Electronic Journal of Statistics

Sparsely observed functional time series: estimation and prediction

Tomáš Rubín and Victor M. Panaretos

Full-text: Open access


Functional time series analysis, whether based on time or frequency domain methodology, has traditionally been carried out under the assumption of complete observation of the constituent series of curves, assumed stationary. Nevertheless, as is often the case with independent functional data, it may well happen that the data available to the analyst are not the actual sequence of curves, but relatively few and noisy measurements per curve, potentially at different locations in each curve’s domain. Under this sparse sampling regime, neither the established estimators of the time series’ dynamics nor their corresponding theoretical analysis will apply. The subject of this paper is to tackle the problem of estimating the dynamics and of recovering the latent process of smooth curves in the sparse regime. Assuming smoothness of the latent curves, we construct a consistent nonparametric estimator of the series’ spectral density operator and use it to develop a frequency-domain recovery approach, that predicts the latent curve at a given time by borrowing strength from the (estimated) dynamic correlations in the series across time. This new methodology is seen to comprehensively outperform a naive recovery approach that would ignore temporal dependence and use only methodology employed in the i.i.d. setting and hinging on the lag zero covariance. Further to predicting the latent curves from their noisy point samples, the method fills in gaps in the sequence (curves nowhere sampled), denoises the data, and serves as a basis for forecasting. Means of providing corresponding confidence bands are also investigated. A simulation study interestingly suggests that sparse observation for a longer time period may provide better performance than dense observation for a shorter period, in the presence of smoothness. The methodology is further illustrated by application to an environmental data set on fair-weather atmospheric electricity, which naturally leads to a sparse functional time series.

Article information

Electron. J. Statist., Volume 14, Number 1 (2020), 1137-1210.

Received: February 2019
First available in Project Euclid: 28 February 2020

Permanent link to this document

Digital Object Identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M15, 60G10

autocovariance operator confidence bands functional data analysis nonparametric regression spectral density operator

Creative Commons Attribution 4.0 International License.


Rubín, Tomáš; Panaretos, Victor M. Sparsely observed functional time series: estimation and prediction. Electron. J. Statist. 14 (2020), no. 1, 1137--1210. doi:10.1214/20-EJS1690. https://projecteuclid.org/euclid.ejs/1582859034

Export citation


  • [1] Aue, A., Norinho, D. D. and Hörmann, S. (2015). On the prediction of stationary functional time series., Journal of the American Statistical Association 110 378–392.
  • [2] Bartlett, M. S. (1950). Periodogram analysis and continuous spectra., Biometrika 37 1–16.
  • [3] Blanke, D. and Bosq, D. (2007). Inference and Prediction in Large Dimensions. In, Wiley Series in Probability and Statistics Dunod.
  • [4] Bosq, D. (2012a)., Linear Processes in Function Spaces: Theory and Applications 149. Springer Science & Business Media.
  • [5] Bosq, D. (2012b)., Nonparametric Statistics for Stochastic Processes: Estimation and Prediction 110. Springer Science & Business Media.
  • [6] Brillinger, D. R. (1981)., Time Series: Data Analysis and Theory 36. Siam.
  • [7] Cai, T. and Yuan, M. (2010). Nonparametric covariance function estimation for functional and longitudinal data., University of Pennsylvania and Georgia inistitute of technology.
  • [8] Degras, D. A. (2011). Simultaneous confidence bands for nonparametric regression with functional data., Statistica Sinica 1735–1765.
  • [9] Fan, J. and Gijbels, I. (1996)., Local Polynomial Modelling and its Applications. Monographs on statistics and applied probability 66. Chapman & Hall, London.
  • [10] Fan, J. and Yao, Q. (2008)., Nonlinear Time Series: Nonparametric and Parametric Methods. Springer Science & Business Media.
  • [11] Ferraty, F. and Vieu, P. (2006)., Nonparametric Functional Data Analysis: Theory and Practice. Springer Science & Business Media.
  • [12] Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis., The annals of statistics 1493–1517.
  • [13] Hansen, B. E. (2008). Uniform convergence rates for kernel estimation with dependent data., Econometric Theory 24 726–748.
  • [14] Hörmann, S., Kidziński, Ł. and Hallin, M. (2015). Dynamic functional principal components., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 319–348.
  • [15] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data., The Annals of Statistics 38 1845–1884.
  • [16] Hörmann, S., Kokoszka, P. and Nisol, G. (2016). Detection of periodicity in functional time series., arXiv preprint arXiv: 1607.02017 .
  • [17] Horváth, L., Kokoszka, P. and Reeder, R. (2013). Estimation of the mean of functional time series and a two-sample problem., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 75 103–122.
  • [18] Horváth, L., Rice, G. and Whipple, S. (2016). Adaptive bandwidth selection in the long run covariance estimator of functional time series., Computational Statistics & Data Analysis 100 676–693.
  • [19] Hsing, T. and Eubank, R. (2015)., Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. John Wiley & Sons.
  • [20] Israelsson, S. and Tammet, H. (2001). Variation of fair weather atmospheric electricity at Marsta Observatory, Sweden, 1993–1998., Journal of atmospheric and solar-terrestrial physics 63 1693–1703.
  • [21] Kowal, D. R. (2018). Dynamic Function-on-Scalars Regression., arXiv preprint arXiv: 1806.01460 .
  • [22] Kowal, D. R., Matteson, D. S. and Ruppert, D. (2017a). Functional autoregression for sparsely sampled data., Journal of Business & Economic Statistics 1–13.
  • [23] Kowal, D. R., Matteson, D. S. and Ruppert, D. (2017b). A Bayesian Multivariate Functional Dynamic Linear Model., Journal of the American Statistical Association 112 733–744.
  • [24] Li, Y. and Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data., The Annals of Statistics 38 3321–3351.
  • [25] Liebscher, E. (1996). Strong convergence of sums of $\alpha $-mixing random variables with applications to density estimation., Stochastic Processes and Their Applications 65 69–80.
  • [26] Masry, E. (1996). Multivariate local polynomial regression for time series: uniform strong consistency and rates., Journal of Time Series Analysis 17 571–599.
  • [27] Mockus, J. (2012)., Bayesian Approach to Global Optimization: Theory and Applications 37. Springer Science & Business Media.
  • [28] Panaretos, V. M. and Tavakoli, S. (2013a). Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series., Stochastic Processes and their Applications 123 2779–2807.
  • [29] Panaretos, V. M. and Tavakoli, S. (2013b). Fourier analysis of stationary time series in function space., The Annals of Statistics 41 568–603.
  • [30] Paul, D. and Peng, J. (2011). Principal components analysis for sparsely observed correlated functional data using a kernel smoothing approach., Electronic Journal of Statistics 5 1960–2003.
  • [31] Peligrad, M. (1992). Properties of uniform consistency of the kernel estimators of density and regression functions under dependence assumptions., Stochastics and Stochastic Reports 40 147–168.
  • [32] Priestley, M. B. (1981)., Spectral Analysis and Time Series. Probability and mathematical statistics. Academic Press, London.
  • [33] Ramsay, J. O. and Silverman, B. W. (2007)., Applied Functional Data Analysis: Methods and Case Studies. Springer.
  • [34] Rice, G. and Shang, H. L. (2017). A Plug-in Bandwidth Selection Procedure for Long-Run Covariance Estimation with Stationary Functional Time Series., Journal of Time Series Analysis 38 591–609.
  • [35] Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves., Journal of the Royal Statistical Society. Series B (Methodological) 233–243.
  • [36] Rio, E. (1995). The functional law of the iterated logarithm for stationary strongly mixing sequences., The Annals of Probability 23 1188–1203.
  • [37] Rosenblatt, M. (1985)., Stationary Sequences and Random Fields. Birkhäuser, Boston - Mass. a.o.
  • [38] Ruppert, D., Sheather, S. J. and Wand, M. P. (1995). An Effective Bandwidth Selector for Local Least Squares Regression., Journal of the American Statistical Association 90 1257–1270.
  • [39] Tammet, H. (2009). A joint dataset of fair-weather atmospheric electricity., Atmospheric Research 91 194–200.
  • [40] van Delft, A. (2019). A note on quadratic forms of stationary functional time series under mild conditions., arXiv e-prints arXiv: 1905.13186.
  • [41] Wang, J.-L., Chiou, J.-M. and Müller, H.-G. (2016). Functional data analysis., Annual Review of Statistics and Its Application 3 257–295.
  • [42] Wong, R. K. and Zhang, X. (2017). Nonparametric Operator-Regularized Covariance Function Estimation for Functional Data., arXiv preprint arXiv: 1701.06263 .
  • [43] Xu, B., Zou, D., Chen, B. Y., Zhang, J. Y. and Xu, G. W. (2013). Periodic variations of atmospheric electric field on fair weather conditions at YBJ, Tibet., Journal of Atmospheric and Solar-Terrestrial Physics 97 85–90.
  • [44] Yao, F., Müller, H.-G. and Wang, J.-L. (2005a). Functional data analysis for sparse longitudinal data., Journal of the American Statistical Association 100 577–590.
  • [45] Yao, F., Müller, H.-G. and Wang, J.-L. (2005b). Functional linear regression analysis for longitudinal data., The Annals of Statistics 33 2873–2903.
  • [46] Yao, F., Müller, H.-G., Clifford, A. J., Dueker, S. R., Follett, J., Lin, Y., Buchholz, B. A. and Vogel, J. S. (2003). Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate., Biometrics 59 676–685.