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Texts in Statistical Science:Introduction to Functional Data Analysis

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Contents 1 First steps in the analysis of functional data 1 1.1 Basis expansions . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Sample mean and covariance . . . . . . . . . . . . . . . . . . 6 1.3 Principal component functions . . . . . . . . . . . . . . . . . 10 1.4 Analysis of BOA stock returns . . . . . . . . . . . . . . . . . 11 1.5 Di usion tensor imaging . . . . . . . . . . . . . . . . . . . . 14 1.6 Chapter 1 problems . . . . . . . . . . . . . . . . . . . . . . . 17 2 Further topics in exploratory FDA 21 2.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Penalized smoothing . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Curve alignment . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Chapter 2 problems . . . . . . . . . . . . . . . . . . . . . . . 34 3 Mathematical framework for functional data 37 3.1 Square integrable functions . . . . . . . . . . . . . . . . . . 38 3.2 Random functions . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Linear transformations . . . . . . . . . . . . . . . . . . . . . 42 4 Scalar{on{function regression 45 4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Review of standard regression theory . . . . . . . . . . . . . 48 4.3 Diculties speci c to functional regression . . . . . . . . . . 51 4.4 Estimation through a basis expansion . . . . . . . . . . . . . 54 4.5 Estimation with a roughness penalty . . . . . . . . . . . . . 56 4.6 Regression on functional principal components . . . . . . . . 58 4.7 Implementation in the refund package . . . . . . . . . . . . 60 4.8 Nonlinear scalar{on{function regression . . . . . . . . . . . . 63 4.9 Chapter 4 problems . . . . . . . . . . . . . . . . . . . . . . . 64 5 Functional response models 67 5.1 Least squares estimation and application to angular motion . 67 5.2 Penalized least squares estimation . . . . . . . . . . . . . . . 69 5.3 Functional regressors . . . . . . . . . . . . . . . . . . . . . . 74 5.4 Penalized estimation in the refund package . . . . . . . . . 77 5.5 Estimation based on functional principal components . . . . 84 xiContents 1 First steps in the analysis of functional data 1 1.1 Basis expansions . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Sample mean and covariance . . . . . . . . . . . . . . . . . . 6 1.3 Principal component functions . . . . . . . . . . . . . . . . . 10 1.4 Analysis of BOA stock returns . . . . . . . . . . . . . . . . . 11 1.5 Di usion tensor imaging . . . . . . . . . . . . . . . . . . . . 14 1.6 Chapter 1 problems . . . . . . . . . . . . . . . . . . . . . . . 17 2 Further topics in exploratory FDA 21 2.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Penalized smoothing . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Curve alignment . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Chapter 2 problems . . . . . . . . . . . . . . . . . . . . . . . 34 3 Mathematical framework for functional data 37 3.1 Square integrable functions . . . . . . . . . . . . . . . . . . 38 3.2 Random functions . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Linear transformations . . . . . . . . . . . . . . . . . . . . . 42 4 Scalar{on{function regression 45 4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Review of standard regression theory . . . . . . . . . . . . . 48 4.3 Diculties speci c to functional regression . . . . . . . . . . 51 4.4 Estimation through a basis expansion . . . . . . . . . . . . . 54 4.5 Estimation with a roughness penalty . . . . . . . . . . . . . 56 4.6 Regression on functional principal components . . . . . . . . 58 4.7 Implementation in the refund package . . . . . . . . . . . . 60 4.8 Nonlinear scalar{on{function regression . . . . . . . . . . . . 63 4.9 Chapter 4 problems . . . . . . . . . . . . . . . . . . . . . . . 64 5 Functional response models 67 5.1 Least squares estimation and application to angular motion . 67 5.2 Penalized least squares estimation . . . . . . . . . . . . . . . 69 5.3 Functional regressors . . . . . . . . . . . . . . . . . . . . . . 74 5.4 Penalized estimation in the refund package . . . . . . . . . 77 5.5 Estimation based on functional principal components . . . . 84 xi

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,Introduction to
Functional
Data Analysis

,CHAPMAN & HALL/CRC
Texts in Statistical Science Series
Series Editors
Francesca Dominici, Harvard School of Public Health, USA
Julian J. Faraway, University of Bath, UK
Martin Tanner, Northwestern University, USA
Jim Zidek, University of British Columbia, Canada
Statistical Theory: A Concise Introduction Problem Solving: A Statistician’s Guide,
F. Abramovich and Y. Ritov Second Edition
Practical Multivariate Analysis, Fifth Edition C. Chatfield
A. Afifi, S. May, and V.A. Clark Statistics for Technology: A Course in Applied
Practical Statistics for Medical Research Statistics, Third Edition
D.G. Altman C. Chatfield

Interpreting Data: A First Course Analysis of Variance, Design, and Regression :
in Statistics Linear Modeling for Unbalanced Data,
A.J.B. Anderson Second Edition
R. Christensen
Introduction to Probability with R
K. Baclawski Bayesian Ideas and Data Analysis: An
Introduction for Scientists and Statisticians
Linear Algebra and Matrix Analysis for
Statistics R. Christensen, W. Johnson, A. Branscum,
S. Banerjee and A. Roy and T.E. Hanson

Modern Data Science with R Modelling Binary Data, Second Edition
B. S. Baumer, D. T. Kaplan, and N. J. Horton D. Collett

Mathematical Statistics: Basic Ideas and Modelling Survival Data in Medical Research,
Selected Topics, Volume I, Third Edition
Second Edition D. Collett
P. J. Bickel and K. A. Doksum Introduction to Statistical Methods for
Mathematical Statistics: Basic Ideas and Clinical Trials
Selected Topics, Volume II T.D. Cook and D.L. DeMets
P. J. Bickel and K. A. Doksum Applied Statistics: Principles and Examples
Analysis of Categorical Data with R D.R. Cox and E.J. Snell
C. R. Bilder and T. M. Loughin Multivariate Survival Analysis and Competing
Statistical Methods for SPC and TQM Risks
D. Bissell M. Crowder
Introduction to Probability Statistical Analysis of Reliability Data
J. K. Blitzstein and J. Hwang M.J. Crowder, A.C. Kimber,
T.J. Sweeting, and R.L. Smith
Bayesian Methods for Data Analysis,
Third Edition An Introduction to Generalized
B.P. Carlin and T.A. Louis Linear Models, Third Edition
A.J. Dobson and A.G. Barnett
Second Edition
R. Caulcutt Nonlinear Time Series: Theory, Methods, and
Applications with R Examples
The Analysis of Time Series: An Introduction,
R. Douc, E. Moulines, and D.S. Stoffer
Sixth Edition
C. Chatfield Introduction to Optimization Methods and
Their Applications in Statistics
Introduction to Multivariate Analysis
B.S. Everitt
C. Chatfield and A.J. Collins

, Extending the Linear Model with R: Graphics for Statistics and Data Analysis with R
Generalized Linear, Mixed Effects and K.J. Keen
Nonparametric Regression Models, Second Mathematical Statistics
Edition K. Knight
J.J. Faraway
Introduction to Functional Data Analysis
Linear Models with R, Second Edition P. Kokoszka and M. Reimherr
J.J. Faraway
Introduction to Multivariate Analysis:
A Course in Large Sample Theory Linear and Nonlinear Modeling
T.S. Ferguson S. Konishi
Multivariate Statistics: A Practical Nonparametric Methods in Statistics with SAS
Approach Applications
B. Flury and H. Riedwyl O. Korosteleva
Readings in Decision Analysis Modeling and Analysis of Stochastic Systems,
S. French Second Edition
Discrete Data Analysis with R: Visualization V.G. Kulkarni
and Modeling Techniques for Categorical and Exercises and Solutions in Biostatistical Theory
Count Data L.L. Kupper, B.H. Neelon, and S.M. O’Brien
M. Friendly and D. Meyer
Exercises and Solutions in Statistical Theory
Markov Chain Monte Carlo: L.L. Kupper, B.H. Neelon, and S.M. O’Brien
Stochastic Simulation for Bayesian Inference,
Second Edition Design and Analysis of Experiments with R
D. Gamerman and H.F. Lopes J. Lawson

Bayesian Data Analysis, Third Edition Design and Analysis of Experiments with SAS
A. Gelman, J.B. Carlin, H.S. Stern, D.B. Dunson, J. Lawson
A. Vehtari, and D.B. Rubin A Course in Categorical Data Analysis
Multivariate Analysis of Variance and T. Leonard
Repeated Measures: A Practical Approach for Statistics for Accountants
Behavioural Scientists S. Letchford
D.J. Hand and C.C. Taylor Introduction to the Theory of Statistical
Practical Longitudinal Data Analysis Inference
D.J. Hand and M. Crowder H. Liero and S. Zwanzig
Logistic Regression Models Statistical Theory, Fourth Edition
J.M. Hilbe B.W. Lindgren
Richly Parameterized Linear Models: Stationary Stochastic Processes: Theory and
Additive, Time Series, and Spatial Models Applications
Using Random Effects G. Lindgren
J.S. Hodges Statistics for Finance
Statistics for Epidemiology E. Lindström, H. Madsen, and J. N. Nielsen
N.P. Jewell The BUGS Book: A Practical Introduction to
Stochastic Processes: An Introduction, Bayesian Analysis
Second Edition D. Lunn, C. Jackson, N. Best, A. Thomas, and
P.W. Jones and P. Smith D. Spiegelhalter
The Theory of Linear Models Introduction to General and Generalized
B. Jørgensen Linear Models
Pragmatics of Uncertainty H. Madsen and P. Thyregod
J.B. Kadane Time Series Analysis
Principles of Uncertainty H. Madsen
J.B. Kadane Pólya Urn Models
H. Mahmoud

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