Abstract
We study the finite sample performance of predictors in the functional (Hilbertian) autoregressive model \({X_{n+1} = \Psi(X_n)+\varepsilon_n}\). Our extensive empirical study based on simulated and real data reveals that predictors of the form \({\hat\Psi(X_n)}\) are practically optimal in a sense that their prediction errors are comparable with those of the infeasible perfect predictor Ψ(X n ). The predictions \({\hat\Psi(X_n)}\) cannot be improved by an improved estimation of Ψ, nor by a more refined prediction approach which uses predictive factors rather than the functional principal components. We also discuss the practical limits of predictions that are feasible using the functional autoregressive model. These findings have not been established by theoretical work currently available, and may serve as a practical reference to the properties of predictors of functional data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Antoniadis A, Sapatinas T (2003) Wavelet methods for continuous time prediction using Hilbert–valued autoregressive processes. J Multivar Anal 87: 133–158
Besse P, Cardot H (1996) Approximation spline de la prévision d’un processus fonctionnel autorégressif d’ordre 1. Can J Stat 24: 467–487
Besse P, Cardot H, Stephenson D (2000) Autoregressive forecasting of some functional climatic variations. Scand J Stat 27: 673–687
Bosq D (2000) Linear processes in function spaces. Springer, New York
Bosq D, Blanke D (2007) Inference and prediction in large dimensions. Wiley, New Jersey
Damon J, Guillas S (2002) The inclusion of exogenous variables in functional autoregressive ozone forecasting. Environmetrics 13: 759–774
Ferraty, F, Romain, Y (eds) (2011) The oxford handbook of functional data analysis. Oxford University Press, Oxford
Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer, New York
Gabrys R, Horváth L, Kokoszka P (2010) Tests for error correlation in the functional linear model. J Am Stat Assoc 105: 1113–1125
Hörmann S, Kokoszka P (2010) Weakly dependent functional data. Ann Stat 38: 1845–1884
Hörmann S, Kokoszka P (2011) Consistency of the mean and the principal components of spatially distributed functional data. Tech. rep., Utah State University, Logan
Horváth L, Kokoszka P (2011+) Inference for functional data with applications. Springer Series in Statistics, Springer, forthcoming
Horváth L, Hušková M, Kokoszka P (2010) Testing the stability of the functional autoregressive process. J Multivar Anal 101: 352–367
Kargin V, Onatski A (2008) Curve forecasting by functional autoregression. J Multivar Anal 99: 2508–2526
Kokoszka P, Zhang X (2010) Improved estimation of the kernel of the functional autoregressive process. Tech. rep., Utah State University, Logan
Ramsay J, Hooker G, Graves S (2009) Functional data analysis with R and MATLAB. Springer, New York
Ramsay JO, Silverman BW (2002) Applied functional data analysis. Springer, New York
Ramsay JO, Silverman BW (2005) Functional data analysis. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Didericksen, D., Kokoszka, P. & Zhang, X. Empirical properties of forecasts with the functional autoregressive model. Comput Stat 27, 285–298 (2012). https://doi.org/10.1007/s00180-011-0256-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-011-0256-2