Contents Preface to the Second Edition iii 1 Introduction 1 1.1 What are functional data? 1 1.2 Functional models for nonfunctional data 5 1.3 Some functional data analyses 5 1.4 The goals of functional data analysis 9 1.5 The first steps in a functional data analysis 11 1.5.1 Data representation: smoothing and interpolation 11 1.5.2 Data regist ration or feature alignment 12 1.5.3 Data display 13 1.5.4 Plotting pairs of derivatives 13 1.6 Exploring variability in functional data 15 1.6.1 Functional descriptive statistics 15 1.6.2 Functional principal components analysis 15 1.6.3 Functional canonical correlation 16 1.7 Funct ional linear models 16 1.8 Using derivatives in functional data analysis 17 1.9 Concluding remarks 18 2 Tools for exploring functional data 19 2.1 Introduction 19 2.2 Some notation 20 2.2.1 Scalars, vectors, functions and matrices 20 2.2.2 Derivatives and integrals 20 2.2.3 Inncr products 21 2.2 4 Functions of functions 21 2.3 Summary statistics for functional data 22 2.3 1 Functional means and variances 22 2.3.2 Covariance and correlation functions 22 2.3.3 Cross-covariance and cross-correlation functions.24 2.4 The anatomy of a function 26 2.4 1 Functional features 26 2.4 2 Data resolution and functional dimensionality 27 2.4.3 The size of a function 28 2.5 Phase-plane plots of periodic effects 29 2.5.1 The log nondurable goods index 29 2.5.2 Phase-plane plots show energy transfer 30 2.5.3 The nondurable goods cycles 33 2.6 Further reading and notes 34 3 From functional data to smooth functions 37 3.1 Introduction 37 3.2 Some properties of functional data 38 3.2.1 What makes discrete data functional? 38 3.2.2 Samples of functional data 39 3.2.3 The interplay between smooth and noisy variation 39 3.2.4 The standard model for error and its limitations 40 3.2.5 The resolving power of data 41 3.2.6 Data resolution and derivative estimation 41 3.3 Representing functions by basis functions 43 3.4 The Fourier basis system for periodic data 45 3.5 The spline basis system for open-ended data 46 3.5.1 Spline functions and degrees of freedom 47 3.5 2 The B-spline basis for spline functions 49 3.6 Other useful basis systems 53 3.6.1 Wavelets 53 3.6.2 Exponential and power bases 54 3.6.3 Polynomial bases 54 3.6 4 The polygonal basis 55 3.6.5 The step-function basis 55 3.6.6 The constant basis 55 3.6.7 Empirical and designer bases 56 3.7 Choosing a scale for t 56 3.8 Further reading and notes 57 4 Smoothing functional data by least squares 59 4.1 Introduction 59 4.2 Fitting data using a basis system by least squares 59 4.2.1 Ordinary or unweighted least squares fits 60 4.2.2 Weighted least squares fits 61 4.3 A performance assessment of least squares smoothing 62 4.4 Least squares fits as linear transformations of the data.63 4.4.1 How linear smoothers work 64 4.4.2 The degrees of freedom of a linear smooth 66 4.5 Choosing the number K of basis functions 67 4.5.1 The bias/ variance trade-off 67 4.5.2 Algorithms for choosing K 69 4.6 Computing sampling variances and confidence limits 70 4.6.1 Sampling variance estimates 70 4.6.2 Estimating Ee 71 4.6.3 Confidence limits 72 4.7 Fitting data by localized least squares 73 4.7.1 Kernel smoothing 74 4.7.2 Localized basis function estimators 76 4.7.3 Local polynomial smoothing 77 4.7.4 Choosing the bandwidth h 78 4.7.5 Summary of localized basis methods 78 4.8 Further reading and notes 79 5 Smoothing functional data with a roughness penalty 81 5.1 Introduction 81 5.2 Spline smoothing 82 5.2.1 Two competing objectives in function estimation 83 5.2.2 Quantifying roughness 84 5.2.3 The penalized sum of squared errors fitting criterion 84 5.2.4 The structure of a smoothing spline 85 5.2.5 How spline smooths are computed 86 5.2.6 Spline smoothing as a linear operation 87 5.2.7 Spline smoothing as an augmented least squares problem 89 5.2.8 Estimating derivatives by spline smoothing 90 5.3 Some extensions 91 5.3.1 Roughness penalties with fewer basis functions 91 5.3.2 More general measures of data fit 92 5.3.3 More general roughness penalties 92 5.3.4 Computing the roughness penalty matrix 93 5.4 Choosing the smoothing parameter 94 5.4.1 Some limits imposed by computational issues 94 5.4.2 The cross-validation or CV method 96 5.4.3 The generalized cross-validation or GCV method 97 5.4.4 Spline smoothing the simulated growth data 99 5.5 Confidence intervals for function values and functional probes 100 5.5.1 Linear functional probes 101 5.5.2 Two linear mappings defining a probe value 102 5.5.3 Computing confidence limits for function values 103 5.5.4 Confidence limits for growth acceleration 104 5.6 A bi-resolution analysis with smoothing splines 104 5.6.1 Complementary bases 105 5.6.2 Specifying the roughness penalty 106 5.6.3 Some properties of the estimates 107 5.6.4 Relationship to the roughness penalty approach 108 5.7 Further reading and notes 109 6 Constrained functions 111 6.1 Introduction 111 6.2 Fitting positive functions 111 6.2.1 A positive smoothing spline 113 6.2.2 Representing a posit ive function by a differential equation 114 6.3 Fitting strictly monotone functions 115 6.3.1 Fitting the growth of a baby's tibia 115 6.3.2 Expressing a strictly monotone function explicitly 115 6.3.3 Expressing a strictly monotone function as a differential equation 116 6.4 The performance of spline smoothing revisited 117 6.5 Fitting probability functions 118 6.6 Estimating probability density functions 119 6.7 Functional data analysis of point processes 121 6.8 Fitting a linear model with estimation of the density of residuals 123 6.9 Further notes and readings 126 7 The registration and display of functional data 127 7.1 Int roduction 127 7.2 Shift registration 129 7.2.1 The least squares criterion for shift alignment 131 7.3 Feature or landmark rcgistration 132 7.4 Using thc warping function h to rcgister x 137 7.5 A morc gcneral warping function h 137 7.6 A continuous fitting criterion for registration 138 7.7 Regist ering the height acceleration curves 140 7.8 Some practical advice 142 7.9 Computational details 142 7.9.1 Shift registration by the Newton-Raphson algorithm 142 7.10 Further reading and notes 144 8 Principal components analysis for functional data 147 8.1 Introduction 147 8.2 Defining functional PCA 148 8.2.1 PCA for multivariate data 148 8.2.2 Defining PCA for functional data 149 8.2.3 Defining an optimal empirical orthonormal basis 151 8.2.4 PCA and eigenanalysis 152 8.3 Visualizing the results 154 8.3.1 Plotting components as perturbations of the mean 154 8.3.2 Plotting principal component scores 156 8.3.3 Rotating principal components 156 8.4 Computational methods for functional PCA 160 8.4.1 Discretizing the functions 161 8.4.2 Basis function expansion of the functions 161 8.4.3 More general numerical quadrature 164 8.5 Bivariate and mult ivariate PCA 166 8.5.1 Defining multivariate functional PCA 167 8.5.2 Visualizing the results 168 8.5.3 Inner product notation: Conc1uding remarks 170 8.6 Further readings and notes 171 9 Regularized principal components analysis 173 9.1 Introduction 173 9.2 The results of smoothing the PCA 175 9.3 The smoothing approach 177 9.3.1 Estimating the leading principal component 177 9.3.2 Estimating subsequent principal components 177 9.3.3 Choosing the smoothing parameter by CV 178 9.4 Finding the regularized PCA in practice 179 9.4.1 The periodic case 179 9.4.2 The nonperiodic case 181 9.5 Alternative approaches 182 9.5.1 Smoothing the data rather than the PCA 182 9.5.2 A stepwise roughness penalty procedure 184 9.5.3 A further approach 185 10 Principal components analysis of mixed data 187 10.1 Int roduction 187 10.2 General approaches to mixed data 189 10.3 The PCA of hybrid data 190 10.3.1 Combining function and vector spaces 190 10.3.2 Finding the principal components in practice 191 10.3.3 Incorporating smoot hing 192 10.3.4 Balance between functional and vector variation 192 10.4 Combining registration and PCA 194 10.4.1 Expressing the observations as mixed data 194 10.4.2 Balancing temperature and time shift effects 194 10.5 The temperature data reconsidered 195 10.5.1 Taking account of effects beyond phase shift 195 10.5.2 Separating out the vector component 198 11 Canonical correlation and discriminant analysis 201 11.1 Introduction 201 11.1.1 The basic problem 201 11.2 Principles of classical CCA 204 11.3 Functional canonical correlation analysis 204 11.3.1 Notation and assumptions 204 11.3.2 The naive approach does not give meaningful results 205 11.3.3 Choice of the smoothing parameter 206 11.3.4 The values of the correlations 207 11.4 Application to the study of lupus nephritis 208 11.5 Why is regularization necessary? 209 11.6 Algorithmic considerations 210 11.6.1 Discretization and basis approaches 210 11.6.2 The roughness of the canonical variates 211 11.7 Penalized optimal scoring and discriminant analysis 213 11.7.1 The optimal scoring problem 213 11.7.2 The discriminant problem 214 11.7.3 The relationship with CCA 214 11.7.4 Applications 215 11.8 Further readings and notes 215 12 Functionallinear models 217 12.1 lntroduction 217 12.2 A functional response and a categorical independent variable 218 12.3 A scalar response and a functional independent variable 219 12.4 A functional response and a functional independent variable 220 12.4.1 Concurrent 220 12.4.2 Annual or total 220 12.4.3 Short-term fccd-forward 220 12.4.4 Local influence 221 12.5 What about predicting derivatives? 221 12.6 Overview 222 13 Modelling functional responses with multivariate covariates 223 13.1 Introduction 223 13.2 Predicting temperature curves from climate zones 223 13.2.1 Fitting the model 225 13.2.2 Assessing the fit 225 13.3 Force plate data for walking horses 229 13.3.1 Structure of the data 229 13.3.2 A functional linear model for the horse data 231 13.3.3 Effects and contrasts 233 13.4 Computational issues 235 13.4.1 The general model 235 13.4.2 Pointwise minimization 236 13.4.3 Functional linear modelling with regularized basis expansions 236 13.4.4 Using the Kronecker product to express B 238 13.4.5 Fitting the raw data 239 13.5 Confidence intervals for regression functions 239 13.5.1 How to compute confidence intervals 239 13.5.2 Confidence intervals for climate zone effects 241 13.5.3 Some cautions on interpreting confidence intervals 243 13.6 Further reading and notes 244 14 Functional responses, functional covariates and the concurrent model 247 14.1 Introduction 247 14.2 Predicting precipitation profiles from temperature curves 248 14.2.1 The model for the daily logarithm of rainfall 248 14.2.2 Preliminary steps 248 14.2.3 Fitting the model and assessing fit 250 14.3 Long-term and seasonal trends in the nondurable goods index 251 14.4 Computational issues 255 14.5 Confidence intervals 257 14.6 Further reading and fiotes 258 15 Functional linear models for scalar responses 261 15.1 Introduction 261 15.2 A naive approach: Discretizing the covariate function 262 15.3 Regularization using restricted basis functions 264 15.4 Regularization with roughness penalties 266 15.5 Computational issues 268 15.5.1 Computing the regularized solution 269 15.5.2 Computing confidence limits 270 15.6 Cross-validation and regression diagnostics 270 15.7 The direct penalty method for computing 271 15.7.1 Functional interpolation 272 15.7.2 The two-stage minimization process 272 15.7.3 Functional interpolation revisited 273 15.8 Functional regression and integral equations 275 15.9 Further reading and notes 276 16 Functional linear models for functional responses 279 16.1 Introduction: Predicting log precipitation from temperature 279 16.1.1 Fitting the model without regularization 280 16.2 Regularizing the fit by restricting the bases 282 16.2.1 Restricting the basis n(8) 282 16.2.2 Restricting the basis θ(t) 283 16.2.3 Restricting both bases 284 16.3 Assessing goodness of fit 285 16.4 Computational details 290 16.4 1 Fitting the model without regularization 291 16 4 2 Fitting the model with regularization 292 16.5 The general case 293 16.6 Further reading and notes 295 17 Derivatives and functional linear models 297 17.1 Introduction 297 17.2 The oil refinery data 298 17.3 The melanoma data 301 17.4 Some comparisons of the refinery a nd melanoma analyses 305 18 Differential equations and operators 307 18.1 Introduction 307 18.2 Exploring a simple linear differential equation 308 18.3 Beyond the constant coefficient first-order linear equation 310 18.3.1 Nonconstant coefficients 310 18 3.2 Higher order equations 311 18.3.3 Systems of equations 312 18 3.4 Beyond linearity 313 18 4 Some applications of linear differential equations and operators 313 18 4.1 Differential operators to produce new functional observations 313 18 4.2 The gross domestic product data 314 18.4.3 Differential operators to regularize or smooth models 316 18.4.4 Differential operators to partition variation 317 18 4.5 Operators to define solutions to problems 319 18.5 Some linear differe ntial equation facts 319 18.5.1 Derivatives are rougher 319 18 5.2 Finding a linear differential operator that a nnihilates known functions 320 18.5.3 Finding the functions j satisfying Lj=0 322 18.6 Initial conditions, boundary conditions and other constraints 323 18.6.1 Why additional constraints are needed to define a solution 323 18.6.2 How L and B partition functions 324 18.6.3 The inner product defined by operators L and B 325 18.7 Further reading and notes 325 19 Principal differential analysis 327 19.1 Introduction 327 19.2 Defining the problem 328 19.3 A principal differential analysis of lip movement 329 19.3.1 The biomechanics of lip movement 330 19.3.2 Visualizing the PDA results 332 19.4 PDA of the pinch force data 334 19.5 Techniques for principal differential analysis 338 19.5.1 PDA by point-wise minimization 338 19.5.2 PDA using the concurrent functionallinear model 339 19.5.3 PDA by iterating the concurrent linear model 340 19.5.4 Assessing fit in PDA 343 19.6 Comparing PDA and PCA 343 19.6.1 PDA and PCA both minimize sums of squared errors 343 19.6.2 PDA and PCA both involve finding linear operators 344 19.6.3 Differences between differential operators (PDA) and projection operators (PCA) 345 19.7 Further readings and notes 348 20 Green's functions and reproducing kernels 349 20.1 Introduction 349 20.2 The Green's function for solving a linear differential equation 350 20.2.1 The definition of the Green's function 351 20.2.2 A matrix analogue of the Green's function 352 20.2.3 A recipe for the Green's function 352 20.3 Reproducing kernels and Green's functions 353 20.3.1 What is a reproducing kernel? 354 20.3.2 The reproducing kernel for ker B 355 20.3.3 The reproducing kernel for ker L 356 20.4 Further reading and notes 357 21 More general roughness penalties 359 21.1 Introduction 359 21.1.1 The lip movement data 360 21.1.2 The weather data 361 21.2 The optimal basis for spline smoothing 363 21.3 An O(n) algorithm for L-spline smoothing 364 21.3.1 The need for a good algorithm 364 21.3.2 Setting up the smoothing procedure 366 21.3.3 The smoothing phase 367 21.3.4 The performance assessment phase 367 21.3.5 Other O(n) algorithms 369 21.4 A compact support basis for L-splines 369 21.5 Sorne case studies 370 2l.5.1 The gross domestic product data 370 21.5.2 The melanoma data 371 21.5.3 The GDP data with seasonal effects 373 21.5.4 Smoothing simulated human growth data 374 22 Some perspectives on FDA 379 22.1 The context of functional data analysis 379 22.1.1 Replication and regularity 379 22.l.2 Some functional aspects elsewhere in statistics 380 22.1.3 Functional analytic treatments 381 22.2 Challenges for the future 382 22.2.1 Probability and inference 382 22.2.2 Asymptotic results 383 22.2.3 Multidimensional arguments 383 22.2.4 Practical methodology and applications 384 22.2.5 Back to the data! 384 Appendix: Some algebraic and functional techniques 385 A.1 Inner products (x ,y) 385 A.1.1 Some specific examples 386 A.1.2 General properties 387 A.1.3 Descriptive statistics in inner product notation 389 A.1.4 Some extended uses of inner product notation 390 A.2 Further aspects of inner product spaces 391 A.2.1 Projections 391 A.2.2 Quadratic optimization 392 A.3 Matrix decompositions and generalized inverses 392 A.3.1 Singular value decompositions 392 A.3.2 Generalized inverses 393 A.3.3 The QR decomposition 393 A.4 Projections 394 A.4.1 Projection matrices 394 A.4.2 Finding an appropriate projection matrix 395 A.4.3 Projections in more general inner product spaces 395 A.5 Constrained maximization of a quadratic function 396 A.5.1 The finite-dimensional case 396 A.5.2 The problem in a more general space 396 A.5.3 Generalized eigenproblems 397 A.6 Kronecker Products 398 A.7 The multivariate linear model 399 A.7.1 Linear models from a transformation perspective 399 A.7.2 The least squares solution for B 400 A.8 Regularizing the multivariate linear model 401 A.8.1 Definition of regularization 401 A.8.2 Hard-edged constraints 401 A.8.3 Soft-edged constra ints 402 References 405 Index 419
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