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Noise reduction for functional time series

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  • Cees Diks
  • Bram Wouters

Abstract

A novel method for noise reduction in the setting of curve time series with error contamination is proposed, based on extending the framework of functional principal component analysis (FPCA). We employ the underlying, finite-dimensional dynamics of the functional time series to separate the serially dependent dynamical part of the observed curves from the noise. Upon identifying the subspaces of the signal and idiosyncratic components, we construct a projection of the observed curve time series along the noise subspace, resulting in an estimate of the underlying denoised curves. This projection is optimal in the sense that it minimizes the mean integrated squared error. By applying our method to similated and real data, we show the denoising estimator is consistent and outperforms existing denoising techniques. Furthermore, we show it can be used as a pre-processing step to improve forecasting.

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  • Cees Diks & Bram Wouters, 2023. "Noise reduction for functional time series," Papers 2307.02154, arXiv.org.
    • Handle: RePEc:arx:papers:2307.02154
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    1. Jushan Bai & Serena Ng, 2002. "Determining the Number of Factors in Approximate Factor Models," Econometrica, Econometric Society, vol. 70(1), pages 191-221, January.
    2. Degui Li & Peter M. Robinson & Han Lin Shang, 2020. "Long-Range Dependent Curve Time Series," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(530), pages 957-971, April.
    3. Shang, Han Lin & Hyndman, Rob.J., 2011. "Nonparametric time series forecasting with dynamic updating," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(7), pages 1310-1324.
    4. Peter Hall & Céline Vial, 2006. "Assessing the finite dimensionality of functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(4), pages 689-705, September.
    5. Pan, Jiazhu & Yao, Qiwei, 2008. "Modelling multiple time series via common factors," LSE Research Online Documents on Economics 22876, London School of Economics and Political Science, LSE Library.
    6. Andreasen, Martin M. & Christensen, Jens H.E. & Rudebusch, Glenn D., 2019. "Term Structure Analysis with Big Data: One-Step Estimation Using Bond Prices," Journal of Econometrics, Elsevier, vol. 212(1), pages 26-46.
    7. Lam, Clifford & Yao, Qiwei & Bathia, Neil, 2011. "Estimation of latent factors for high-dimensional time series," LSE Research Online Documents on Economics 31549, London School of Economics and Political Science, LSE Library.
    8. Cho, Haeran & Goude, Yannig & Brossat, Xavier & Yao, Qiwei, 2013. "Modeling and forecasting daily electricity load curves: a hybrid approach," LSE Research Online Documents on Economics 49634, London School of Economics and Political Science, LSE Library.
    9. Cheng Chen & Shaojun Guo & Xinghao Qiao, 2022. "Functional Linear Regression: Dependence and Error Contamination," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(1), pages 444-457, January.
    10. Bai, Jushan & Ng, Serena, 2007. "Determining the Number of Primitive Shocks in Factor Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 52-60, January.
    11. JoÃo Caldeira & Hudson Torrent, 2017. "Forecasting the US Term Structure of Interest Rates Using Nonparametric Functional Data Analysis," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 36(1), pages 56-73, January.
    12. Hyndman, Rob J. & Shahid Ullah, Md., 2007. "Robust forecasting of mortality and fertility rates: A functional data approach," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4942-4956, June.
    13. Serge Guillas & Ming-Jun Lai, 2010. "Bivariate splines for spatial functional regression models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 477-497.
    14. Gianluca Cubadda & Alain Hecq, 2022. "Dimension Reduction for High‐Dimensional Vector Autoregressive Models," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 84(5), pages 1123-1152, October.
    15. Haeran Cho & Yannig Goude & Xavier Brossat & Qiwei Yao, 2013. "Modeling and Forecasting Daily Electricity Load Curves: A Hybrid Approach," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 7-21, March.
    16. Yao, Fang & Muller, Hans-Georg & Wang, Jane-Ling, 2005. "Functional Data Analysis for Sparse Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 577-590, June.
    17. Yang, Yang & Yang, Yanrong & Shang, Han Lin, 2022. "Feature extraction for functional time series: Theory and application to NIR spectroscopy data," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    18. Clifford Lam & Qiwei Yao & Neil Bathia, 2011. "Estimation of latent factors for high-dimensional time series," Biometrika, Biometrika Trust, vol. 98(4), pages 901-918.
    19. Hallin, Marc & Liska, Roman, 2007. "Determining the Number of Factors in the General Dynamic Factor Model," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 603-617, June.
    20. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
    21. Alexander Aue & Diogo Dubart Norinho & Siegfried Hörmann, 2015. "On the Prediction of Stationary Functional Time Series," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 378-392, March.
    22. Jiazhu Pan & Qiwei Yao, 2008. "Modelling multiple time series via common factors," Biometrika, Biometrika Trust, vol. 95(2), pages 365-379.
    23. Siegfried Hörmann & Łukasz Kidziński & Marc Hallin, 2015. "Dynamic functional principal components," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(2), pages 319-348, March.
    24. Rituparna Sen & Claudia Klüppelberg, 2019. "Time series of functional data with application to yield curves," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 35(4), pages 1028-1043, July.
    25. Gianluca Cubadda & Alain Hecq, 2020. "Dimension Reduction for High Dimensional Vector Autoregressive Models," Papers 2009.03361, arXiv.org, revised Feb 2022.
    26. Lam, Clifford & Yao, Qiwei, 2012. "Factor modeling for high-dimensional time series: inference for the number of factors," LSE Research Online Documents on Economics 45684, London School of Economics and Political Science, LSE Library.
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