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

TL;DR

This paper develops an adaptive, online predictor for nonstationary, time-varying autoregressive processes by aggregating multiple predictors, achieving minimax optimality and adapting to unknown smoothness with practical recursive computation.

Contribution

It introduces a novel aggregation method for nonstationary processes that achieves minimax rates and adapts to unknown smoothness in an online setting.

Findings

The aggregated predictor attains minimax convergence rates.

The method is applicable in real-time online prediction.

Numerical experiments validate the theoretical results.

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.