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Kernel-based functional principal components

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  • Boente, Graciela
  • Fraiman, Ricardo

Abstract

In this paper, we propose kernel-based smooth estimates of the functional principal components when data are continuous trajectories of stochastic processes. Strong consistency and the asymptotic distribution are derived under mild conditions.

Suggested Citation

  • Boente, Graciela & Fraiman, Ricardo, 2000. "Kernel-based functional principal components," Statistics & Probability Letters, Elsevier, vol. 48(4), pages 335-345, July.
  • Handle: RePEc:eee:stapro:v:48:y:2000:i:4:p:335-345
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    References listed on IDEAS

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    1. Philippe Besse & J. Ramsay, 1986. "Principal components analysis of sampled functions," Psychometrika, Springer;The Psychometric Society, vol. 51(2), pages 285-311, June.
    2. J. Ramsay, 1982. "When the data are functions," Psychometrika, Springer;The Psychometric Society, vol. 47(4), pages 379-396, December.
    3. 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.
    4. Fraiman, Ricardo & Iribarren, Gonzalo Pérez, 1991. "Nonparametric regression estimation in models with weak error's structure," Journal of Multivariate Analysis, Elsevier, vol. 37(2), pages 180-196, May.
    5. Yurinskii, V. V., 1976. "Exponential inequalities for sums of random vectors," Journal of Multivariate Analysis, Elsevier, vol. 6(4), pages 473-499, December.
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    Cited by:

    1. Park, Juhyun & Gasser, Theo & Rousson, Valentin, 2009. "Structural components in functional data," Computational Statistics & Data Analysis, Elsevier, vol. 53(9), pages 3452-3465, July.
    2. Davy Paindaveine & Julien Remy & Thomas Verdebout, 2017. "Testing for Principal Component Directions under Weak Identifiability," Working Papers ECARES ECARES 2017-37, ULB -- Universite Libre de Bruxelles.
    3. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2004. "An anova test for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 111-122, August.
    4. van der Linde, Angelika, 2008. "Variational Bayesian functional PCA," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 517-533, December.
    5. Lakraj, Gamage Pemantha & Ruymgaart, Frits, 2017. "Some asymptotic theory for Silverman’s smoothed functional principal components in an abstract Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 122-132.
    6. Bali, Juan Lucas & Boente, Graciela, 2014. "Consistency of a numerical approximation to the first principal component projection pursuit estimator," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 181-191.
    7. Peter Hall & You-Jun Yang, 2010. "Ordering and selecting components in multivariate or functional data linear prediction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(1), pages 93-110.
    8. Fang Yao & Yichao Wu & Jialin Zou, 2016. "Probability-enhanced effective dimension reduction for classifying sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 1-22, March.
    9. Boente, Graciela & Rodriguez, Daniela & Sued, Mariela, 2010. "Inference under functional proportional and common principal component models," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 464-475, February.
    10. Peter Hall & Hans-Georg Müller & Fang Yao, 2008. "Modelling sparse generalized longitudinal observations with latent Gaussian processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(4), pages 703-723.
    11. Poskitt, D.S. & Sengarapillai, Arivalzahan, 2013. "Description length and dimensionality reduction in functional data analysis," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 98-113.
    12. Fraiman, Ricardo & Pateiro-López, Beatriz, 2012. "Quantiles for finite and infinite dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 108(C), pages 1-14.
    13. Benhenni, K. & Hedli-Griche, S. & Rachdi, M., 2010. "Estimation of the regression operator from functional fixed-design with correlated errors," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 476-490, February.
    14. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2006. "On the use of the bootstrap for estimating functions with functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 1063-1074, November.
    15. Martínez-Camblor, Pablo & Corral, Norberto, 2011. "Repeated measures analysis for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3244-3256, December.
    16. Berrendero, J.R. & Justel, A. & Svarc, M., 2011. "Principal components for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(9), pages 2619-2634, September.
    17. Huang, Lele & Wang, Huiwen & Zheng, Andi, 2014. "The M-estimator for functional linear regression model," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 165-173.
    18. Michio Yamamoto, 2012. "Clustering of functional data in a low-dimensional subspace," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 6(3), pages 219-247, October.
    19. Said Attaoui, 2014. "Strong uniform consistency rates and asymptotic normality of conditional density estimator in the single functional index modeling for time series data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 98(3), pages 257-286, July.
    20. Idir Ouassou & Mustapha Rachdi, 2012. "Regression operator estimation by delta-sequences method for functional data and its applications," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 96(4), pages 451-465, October.
    21. Tomasz Górecki & Łukasz Smaga, 2015. "A comparison of tests for the one-way ANOVA problem for functional data," Computational Statistics, Springer, vol. 30(4), pages 987-1010, December.

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