Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Alquier, P., & Li, X. (2012). Prediction of quantiles by statistical learning and application to GDP forecasting. In J.-G. Ganascia, P. Lenca, & J.-M. Petit (Eds.), Discovery science (Volume 7569 of Lecture notes in computer science, pp. 22–36). Berlin/Heidelberg: Springer.
Alquier, P., & Wintenberger, O. (2012). Model selection for weakly dependent time series forecasting. Bernoulli, 18(3), 883–913.
Andrieu, C., & Doucet, A. (1999). An improved method for uniform simulation of stable minimum phase real ARMA (p,q) processes. IEEE Signal Processing Letters, 6(6), 142–144.
Atchadé, Y. F. (2006). An adaptive version for the Metropolis adjusted Langevin algorithm with a truncated drift. Methodology and Computing in Applied Probability, 8(2), 235–254.
Audibert, J.-Y. (2004). PAC-bayesian statistical learning theory. PhD thesis, Université Pierre et Marie Curie-Paris VI.
Beadle, E. R., & Djurić, P. M. (1999). Uniform random parameter generation of stable minimum-phase real ARMA (p,q) processes. IEEE Signal Processing Letters, 4(9), 259–261.
Brockwell, P. J., & Davis, R. A. (2006). Time series: Theory and methods (Springer series in statistics). New York: Springer. Reprint of the second (1991) edition.
Catoni, O. (2004). Statistical learning theory and stochastic optimization (Volume 1851 of Lecture notes in mathematics). Berlin: Springer. Lecture notes from the 31st Summer School on Probability Theory held in Saint-Flour, 8–25 July 2001.
Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, learning, and games. Cambridge: Cambridge University Press.
Coulon-Prieur, C., & Doukhan, P. (2000). A triangular central limit theorem under a new weak dependence condition. Statistics and Probability Letters, 47(1), 61–68.
Dalalyan, A. S., & Tsybakov, A. B. (2008). Aggregation by exponential weighting, sharp PAC-bayesian bounds and sparsity. Machine Learning, 72(1–2), 39–61.
Dedecker, J., Doukhan, P., Lang, G., León R, J. R., Louhichi, S., & Prieur, C. (2007). Weak dependence: With examples and applications (Volume 190 of Lecture notes in statistics). New York: Springer.
Dedecker, J., & Prieur, C. (2005). New dependence coefficients. Examples and applications to statistics. Probability Theory and Related Fields, 132(2), 203–236.
Künsch, H. R. (1995). A note on causal solutions for locally stationary AR-processes. Note from ETH Zürich, available on line here: ftp://ftp.stat.math.ethz.ch/U/hkuensch/localstat-ar.pdf.
Łatuszyński, K., Miasojedow, B., & Niemiro, W. (2013). Nonasymptotic bounds on the estimation error of MCMC algorithms. Bernoulli, 19, 2033–2066.
Łatuszyński, K., & Niemiro, W. (2011). Rigorous confidence bounds for MCMC under a geometric drift condition. Journal of Complexity, 27(1), 23–38.
Leung, G., & Barron, A. R. (2006). Information theory and mixing least-squares regressions. IEEE Transactions on Information Theory, 52(8), 3396–3410.
Mengersen, K. L., & Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. The Annals of Statistics, 24(1), 101–121.
Moulines, E., Priouret, P., & Roueff, F. (2005). On recursive estimation for time varying autoregressive processes. The Annals of Statistics, 33(6), 2610–2654.
Rio, E. (2000). Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes. Comptes Rendus de l’Academie des Sciences Paris Series I Mathematics, 330(10), 905–908.
Roberts, G. O., & Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probability Surveys, 1, 20–71.
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Sanchez-Perez, A. (2015). Time Series Prediction via Aggregation: An Oracle Bound Including Numerical Cost. In: Antoniadis, A., Poggi, JM., Brossat, X. (eds) Modeling and Stochastic Learning for Forecasting in High Dimensions. Lecture Notes in Statistics(), vol 217. Springer, Cham. https://doi.org/10.1007/978-3-319-18732-7_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-18732-7_13
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18731-0
Online ISBN: 978-3-319-18732-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)