Time Series Prediction via Aggregation: An Oracle Bound Including Numerical Cost
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We address the problem of forecasting a time series meeting the Causal Bernoulli Shift model, using a parametric set of predictors. The aggregation technique provides a predictor with well established and quite satisfying theoretical properties expressed by an oracle inequality for the prediction risk. The numerical computation of the aggregated predictor usually relies on a Markov chain Monte Carlo method whose convergence should be evaluated. In particular, it is crucial to bound the number of simulations needed to achieve a numerical precision of the same order as the prediction risk. In this direction we present a fairly general result which can be seen as an oracle inequality including the numerical cost of the predictor computation. The numerical cost appears by letting the oracle inequality depend on the number of simulations required in the Monte Carlo approximation. Some numerical experiments are then carried out to support our findings.
KeywordsOracle Inequalities Markov Chain Monte Carlo (MCMC) MCMC Method Gibbs Estimator MCMC Approximation
The author is specially thankful to François Roueff, Christophe Giraud, Peter Weyer-Brown and the two referees for their extremely careful readings and highly pertinent remarks which substantially improved the paper. This work has been partially supported by the Conseil régional d’Île-de-France under a doctoral allowance of its program Réseau de Recherche Doctoral en Mathématiques de l’Île de France (RDM-IdF) for the period 2012–2015 and by the Labex LMH (ANR-11-IDEX-003-02).
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