On the definition of phase and amplitude variability in functional data analysis
We introduce a modeling and mathematical framework in which the problem of registering a functional data set can be consistently set. In detail, we show that the introduction, in a functional data analysis, of a metric/semi-metric and of a group of warping functions, with respect to which the metric/semi-metric is invariant, enables a sound and not ambiguous definition of phase and amplitude variability. Indeed, in this framework, we prove that the analysis of a registered functional data set can be re-interpreted as the analysis of a set of suitable equivalence classes associated to original functions and induced by the group of the warping functions. Moreover, an amplitude-to-total variability index is proposed. This index turns out to be useful in practical situations for measuring to what extent phase variability affects the data and for comparing the effectiveness of different registration methods. Copyright Sociedad de Estadística e Investigación Operativa 2012
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Volume (Year): 21 (2012)
Issue (Month): 4 (December)
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- Mariano Valderrama, 2007. "An overview to modelling functional data," Computational Statistics, Springer, vol. 22(3), pages 331-334, September.
- Antonio Cuevas & Manuel Febrero & Ricardo Fraiman, 2007. "Robust estimation and classification for functional data via projection-based depth notions," Computational Statistics, Springer, vol. 22(3), pages 481-496, September.
- Ferraty, F., 2010. "High-dimensional data: a fascinating statistical challenge," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 305-306, February.
- Xueli Liu & Hans-Georg Muller, 2004. "Functional Convex Averaging and Synchronization for Time-Warped Random Curves," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 687-699, January.
- Sangalli, Laura M. & Secchi, Piercesare & Vantini, Simone & Vitelli, Valeria, 2010. "k-mean alignment for curve clustering," Computational Statistics & Data Analysis, Elsevier, vol. 54(5), pages 1219-1233, May.
- Manteiga, Wenceslao Gonzalez & Vieu, Philippe, 2007. "Statistics for Functional Data," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4788-4792, June.
- Kneip, Alois & Ramsay, James O, 2008. "Combining Registration and Fitting for Functional Models," Journal of the American Statistical Association, American Statistical Association, vol. 103(483), pages 1155-1165.
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