kinds of aggregations are considered giving rise to different moment conditions on the noise and more or less sharp oracle inequalities. We apply this approach for deriving an adaptive predictor for locally stationary time varying autoregressive (TVAR) processes. It is obtained by aggregating a finite number of well chosen predictors, each of them enjoying an optimal minimax convergence rate under specific smoothness conditions on the TVAR coefficients. We show that the obtained aggregated predictor achieves a minimax rate while adapting to the unknown smoothness. To prove this result, a lower bound is established for the minimax rate of the prediction risk for the TVAR process. Numerical experiments complete this study. An important feature of this approach is that the aggregated predictor can be computed recursively and is thus applicable in an online prediction context.

Article information

Source
Ann. Statist., Volume 43, Number 6 (2015), 2412-2450.

Dates
Received: May 2014
Revised: May 2015
First available in Project Euclid: 7 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1444222080

Digital Object Identifier
doi:10.1214/15-AOS1345

Mathematical Reviews number (MathSciNet)
MR3405599

Zentralblatt MATH identifier
1327.62478

Subjects
Primary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 62G99: None of the above, but in this section 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 68W27: Online algorithms

Keywords
Nonstationary time series exponential weighted aggregation online learning time varying autoregressive processes adaptive prediction

Citation

Giraud, Christophe; Roueff, François; Sanchez-Perez, Andres. Aggregation of predictors for nonstationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes. Ann. Statist. 43 (2015), no. 6, 2412--2450. doi:10.1214/15-AOS1345. https://projecteuclid.org/euclid.aos/1444222080


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Supplemental materials

  • Supplementary material for: Aggregation of predictors for nonstationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes. We explain how to build nonadaptive minimax predictors which can be used in the aggregation step. The document also contains some technical proofs and provides additional results with improved aggregation rates.