Functional Modelling and Classification of Longitudinal Data.

We review and extend some statistical tools that have proved useful for analysing functional data. Functional data analysis primarily is designed for the analysis of random trajectories and infinite-dimensional data, and there exists a need for the development of adequate statistical estimation and inference techniques. While this field is in flux, some methods have proven useful. These include warping methods, functional principal component analysis, and conditioning under Gaussian assumptions for the case of sparse data. The latter is a recent development that may provide a bridge between functional and more classical longitudinal data analysis. Besides presenting a brief review of functional principal components and functional regression, we develop some concepts for estimating functional principal component scores in the sparse situation. An extension of the so-called generalized functional linear model to the case of sparse longitudinal predictors is proposed. This extension includes functional binary regression models for longitudinal data and is illustrated with data on primary biliary cirrhosis.

[1]  John A. Rice,et al.  FUNCTIONAL AND LONGITUDINAL DATA ANALYSIS: PERSPECTIVES ON SMOOTHING , 2004 .

[2]  Jane-ling Wang,et al.  Functional linear regression analysis for longitudinal data , 2005, math/0603132.

[3]  T. Gasser,et al.  Synchronizing sample curves nonparametrically , 1999 .

[4]  U. Grenander.  Stochastic processes and statistical inference , 1950 .

[5]  J L Wang,et al.  Early mortality surge in protein-deprived females causes reversal of sex differential of life expectancy in Mediterranean fruit flies. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Gillian Z Heller,et al.  Functional data analysis with application to periodically stimulated foetal heart rate data. I: Functional regression , 2002, Statistics in medicine.

[7]  H. Müller,et al.  Dynamical Correlation for Multivariate Longitudinal Data , 2005 .

[8]  J. Faraway.  Regression analysis for a functional response , 1997 .

[9]  A. Cuevas,et al.  Linear functional regression: The case of fixed design and functional response , 2002 .

[10]  Catherine A. Sugar,et al.  Principal component models for sparse functional data , 1999 .

[11]  H. Müller,et al.  Functional Convex Averaging and Synchronization for Time-Warped Random Curves , 2004 .

[12]  B. Silverman,et al.  Functional Data Analysis , 1997 .

[13]  H. Hotelling.  Analysis of a complex of statistical variables into principal components. , 1933 .

[14]  Marie Davidian,et al.  Nonlinear Models for Repeated Measurement Data , 1995 .

[15]  Theo Gasser,et al.  Asymptotic and bootstrap confidence bounds for the structural average of curves , 1998 .

[16]  Colin O. Wu,et al.  Nonparametric Mixed Effects Models for Unequally Sampled Noisy Curves , 2001, Biometrics.

[17]  George M. Church,et al.  Aligning gene expression time series with time warping algorithms , 2001, Bioinform..

[18]  P. Sarda,et al.  Functional linear model , 1999 .

[19]  H. Muller,et al.  Generalized functional linear models , 2005, math/0505638.

[20]  Fang Yao,et al.  Functional Linear Regression Analysis for Longitudinal Data 1 This Reprint Differs from the Original in Pagination and Typographic Detail. 1 2 , 2005 .

[21]  Ruben H. Zamar,et al.  Comparing the shapes of regression functions , 2000 .

[22]  H. Hotelling.  Relations Between Two Sets of Variates , 1936 .

[23]  Jane-Ling Wang,et al.  Functional canonical analysis for square integrable stochastic processes , 2003 .

[24]  G. He,et al.  EXTENDING CORRELATION AND REGRESSION FROM MULTIVARIATE TO FUNCTIONAL DATA , 2000 .

[25]  T. Gasser,et al.  Alignment of curves by dynamic time warping , 1997 .

[26]  Jane-Ling Wang,et al.  Functional quasi‐likelihood regression models with smooth random effects , 2003 .

[27]  André Mas,et al.  Testing hypotheses in the functional linear model , 2003 .

[28]  M. Benko,et al.  Functional Data Analysis with Applications in , 2006 .

[29]  Jeng-Min Chiou,et al.  Quasi‐Likelihood Regression with Multiple Indices and Smooth Link and Variance Functions , 2004 .

[30]  T. Gasser,et al.  Searching for Structure in Curve Samples , 1995 .

[31]  Francisco P. Chavez,et al.  Relationship between physical and biological variables during the upwelling period in Monterey Bay, CA , 1998 .

[32]  K. Karhunen.  Zur Spektraltheorie stochastischer prozesse , 1946 .

[33]  Xueli Liu,et al.  Modes and clustering for time-warped gene expression profile data , 2003, Bioinform..

[34]  William B. Capra,et al.  An Accelerated-Time Model for Response Curves , 1997 .

[35]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[36]  P. Grambsch,et al.  Primary biliary cirrhosis: Prediction of short‐term survival based on repeated patient visits , 1994, Hepatology.

[37]  H. Müller,et al.  Methods of canonical analysis for functional data , 2004 .

[38]  Peter Hall,et al.  Nonparametric estimation of a periodic function , 2000 .

[39]  H. Müller,et al.  FUNCTIONAL RESPONSE MODELS , 2004 .

[40]  Robert E. Weiss,et al.  An Analysis of Paediatric Cd4 Counts for Acquired Immune Deficiency Syndrome Using Flexible Random Curves , 1996 .

[41]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[42]  T. Gasser,et al.  Statistical Tools to Analyze Data Representing a Sample of Curves , 1992 .

[43]  H. Müller,et al.  Shrinkage Estimation for Functional Principal Component Scores with Application to the Population Kinetics of Plasma Folate , 2003, Biometrics.

[44]  T. Gasser,et al.  Shape-invariant modelling of human growth. , 1980, Annals of human biology.

[45]  B. Silverman,et al.  Estimating the mean and covariance structure nonparametrically when the data are curves , 1991 .

[46]  H. Cardot.  Nonparametric estimation of smoothed principal components analysis of sampled noisy functions , 2000 .

[47]  Douglas C. Montgomery,et al.  Using Control Charts to Monitor Process and Product Quality Profiles , 2004 .

[48]  Gareth M. James,et al.  Functional Adaptive Model Estimation , 2005 .

[49]  Gareth M. James.  Generalized linear models with functional predictors , 2002 .

[50]  C. R. Rao,et al.  Some statistical methods for comparison of growth curves. , 1958 .

[51]  Ying Zhang,et al.  Time‐Varying Functional Regression for Predicting Remaining Lifetime Distributions from Longitudinal Trajectories , 2005, Biometrics.

[52]  B. Silverman,et al.  Canonical correlation analysis when the data are curves. , 1993 .

[53]  L. Ferré,et al.  Functional sliced inverse regression analysis , 2003 .

[54]  J. Ramsay,et al.  Some Tools for Functional Data Analysis , 1991 .

[55]  Xin Zhao,et al.  The functional data analysis view of longitudinal data , 2004 .

[56]  Peter Hall,et al.  A Functional Data—Analytic Approach to Signal Discrimination , 2001, Technometrics.

[57]  Birgitte B. Rønn,et al.  Nonparametric maximum likelihood estimation for shifted curves , 2001 .

[58]  H. Müller,et al.  Functional Data Analysis for Sparse Longitudinal Data , 2005 .

[59]  Denis Bosq,et al.  Modelization, Nonparametric Estimation and Prediction for Continuous Time Processes , 1991 .

[60]  Catherine A. Sugar,et al.  Clustering for Sparsely Sampled Functional Data , 2003 .

[61]  J. Dauxois,et al.  Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference , 1982 .

[62]  M. Kirkpatrick,et al.  A quantitative genetic model for growth, shape, reaction norms, and other infinite-dimensional characters , 1989, Journal of mathematical biology.

[63]  K. J. Utikal,et al.  Inference for Density Families Using Functional Principal Component Analysis , 2001 .

[64]  Jianqing Fan,et al.  Test of Significance When Data Are Curves , 1998 .

[65]  James Stephen Marron,et al.  Discussion of nonparametric and semiparametric regression , 2004 .

[66]  E. A. Sylvestre,et al.  Principal modes of variation for processes with continuous sample curves , 1986 .

[67]  R. Fraiman,et al.  Kernel-based functional principal components ( , 2000 .

[68]  H. Müller,et al.  Nonparametric Regression Analysis of Growth Curves , 1984 .

[69]  J. Ramsay,et al.  Curve registration by local regression , 2000 .