Description Length and Dimensionality Reduction in Functional Data Analysis
In this paper we investigate the use of description length principles to select an appropriate number of basis functions for functional data. We provide a flexible definition of the dimension of a random function that is constructed directly from the Karhunen-Loève expansion of the observed process. Our results show that although the classical, principle component variance decomposition technique will behave in a coherent manner, in general, the dimension chosen by this technique will not be consistent. We describe two description length criteria, and prove that they are consistent and that in low noise settings they will identify the true finite dimension of a signal that is embedded in noise. Two examples, one from mass-spectroscopy and the one from climatology, are used to illustrate our ideas. We also explore the application of different forms of the bootstrap for functional data and use these to demonstrate the workings of our theoretical results.
|Date of creation:||12 Nov 2009|
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- 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.
- 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.
- Hansen M. H & Yu B., 2001. "Model Selection and the Principle of Minimum Description Length," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 746-774, June.
- Jeng-Min Chiou & Pai-Ling Li, 2007. "Functional clustering and identifying substructures of longitudinal data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(4), pages 679-699.
- Peter Hall & Mohammad Hosseini-Nasab, 2006. "On properties of functional principal components analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 109-126.
- Ferraty, F. & Vieu, P., 2003. "Curves discrimination: a nonparametric functional approach," Computational Statistics & Data Analysis, Elsevier, vol. 44(1-2), pages 161-173, October.
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