The Annals of Statistics

Identifying the finite dimensionality of curve time series

Neil Bathia, Qiwei Yao, and Flavio Ziegelmann

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The curve time series framework provides a convenient vehicle to accommodate some nonstationary features into a stationary setup. We propose a new method to identify the dimensionality of curve time series based on the dynamical dependence across different curves. The practical implementation of our method boils down to an eigenanalysis of a finite-dimensional matrix. Furthermore, the determination of the dimensionality is equivalent to the identification of the nonzero eigenvalues of the matrix, which we carry out in terms of some bootstrap tests. Asymptotic properties of the proposed method are investigated. In particular, our estimators for zero-eigenvalues enjoy the fast convergence rate n while the estimators for nonzero eigenvalues converge at the standard √n-rate. The proposed methodology is illustrated with both simulated and real data sets.

Article information

Ann. Statist., Volume 38, Number 6 (2010), 3352-3386.

First available in Project Euclid: 30 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]
Secondary: 60G99: None of the above, but in this section

Autocovariance curve time series dimension reduction eigenanalysis Karhunen–Loéve expansion n convergence rate root-n convergence rate


Bathia, Neil; Yao, Qiwei; Ziegelmann, Flavio. Identifying the finite dimensionality of curve time series. Ann. Statist. 38 (2010), no. 6, 3352--3386. doi:10.1214/10-AOS819.

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  • Benko, M., Hardle, W. and Kneip, A. (2009). Common functional principal components. Ann. Statist. 37 1–34.
  • Besse, P. and Ramsay, J. O. (1986). Principal components analysis of sampled functions. Psychometrika 51 285–311.
  • Bosq, D. (2000). Linear Processes in Function Spaces. Springer, New York.
  • Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136–154.
  • Dunford, N. and Schwartz, J. T. (1988). Linear Operators. Wiley, New York.
  • Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis. Springer, New York.
  • Guillas, S. and Lai, M. J. (2010). Bivariate splines for spatial functional regression models. J. Nonparametr. Stat. 22 477–497.
  • Hall, P. and Vial, C. (2006). Assessing the finite dimensionality of functional data. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 689–705.
  • Hyndman, R. J. and Ullah, M. S. (2007). Robust forecasting of mortality and fertility rates: A functional data approach. Comput. Statist. Data Anal. 51 4942–4956.
  • Lee, A. J. (1990). U-Statistics. Dekker, New York.
  • Li, W. K. and McLeod, A. I. (1981). Distribution of the residual autocorrelations in multivariate ARMA time series models. J. Roy. Statist. Soc. Ser. B 43 231–239.
  • Kneip, A. and Utikal, K. J. (2001). Inference for density families using functional principal component analysis (with discussion). J. Amer. Statist. Assoc. 96 519–542.
  • Mas, A. and Menneteau, L. (2003). Perturbation approach applied to the asymptotic study of random operators. In High Dimensional Probability III (J. Hoffmann-Jorgensen, M. B. Marcus and J. A. Wellner, eds.) 127–134. Birkhäuser, Boston.
  • Pan, J. and Yao, Q. (2008). Modelling multiple time series via common factors. Biometrika 95 365–379.
  • Peña, D. and Box, G. E. P. (1987). Identifying a simplifying structure in time series. J. Amer. Statist. Assoc. 82 836–843.
  • Ramsay, J. O. and Dalzell, C. J. (1991). Some tools for functional data analysis (with discussion). J. Roy. Statist. Soc. Ser. B 53 539–572.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis. Springer, New York.
  • Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233–243.
  • Sen, P. K. (1972). Limiting behavior of regular functionals of empirical distributions for stationary ∗-mixing processes. Probab. Theory Related Fields 25 71–82.
  • Tiao, G. C. and Tsay, R. S. (1989). Model specification in multivariate time series (with discussion). J. Roy. Statist. Soc. Ser. B 51 157–213.