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Xiao, Luo; Li, Cai; Checkley, William; Crainiceanu, Ciprian

Fast covariance estimation for sparse functional data. (English) Zbl 1384.62142

Stat. Comput. 28, No. 3, 511-522 (2018).
Summary: Smoothing of noisy sample covariances is an important component in functional data analysis. We propose a novel covariance smoothing method based on penalized splines and associated software. The proposed method is a bivariate spline smoother that is designed for covariance smoothing and can be used for sparse functional or longitudinal data. We propose a fast algorithm for covariance smoothing using leave-one-subject-out cross-validation. Our simulations show that the proposed method compares favorably against several commonly used methods. The method is applied to a study of child growth led by one of coauthors and to a public dataset of longitudinal CD4 counts.

Cited in 12 Documents

MSC:

62G08 Nonparametric regression and quantile regression
62H25 Factor analysis and principal components; correspondence analysis

Keywords:

bivariate smoothing; FACEs; fPCA

Software:

face; mgcv; SemiPar; refund
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\textit{L. Xiao} et al., Stat. Comput. 28, No. 3, 511--522 (2018; Zbl 1384.62142)
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References:

[1] Besse, P; Ramsay, JO, Principal components analysis of sampled functions, Psychometrika, 51, 285-311, (1986) · Zbl 0623.62048
[2] Besse, P; Cardot, H; Ferraty, F, Simultaneous nonparametric regressions of unbalanced longitudinal data, Comput. Stat. Data Anal., 24, 255-270, (1997) · Zbl 0900.62199
[3] Cai, T., Yuan, M.: Nonparametric Covariance Function Estimation for Functional and Longitudinal Data. Technical report, University of Pennsylvania, Philadelphia, PA (2012) · Zbl 1117.62451
[4] Cederbaum, J; Pouplier, M; Hoole, P; Greven, S, Functional linear mixed models for irregularly or sparsely sampled data, Stat. Model., 16, 67-88, (2016)
[5] Chen, H; Wang, Y, A penalized spline approach to functional mixed effects model analysis, Biometrics, 67, 861-870, (2011) · Zbl 1226.62030
[6] de Boor, C.: A Practical Guide to Splines. Springer, Berlin (1978) · Zbl 0406.41003
[7] Diggle, P., Heagerty, P., Liang, K.-Y., Zeger, S.: Analysis of Longitudinal Data. Oxford University Press, Oxford (1994) · Zbl 1268.62001
[8] Durban, M; Harezlak, J; Wand, MP; Carroll, RJ, Simple Fitting of subject-specific curves for longitudinal data, Stat. Med., 24, 1153-1167, (2005)
[9] Eilers, P; Marx, B, Flexible smoothing with B-splines and penalties (with discussion), Stat. Sci., 11, 89-121, (1996) · Zbl 0955.62562
[10] Eilers, P; Marx, B, Multivariate calibration with temperature interaction using two-dimensional penalized signal regression, Chemom. Intell. Lab. Syst., 66, 159-174, (2003)
[11] Fan, J., Gijbels, I.: Local Polynomial Modelling and its Applications. Chapman & Hall, London (1996) · Zbl 0873.62037
[12] Goldsmith, J; Bobb, J; Crainiceanu, C; Caffo, B; Reich, D, Penalized functional regression, J. Comput. Graph. Stat., 20, 830-851, (2010)
[13] Goldsmith, J; Greven, S; Crainiceanu, C, Corrected confidence bands for functional data using principal components, Biometrics, 69, 41-51, (2013) · Zbl 1274.62776
[14] Huang, L., Scheipl, F., Goldsmith, J., Gellar, J., Harezlak, J., Mclean, M., Swihart, B., Xiao, L., Crainiceanu, C., Reiss, P., Chen, Y., Greven, S., Huo, L., Kundu, M., Wrobel, J.: R package mgcv: Methodology for regression with functional data (version 0.1-13). https://cran.r-project.org/web/packages/refund/index.html (2015)
[15] James, G; Hastie, T; Sugar, C, Principal component models for sparse functional data, Biometrika, 87, 587-602, (2000) · Zbl 0962.62056
[16] Kaslow, RA; Ostrow, DG; Detels, R; Phair, JP; Polk, BF; Rinaldo, CR, The multicenter aids cohort study: rationale, organization, and selected characteristics of the participants, Am. J. Epidemiol., 126, 310-318, (1987)
[17] Kneip, A, Nonparametric estimation of common regressors for similar curve data, Ann. Stat., 22, 1386-1427, (1994) · Zbl 0817.62029
[18] Peng, J; Paul, D, A geometric approach to maximum likelihood estimation of functional principal components from sparse longitudinal data, J. Comput. Graph. Stat., 18, 995-1015, (2009)
[19] Ramsay, J; Dalzell, CJ, Some tools for functional data analysis (with discussion), J. R. Stat. Soc. B, 53, 539-572, (1991) · Zbl 0800.62314
[20] Reiss, P.T., Todd Ogden, R.: Smoothing parameter selection for a class of semiparametric linear models. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 71(2), 505-523 (2009) · Zbl 1248.62057
[21] Reiss, P; Huang, L; Mennes, M, Fast function-on-scalar regression with penalized basis expansions, Int. J. Biostat., 6, 28, (2010)
[22] Rodríguez-Álvarez, MX; Lee, D-J; Kneib, T; Durbán, M; Eilers, P, Fast smoothing parameter separation in multidimensional generalized p-splines: the sap algorithm, Stat. Comput., 25, 941-957, (2015) · Zbl 1332.62139
[23] Rodríguez-Álvarez, M. X., Durbán, M., Lee, D.-J., Eilers, P.: Fast estimation of multidimensional adaptive P-spline models. http://arxiv.org/pdf/1610.06861.pdf (2016) · Zbl 0962.62056
[24] Ruppert, D., Wand, M., Carroll, R.: Semiparametric Regression. Cambridge University Press, Cambridge (2003) · Zbl 1038.62042
[25] Scheipl, F; Staicu, A-M; Greven, S, Functional additive mixed models, J. Comput. Graph. Stat., 24, 477-501, (2015)
[26] Seber, G.: A Matrix Handbook for Statisticians. Wiley-Interscience, Hoboken (2007)
[27] Staniswalis, J; Lee, J, Nonparametric regression analysis of longitudinal data, J. Am. Stat. Assoc., 93, 1403-1418, (1998) · Zbl 1064.62522
[28] Wood, S, Thin plate regression splines, J. R. Stat. Soc. B, 65, 95-114, (2003) · Zbl 1063.62059
[29] Wood, S.N.: Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 73(1), 3-36 (2011) · Zbl 1411.62089
[30] Wood, S.: R package mgcv: mixed GAM computation vehicle with GCV/AIC/REML, smoothese estimation (version 1.7-24). http://cran.r-project.org/web/packages/mgcv/index.html (2013) · Zbl 1064.62522
[31] Wood, S.N., Fasiolo, M.: A generalized Fellner-Schall method for smoothing parameter optimization with application to tweedie location, scale and shape models. Biometrics (2017) doi:10.1111/biom.12666 · Zbl 1405.62216
[32] Xiao, L., Li, Y., Ruppert, D.: Fast bivariate P-splines: the sandwich smoother. J. R. Stat. Soc. B 75, 577-599 (2013) · Zbl 1411.62109
[33] Xiao, L; Huang, L; Schrack, J; Ferrucci, L; Zipunnikov, V; Crainiceanu, C, Quantifying the life-time Circadian rhythm of physical activity: a covariate-dependent functional approach, Biostatistics, 16, 352-367, (2015)
[34] Xiao, L; Ruppert, D; Zipunnikov, V; Crainiceanu, C, Fast covariance function estimation for high-dimensional functional data, Stat. Comput., 26, 409-421, (2016) · Zbl 1342.62094
[35] Xiao, L., Li, C., Checkley, W., Crainiceanu, C.: R package face: fast covariance estimation for sparse functional data (version 0.1-3). https://cran.r-project.org/web/packages/face/index.html (2017) · Zbl 1332.62139
[36] Xu, G; Huang, J, Asymptotic optimality and efficient computation of the leave-subject-out cross-validation, Ann. Stat., 40, 3003-3030, (2012) · Zbl 1296.62096
[37] Yao, F; Müller, H; Clifford, A; Dueker, S; Follett, J; Lin, Y; Buchholz, B; Vogel, J, Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate, Biometrics, 20, 852-873, (2003) · Zbl 1210.62076
[38] Yao, F; Müller, H; Wang, J, Functional data analysis for sparse longitudinal data, J. Am. Stat. Assoc., 100, 577-590, (2005) · Zbl 1117.62451
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