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Tensor product splines and functional principal components. (English) Zbl 1435.62451
Summary: Functional principal component analysis for sparse longitudinal data usually proceeds by first smoothing the covariance surface, and then obtaining an eigendecomposition of the associated covariance operator. Here we consider the use of penalized tensor product splines for the initial smoothing step. Drawing on a result regarding finite-rank symmetric integral operators, we derive an explicit spline representation of the estimated eigenfunctions, and show that the effect of penalization can be notably disparate for alternative approaches to tensor product smoothing. The latter phenomenon is illustrated with two data sets derived from magnetic resonance imaging of the human brain.
Reviewer: Reviewer (Berlin)
MSC:
62R10 Functional data analysis
62H25 Factor analysis and principal components; correspondence analysis
65D07 Numerical computation using splines
47A80 Tensor products of linear operators
62H35 Image analysis in multivariate analysis
62P10 Applications of statistics to biology and medical sciences; meta analysis
60L90 Applications of rough analysis
Software:
fda (R); gamair; R; refund; SemiPar
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[1] Cederbaum, J.; Scheipl, F.; Greven, S., Fast symmetric additive covariance smoothing, Comput. Statist. Data Anal., 120, 25-41 (2018) · Zbl 06920202
[2] Chen, S. T.; Xiao, L.; Staicu, A.-M., A smoothing-based goodness-of-fit test of covariance for functional data, Biometrics, 75, 562-571 (2019) · Zbl 1436.62525
[3] Currie, I. D.; Durban, M.; Eilers, P. H.C., Generalized linear array models with applications to multidimensional smoothing, J. R. Stat. Soc. Ser. B Stat. Methodol., 68, 2, 259-280 (2006) · Zbl 1110.62090
[4] Delaigle, A.; Hall, P., Approximating fragmented functional data by segments of Markov chains, Biometrika, 103, 4, 779-799 (2016) · Zbl 07072155
[5] Di, C.; Crainiceanu, C. M.; Caffo, B. S.; Punjabi, N. M., Multilevel functional principal component analysis, Ann. Appl. Stat., 3, 1, 458 (2009) · Zbl 1160.62061
[6] Di, C.; Crainiceanu, C. M.; Jank, W. S., Multilevel sparse functional principal component analysis, Stat, 3, 1, 126-143 (2014)
[7] Eilers, P. H.C.; Marx, B. D., Multivariate calibration with temperature interaction using two-dimensional penalized signal regression, Chemometr. Intell. Lab. Syst., 66, 2, 159-174 (2003)
[8] Giedd, J. N.; Raznahan, A.; Alexander-Bloch, A.; Schmitt, E.; Gogtay, N.; Rapoport, J. L., Child Psychiatry Branch of the National Institute of Mental Health Longitudinal Structural Magnetic Resonance Imaging Study of Human Brain Development, Neuropsychopharmacology, 40, 1, 43-49 (2015)
[9] Goldsmith, J.; Bobb, J.; Crainiceanu, C. M.; Caffo, B.; Reich, D., Penalized functional regression, J. Comput. Graph. Statist., 20, 4, 830-851 (2011)
[10] Goldsmith, J.; Greven, S.; Crainiceanu, C., Corrected confidence bands for functional data using principal components, Biometrics, 69, 1, 41-51 (2013) · Zbl 1274.62776
[11] Goldsmith, J.; Scheipl, F.; Huang, L.; Wrobel, J.; Gellar, J.; Harezlak, J.; McLean, M. W.; Swihart, B.; Xiao, L.; Crainiceanu, C.; Reiss, P. T., Refund: Regression with functional data. R package version 0.1-17 (2018), URL https://CRAN.R-project.org/package=refund
[12] Green, P. J.; Silverman, B. W., Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach (1994), Chapman & Hall: Chapman & Hall Boca Raton, Florida · Zbl 0832.62032
[13] Hsing, T.; Eubank, R., Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators (2015), John Wiley & Sons: John Wiley & Sons Chichester, UK · Zbl 1338.62009
[14] R: A Language and Environment for Statistical Computing (2018), R Foundation for Statistical Computing: R Foundation for Statistical Computing Vienna, Austria, URL https://www.R-project.org/
[15] Ramsay, J. O.; Dalzell, C. J., Some tools for functional data analysis (with discussion), J. R. Stat. Soc. Ser. B Stat. Methodol., 53, 539-572 (1991) · Zbl 0800.62314
[16] Ramsay, J. O.; Silverman, B. W., Functional Data Analysis (2005), Springer: Springer New York · Zbl 1079.62006
[17] Reiss, P. T.; Crainiceanu, C. M.; Thompson, W. K.; Huo, L., Modeling change in the brain: methods for cross-sectional and longitudinal data, (Ombao, H.; Lindquist, M.; Thompson, W.; Aston, J., Handbook of Neuroimaging Data Analysis (2016), Chapman and Hall/CRC), 467-494
[18] Reiss, P. T.; Huang, L.; Chen, H.; Colcombe, S., Varying-smoother models for functional responses (2014), arXiv preprint arXiv:1412.0778
[19] Reiss, P. T.; Ogden, R. T., 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
[20] Rice, J. A.; Silverman, B. W., Estimating the mean and covariance structure nonparametrically when the data are curves, J. R. Stat. Soc. Ser. B Stat. Methodol., 53, 1, 233-243 (1991) · Zbl 0800.62214
[21] Ruppert, D.; Wand, M. P.; Carroll, R. J., Semiparametric Regression (2003), Cambridge University Press: Cambridge University Press New York · Zbl 1038.62042
[22] Schott, J. R., Matrix Analysis for Statistics (2016), John Wiley & Sons: John Wiley & Sons Hoboken, New Jersey
[23] Silverman, B. W., Smoothed functional principal components analysis by choice of norm, Ann. Statist., 24, 1, 1-24 (1996) · Zbl 0853.62044
[24] Staniswalis, J.; Lee, J., Nonparametric regression analysis of longitudinal data, J. Amer. Statist. Assoc., 444, 1403-1418 (1998) · Zbl 1064.62522
[25] Thompson, W. K.; Hallmayer, J.; O’Hara, R., Design considerations for characterizing psychiatric trajectories across the lifespan: Application to effects of APOE-e4 on cerebral cortical thickness in Alzheimer’s disease, Am. J. Psychiatry, 168, 9, 894-903 (2011)
[26] Wood, S. N., Low-rank scale-invariant tensor product smooths for generalized additive mixed models, Biometrics, 62, 4, 1025-1036 (2006) · Zbl 1116.62076
[27] 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
[28] Wood, S. N., Generalized Additive Models: An Introduction with R (2017), CRC Press: CRC Press Boca Raton, Florida · Zbl 1368.62004
[29] Xiao, L.; Li, C.; Checkley, W.; Crainiceanu, C., Fast covariance estimation for sparse functional data, Stat. Comput., 28, 3, 511-522 (2018) · Zbl 1384.62142
[30] Yao, F.; Müller, H.; Wang, J., Functional data analysis for sparse longitudinal data, J. Amer. Statist. Assoc., 100, 577-590 (2005) · Zbl 1117.62451
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