1932

Abstract

With the advance of modern technology, more and more data are being recorded continuously during a time interval or intermittently at several discrete time points. These are both examples of functional data, which has become a commonly encountered type of data. Functional data analysis (FDA) encompasses the statistical methodology for such data. Broadly interpreted, FDA deals with the analysis and theory of data that are in the form of functions. This paper provides an overview of FDA, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is functional principal component analysis (FPCA). FPCA is an important dimension reduction tool, and in sparse data situations it can be used to impute functional data that are sparsely observed. Other dimension reduction approaches are also discussed. In addition, we review another core technique, functional linear regression, as well as clustering and classification of functional data. Beyond linear and single- or multiple- index methods, we touch upon a few nonlinear approaches that are promising for certain applications. They include additive and other nonlinear functional regression models and models that feature time warping, manifold learning, and empirical differential equations. The paper concludes with a brief discussion of future directions.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-statistics-041715-033624
2016-06-01
2024-04-21
Loading full text...

Full text loading...

/deliver/fulltext/statistics/3/1/annurev-statistics-041715-033624.html?itemId=/content/journals/10.1146/annurev-statistics-041715-033624&mimeType=html&fmt=ahah

Literature Cited

  1. Abraham C, Cornillon PA, Matzner-Lober E, Molinari N. 2003. Unsupervised curve clustering using B-splines. Scand. J. Stat. 30:581–95 [Google Scholar]
  2. Amini AA, Wainwright MJ. 2012. Sampled forms of functional PCA in reproducing kernel Hilbert spaces. Ann. Stat. 40:2483–510 [Google Scholar]
  3. Angelini C, De Canditiis D, Pensky M. 2012. Clustering time-course microarray data using functional Bayesian infinite mixture model. J. Appl. Stat. 39:129–49 [Google Scholar]
  4. Araki Y, Konishi S, Kawano S, Matsui H. 2009. Functional logistic discrimination via regularized basis expansions. Commun. Stat. Theory Methods 38:2944–57 [Google Scholar]
  5. Arribas-Gil A, Müller HG. 2014. Pairwise dynamic time warping for event data. Comput. Stat. Data Anal. 69:255–68 [Google Scholar]
  6. Ash RB, Gardner MF. 1975. Topics in Stochastic Processes New York: Academic
  7. Bali JL, Boente G, Tyler DE, Wang JL. 2011. Robust functional principal components: a projection-pursuit approach. Ann. Stat. 39:2852–82 [Google Scholar]
  8. Banfield JD, Raftery AE. 1993. Model-based Gaussian and non-Gaussian clustering. Biometrics 49:803–21 [Google Scholar]
  9. Besse P, Ramsay JO. 1986. Principal components analysis of sampled functions. Psychometrika 51:285–311 [Google Scholar]
  10. Bickel P, Li B. 2007. Local polynomial regression on unknown manifolds. Complex Datasets and Inverse Problems: Tomography, Networks and Beyond R Liu, W Strawderman, C-H Zhang 177–86 Beachwood, OH: Inst Math. Stat.
  11. Bickel PJ, Rosenblatt M. 1973. On some global measures of the deviations of density function estimates. Ann. Stat. 1:1071–95 [Google Scholar]
  12. Boente G, Fraiman R. 2000. Kernel-based functional principal components. Stat. Probab. Lett. 48:335–45 [Google Scholar]
  13. Boente G, Rodriguez D, Sued M. 2011. Testing the equality of covariance operators. Recent Advances in Functional Data Analysis and Related Topics F Ferraty 49–53 New York: Springer [Google Scholar]
  14. Boente G, Salibián-Barrera M. 2014. S-estimators for functional principal component analysis. J. Am. Stat. Assoc. 110:1100–11 [Google Scholar]
  15. Bosq D. 2000. Linear Processes in Function Spaces: Theory and Applications New York: Springer
  16. Brumback B, Rice J. 1998. Smoothing spline models for the analysis of nested and crossed samples of curves. J. Am. Stat. Assoc. 93:961–76 [Google Scholar]
  17. Cai T, Hall P. 2006. Prediction in functional linear regression. Ann. Stat. 34:2159–79 [Google Scholar]
  18. Cai TT, Yuan M. 2011. Optimal estimation of the mean function based on discretely sampled functional data: phase transition. Ann. Stat. 39:2330–55 [Google Scholar]
  19. Cao G, Yang L, Todem D. 2012. Simultaneous inference for the mean function based on dense functional data. J. Nonparametric Stat. 24:359–77 [Google Scholar]
  20. Cardot H. 2000. Nonparametric estimation of smoothed principal components analysis of sampled noisy functions. J. Nonparametric Stat. 12:503–38 [Google Scholar]
  21. Cardot H. 2007. Conditional functional principal components analysis. Scand. J. Stat. 34:317–35 [Google Scholar]
  22. Cardot H, Ferraty F, Sarda P. 1999. Functional linear model. Stat. Probab. Lett. 45:11–22 [Google Scholar]
  23. Cardot H, Ferraty F, Sarda P. 2003. Spline estimators for the functional linear model. Stat. Sin. 13:571–92 [Google Scholar]
  24. Cardot H, Sarda P. 2005. Estimation in generalized linear models for functional data via penalized likelihood. J. Multivar. Anal. 92:24–41 [Google Scholar]
  25. Cardot H, Sarda P. 2006. Linear regression models for functional data. The Art of Semiparametrics S Sperlich, G Aydinli 49–66 Heidelberg, Ger.: Springer [Google Scholar]
  26. Carroll RJ, Maity A, Mammen E, Yu K. 2009. Nonparametric additive regression for repeatedly measured data. Biometrika 96:383–98 [Google Scholar]
  27. Castro PE, Lawton WH, Sylvestre EA. 1986. Principal modes of variation for processes with continuous sample curves. Technometrics 28:329–37 [Google Scholar]
  28. Chang C, Chen Y, Ogden RT. 2014. Functional data classification: a wavelet approach. Comput. Stat. 291497–1513
  29. Chen D, Hall P, Müller HG. 2011. Single and multiple index functional regression models with nonparametric link. Ann. Stat. 39:1720–47 [Google Scholar]
  30. Chen D, Müller HG. 2012. Nonlinear manifold representations for functional data. Ann. Stat. 40:1–29 [Google Scholar]
  31. Chen K, Chen K, Müller HG, Wang JL. 2011. Stringing high-dimensional data for functional analysis. J. Am. Stat. Assoc. 106:275–84 [Google Scholar]
  32. Chiou JM. 2012. Dynamical functional prediction and classification, with application to traffic flow prediction. Ann. Appl. Stat. 6:1588–614 [Google Scholar]
  33. Chiou JM, Li PL. 2007. Functional clustering and identifying substructures of longitudinal data. J. R. Stat. Soc. Ser. B 69:679–99 [Google Scholar]
  34. Chiou JM, Li PL. 2008. Correlation-based functional clustering via subspace projection. J. Am. Stat. Assoc. 103:1684–92 [Google Scholar]
  35. Chiou JM, Müller HG. 2007. Diagnostics for functional regression via residual processes. Comput. Stat. Data Anal. 51:4849–63 [Google Scholar]
  36. Chiou JM, Müller HG. 2009. Modeling hazard rates as functional data for the analysis of cohort lifetables and mortality forecasting. J. Am. Stat. Assoc. 104:572–85 [Google Scholar]
  37. Chiou JM, Müller HG, Wang JL. 2003. Functional quasi-likelihood regression models with smooth random effects. J. R. Stat. Soc. Ser. B 65:405–23 [Google Scholar]
  38. Coffey N, Hinde J, Holian E. 2014. Clustering longitudinal profiles using P-splines and mixed effects models applied to time-course gene expression data. Comput. Stat. Data Anal. 71:14–29 [Google Scholar]
  39. Conway JB. 1994. A Course in Functional Analysis New York: Springer. , 2nd ed.. [Google Scholar]
  40. Cook RD, Forzani L, Yao AF. 2010. Necessary and sufficient conditions for consistency of a method for smoothed functional inverse regression. Stat. Sin. 20:235–38 [Google Scholar]
  41. Crambes C, Delsol L, Laksaci A. 2008. Robust nonparametric estimation for functional data. J. Nonparametric Stat. 20:573–98 [Google Scholar]
  42. Cuevas A. 2014. A partial overview of the theory of statistics with functional data. J. Stat. Plan. Inference 147:1–23 [Google Scholar]
  43. Cuevas A, Febrero M, Fraiman R. 2004. An ANOVA test for functional data. Comput. Stat. Data Anal. 47:111–22 [Google Scholar]
  44. Dauxois J, Pousse A. 1976. Les analyses factorielles en calcul des probabilités et en statistique: essai d'étude syntéthique PhD Thesis, Université de Toulouse
  45. Dauxois J, Pousse A, Romain Y. 1982. Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference. J. Multivariate Anal. 12:136–54 [Google Scholar]
  46. de Boor C. 2001. A Practical Guide to Splines New York: Springer Verlag
  47. Degras DA. 2008. Asymptotics for the nonparametric estimation of the mean function of a random process. Stat. Probab. Lett. 78:2976–80 [Google Scholar]
  48. Degras DA. 2011. Simultaneous confidence bands for nonparametric regression with functional data. Stat. Sin. 21:1735–65 [Google Scholar]
  49. Delaigle A, Hall P. 2010. Defining probability density for a distribution of random functions. Ann. Stat. 38:1171–93 [Google Scholar]
  50. Delaigle A, Hall P. 2012. Achieving near perfect classification for functional data. J. R. Stat. Soc. Ser. B 74:267–86 [Google Scholar]
  51. Delaigle A, Hall P. 2013. Classification using censored functional data. J. Am. Stat. Assoc. 108:1269–83 [Google Scholar]
  52. Donoho DL, Grimes C. 2005. Image manifolds which are isometric to Euclidean space. J. Math. Imaging Vis. 23:5–24 [Google Scholar]
  53. Dou WW, Pollard D, Zhou HH. 2012. Estimation in functional regression for general exponential families. Ann. Stat. 40:2421–51 [Google Scholar]
  54. Duan N, Li KC. 1991. Slicing regression: a link-free regression method. Ann. Stat. 19:505–30 [Google Scholar]
  55. Dubin JA, Müller HG. 2005. Dynamical correlation for multivariate longitudinal data. J. Am. Stat. Assoc. 100:872–81 [Google Scholar]
  56. Eggermont PPB, Eubank RL, LaRiccia VN. 2010. Convergence rates for smoothing spline estimators in varying coefficient models. J. Stat. Plann. Inference 140:369–81 [Google Scholar]
  57. Eubank RL. 1999. Nonparametric Regression and Spline Smoothing New York: CRC, 2nd ed..
  58. Eubank RL, Hsing T. 2008. Canonical correlation for stochastic processes. Stoch. Process. Appl. 118:1634–61 [Google Scholar]
  59. Fan J, Gijbels I. 1996. Local Polynomial Modelling and Its Applications London: Chapman and Hall
  60. Fan J, Lin SK. 1998. Test of significance when data are curves. J. Am. Stat. Assoc. 93:1007–21 [Google Scholar]
  61. Fan J, Zhang W. 1999. Statistical estimation in varying coefficient models. Ann. Stat. 27:1491–518 [Google Scholar]
  62. Fan J, Zhang W. 2008. Statistical methods with varying coefficient models. Stat. Interface 1:179–95 [Google Scholar]
  63. Fan Y, Foutz N, James GM, Jank W. 2014. Functional response additive model estimation with online virtual stock markets. Ann. Appl. Stat. 8:2435–60 [Google Scholar]
  64. Ferraty F, Hall P, Vieu P. 2010. Most-predictive design points for functional data predictors. Biometrika 97:807–24 [Google Scholar]
  65. Ferraty F, Vieu P. 2003. Curves discrimination: a nonparametric functional approach. Comput. Stat. Data Anal. 44:161–73 [Google Scholar]
  66. Ferraty F, Vieu P. 2006. Nonparametric Functional Data Analysis New York: Springer
  67. Ferré L, Yao AF. 2003. Functional sliced inverse regression analysis. Statistics 37:475–88 [Google Scholar]
  68. Ferré L, Yao AF. 2005. Smoothed functional inverse regression. Stat. Sin. 15:665–83 [Google Scholar]
  69. Garcia-Escudero LA, Gordaliza A. 2005. A proposal for robust curve clustering. J. Classif. 22:185–201 [Google Scholar]
  70. Gasser T, Kneip A. 1995. Searching for structure in curve samples. J. Am. Stat. Assoc. 90:1179–88 [Google Scholar]
  71. Gasser T, Müller HG, Köhler W, Molinari L, Prader A. 1984. Nonparametric regression analysis of growth curves. Ann. Stat. 12:210–29 [Google Scholar]
  72. Gervini D. 2008. Robust functional estimation using the median and spherical principal components. Biometrika 95:587–600 [Google Scholar]
  73. Gervini D. 2015. Warped functional regression. Biometrika 102:1–14 [Google Scholar]
  74. Giacofci M, Lambert-Lacroix S, Marot G, Picard F. 2013. Wavelet-based clustering for mixed-effects functional models in high dimension. Biometrics 69:31–40 [Google Scholar]
  75. Grenander U. 1950. Stochastic processes and statistical inference. Arkiv Matematik 1:195–277 [Google Scholar]
  76. Grenander U. 1981. Abstract Inference New York: Wiley
  77. Hadjipantelis PZ, Aston JA, Müller HG, Evans JP. 2015. Unifying amplitude and phase analysis: a compositional data approach to functional multivariate mixed-effects modeling of Mandarin Chinese. J. Am. Stat. Assoc. 110:545–59 [Google Scholar]
  78. Hall P, Horowitz JL. 2007. Methodology and convergence rates for functional linear regression. Ann. Stat. 35:70–91 [Google Scholar]
  79. Hall P, Hosseini-Nasab M. 2006. On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B 68:109–26 [Google Scholar]
  80. Hall P, Müller HG, Wang JL. 2006. Properties of principal component methods for functional and longitudinal data analysis. Ann. Stat. 34:1493–517 [Google Scholar]
  81. Hall P, Müller HG, Yao F. 2009. Estimation of functional derivatives. Ann. Stat. 37:3307–29 [Google Scholar]
  82. Hall P, Poskitt DS, Presnell B. 2001. A functional data analytic approach to signal discrimination. Technometrics 43:1–9 [Google Scholar]
  83. Hall P, Van Keilegom I. 2007. Two-sample tests in functional data analysis starting from discrete data. Stat. Sin. 17:1511 [Google Scholar]
  84. Hastie T, Tibshirani R. 1986. Generalized additive models. Stat. Sci. 1:297–310 [Google Scholar]
  85. He G, Müller HG, Wang JL. 2000. Extending correlation and regression from multivariate to functional data. Asymptotics in Statistics and Probability ML Puri, pp. 197–210 Leiden, Neth: VSP Int. [Google Scholar]
  86. He G, Müller HG, Wang JL. 2003. Functional canonical analysis for square integrable stochastic processes. J. Multivariate Anal. 85:54–77 [Google Scholar]
  87. He G, Müller HG, Wang JL, Yang W. 2010. Functional linear regression via canonical analysis. Bernoulli 16:705–29 [Google Scholar]
  88. Heckman NE. 1986. Spline smoothing in a partly linear model. J. R. Stat. Soc. Ser. B 48:244–48 [Google Scholar]
  89. Heinzl F, Tutz G. 2013. Clustering in linear mixed models with approximate Dirichlet process mixtures using EM algorithm. Stat. Model. 13:41–67 [Google Scholar]
  90. Heinzl F, Tutz G. 2014. Clustering in linear-mixed models with a group fused lasso penalty. Biometrical J. 56:44–68 [Google Scholar]
  91. Hilgert N, Mas A, Verzelen N. 2013. Minimax adaptive tests for the functional linear model. Ann. Stat. 41:838–69 [Google Scholar]
  92. Hoover D, Rice J, Wu C, Yang L. 1998. Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85:809–22 [Google Scholar]
  93. Horváth L, Kokoszka P. 2012. Inference for Functional Data with Applications New York: Springer
  94. Horváth L, Reeder R. 2013. A test of significance in functional quadratic regression. Bernoulli 19:2120–51 [Google Scholar]
  95. Hsing T, Eubank R. 2015. Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators New York: Wiley
  96. Hu Z, Wang N, Carroll RJ. 2004. Profile-kernel versus backfitting in the partially linear models for longitudinal/clustered data. Biometrika 91:251–62 [Google Scholar]
  97. Huang J, Wu C, Zhou L. 2002. Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika 89:111–28 [Google Scholar]
  98. Huang J, Wu C, Zhou L. 2004. Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Stat. Sin. 14:763–88 [Google Scholar]
  99. Hyndman RJ, Shang HL. 2010. Rainbow plots, bagplots, and boxplots for functional data. J. Comput. Graph. Stat. 19:29–45 [Google Scholar]
  100. Int. Year Stat. 2013. Statistics and Science: A Report of the London Workshop on the Future of the Statistical Sciences World Stat. http://www.worldofstatistics.org/wos/pdfs/Statistics&Science-TheLondonWorkshopReport.pdf
  101. Jacques J, Preda C. 2013. Funclust: a curves clustering method using functional random variables density approximation. Neurocomputing 112:164–71 [Google Scholar]
  102. Jacques J, Preda C. 2014. Model-based clustering for multivariate functional data. Comput. Stat. Data Anal. 71:92–106 [Google Scholar]
  103. James GM. 2002. Generalized linear models with functional predictors. J. R. Stat. Soc. Ser. B 64:411–32 [Google Scholar]
  104. James GM, Hastie TJ. 2001. Functional linear discriminant analysis for irregularly sampled curves. J. R. Stat. Soc. Ser. B 63:533–50 [Google Scholar]
  105. James GM, Hastie TJ, Sugar C. 2000. Principal component models for sparse functional data. Biometrika 87:587–602 [Google Scholar]
  106. James GM, Silverman BW. 2005. Functional adaptive model estimation. J. Am. Stat. Assoc. 100:565–76 [Google Scholar]
  107. James GM, Sugar CA. 2003. Clustering for sparsely sampled functional data. J. Am. Stat. Assoc. 98:397–408 [Google Scholar]
  108. Jiang C, Aston JA, Wang JL. 2009. Smoothing dynamic positron emission tomography time courses using functional principal components. NeuroImage 47:184–93 [Google Scholar]
  109. Jiang C, Wang JL. 2010. Covariate adjusted functional principal components analysis for longitudinal data. Ann. Stat. 38:1194–226 [Google Scholar]
  110. Jiang C, Wang JL. 2011. Functional single index models for longitudinal data. Ann. Stat. 39:362–88 [Google Scholar]
  111. Jiang C, Yu W, Wang JL. 2014. Inverse regression for longitudinal data. Ann. Stat. 42:563–91 [Google Scholar]
  112. Jolliffe IT. 2002. Principal Component Analysis New York: Springer, 2nd ed..
  113. Jones MC, Rice JA. 1992. Displaying the important features of large collections of similar curves. Am. Stat. 46:140–45 [Google Scholar]
  114. Karhunen K. 1946. Zur Spektraltheorie stochastischer Prozesse. Ann. Acad. Sci. Fenn. Ser. A. I Math. 34:1–7 [Google Scholar]
  115. Kato T. 1980. Perturbation Theory for Linear Operators New York: Springer, 2nd ed..
  116. Kayano M, Dozono K, Konishi S. 2010. Functional cluster analysis via orthonormalized Gaussian basis expansions and its application. J. Classif. 27:211–30 [Google Scholar]
  117. Kirkpatrick M, Heckman N. 1989. A quantitative genetic model for growth, shape, reaction norms, and other infinite-dimensional characters. J. Math. Biol. 27:429–50 [Google Scholar]
  118. Kleffe J. 1973. Principal components of random variables with values in a separable Hilbert space. Math. Oper. Stat. 4:391–406 [Google Scholar]
  119. Kneip A, Gasser T. 1988. Convergence and consistency results for self-modeling nonlinear regression. Ann. Stat. 16:82–112 [Google Scholar]
  120. Kneip A, Gasser T. 1992. Statistical tools to analyze data representing a sample of curves. Ann. Stat. 20:1266–305 [Google Scholar]
  121. Kneip A, Ramsay JO. 2008. Combining registration and fitting for functional models. J. Am. Stat. Assoc. 103:1155–65 [Google Scholar]
  122. Kneip A, Sarda P. 2011. Factor models and variable selection in high-dimensional regression analysis. Ann. Stat. 39:2410–47 [Google Scholar]
  123. Kneip A, Utikal KJ. 2001. Inference for density families using functional principal component analysis. J. Am. Stat. Assoc. 96:519–42 [Google Scholar]
  124. Kong D, Xue K, Yao F, Zhang HH. 2015. Partially functional linear regression in high dimensions. Biometrika 103:147–59 [Google Scholar]
  125. Kraus D, Panaretos VM. 2012. Dispersion operators and resistant second-order functional data analysis. Biometrika 99:813–32 [Google Scholar]
  126. Lai RCS, Huang HC, Lee TCM. 2012. Fixed and random effects selection in nonparametric additive mixed models. Electron. J. Stat. 6:810–42 [Google Scholar]
  127. Lawton WH, Sylvestre EA. 1971. Self modeling curve resolution. Technometrics 13:617–33 [Google Scholar]
  128. Leng X, Müller HG. 2006. Time ordering of gene co-expression. Biostatistics 7:569–84 [Google Scholar]
  129. Leurgans SE, Moyeed RA, Silverman BW. 1993. Canonical correlation analysis when the data are curves. J. R. Stat. Soc. Ser. B 55:725–40 [Google Scholar]
  130. Li KC. 1991. Sliced inverse regression for dimension reduction. J. Am. Stat. Assoc. 86:316–27 [Google Scholar]
  131. Li PL, Chiou JM. 2011. Identifying cluster number for subspace projected functional data clustering. Comput. Stat. Data Anal. 55:2090–103 [Google Scholar]
  132. Li Y, Hsing T. 2010. Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Stat. 38:3321–51 [Google Scholar]
  133. Lin X, Zhang D. 1999. Inference in generalized additive mixed models by using smoothing splines. J. R. Stat. Soc. Ser. B 61:381–400 [Google Scholar]
  134. Little RJ, Rubin DB. 2014. Statistical Analysis with Missing Data New York: Wiley
  135. Liu B, Müller HG. 2009. Estimating derivatives for samples of sparsely observed functions, with application to on-line auction dynamics. J. Am. Stat. Assoc. 104:704–14 [Google Scholar]
  136. Liu X, Müller HG. 2004. Functional convex averaging and synchronization for time-warped random curves. J. Am. Stat. Assoc. 99:687–99 [Google Scholar]
  137. Loève M. 1946. Fonctions aléatoires à décomposition orthogonale exponentielle. La Rev. Sci. 84:159–62 [Google Scholar]
  138. Ma S, Yang L, Carroll RJ. 2012. A simultaneous confidence band for sparse longitudinal regression. Stat. Sin. 22:95 [Google Scholar]
  139. Malfait N, Ramsay JO. 2003. The historical functional linear model. Can. J. Stat. 31:115–28 [Google Scholar]
  140. Matsui H, Araki T, Konishi S. 2011. Multiclass functional discriminant analysis and its application to gesture recognition. J. Classification 28:227–43 [Google Scholar]
  141. McCullagh P, Nelder JA. 1983. Generalized Linear Models. London: Chapman & Hall
  142. McLean MW, Hooker G, Staicu AM, Scheipl F, Ruppert D. 2014. Functional generalized additive models. J. Comput. Graph. Stat. 23:249–69 [Google Scholar]
  143. Morris JS. 2015. Functional regression. Annu. Rev. Stat. Appl. 2:321–59 [Google Scholar]
  144. Müller HG. 2005. Functional modelling and classification of longitudinal data. Scand. J. Stat. 32:223–40 [Google Scholar]
  145. Müller HG. 2008. Functional modeling of longitudinal data. Longitudinal Data Analysis G Fitzmaurice, M Davidian, G Verbeke, G Molenberghs 223–52 Boca Raton, FL: Chapman & Hall [Google Scholar]
  146. Müller HG. 2011. Functional data analysis. International Encyclopedia of Statistical Science M Lovric 554–55 Heidelberg, Ger.: Springer [Google Scholar]
  147. Müller HG, Carey JR, Wu D, Liedo P, Vaupel JW. 2001. Reproductive potential predicts longevity of female Mediterranean fruit flies. Proc. R. Soc. B 268:445–50 [Google Scholar]
  148. Müller HG, Stadtmüller U. 2005. Generalized functional linear models. Ann. Stat. 33:774–805 [Google Scholar]
  149. Müller HG, Wu S, Diamantidis AD, Papadopoulos NT, Carey JR. 2009. Reproduction is adapted to survival characteristics across geographically isolated medfly populations. Proc. R. Soc. B 276:4409–16 [Google Scholar]
  150. Müller HG, Wu Y, Yao F. 2013. Continuously additive models for nonlinear functional regression. Biometrika 100:607–22 [Google Scholar]
  151. Müller HG, Yao F. 2008. Functional additive models. J. Am. Stat. Assoc. 103:1534–44 [Google Scholar]
  152. Müller HG, Yao F. 2010a. Additive modelling of functional gradients. Biometrika 97:791–805 [Google Scholar]
  153. Müller HG, Yao F. 2010b. Empirical dynamics for longitudinal data. Ann. Stat. 38:3458–86 [Google Scholar]
  154. Opgen-Rhein R, Strimmer K. 2006. Inferring gene dependency networks from genomic longitudinal data: a functional data approach. REVSTAT 4:53–65 [Google Scholar]
  155. Panaretos VM, Kraus D, Maddocks JH. 2010. Second-order comparison of Gaussian random functions and the geometry of DNA minicircles. J. Am. Stat. Assoc. 105:670–82 [Google Scholar]
  156. Panaretos VM, Tavakoli S. 2013. Fourier analysis of stationary time series in function space. Ann. Stat. 41:568–603 [Google Scholar]
  157. Paul D, Peng J. 2009. Consistency of restricted maximum likelihood estimators of principal components. Ann. Stat. 37:1229–71 [Google Scholar]
  158. Peng J, Müller HG. 2008. Distance-based clustering of sparsely observed stochastic processes, with applications to online auctions. Ann. Appl. Stat. 2:1056–77 [Google Scholar]
  159. Peng J, Paul D, Müller HG. 2014. Time-warped growth processes, with applications to the modeling of boom–bust cycles in house prices. Ann. Appl. Stat. 8:1561–82 [Google Scholar]
  160. Petrone S, Guindani M, Gelfand AE. 2009. Hybrid Dirichlet mixture models for functional data. J. R. Stat. Soc. Ser. B 71:755–82 [Google Scholar]
  161. Pezzulli S, Silverman B. 1993. Some properties of smoothed principal components analysis for functional data. Comput. Stat. 8:1–16 [Google Scholar]
  162. R Core Team. 2013. R: a language and environment for statistical computing. Vienna: R Found. Stat. Comput.
  163. Ramsay JO. 1982. When the data are functions. Psychometrika 47:379–96 [Google Scholar]
  164. Ramsay JO, Dalzell C. 1991. Some tools for functional data analysis. J. R. Stat. Soc. Ser. B 53:539–72 [Google Scholar]
  165. Ramsay JO, Hooker G, Graves S. 2009. Functional Data Analysis with R and Matlab New York: Springer
  166. Ramsay JO, Li X. 1998. Curve registration. J. R. Stat. Soc. Ser. B 60:351–63 [Google Scholar]
  167. Ramsay JO, Silverman BW. 2002. Applied Functional Data Analysis: Methods and Case Studies New York: Springer-Verlag
  168. Ramsay JO, Silverman BW. 2005. Functional Data Analysis New York: Springer, 2nd ed..
  169. Rao CR. 1958. Some statistical methods for comparison of growth curves. Biometrics 14:1–17 [Google Scholar]
  170. Rice J, Silverman B. 1991. Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B 53:233–43 [Google Scholar]
  171. Rice JA. 2004. Functional and longitudinal data analysis: perspectives on smoothing. Stat. Sin. 14:631–47 [Google Scholar]
  172. Rice JA, Wu CO. 2001. Nonparametric mixed effects models for unequally sampled noisy curves. Biometrics 57:253–59 [Google Scholar]
  173. Rincón M, Ruiz-Medina MD. 2012. Wavelet-RKHS-based functional statistical classification. Adv. Data Anal. Classif. 6:201–17 [Google Scholar]
  174. Rodriguez A, Dunson DB, Gelfand AE. 2009. Bayesian nonparametric functional data analysis through density estimation. Biometrika 96:149–62 [Google Scholar]
  175. Roweis ST, Saul LK. 2000. Nonlinear dimensionality reduction by locally linear embedding. Science 290:2323–26 [Google Scholar]
  176. Sakoe H, Chiba S. 1978. Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans. Acoust. Speech Signal Process. 26:43–49 [Google Scholar]
  177. Şentürk D, Müller HG. 2005. Covariate adjusted correlation analysis via varying coefficient models. Scand. J. Stat. 32:365–83 [Google Scholar]
  178. Şentürk D, Müller HG. 2008. Generalized varying coefficient models for longitudinal data. Biometrika 95:653–66 [Google Scholar]
  179. Serban N, Wasserman L. 2005. CATS: clustering after transformation and smoothing. J. Am. Stat. Assoc. 100:990–99 [Google Scholar]
  180. Shi M, Weiss RE, Taylor JM. 1996. An analysis of paediatric CD4 counts for acquired immune deficiency syndrome using flexible random curves. Appl. Stat. 45:151–63 [Google Scholar]
  181. Shin H, Lee S. 2015. Canonical correlation analysis for irregularly and sparsely observed functional data. J. Multivar. Anal. 134:1–18 [Google Scholar]
  182. Silverman BW. 1995. Incorporating parametric effects into functional principal components analysis. J. R. Stat. Soc. Ser. B 57:673–89 [Google Scholar]
  183. Silverman BW. 1996. Smoothed functional principal components analysis by choice of norm. Ann. Stat. 24:1–24 [Google Scholar]
  184. Sood A, James G, Tellis GJ. 2009. Functional regression: a new model for predicting market penetration of new products. Mark. Sci. 28:36–51 [Google Scholar]
  185. Speckman P. 1988. Kernel smoothing in partial linear models. J. R. Stat. Soc. Ser. B 50:413–36 [Google Scholar]
  186. Staniswalis JG, Lee JJ. 1998. Nonparametric regression analysis of longitudinal data. J. Am. Stat. Assoc. 93:1403–18 [Google Scholar]
  187. Stone CJ. 1985. Additive regression and other nonparametric models. Ann. Stat. 13:689–705 [Google Scholar]
  188. Sun Y, Genton MG. 2011. Functional boxplots. J. Comput. Graphical Stat. 20:316–34 [Google Scholar]
  189. Tang R, Müller HG. 2008. Pairwise curve synchronization for functional data. Biometrika 95:875–89 [Google Scholar]
  190. Tenenbaum JB, De Silva V, Langford JC. 2000. A global geometric framework for nonlinear dimensionality reduction. Science 290:2319–23 [Google Scholar]
  191. Tucker JD, Wu W, Srivastava A. 2013. Generative models for functional data using phase and amplitude separation. Comput. Stat. Data Anal. 61:50–66 [Google Scholar]
  192. Verzelen N, Tao W, Müller HG. 2012. Inferring stochastic dynamics from functional data. Biometrika 99:533–50 [Google Scholar]
  193. Wand MP, Jones CM. 1995. Kernel Smoothing New York: Chapman & Hall
  194. Wang J, Yang L. 2009. Polynomial spline confidence bands for regression curves. Stat. Sin. 19:325–42 [Google Scholar]
  195. Wang K, Gasser T. 1997. Alignment of curves by dynamic time warping. Ann. Stat. 25:1251–76 [Google Scholar]
  196. Wang S, Qian L, Carroll RJ. 2010. Generalized empirical likelihood methods for analyzing longitudinal data. Biometrika 97:79–93 [Google Scholar]
  197. Wang XH, Ray S, Mallick BK. 2007. Bayesian curve classification using wavelets. J. Am. Stat. Assoc. 102:962–73 [Google Scholar]
  198. Wu CO, Chiang CT. 2000. Kernel smoothing on varying coefficient models with longitudinal dependent variable. Stat. Sin. 10:433–56 [Google Scholar]
  199. Wu CO, Chiang CT, Hoover DR. 1998. Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data. J. Am. Stat. Assoc. 93:1388–402 [Google Scholar]
  200. Wu H, Zhang JT. 2006. Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling Approaches New York: Wiley
  201. Wu P, Müller HG. 2010. Functional embedding for the classification of gene expression profiles. Bioinformatics 26:509–17 [Google Scholar]
  202. Wu W, Srivastava A. 2014. Analysis of spike train data: alignment and comparisons using the extended Fisher-Rao metric. Electron. J. Stat. 8:1776–85 [Google Scholar]
  203. Xia Y, Tong H, Li W, Zhu LX. 2002. An adaptive estimation of dimension reduction space. J. R. Stat. Soc. Ser. B 64:363–410 [Google Scholar]
  204. Yang W, Müller HG, Stadtmüller U. 2011. Functional singular component analysis. J. R. Stat. Soc. Ser. B 73:303–24 [Google Scholar]
  205. Yao F, Lee T. 2006. Penalized spline models for functional principal component analysis. J. R. Stat. Soc. Ser. B 68:3–25 [Google Scholar]
  206. Yao F, Müller HG. 2010. Functional quadratic regression. Biometrika 97:49–64 [Google Scholar]
  207. Yao F, Müller HG, Wang JL. 2005a. Functional data analysis for sparse longitudinal data. J. Am. Stat. Assoc. 100:577–90 [Google Scholar]
  208. Yao F, Müller HG, Wang JL. 2005b. Functional linear regression analysis for longitudinal data. Ann. Stat. 33:2873–903 [Google Scholar]
  209. You J, Zhou H. 2007. Two-stage efficient estimation of longitudinal nonparametric additive models. Stat. Probability Lett. 77:1666–75 [Google Scholar]
  210. Zhang JT. 2013. Analysis of Variance for Functional Data Boca Raton, FL: CRC
  211. Zhang X, Park BU, Wang JL. 2013. Time-varying additive models for longitudinal data. J. Am. Stat. Assoc. 108:983–98 [Google Scholar]
  212. Zhang X, Wang JL. 2015. Varying-coefficient additive models for functional data. Biometrika 102:15–32 [Google Scholar]
  213. Zhang X, Wang JL. 2016. From sparse to dense functional data and beyond. Ann. Stat. In press [Google Scholar]
  214. Zhang Z, Müller HG. 2011. Functional density synchronization. Comput. Stat. Data Anal. 55:2234–49 [Google Scholar]
  215. Zhao X, Marron JS, Wells MT. 2004. The functional data analysis view of longitudinal data. Stat. Sin. 14:789–808 [Google Scholar]
  216. Zhu H, Fan J, Kong L. 2014. Spatially varying coefficient model for neuroimaging data with jump discontinuities. J. Am. Stat. Assoc. 109:1084–98 [Google Scholar]
  217. Zhu HX, Brown PJ, Morris JS. 2012. Robust classification of functional and quantitative image data using functional mixed models. Biometrics 68:1260–68 [Google Scholar]
  218. Zhu HX, Vannucci M, Cox DD. 2010. A Bayesian hierarchical model for classification with selection of functional predictors. Biometrics 66:463–73 [Google Scholar]
/content/journals/10.1146/annurev-statistics-041715-033624
Loading
/content/journals/10.1146/annurev-statistics-041715-033624
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error