Prediction of time series by statistical learning: general losses and fast rates

Pierre Alquier; Xiaoyin Li; Olivier Wintenberger

Dependence Modeling (2013)

  • Volume: 1, page 65-93
  • ISSN: 2300-2298

Abstract

top
We establish rates of convergences in statistical learning for time series forecasting. Using the PAC-Bayesian approach, slow rates of convergence √ d/n for the Gibbs estimator under the absolute loss were given in a previous work [7], where n is the sample size and d the dimension of the set of predictors. Under the same weak dependence conditions, we extend this result to any convex Lipschitz loss function. We also identify a condition on the parameter space that ensures similar rates for the classical penalized ERM procedure. We apply this method for quantile forecasting of the French GDP. Under additional conditions on the loss functions (satisfied by the quadratic loss function) and for uniformly mixing processes, we prove that the Gibbs estimator actually achieves fast rates of convergence d/n. We discuss the optimality of these different rates pointing out references to lower bounds when they are available. In particular, these results bring a generalization the results of [29] on sparse regression estimation to some autoregression.

How to cite

top

Pierre Alquier, Xiaoyin Li, and Olivier Wintenberger. "Prediction of time series by statistical learning: general losses and fast rates." Dependence Modeling 1 (2013): 65-93. <http://eudml.org/doc/267337>.

@article{PierreAlquier2013,
abstract = {We establish rates of convergences in statistical learning for time series forecasting. Using the PAC-Bayesian approach, slow rates of convergence √ d/n for the Gibbs estimator under the absolute loss were given in a previous work [7], where n is the sample size and d the dimension of the set of predictors. Under the same weak dependence conditions, we extend this result to any convex Lipschitz loss function. We also identify a condition on the parameter space that ensures similar rates for the classical penalized ERM procedure. We apply this method for quantile forecasting of the French GDP. Under additional conditions on the loss functions (satisfied by the quadratic loss function) and for uniformly mixing processes, we prove that the Gibbs estimator actually achieves fast rates of convergence d/n. We discuss the optimality of these different rates pointing out references to lower bounds when they are available. In particular, these results bring a generalization the results of [29] on sparse regression estimation to some autoregression.},
author = {Pierre Alquier, Xiaoyin Li, Olivier Wintenberger},
journal = {Dependence Modeling},
keywords = {Statistical learning theory; time series forecasting; PACBayesian bounds; weak dependence; mixing; oracle inequalities; fast rates; GDP forecasting; statistical learning theory},
language = {eng},
pages = {65-93},
title = {Prediction of time series by statistical learning: general losses and fast rates},
url = {http://eudml.org/doc/267337},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Pierre Alquier
AU - Xiaoyin Li
AU - Olivier Wintenberger
TI - Prediction of time series by statistical learning: general losses and fast rates
JO - Dependence Modeling
PY - 2013
VL - 1
SP - 65
EP - 93
AB - We establish rates of convergences in statistical learning for time series forecasting. Using the PAC-Bayesian approach, slow rates of convergence √ d/n for the Gibbs estimator under the absolute loss were given in a previous work [7], where n is the sample size and d the dimension of the set of predictors. Under the same weak dependence conditions, we extend this result to any convex Lipschitz loss function. We also identify a condition on the parameter space that ensures similar rates for the classical penalized ERM procedure. We apply this method for quantile forecasting of the French GDP. Under additional conditions on the loss functions (satisfied by the quadratic loss function) and for uniformly mixing processes, we prove that the Gibbs estimator actually achieves fast rates of convergence d/n. We discuss the optimality of these different rates pointing out references to lower bounds when they are available. In particular, these results bring a generalization the results of [29] on sparse regression estimation to some autoregression.
LA - eng
KW - Statistical learning theory; time series forecasting; PACBayesian bounds; weak dependence; mixing; oracle inequalities; fast rates; GDP forecasting; statistical learning theory
UR - http://eudml.org/doc/267337
ER -

References

top
  1. [1] A. Agarwal and J. C. Duchi, The generalization ability of online algorithms for dependent data, IEEE Trans. Inform. Theory 59 (2011), no. 1, 573–587. 
  2. [2] H. Akaike, Information theory and an extension of the maximum likelihood principle, 2nd International Symposium on Information Theory (B. N. Petrov and F. Csaki, eds.), Budapest: Akademia Kiado, 1973, pp. 267–281. Zbl0283.62006
  3. [3] P. Alquier and P. Lounici, PAC-Bayesian bounds for sparse regression estimation with exponential weights, Electron. J. Stat. 5 (2011), 127–145. [Crossref] Zbl1274.62463
  4. [4] P. Alquier, PAC-Bayesian bounds for randomized empirical risk minimizers, Math. Methods Statist. 17 (2008), no. 4, 279–304. [Crossref] Zbl1260.62038
  5. [5] K. B. Athreya and S. G. Pantula, Mixing properties of Harris chains and autoregressive processes, J. Appl. Probab. 23 (1986), no. 4, 880–892. MR 867185 (88c:60127) Zbl0623.60087
  6. [6] J.-Y. Audibert, Fast rates in statistical inference through aggregation, Ann. Statist. 35 (2007), no. 2, 1591–1646. Zbl05582005
  7. [7] P. Alquier and O. Wintenberger, Model selection for weakly dependent time series forecasting, Bernoulli 18 (2012), no. 3, 883–193. [Crossref] Zbl1243.62117
  8. [8] G. Biau, O. Biau, and L. Rouvière, Nonparametric forecasting of the manufacturing output growth with firm-level survey data, Journal of Business Cycle Measurement and Analysis 3 (2008), 317–332. 
  9. [9] A. Belloni and V. Chernozhukov, L1-penalized quantile regression in high-dimensional sparse models, Ann. Statist. 39 (2011), no. 1, 82–130. [Crossref] Zbl1209.62064
  10. [10] P. Brockwell and R. Davis, Time series: Theory and methods (2nd edition), Springer, 2009. Zbl1169.62074
  11. [11] E. Britton, P. Fisher, and J. Whitley, The inflation report projections: Understanding the fan chart, Bank of England Quarterly Bulletin 38 (1998), no. 1, 30–37. 
  12. [12] L. Birgé and P. Massart, Gaussian model selection, J. Eur. Math. Soc. 3 (2001), no. 3, 203–268. Zbl1037.62001
  13. [13] G. Biau and B. Patra, Sequential quantile prediction of time series, IEEE Trans. Inform. Theory 57 (2011), 1664– 1674. [Crossref] 
  14. [14] F. Bunea, A. B. Tsybakov, and M. H. Wegkamp, Aggregation for gaussian regression, Ann. Statist. 35 (2007), no. 4, 1674–1697. [Crossref] Zbl1209.62065
  15. [15] O. Catoni, A PAC-Bayesian approach to adaptative classification, preprint (2003). 
  16. [16] O. Catoni, Statistical learning theory and stochastic optimization, Springer Lecture Notes in Mathematics, 2004. Zbl1076.93002
  17. [17] O. Catoni, PAC-Bayesian supervised classification (the thermodynamics of statistical learning), Lecture Notes- Monograph Series, vol. 56, IMS, 2007. Zbl1277.62015
  18. [18] N. Cesa-Bianchi and G. Lugosi, Prediction, learning, and games, Cambridge University Press, New York, 2006. Zbl1114.91001
  19. [19] L. Clavel and C. Minodier, A monthly indicator of the french business climate, Documents de Travail de la DESE, 2009. 
  20. [20] M. Cornec, Constructing a conditional gdp fan chart with an application to french business survey data, 30th CIRET Conference, New York, 2010. 
  21. [21] N. V. Cuong, L. S. Tung Ho, and V. Dinh, Generalization and robustness of batched weighted average algorithm with v-geometrically ergodic markov data, Proceedings of ALT’13 (Jain S., R. Munos, F. Stephan, and T. Zeugmann, eds.), Springer, 2013, pp. 264–278. Zbl06223345
  22. [22] J. C. Duchi, A. Agarwal, M. Johansson, and M. I. Jordan, Ergodic mirror descent, SIAM J. Optim. 22 (2012), no. 4, 1549–1578. Zbl1262.90114
  23. [23] J. Dedecker, P. Doukhan, G. Lang, J. R. León, S. Louhichi, and C. Prieur, Weak dependence, examples and applications, Lecture Notes in Statistics, vol. 190, Springer-Verlag, Berlin, 2007. Zbl1165.62001
  24. [24] M. Devilliers, Les enquêtes de conjoncture, Archives et Documents, no. 101, INSEE, 1984. 
  25. [25] E. Dubois and E. Michaux, étalonnages à l’aide d’enquêtes de conjoncture: de nouvaux résultats, Économie et Prévision, no. 172, INSEE, 2006. 
  26. [26] P. Doukhan, Mixing, Lecture Notes in Statistics, Springer, New York, 1994. 
  27. [27] K. Dowd, The inflation fan charts: An evaluation, Greek Economic Review 23 (2004), 99–111. 
  28. [28] A. Dalalyan and J. Salmon, Sharp oracle inequalities for aggregation of affine estimators, Ann. Statist. 40 (2012), no. 4, 2327–2355. [Crossref] Zbl1257.62038
  29. [29] A. Dalalyan and A. Tsybakov, Aggregation by exponential weighting, sharp PAC-Bayesian bounds and sparsity, Mach. Learn. 72 (2008), 39–61. 
  30. [30] F. X. Diebold, A. S. Tay, and K. F. Wallis, Evaluating density forecasts of inflation: the survey of professional forecasters, Discussion Paper No.48, ESRC Macroeconomic Modelling Bureau, University of Warwick and Working Paper No.6228, National Bureau of Economic Research, Cambridge, Mass., 1997. 
  31. [31] M. D. Donsker and S. S. Varadhan, Asymptotic evaluation of certain markov process expectations for large time. iii., Comm. Pure Appl. Math. 28 (1976), 389–461. Zbl0348.60032
  32. [32] P. Doukhan and O. Wintenberger, Weakly dependent chain with infinite memory, Stochastic Process. Appl. 118 (2008), no. 11, 1997–2013. Zbl1166.60031
  33. [33] R. F. Engle, Autoregressive conditional heteroscedasticity with estimates of variance of united kingdom inflation, Econometrica 50 (1982), 987–1008. [Crossref] Zbl0491.62099
  34. [34] C. Francq and J.-M. Zakoian, Garch models: Structure, statistical inference and financial applications, Wiley- Blackwell, 2010. 
  35. [35] S. Gerchinovitz, Sparsity regret bounds for individual sequences in online linear regression, Proceedings of COLT’11, 2011. Zbl1320.62165
  36. [36] J. Hamilton, Time series analysis, Princeton University Press, 1994. Zbl0831.62061
  37. [37] H. Hang and I. Steinwart, Fast learning from α-mixing observations, Technical report, Fakultät für Mathematik und Physik, Universität Stuttgart, 2012. 
  38. [38] I. A. Ibragimov, Some limit theorems for stationary processes, Theory Probab. Appl. 7 (1962), no. 4, 349–382. Zbl0119.14204
  39. [39] A. B. Juditsky, A. V. Nazin, A. B. Tsybakov, and N. Vayatis, Recursive aggregation of estimators bythe mirror descent algorithm with averaging, Probl. Inf. Transm. 41 (2005), no. 4, 368–384. [Crossref] Zbl1123.62044
  40. [40] A. B. Juditsky, P. Rigollet, and A. B. Tsybakov, Learning my mirror averaging, Ann. Statist. 36 (2008), no. 5, 2183–2206. [Crossref] Zbl1274.62288
  41. [41] R. Koenker and G. Jr. Bassett, Regression quantiles, Econometrica 46 (1978), 33–50. [Crossref] 
  42. [42] R. Koenker, Quantile regression, Cambridge University Press, Cambridge, 2005. Zbl1111.62037
  43. [43] S. Kullback, Information theory and statistics, Wiley, New York, 1959. 
  44. [44] N. Littlestone and M.K. Warmuth, The weighted majority algorithm, Information and Computation 108 (1994), 212–261. Zbl0804.68121
  45. [45] P. Massart, Concentration inequalities and model selection - ecole d’été de probabilités de saint-flour xxxiii - 2003, Lecture Notes in Mathematics - J. Picard Editor, vol. 1896, Springer, 2007. Zbl1170.60006
  46. [46] D. A. McAllester, PAC-Bayesian model averaging, Procs. of of the 12th Annual Conf. On Computational Learning Theory, Santa Cruz, California (Electronic), ACM, New-York, 1999, pp. 164–170. 
  47. [47] R. Meir, Nonparametric time series prediction through adaptive model selection, Mach. Learn. 39 (2000), 5–34. Zbl0954.68124
  48. [48] C. Minodier, Avantages comparés des séries premières valeurs publiées et des séries des valeurs révisées, Documents de Travail de la DESE, 2010. 
  49. [49] D. S. Modha and E. Masry, Memory-universal prediction of stationary random processes, IEEE Trans. Inform. Theory 44 (1998), no. 1, 117–133. [Crossref] Zbl0938.62106
  50. [50] S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Communications and Control Engineering Series, Springer-Verlag London Ltd., London, 1993. MR 1287609 (95j:60103) Zbl0925.60001
  51. [51] A. Nemirovski, Topics in nonparametric statistics, Lectures on Probability Theory and Statistics - Ecole d’ét’e de probagilités de Saint-Flour XXVIII (P. Bernard, ed.), Springer, 2000, pp. 85–277. 
  52. [52] R Development Core Team, R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2008. 
  53. [53] E. Rio, Ingalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes, C. R. Math. Acad. Sci. Paris 330 (2000), 905–908. 
  54. [54] P.-M. Samson, Concentration of measure inequalities for markov chains and φ-mixing processes, Ann. Probab. 28 (2000), no. 1, 416–461. Zbl1044.60061
  55. [55] I. Steinwart and A. Christmann, Fast learning from non-i.i.d. observations, Advances in Neural Information Processing Systems 22 (Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta, eds.), 2009, pp. 1768–1776. 
  56. [56] I. Steinwart, D. Hush, and C. Scovel, Learning from dependent observations, J. Multivariate Anal. 100 (2009), 175–194. [Crossref] Zbl1158.68040
  57. [57] Y. Seldin, F. Laviolette, N. Cesa-Bianchi, J. Shawe-Taylor, J. Peters, and P. Auer, Pac-bayesian inequalities for martingales, IEEE Trans. Inform. Theory 58 (2012), no. 12, 7086–7093. [Crossref] 
  58. [58] A. Sanchez-Perez, Time series prediction via aggregation : an oracle bound including numerical cost, Preprint arXiv:1311.4500, 2013. 
  59. [59] G. Stoltz, Agrégation séquentielle de prédicteurs : méthodologie générale et applications à la prévision de la qualité de l’air et à celle de la consommation électrique, Journal de la SFDS 151 (2010), no. 2, 66–106. 
  60. [60] J. Shawe-Taylor and R. Williamson, A PAC analysis of a bayes estimator, Proceedings of the Tenth Annual Conference on Computational Learning Theory, COLT’97, ACM, 1997, pp. 2–9. 
  61. [61] N. N. Taleb, Black swans and the domains of statistics, Amer. Statist. 61 (2007), no. 3, 198–200. 
  62. [62] A. S. Tay and K. F. Wallis, Density forecasting: a survey, J. Forecast 19 (2000), 235–254. 
  63. [63] V. Vapnik, The nature of statistical learning theory, Springer, 1999. Zbl0934.62009
  64. [64] V.G. Vovk, Aggregating strategies, Proceedings of the 3rd Annual Workshop on Computational Learning Theory (COLT), 1990, pp. 372–283. 
  65. [65] O. Wintenberger, Deviation inequalities for sums of weakly dependent time series, Electron. Commun. Probab. 15 (2010), 489–503. Zbl1225.60034
  66. [66] Y.-L. Xu and D.-R. Chen, Learning rate of regularized regression for exponentially strongly mixing sequence, J. Statist. Plann. Inference 138 (2008), 2180–2189. Zbl1134.62050
  67. [67] B. Zou, L. Li, and Z. Xu, The generalization performance of erm algorithm with strongly mixing observations, Mach. Learn. 75 (2009), 275–295. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.