Abstract
In implementations of the functional data methods, the effect of the initial choice of an orthonormal basis has not been properly studied. Typically, several standard bases such as Fourier, wavelets, splines, etc. are considered to transform observed functional data and a choice is made without any formal criteria indicating which of the bases is preferable for the initial transformation of the data. In an attempt to address this issue, we propose a strictly data-driven method of orthonormal basis selection. The method uses B-splines and utilizes recently introduced efficient orthornormal bases called the splinets. The algorithm learns from the data in the machine learning style to efficiently place knots. The optimality criterion is based on the average (per functional data point) mean square error and is utilized both in the learning algorithms and in comparison studies. The latter indicate efficiency that could be used to analyze responses to a complex physical system.
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References
Andrén, P.: Power spectral density approximations of longitudinal road profiles. Int. J. Vehicle Design 40, 2–14 (2006)
De Boor, C.: A practical guide to splines, Applied Mathematical Sciences, vol. 27, revised edn. Springer-Verlag New York (2001)
Ferraty, F., Vieu, P.: Nonparametric functional data analysis: theory and practice. Springer Science and Business Media (2006)
Guo J., H.J.J.B.Y., Zhang, Z.: Spline-lasso in high-dimensional linear regression. Journal of the American Statistical Association 111(513), 288–297 ((2016))
Hastie, T., Tibshirani, R., Friedman, J.: The elements of statistical learning: data mining, inference and prediction, 2 edn. Springer (2009). URL http://www-stat.stanford.edu/~tibs/ElemStatLearn/
Hsing, T., Eubank, R.: Theoretical foundations of functional data analysis, with an introduction to linear operators. John Wiley and Sons (2015)
Johannesson, P., Speckert, M. (eds.): Guide to Load Analysis for Durability in Vehicle Engineering. Wiley, Chichester. (2013)
Karhunen, K.: Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae. Ser. A. 37, 1–79 (1947)
Liu, X., Nassar, H., Podgórski, K.: Splinets—efficient orthonormalization of the b-splines. ArXiv abs/1910.07341 (2019)
Podgórski, K., Rychlik, I., Wallin, J.: Slepian noise approach for gaussian and Laplace moving average processes. Extremes 18(4), 665–695 (2015). URL https://doi.org/10.1007/s10687-015-0227-z
Ramsay, J.O.: Functional data analysis. Encyclopedia of Statistical Sciences 4 (2004)
Schumaker, L.: Spline functions: basic theory. Cambridge University Press (2007)
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Nassar, H., Podgórski, K. (2021). Empirically Driven Orthonormal Bases for Functional Data Analysis. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_76
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DOI: https://doi.org/10.1007/978-3-030-55874-1_76
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