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Abstract

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay & Silverman's (1997) textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article focuses on functional regression, the area of FDA that has received the most attention in applications and methodological development. First, there is an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: () functional predictor regression (scalar-on-function), () functional response regression (function-on-scalar), and () function-on-function regression. For each, the role of replication and regularization is discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. The review concludes with a brief discussion describing potential areas of future development in this field.

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2015-04-10
2024-04-21
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Literature Cited

  1. Abramovich F, Angelini C. 2006. Testing in mixed-effects FANOVA models. J. Stat. Plan. Inf. 136:4326–48 [Google Scholar]
  2. Antoniadis A, Sapatinas T. 2007. Estimation and inference in functional mixed-effect models. Comp. Stat. Data Anal. 51:4793–813 [Google Scholar]
  3. Armagan A, Dunson D, Clyde M. 2011. Generalized beta mixtures of Gaussians. Advances in Neural Information Processing Systems 24 J Shawe-Taylor, R Zemel, P Bartlett, F Pereira, K Weinberger 523–31 La Jolla, CA: Neural Inf. Process. Syst. Found http://papers.nips.cc/paper/4439-generalized-beta-mixtures-of-gaussians [Google Scholar]
  4. Aston JA, Chiou J, Evans J. 2010. Linguistic pitch analysis using functional principal component mixed effect models. J. R. Stat. Soc. C 59:297–317 [Google Scholar]
  5. Baik J, Silverstein JW. 2006. Eigenvalues of large sample covariance matrices of spiked population models. J. Multivar. Anal. 97:1382–408 [Google Scholar]
  6. Baladandayuthapani V, Mallick BK, Hong MY, Lupton JR, Turner ND, Carroll RJ. 2008. Bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis. Biometrics 64:321–22 [Google Scholar]
  7. Barry D. 1995. A Bayesian model for growth curve analysis. Biometrics 51:2639–55 [Google Scholar]
  8. Barry D. 1996. An empirical Bayes approach to growth curve analysis. Statistician 45:3–19 [Google Scholar]
  9. Berhane K, Molitor NT. 2008. A Bayesian approach to functional-based multilevel modeling of longitudinal data: applications to environmental epidemiology. Biostatistics 4:686–99 [Google Scholar]
  10. Besse PC, Cardot H. 1996. Approximation spline de la prévision d'un processus fonctionnel autorégressif d'ordre 1. Can. J. Stat. 24:467–87 [Google Scholar]
  11. Bigelow JL, Dunson DB. 2007. Bayesian adaptive regression splines for hierarchical data. Biometrics 63:724–32 [Google Scholar]
  12. Brown PJ, Fearn T, Vannucci M. 2001. Bayesian wavelet regression on curves with application to a spectroscopic calibration problem. J. Am. Stat. Assoc. 96:398–408 [Google Scholar]
  13. Brumback BA, Rice JA. 1998. Smoothing spline models for the analysis of nested and crossed samples of curves. J. Am. Stat. Assoc. 93:961–76 [Google Scholar]
  14. Bunea F, Ivanescu AE, Wegkamp MH. 2011. Adaptive inference for the mean of a Gaussian process in functional data. J. R. Stat. Soc. B 73:4531–58 [Google Scholar]
  15. Cardot H, Ferraty F, Mas A, Sarda P. 2003. Testing hypotheses in the functional linear model. Scand. J. Stat. 30:241–55 [Google Scholar]
  16. Cardot H, Ferraty F, Sarda P. 1999. Functional linear model. Stat. Prob. Lett. 45:11–22 [Google Scholar]
  17. Carroll RJ, Ruppert D, Stefanski LA. 1995. Measurement Error in Nonlinear Models: A Modern Perspective New York: Springer-Verlag
  18. Carvahlo CM, Polson NG, Scott JG. 2010. The horseshoe estimator for sparse signals. Biometrika 97:2465–80 [Google Scholar]
  19. Chen H, Wang Y. 2011. A penalized spline approach to functional mixed effect model analysis. Biometrics 67:861–70 [Google Scholar]
  20. Crainiceanu CM, Goldsmith AJ. 2010. Bayesian functional data analysis using WinBUGS. J. Stat. Softw. 32:11i11 [Google Scholar]
  21. Crainiceanu CM, Ruppert D. 2004. Likelihood ratio tests in linear mixed models with one variance component. J. R. Stat. Soc. B 66:1165–85 [Google Scholar]
  22. Crainiceanu CM, Staicu AM, Di CZ. 2009. Generalized multilevel functional regression. J. Am. Stat. Assoc. 104:1550–61 [Google Scholar]
  23. Crainiceanu CM, Staicu AM, Ray S, Punjabi N. 2012. Bootstrap-based inference on the difference in the means of two correlated functional responses. Stat. Med. 31:3223–40 [Google Scholar]
  24. Crambes C, Kneip A, Sarda P. 2009. Smoothing spline estimators for functional linear regression. Ann. Stat. 37:135–72 [Google Scholar]
  25. Degras D. 2011. Simultaneous confidence bands for nonparametric regression with functional data. Stat. Sin. 21:1735–65 [Google Scholar]
  26. Demmler A, Reinsch C. 1975. Oscillation matrices with spline smoothing. Numer. Math. 24:375–82 [Google Scholar]
  27. Di CZ, Crainiceanu CM, Caffo BM, Punjabi NM. 2009. Multilevel functional principal component analysis. Ann. Appl. Stat. 3:458–88 [Google Scholar]
  28. Di CZ, Crainiceanu CM, Jank WS. 2014. Multilevel sparse functional principal component analysis. Stat 3:1126–43 [Google Scholar]
  29. Diggle PJ, Wasel A. 1997. Spectral analysis of replicated biomedical time series. J. R. Stat. Soc. C 46:131–71 [Google Scholar]
  30. Do KA, Kirk K. 1999. Discriminant analysis of event-related potential curves using smoothed principal components. Biometrics 55:174–81 [Google Scholar]
  31. Donoho DL, Johnstone IM. 1995. Minimax estimation via wavelet shrinkage. Ann. Stat. 26:3879–921 [Google Scholar]
  32. Eilers PHC, Marx BD. 1986. Flexible smoothing with B-splines and penalties. Stat. Sci. 11:289–121 [Google Scholar]
  33. Eilers PHC, Marx BD. 2002. Generalized linear additive smooth structures. J. Comput. Graph. Stat. 11:758–83 [Google Scholar]
  34. Fan J, Zhang JT. 2000. Two-step estimation of functional linear models with applications to longitudinal data. J. R. Stat. Soc. Ser. B 62:303–22 [Google Scholar]
  35. Fan Y, James GM, Radchenko P. 2014. Functional additive regression Tech. Rep. Marshall Sch. Bus., Univ. South. Calif., Los Angeles, CA. http://www-bcf.usc.edu/∼gareth/research/FAR.pdf
  36. Faraway JJ. 1997. Regression analysis for a functional response. Technometrics 39:254–61 [Google Scholar]
  37. Fazio MA, Grytz R, Morris JS, Bruno L, Gardiner S. et al. 2013. Age-related changes in human peripapillary scleral stiffness. Biomech. Model. Mechanobiol. 13:3551–63 [Google Scholar]
  38. Fox EB, Dunson DB. 2012. Multiresolution Gaussian processes. Adv. Neural Inf. Process. Syst. 25:746–54 [Google Scholar]
  39. Freyermuth JM, Ombao H, von Sachs R. 2010. Tree-structured wavelet estimation in a mixed effects model for spectra of replicated time series. J. Am. Stat. Assoc. 105:490634–46 [Google Scholar]
  40. George EI, McCulloch RE. 1993. Variable selection via Gibbs sampling. J. Am. Stat. Assoc. 88:881–89 [Google Scholar]
  41. Gertheiss J, Goldsmith J, Crainiceanu C, Greven S. 2013. Longitudinal scalar-on-function regression with application to tractography data. Biostatistics 14:3447–61 [Google Scholar]
  42. Gervini D. 2006. Free-knot spline smoothing for functional data. J. R. Stat. Soc. Ser. B 68:4671–88 [Google Scholar]
  43. Goldsmith J, Bobb J, Crainiceanu C, Caffo B, Reich D. 2011a. Penalized functional regression. J. Comput. Gr. Stat. 20:830–51 [Google Scholar]
  44. Goldsmith J, Crainiceanu CM, Caffo B, Reich D. 2012. Longitudinal penalized functional regression for cognitive outcomes on neuronal tract measurements. J. R. Stat. Soc. C 61:3453–69 [Google Scholar]
  45. Goldsmith J, Huang L, Crainiceanu CM. 2014. Smooth scalar-on-image regression via spatial Bayesian variable selection. J. Comput. Gr. Stat. 23:146–64 [Google Scholar]
  46. Goldsmith J, Kitago K. 2013. Assessing systematic effects of stroke on motor control using hierarchical scalar-on-function regression Tech. Rep., Columbia Univ., New York, NY. http://jeffgoldsmith.com/Downloads/BayesPFSR_Paper.pdf
  47. Goldsmith J, Wand MP, Crainiceanu C. 2011b. Functional regression via variational Bayes. Electron. J. Stat. 5:571–602 [Google Scholar]
  48. Greven S, Crainiceanu C, Caffo B, Reich D. 2010. Longitudinal functional principal components analysis. Electron. J. Stat. 4:1022–54 [Google Scholar]
  49. Griffin JE, Brown PJ. 2010. Inference with normal-gamma prior distributions in regression problems. Bayesian Anal. 5:171–88 [Google Scholar]
  50. Guo W. 2002. Functional mixed effect models. Biometrics 58:121–28 [Google Scholar]
  51. Guo W. 2003. Smoothing spline ANOVA for time-dependent spectral analysis. J. Am. Stat. Assoc. 98:463643–52 [Google Scholar]
  52. Hall P, Marron JS, Neeman A. 2005. Geometric representation of high dimension, low sample size data. J. R. Stat. Soc. Ser. B 67:427–44 [Google Scholar]
  53. Hall P, Poskitt DS, Presnell B. 2001. A functional data-analytic approach to signal discrimination. Technometrics 43:1–9 [Google Scholar]
  54. Harezlak J, Coull BA, Laird NM, Magari SR, Christiani DC. 2007. Penalized solutions to functional regression problems. Comput. Stat. Stat. Anal. 51:104911–25 [Google Scholar]
  55. Hart JD, Wehrly TE. 1986. Kernel regression estimation using repeated measurements data. J. Am. Stat. Assoc. 81:1080–88 [Google Scholar]
  56. Hastie T, Mallows C. 1993. [A statistical view of some chemometrics regression tools]: Discussion. Technology 35:2140–43 [Google Scholar]
  57. Hastie T, Tibshirani R. 1986. Generalized additive models. Stat. Sci. 1:3297–318 [Google Scholar]
  58. Hastie T, Tibshirani R. 1993. Varying coefficient models. J. R. Stat. Soc. Ser. B 55:757–96 [Google Scholar]
  59. Herrick RC, Morris JS. 2006. Wavelet-based functional mixed model analysis: computational considerations. Joint Stat. Meet. 2006 Proc., ASA Sect. Stat. Comput., Seattle2051–53 Alexandria, VA: Am. Stat. Assoc. [Google Scholar]
  60. Hodges JS. 2013. Richly Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects New York: Chapman & Hall
  61. Holan SH, Wikle CK, Sullivan-Beckers LE, Cocroft RB. 2010. Modeling complex phenotypes: generalized linear models using spectrogram predictors of animal communication signals. Biometrics 66:3914–24 [Google Scholar]
  62. Hoover DR, Rice JA, Wu CO, Yang LP. 1998. Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85:809–22 [Google Scholar]
  63. Huang JZ, Wu CO, Zhou L. 2004. Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Stat. Sin. 14:763–88 [Google Scholar]
  64. Ivanescu AE, Staicu AM, Greven S, Scheipl F, Crainiceanu CM. 2012. Penalized function-on-function regression. Tech. Rep. In press. doi: 10.1007/s00180-014-0548-4
  65. James GM. 2002. Generalized linear models with functional predictors. J. R. Stat. Soc. Ser. B 64:411–32 [Google Scholar]
  66. James GM, Hastie TJ. 2001. Functional linear discriminant analysis for irregularly sampled curves. J. R. Stat. Soc. Ser. B 63:533–50 [Google Scholar]
  67. James GM, Hastie TJ, Sugar CA. 2000. Principal component models for sparse functional data. Biometrika 87:3587–602 [Google Scholar]
  68. James GM, Silverman BW. 2005. Functional adaptive model estimation. J. Am. Stat. Assoc. 100:565–76 [Google Scholar]
  69. James GM, Wang J, Zhu J. 2009. Functional linear regression that's interpretable. Ann. Stat. 37:5A2083–108 [Google Scholar]
  70. Johnstone IM, Lu AY. 2009. On consistency and sparsity for principal component analysis in high dimensions. J. Am. Stat. Assoc. 104:486683–93 [Google Scholar]
  71. Joliffe IT. 1982. A note on the use of principal components in regression. J. R. Stat. Soc. Ser. C 31:3300–3 [Google Scholar]
  72. Jung S, Marron J. 2009. PCA consistency in high dimension, low sample size context. Ann. Stat. 37:4104–30 [Google Scholar]
  73. Kendall MG. 1957. A Course in Multivariate Analysis London: Griffin
  74. Kim K, Senturk D, Li R. 2011. Recent history functional linear models for sparse longitudinal data. J. Stat. Plan. Inference 141:41554–66 [Google Scholar]
  75. Koomen JM, Shih LN, Coombes KR, Li D, Xiao LC. et al. 2005. Plasma protein profiling for diagnosis of pancreatic cancer reveals the presence of host response proteins. Clin. Cancer Res. 11:31110–18 [Google Scholar]
  76. Krafty RT, Gimotty PA, Holtz D, Coukos G, Guo W. 2008. Varying coefficient model with unknown within-subject covariance for analysis of tumor growth curves. Biometrics 64:41023–31 [Google Scholar]
  77. Krafty RT, Hall M, Guo W. 2011. Functional mixed effects spectral analysis. Biometrika 98:3583–98 [Google Scholar]
  78. Kundu MG, Harezlak J, Randolph TW. 2012. Longitudinal functional models with structured penalties. arXiv:1211.4763 [stat.AP]
  79. Laird NM, Ware JH. 1982. Random-effects models for longitudinal data. Biometrics 38:4963–74 [Google Scholar]
  80. Lang S, Bretzger A. 2004. Bayesian P-splines. J. Comp. Graph. Stat. 13:183–212 [Google Scholar]
  81. Lee ER, Park BU. 2012. Sparse estimation in functional linear regression. J. Multivar. Anal. 105:1–17 [Google Scholar]
  82. Lee S, Zou F, Wright FA. 2010. Convergence and prediction of principal component scores in high-dimensional settings. Ann. Stat. 38:3605–29 [Google Scholar]
  83. Lee W, Baladandayuthapani V, Fazio M, Downs C, Morris JS. 2014. Semiparametric functional mixed models for longitudinally observed functional data, with application to glaucoma data Tech. Rep., Univ. Tex. MD Anderson Cancer Cent., Houston, Tex.
  84. Lee W, Morris JS. 2014. Identification of differentially expressed methylated loci using wavelet-based functional mixed models Tech. Rep., Univ. Tex. MD Anderson Cancer Cent., Houston, Tex.
  85. Li B, Marx BD. 2008. Sharpening P-spline signal regression. Stat. Model. 8:367–83 [Google Scholar]
  86. Li B, Yu Q. 2008. Classification of functional data: a segmentation approach. Comp. Stat. Data. Anal. 52:4790–800 [Google Scholar]
  87. Li Y, Hsing T. 2010. Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Stat. 38:63321–51 [Google Scholar]
  88. Li Y, Wang N, Carroll RJ. 2010. Generalized functional linear models with semiparametric single-index interactions. J. Am. Stat. Assoc. 105:490621–33 [Google Scholar]
  89. Liang H, Wu H, Carroll RJ. 2003. The relationship between virology and immunologic responses in AIDS clinical research using mixed-effects varying-coefficient models with measurement error. Biostatistics 4:2297–312 [Google Scholar]
  90. Lin X, Wang N, Welsh AH, Carroll RJ. 2004. Equivalent kernels of smoothing splines in nonparametric regression for longitudinal/clustered data. Biometrika 91:1177–93 [Google Scholar]
  91. Ma Y, Yang L, Carroll RJ. 2012. A simultaneous confidence band for sparse longitudinal regression. Stat. Sin. 22:95–122 [Google Scholar]
  92. Malfait N, Ramsay JO. 2003. The historical functional linear model. Can. J. Stat. 31:115–28 [Google Scholar]
  93. Malloy EJ, Morris JS, Adar SD, Suh HH, Gold DR, Coull BA. 2010. Wavelet-based functional linear mixed models: an application to measurement error–corrected distributed lag models. Biostatistics 11:3432–52 [Google Scholar]
  94. Maronna RA, Yohai VJ. 2013. Robust functional linear regression based on splines. Comput. Stat. Data Anal. 65:46–55 [Google Scholar]
  95. Martinez JG, Fiang F, Zhou L, Carroll RJ. 2010. Longitudinal functional principal component modeling via stochastic approximation Monte Carlo. Can. J. Stat. 38:2256–70 [Google Scholar]
  96. Martinez JG, Bohn KM, Carroll RJ, Morris JS. 2013. A study of Mexican free-tailed bat syllables: Bayesian functional mixed modeling of nonstationary acoustic time series. J. Am. Stat. Assoc. 108:502514–26 [Google Scholar]
  97. Marx BD, Eilers PHC. 1999. Generalized linear regression on sampled signals and curves: a P-spline approach. Technometrics 41:11–13 [Google Scholar]
  98. Marx BD, Eilers PHC. 2002. Multivariate calibration stability: a comparison of methods. J. Chemom. 16:129–40 [Google Scholar]
  99. Marx BD, Eilers PHC. 2005. Multidimensional penalized signal regression. Technometrics 47:13–22 [Google Scholar]
  100. Marx BD, Eilers PHC, Li B. 2011. Multidimensional single-index signal regression. Chemom. Intell. Lab. Syst. 109:120–30 [Google Scholar]
  101. McLean MW, Hooker G, Staicu AM, Scheipl F, Ruppert D. 2012. Functional generalized additive models. J. Comput. Gr. Stat. 23:249–69 [Google Scholar]
  102. McLean MW, Scheipl F, Hooker G, Greven S, Ruppert D. 2013. Bayesian functional generalized additive models with sparsely observed covariates arXiv:1305.3585 [stat.ME]
  103. Meyer MJ, Coull BA, Versace F, Cinciripini P, Morris JS. 2015. Bayesian function-on-function regression for multi-level functional data. Biometrics In press
  104. Morris JS. 2012. Statistical methods for proteomic biomarker discovery using feature extraction or functional data analysis approaches. Stat. Interface 5:1117–36 [Google Scholar]
  105. Morris JS, Arroyo C, Coull BA, Ryan LM, Herrick RC, Gortmaker SL. 2006. Using wavelet-based functional mixed models to characterize population heterogeneity in accelerometer profiles: a case study. J. Am. Stat. Assoc. 101:4761352–64 [Google Scholar]
  106. Morris JS, Baladandayuthapani V, Herrick RC, Sanna PP, Gutstein H. 2011. Automated analysis of quantitative image data using isomorphic functional mixed models, with application to proteomics data. Ann. Appl. Stat. 5:894–923 [Google Scholar]
  107. Morris JS, Brown PJ, Herrick RC, Baggerly KA, Coombes KR. 2008. Bayesian analysis of mass spectrometry data using wavelet-based functional mixed models. Biometrics 12:479–89 [Google Scholar]
  108. Morris JS, Carroll RJ. 2006. Wavelet-based functional mixed models. J. R. Stat. Soc. B 68:2179–99 [Google Scholar]
  109. Morris JS, Vannucci M, Brown PJ, Carroll RJ. 2003. Wavelet-based nonparametric modeling of hierarchical functions in colon carcinogenesis. J. Am. Stat. Assoc. 98:573–83 [Google Scholar]
  110. Mostacci E, Truntzner C, Cardot H, Ducoroy P. 2010. Multivariate denoising methods combining wavelets and principal component analysis for mass spectrometry data. Proteomics 10:2564–72 [Google Scholar]
  111. Müller HG, Stadtmüller U. 2005. Generalized functional linear models. Ann. Stat. 33:774–805 [Google Scholar]
  112. Ogden RT, Greene E. 2010. Wavelet modeling of functional random effects with application to human vision data. J. Stat. Plann. Inference 140:3797–808 [Google Scholar]
  113. Ogden RT, Miller CE, Takezawa K, Ninomiya S. 2002. Functional regression in crop lodging assessment with digital images. J. Agric. Biol. Environ. Stat. 7:3389–402 [Google Scholar]
  114. Park T, Casella G. 2008. The Bayesian lasso. J. Am. Stat. Assoc. 103:672–80 [Google Scholar]
  115. Qin L, Guo W. 2006. Functional mixed-effects model for periodic data. Biostatistics 7:225–34 [Google Scholar]
  116. Qin L, Guo W, Litt B. 2009. A time-frequency functional model for locally stationary time series. J. Comput. Graph. Stat. 18:3675–93 [Google Scholar]
  117. Ramsay JO, Dalzell CJ. 1991. Some tools for functional data analysis. J. R. Stat. Soc. Ser. B 53:539–72 [Google Scholar]
  118. Ramsay JO, Hooker G, Graves S. 2009. Functional Data Analysis with R and MATLAB New York: Springer-Verlag
  119. Ramsay JO, Silverman BW. 1997. Functional Data Analysis New York: Springer-Verlag 1st ed.
  120. Ramsay JO, Silverman BW. 2005. Functional Data Analysis New York: Springer-Verlag 2nd ed.
  121. Randolph TW, Harezlak J, Feng Z. 2012. Structured penalties for functional linear models—partially empirical eigenvectors for regression. Elec. J. Stat. 6:323–53 [Google Scholar]
  122. Rao CR. 1958. Some statistical methods for comparison of growth curves. Biometrics 14:11–17 [Google Scholar]
  123. Ratliffe SJ, Heller GZ, Leader LR. 2002a. Functional data analysis with application to periodically stimulated foetal heart rate data. I: Functional regression. Stat. Med. 21:81103–14 [Google Scholar]
  124. Ratliffe SJ, Heller GZ, Leader LR. 2002b. Functional data analysis with application to periodically stimulated foetal heart rate data. II: Functional logistic regression. Stat. Med. 21:81115–27 [Google Scholar]
  125. Reiss PT, Huang L, Mennes M. 2010. Fast function-on-scalar regression with penalized basis expansions. Int. J. Biostat. 6:128 [Google Scholar]
  126. Reiss PT, Ogden RT. 2007. Functional principal component regression and functional partial least squares. J. Am. Stat. Assoc. 102:479984–96 [Google Scholar]
  127. Reiss PT, Ogden RT. 2010. Functional generalized linear models with images as predictors. Biometrics 66:161–69 [Google Scholar]
  128. Rice JA, Silverman BW. 1991. Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. B 53:1233–43 [Google Scholar]
  129. Rice JA, Wu CO. 2001. Nonparametric mixed effects models for unequally sampled noisy curves. Biometrics 57:253–59 [Google Scholar]
  130. Røislien J, Winje B. 2013. Feature extraction across individual time series observations with spikes using wavelet principal component analysis. Stat. Med. 32:213660–69 [Google Scholar]
  131. Ruppert D, Wand MP, Carroll RJ. 2003. Semiparametric Regression New York: Cambridge Univ. Press
  132. Scarpa B, Dunson DB. 2009. Bayesian hierarchical functional data analysis via contaminated information priors. Biometrics 65:772–80 [Google Scholar]
  133. Scheipl F, Greven S. 2012. Identifiability in penalized function-on-function regression models Tech. Rep. No. 125, Dept. Stat., Univ. Munich, Ger.
  134. Scheipl F, Staicu AM, Greven S. 2014. Functional additive mixed models. J. Comput. Gr. Stat. In press
  135. Shen D, Shen H, Marron JS. 2013. Consistency of sparse PCA in high dimension, low sample size contexts. J. Multivar. Anal. 115:317–33 [Google Scholar]
  136. Shi M, Weiss RE, Taylor JMG. 1996. An analysis of paediatric CD4 counts for acquired immune deficiency syndrome using flexible random curves. J. R. Stat. Soc. Ser. C 45:151–63 [Google Scholar]
  137. Silverman BW. 1996. Smoothed principal component analysis by choice of norm. Ann. Stat. 24:1–24 [Google Scholar]
  138. Spitzner DJ, Marron JS, Essick GK. 2003. Mixed model functional ANOVA. J. Am. Stat. Assoc. 98:263–72 [Google Scholar]
  139. Staicu AM, Crainiceanu CM, Carroll RJ. 2010. Fast methods for spatially correlated multilevel functional data. Biostatistics 11:2177–94 [Google Scholar]
  140. Staicu AM, Crainiceanu CM, Reich DS, Ruppert D. 2012. Modeling functional data with spatially heterogeneous shape characteristics. Biometrics 68:331–43 [Google Scholar]
  141. Staniswallis JG, Lee JJ. 1998. Nonparametric regression analysis of longitudinal data. J. Am. Stat. Assoc. 93:1403–18 [Google Scholar]
  142. Stingo F, Vannucci M, Downey G. 2012. Bayesian wavelet-based curve classification via discriminant analysis with Markov tree priors. Stat. Sin. 22:465–88 [Google Scholar]
  143. Storlie CB, Fugate ML, Higdon DM, Huzurbazar AV, Francois EG, McHugh DC. 2013. Methods for characterizing and comparing populations of shock wave curves. Technometrics 55:4436–49 [Google Scholar]
  144. Swihart BJ, Goldsmith J, Crainiceanu CM. 2013. Restricted likelihood ratio tests for functional effects in the functional linear model Work. Pap. 247, Dept. Biostat., Johns Hopkins Univ., Baltimore, MD
  145. Thompson WK, Rosen O. 2008. A Bayesian model for sparse functional data. Biometrics 64:154–63 [Google Scholar]
  146. Tibshirani R. 1996. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58:267–88 [Google Scholar]
  147. Wang XF, Ray S, Mallick BK. 2007. Bayesian curve classification using wavelets. J. Am. Stat. Assoc. 102:479962–73 [Google Scholar]
  148. Wang XF, Yang Q, Fan Z, Sun CK, Yue GH. 2009. Assessing time-dependent association between scalp EEG and muscle activation: a functional random-effects model approach. J. Neurosci. Methods 177:232–40 [Google Scholar]
  149. Wold H. 1966. Estimation of principal components and related models by iterative least squares. Multivariate Analysis PR Krishnaiah 383–487 New York: Acad. Press [Google Scholar]
  150. Wood SN. 2006. Generalized Additive Models: An Introduction with R New York: Chapman & Hall
  151. Wood SN. 2011. Fast stable restricted maximum likelihood and margin likelihood estimation of semiparametric generalized linear models. J. R. Stat. Soc. B 73:13–36 [Google Scholar]
  152. Wu CO, Chiang CT. 2000. Kernel smoothing on varying coefficient models with longitudinal dependent variable. Stat. Sin. 10:433–56 [Google Scholar]
  153. Wu CO, Chiang CT, Hoover DR. 1998. Asymptotic confidence regions for kernel smoothing of the varying coefficients model. J. Am. Stat. Assoc. 93:1388–402 [Google Scholar]
  154. Wu H, Liang H. 2004. Backfitting random varying-coefficients models with time-dependent smoothing covariates. Scand. J. Stat. 31:3–19 [Google Scholar]
  155. Wu H, Zhang JT. 2002. Local polynomical mixed-effects models for longitudinal data. J. Am. Stat. Assoc. 97:883–97 [Google Scholar]
  156. Wu S, Müller HG. 2011. Response-adaptive regression for longitudinal data. Biometrics 67:852–60 [Google Scholar]
  157. Yang WH, Wikle CK, Holan SH, Wildhaber ML. 2013. Ecological prediction with nonlinear multivariate time-frequency functional data models. J. Agric. Biol. Environ. Stat. 18:3450–74 [Google Scholar]
  158. Yao F, Müller HG. 2010. Functional quadratic regression. Biometrika 97:149–64 [Google Scholar]
  159. Yao F, Müller HG, Wang JL. 2005a. Functional data analysis for sparse longitudinal data. J. Am. Stat. Assoc. 100:577–90 [Google Scholar]
  160. Yao F, Müller HG, Wang JL. 2005b. Functional linear regression for longitudinal data. Ann. Stat. 33:2873–903 [Google Scholar]
  161. Yuan M, Cai TT. 2010. A reproducing kernel Hilbert space approach to functional linear regression. Ann. Stat. 28:63412–44 [Google Scholar]
  162. Zellner A. 1962. An efficient method of estimating seemingly unrelated regression equations and tests for aggregation bias. J. Am. Stat. Assoc. 57:348–86 [Google Scholar]
  163. Zhang D, Lin X, Sowers MF. 2000. Semiparametric regression for periodic longitudinal hormone data from multiple menstrual cycles. Biometrics 56:31–39 [Google Scholar]
  164. Zhang D, Lin X, Sowers MF. 2007. Two-stage functional mixed models for evaluating the effect of longitudinal covariate profiles on a scalar outcome. Biometrics 63:351–62 [Google Scholar]
  165. Zhang L, Baladandayuthapani V, Zhu H, Baggerly KA, Majewski TA. et al. 2015. Functional CAR models for large spatially correlated functional datasets. J. Am. Stat. Assoc. In press
  166. Zhao Y, Ogden RT, Reiss PT. 2012. Wavelet-based LASSO in functional linear regression. J. Comput. Gr. Stat. 21:3600–17 [Google Scholar]
  167. Zhou L, Huang JZ, Carroll RJ. 2008. Joint modeling of paired sparse functional data using principal components. Biometrika 95:3601–19 [Google Scholar]
  168. Zhou L, Huang JZ, Martinez JG, Maity A, Baladandayuthapani V, Carroll RJ. 2010. Reduced rank mixed effects models for spatially correlated hierarchical functional data. J. Am. Stat. Assoc. 105:390–400 [Google Scholar]
  169. Zhu H, Brown PJ, Morris JS. 2011. Robust, adaptive functional regression in functional mixed model framework. J. Am. Stat. Assoc. 106:4951167–79 [Google Scholar]
  170. Zhu H, Brown PJ, Morris JS. 2012. Robust classification of functional and quantitative image data using functional mixed models. Biometrics 68:1260–68 [Google Scholar]
  171. Zhu H, Cox DD. 2009. A functional generalized linear model with curve selection in cervical pre-cancer diagnosis using fluorescence spectroscopy. Optimality: Third Erich L. Lehmann Symp. 57:173–89 [Google Scholar]
  172. Zhu H, Vannucci M, Cox DD. 2010. A Bayesian hierarchical model for classification with selection of functional predictors. Biometrics 66:463–73 [Google Scholar]
  173. Zhu H, Yao F, Zhang HH. 2014. Structured functional additive regression in reproducing kernel Hilbert spaces. J. R. Stat. Soc. B 76:3581–603 doi: 10.1111/rssb.12036 [Google Scholar]
  174. Zipunnikinov V, Caffo B, Yousem D, Davitzikos C, Schwartz BS, Crainiceanu C. 2011. Multilevel functional principal component analysis for high-dimensional data. J. Comput. Gr. Stat. 20:852–73 [Google Scholar]
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