Functional Data Analysis

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ISBN: 9780387400808 | 038740080X
Cover: Hardcover
Copyright: 6/30/2005

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Scientists today collect samples of curves and other functional observations. This monograph presents many ideas and techniques for such data. Included are expressions in the functional domain of such classics as linear regression, principal components analysis, linear modelling, and canonical correlation analysis, as well as specifically functional techniques such as curve registration and principal differential analysis. Data arising in real applications are used throughout for both motivation and illustration, showing how functional approaches allow us to see new things, especially by exploiting the smoothness of the processes generating the data. The data sets exemplify the wide scope of functional data analysis; they are drwan from growth analysis, meterology, biomechanics, equine science, economics, and medicine.The book presents novel statistical technology while keeping the mathematical level widely accessible. It is designed to appeal to students, to applied data analysts, and to experienced researchers; it will have value both within statistics and across a broad spectrum of other fields. Much of the material is based on the authors' own work, some of which appears here for the first time.Jim Ramsay is Professor of Psychology at McGill University and is an international authority on many aspects of multivariate analysis. He draws on his collaboration with researchers in speech articulation, motor control, meteorology, psychology, and human physiology to illustrate his technical contributions to functional data analysis in a wide range of statistical and application journals.Bernard Silverman, author of the highly regarded "Density Estimation for Statistics and Data Analysis," and coauthor of "Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach," is Professor of Statistics at Bristol University. His published work on smoothing methods and other aspects of applied, computational, and theoretical statistics has been recognized by the Presidents' Award of the Committee of Presidents of Statistical Societies, and the award of two Guy Medals by the Royal Statistical Society.

Preface to the Second Edition

1 Introduction

(18)

1.1 What are functional data?

(4)

1.2 Functional models for nonfunctional data

(1)

1.3 Some functional data analyses

(4)

1.4 The goals of functional data analysis

(2)

1.5 The first steps in a functional data analysis

(4)

1.5.1 Data representation: smoothing and interpolation

(1)

1.5.2 Data registration or feature alignment

(1)

1.5.3 Data display

(1)

1.5.4 Plotting pairs of derivatives

(2)

1.6 Exploring variability in functional data

(1)

1.6.1 Functional descriptive statistics

(1)

1.6.2 Functional principal components analysis

(1)

1.6.3 Functional canonical correlation

(1)

1.7 Functional linear models

(1)

1.8 Using derivatives in functional data analysis

(1)

1.9 Concluding remarks

(1)

2 Tools for exploring functional data

(18)

2.1 Introduction

(1)

2.2 Some notation

(2)

2.2.1 Scalars, vectors, functions and matrices

(1)

2.2.2 Derivatives and integrals

(1)

2.2.3 Inner products

(1)

2.2.4 Functions of functions

(1)

2.3 Summary statistics for functional data

(4)

2.3.1 Functional means and variances

(1)

2.3.2 Covariance and correlation functions

(2)

2.3.3 Cross-covariance and cross-correlation functions

(2)

2.4 The anatomy of a function

(3)

2.4.1 Functional features

(1)

2.4.2 Data resolution and functional dimensionality

(1)

2.4.3 The size of a function

(1)

2.5 Phase-plane plots of periodic effects

(5)

2.5.1 The log nondurable goods index

(1)

2.5.2 Phase-plane plots show energy transfer

(3)

2.5.3 The nondurable goods cycles

(1)

2.6 Further reading and notes

(3)

3 From functional data to smooth functions

(22)

3.1 Introduction

(1)

3.2 Some properties of functional data

(5)

3.2.1 What makes discrete data functional?

(1)

3.2.2 Samples of functional data

(1)

3.2.3 The interplay between smooth and noisy variation

(1)

3.2.4 The standard model for error and its limitations

(1)

3.2.5 The resolving power of data

(1)

3.2.6 Data resolution and derivative estimation

(2)

3.3 Representing functions by basis functions

(2)

3.4 The Fourier basis system for periodic data

(1)

3.5 The spline basis system for open-ended data

(7)

3.5.1 Spline functions and degrees of freedom

(2)

3.5.2 The B-spline basis for spline functions

(4)

3.6 Other useful basis systems

(3)

3.6.1 Wavelets

(1)

3.6.2 Exponential and power bases

(1)

3.6.3 Polynomial bases

(1)

3.6.4 The polygonal basis

(1)

3.6.5 The step-function basis

(1)

3.6.6 The constant basis

(1)

3.6.7 Empirical and designer bases

(1)

3.7 Choosing a scale for t

(1)

3.8 Further reading and notes

(2)

4 Smoothing functional data by least squares

(22)

4.1 Introduction

(1)

4.2 Fitting data using a basis system by least squares

(3)

4.2.1 Ordinary or unweighted least squares fits

(1)

4.2.2 Weighted least squares fits

(1)

4.3 A performance assessment of least squares smoothing

(1)

4.4 Least squares fits as linear transformations of the data

(4)

4.4.1 How linear smoothers work

(2)

4.4.2 The degrees of freedom of a linear smooth

(1)

4.5 Choosing the number K of basis functions

(3)

4.5.1 The bias/variance trade-off

(2)

4.5.2 Algorithms for choosing K

(1)

4.6 Computing sampling variances and confidence limits

(3)

4.6.1 Sampling variance estimates

(1)

4.6.2 Estimating Σe

(1)

4.6.3 Confidence limits

(1)

4.7 Fitting data by localized least squares

(6)

4.7.1 Kernel smoothing

(2)

4.7.2 Localized basis function estimators

(1)

4.7.3 Local polynomial smoothing

(1)

4.7.4 Choosing the bandwidth h

(1)

4.7.5 Summary of localized basis methods

(1)

4.8 Further reading and notes

(2)

5 Smoothing functional data with a roughness penalty

(30)

5.1 Introduction

(1)

5.2 Spline smoothing

(9)

5.2.1 Two competing objectives in function estimation

(1)

5.2.2 Quantifying roughness

(1)

5.2.3 The penalized sum of squared errors fitting criterion

(1)

5.2.4 The structure of a smoothing spline

(1)

5.2.5 How spline smooths are computed

(1)

5.2.6 Spline smoothing as a linear operation

(2)

5.2.7 Spline smoothing as an augmented least squares problem

(1)

5.2.8 Estimating derivatives by spline smoothing

(1)

5.3 Some extensions

(3)

5.3.1 Roughness penalties with fewer basis functions

(1)

5.3.2 More general measures of data fit

(1)

5.3.3 More general roughness penalties

(1)

5.3.4 Computing the roughness penalty matrix

(1)

5.4 Choosing the smoothing parameter

(6)

5.4.1 Some limits imposed by computational issues

(2)

5.4.2 The cross-validation or CV method

(1)

5.4.3 The generalized cross-validation or GCV method

(2)

5.4.4 Spline smoothing the simulated growth data

(1)

5.5 Confidence intervals for function values and functional probes

100

(4)

5.5.1 Linear functional probes

101

(1)

5.5.2 Two linear mappings defining a probe value

102

(1)

5.5.3 Computing confidence limits for function values

103

(1)

5.5.4 Confidence limits for growth acceleration

104

(1)

5.6 A bi-resolution analysis with smoothing splines

104

(5)

5.6.1 Complementary bases

105

(1)

5.6.2 Specifying the roughness penalty

106

(1)

5.6.3 Some properties of the estimates

107

(1)

5.6.4 Relationship to the roughness penalty approach

108

(1)

5.7 Further reading and notes

109

(2)

6 Constrained functions

111

(16)

6.1 Introduction

111

(1)

6.2 Fitting positive functions

111

(4)

6.2.1 A positive smoothing spline

113

(1)

6.2.2 Representing a positive function by a differential equation

114

(1)

6.3 Fitting strictly monotone functions

115

(2)

6.3.1 Fitting the growth of a baby's tibia

115

(1)

6.3.2 Expressing a strictly monotone function explicitly

115

(1)

6.3.3 Expressing a strictly monotone function as a differential equation

116

(1)

6.4 The performance of spline smoothing revisited

117

(1)

6.5 Fitting probability functions

118

(1)

6.6 Estimating probability density functions

119

(2)

6.7 Functional data analysis of point processes

121

(2)

6.8 Fitting a linear model with estimation of the density of residuals

123

(3)

6.9 Further notes and readings

126

(1)

7 The registration and display of functional data

127

(20)

7.1 Introduction

127

(2)

7.2 Shift registration

129

(3)

7.2.1 The least squares criterion for shift alignment

131

(1)

7.3 Feature or landmark registration

132

(5)

7.4 Using the warping function h to register x

137

(1)

7.5 A more general warping function h

137

(1)

7.6 A continuous fitting criterion for registration

138

(2)

7.7 Registering the height acceleration curves

140

(2)

7.8 Some practical advice

142

(1)

7.9 Computational details

142

(2)

7.9.1 Shift registration by the Newton-Raphson algorithm

142

(2)

7.10 Further reading and notes

144

(3)

8 Principal components analysis for functional data

147

(26)

8.1 Introduction

147

(1)

8.2 Defining functional PCA

148

(6)

8.2.1 PCA for multivariate data

148

(1)

8.2.2 Defining PCA for functional data

149

(2)

8.2.3 Defining an optimal empirical orthonormal basis

151

(1)

8.2.4 PCA and eigenanalysis

152

(2)

8.3 Visualizing the results

154

(6)

8.3.1 Plotting components as perturbations of the mean

154

(2)

8.3.2 Plotting principal component scores

156

(1)

8.3.3 Rotating principal components

156

(4)

8.4 Computational methods for functional PCA

160

(6)

8.4.1 Discretizing the functions

161

(1)

8.4.2 Basis function expansion of the functions

161

(3)

8.4.3 More general numerical quadrature

164

(2)

8.5 Bivariate and multivariate PCA

166

(5)

8.5.1 Defining multivariate functional PCA

167

(1)

8.5.2 Visualizing the results

168

(2)

8.5.3 Inner product notation: Concluding remarks

170

(1)

8.6 Further readings and notes

171

(2)

9 Regularized principal components analysis

173

(14)

9.1 Introduction

173

(2)

9.2 The results of smoothing the, PCA

175

(2)

9.3 The smoothing approach

177

(2)

9.3.1 Estimating the leading principal component

177

(1)

9.3.2 Estimating subsequent principal components

177

(1)

9.3.3 Choosing the smoothing parameter by CV

178

(1)

9.4 Finding the regularized PCA in practice

179

(3)

9.4.1 The periodic case

179

(2)

9.4.2 The nonperiodic case

181

(1)

9.5 Alternative approaches

182

(5)

9.5.1 Smoothing the data rather than the PCA

182

(2)

9.5.2 A stepwise roughness penalty procedure

184

(1)

9.5.3 A further approach

185

(2)

10 Principal components analysis of mixed data

187

(14)

10.1 Introduction

187

(2)

10.2 General approaches to mixed data

189

(1)

10.3 The PCA of hybrid data

190

(4)

10.3.1 Combining function and vector spaces

190

(1)

10.3.2 Finding the principal components in practice

191

(1)

10.3.3 Incorporating smoothing

192

(1)

10.3.4 Balance between functional and vector variation

192

(2)

10.4 Combining registration and PCA

194

(1)

10.4.1 Expressing the observations as mixed data

194

(1)

10.4.2 Balancing temperature and time shift effects

194

(1)

10.5 The temperature data reconsidered

195

(6)

10.5.1 Taking account of effects beyond phase shift

195

(3)

10.5.2 Separating out the vector component

198

(3)

11 Canonical correlation and discriminant analysis

201

(16)

11.1 Introduction

201

(3)

11.1.1 The basic problem

201

(3)

11.2 Principles of classical CCA

204

(1)

11.3 Functional canonical correlation analysis

204

(4)

11.3.1 Notation and assumptions

204

(1)

11.3.2 The naive approach does not give meaningful results

205

(1)

11.3.3 Choice of the smoothing parameter

206

(1)

11.3.4 The values of the correlations

207

(1)

11.4 Application to the study of lupus nephritis

208

(1)

11.5 Why is regularization necessary?

209

(1)

11.6 Algorithmic considerations

210

(3)

11.6.1 Discretization and basis approaches

210

(1)

11.6.2 The roughness of the canonical variates

211

(2)

11.7 Penalized optimal scoring and discriminant analysis

213

(2)

11.7.1 The optimal scoring problem

213

(1)

11.7.2 The discriminant problem

214

(1)

11.7.3 The relationship with CCA

214

(1)

11.7.4 Applications

215

(1)

11.8 Further readings and notes

215

(2)

12 Functional linear models

217

(6)

12.1 Introduction

217

(1)

12.2 A functional response and a categorical independent variable

218

(1)

12.3 A scalar response and a functional independent variable

219

(1)

12.4 A functional response and a functional independent variable

220

(1)

12.4.1 Concurrent

220

(1)

12.4.2 Annual or total

220

(1)

12.4.3 Short-term feed-forward

220

(1)

12.4.4 Local influence

221

(1)

12.5 What about predicting derivatives?

221

(1)

12.6 Overview

222

(1)

13 Modelling functional responses with multivariate covariates

223

(24)

13.1 Introduction

223

(1)

13.2 Predicting temperature curves from climate zones

223

(6)

13.2.1 Fitting the model

225

(1)

13.2.2 Assessing the fit

225

(4)

13.3 Force plate data for walking horses

229

(6)

13.3.1 Structure of the data

229

(2)

13.3.2 A functional linear model for the horse data

231

(2)

13.3.3 Effects and contrasts

233

(2)

13.4 Computational issues

235

(4)

13.4.1 The general model

235

(1)

13.4.2 Pointwise minimization

236

(1)

13.4.3 Functional linear modelling with regularized basis expansions

236

(2)

13.4.4 Using the Kronecker product to express B

238

(1)

13.4.5 Fitting the raw data

239

(1)

13.5 Confidence intervals for regression functions

239

(5)

13.5.1 How to compute confidence intervals

239

(2)

13.5.2 Confidence intervals for climate zone effects

241

(2)

13.5.3 Some cautions on interpreting confidence intervals

243

(1)

13.6 Further reading and notes

244

(3)

14 Functional responses, functional covariates and the con-current model

247

(14)

14.1 Introduction

247

(1)

14.2 Predicting precipitation profiles from temperature curves

248

(3)

14.2.1 The model for the daily logarithm of rainfall

248

(1)

14.2.2 Preliminary steps

248

(2)

14.2.3 Fitting the model and assessing fit

250

(1)

14.3 Long-term and seasonal trends in the nondurable goods index

251

(4)

14.4 Computational issues

255

(2)

14.5 Confidence intervals

257

(1)

14.6 Further reading and notes

258

(3)

15 Functional linear models for scalar responses

261

(18)

15.1 Introduction

261

(1)

15.2 A naive approach: Discretizing the covariate function

262

(2)

15.3 Regularization using restricted basis functions

264

(2)

15.4 Regularization with roughness penalties

266

(2)

15.5 Computational issues

268

(2)

15.5.1 Computing the regularized solution

269

(1)

15.5.2 Computing confidence limits

270

(1)

15.6 Cross-validation and regression diagnostics

270

(1)

15.7 The direct penalty method for computing β

271

(4)

15.7.1 Functional interpolation

272

(1)

15.7.2 The two-stage minimization process

272

(1)

15.7.3 Functional interpolation revisited

273

(2)

15.8 Functional regression and integral equations

275

(1)

15.9 Further reading and notes

276

(3)

16 Functional linear models for functional responses

279

(18)

16.1 Introduction: Predicting log precipitation from temperature

279

(3)

16.1.1 Fitting the model without regularization

280

(2)

16.2 Regularizing the fit by restricting the bases

282

(3)

16.2.1 Restricting the basis η(s)

282

(1)

16.2.2 Restricting the basis theta(t)

283

(1)

16.2.3 Restricting both bases

284

(1)

16.3 Assessing goodness of fit

285

(5)

16.4 Computational details

290

(3)

16.4.1 Fitting the model without regularization

291

(1)

16.4.2 Fitting the model with regularization

292

(1)

16.5 The general case

293

(2)

16.6 Further reading and notes

295

(2)

17 Derivatives and functional linear models

297

(10)

17.1 Introduction

297

(1)

17.2 The oil refinery data

298

(3)

17.3 The melanoma data

301

(4)

17.4 Some comparisons of the refinery and melanoma analyses

305

(2)

18 Differential equations and operators

307

(20)

18.1 Introduction

307

(1)

18.2 Exploring a simple linear differential equation

308

(2)

18.3 Beyond the constant coefficient first-order linear equation

310

(3)

18.3.1 Nonconstant coefficients

310

(1)

18.3.2 Higher order equations

311

(1)

18.3.3 Systems of equations

312

(1)

18.3.4 Beyond linearity

313

(1)

18.4 Some applications of linear differential equations and operators

313

(6)

18.4.1 Differential operators to produce new functional observations

313

(1)

18.4.2 The gross domestic product data

314

(2)

18.4.3 Differential operators to regularize or smooth models

316

(1)

18.4.4 Differential operators to partition variation

317

(2)

18.4.5 Operators to define solutions to problems

319

(1)

18.5 Some linear differential equation facts

319

(4)

18.5.1 Derivatives are rougher

319

(1)

18.5.2 Finding a linear differential operator that annihilates known functions

320

(2)

18.5.3 Finding the functions ξj satisfying Lξj = 0

322

(1)

18.6 Initial conditions, boundary conditions and other constraints

323

(2)

18.6.1 Why additional constraints are needed to define a solution

323

(1)

18.6.2 How L and B partition functions

324

(1)

18.6.3 The inner product defined by operators L and B

325

(1)

18.7 Further reading and notes

325

(2)

19 Principal differential analysis

327

(22)

19.1 Introduction

327

(1)

19.2 Defining the problem

328

(1)

19.3 A principal differential analysis of lip movement

329

(5)

19.3.1 The biomechanics of lip movement

330

(2)

19.3.2 Visualizing the PDA results

332

(2)

19.4 PDA of the pinch force data

334

(4)

19.5 Techniques for principal differential analysis

338

(5)

19.5.1 PDA by point-wise minimization

338

(1)

19.5.2 PDA using the concurrent functional linear model

339

(1)

19.5.3 PDA by iterating the concurrent linear model

340

(3)

19.5.4 Assessing fit in PDA

343

(1)

19.6 Comparing PDA and PCA

343

(5)

19.6.1 PDA and PCA both minimize sums of squared errors

343

(1)

19.6.2 PDA and PCA both involve finding linear operators

344

(1)

19.6.3 Differences between differential operators (PDA) and projection operators (PCA)

345

(3)

19.7 Further readings and notes

348

(1)

20 Green's functions and reproducing kernels

349

(10)

20.1 Introduction

349

(1)

20.2 The Green's function for solving a linear differential equation

350

(3)

20.2.1 The definition of the Green's function

351

(1)

20.2.2 A matrix analogue of the Green's function

352

(1)

20.2.3 A recipe for the Green's function

352

(1)

20.3 Reproducing kernels and Green's functions

353

(4)

20.3.1 What is a reproducing kernel?

354

(1)

20.3.2 The reproducing kernel for ker B

355

(1)

20.3.3 The reproducing kernel for ker L

356

(1)

20.4 Further reading and notes

357

(2)

21 More general roughness penalties

359

(20)

21.1 Introduction

359

(4)

21.1.1 The lip movement data

360

(1)

21.1.2 The weather data

361

(2)

21.2 The optimal basis for spline smoothing

363

(1)

21.3 An O(n) algorithm for L-spline smoothing

364

(5)

21.3.1 The need for a good algorithm

364

(2)

21.3.2 Setting up the smoothing procedure

366

(1)

21.3.3 The smoothing phase

367

(1)

21.3.4 The performance assessment phase

367

(2)

21.3.5 Other O(n) algorithms

369

(1)

21.4 A compact support basis for L-splines

369

(1)

21.5 Some case studies

370

(9)

21.5.1 The gross domestic product data

370

(1)

21.5.2 The melanoma data

371

(2)

21.5.3 The GDP data with seasonal effects

373

(1)

21.5.4 Smoothing simulated human growth data

374

(5)

22 Some perspectives on FDA

379

(6)

22.1 The context of functional data analysis

379

(3)

22.1.1 Replication and regularity

379

(1)

22.1.2 Some functional aspects elsewhere in statistics

380

(1)

22.1.3 Functional analytic treatments

381

(1)

22.2 Challenges for the future

382

(3)

22.2.1 Probability and inference

382

(1)

22.2.2 Asymptotic results

383

(1)

22.2.3 Multidimensional arguments

383

(1)

22.2.4 Practical methodology and applications

384

(1)

22.2.5 Back to the data!

384

(1)

Appendix: Some algebraic and functional techniques

385

(20)

A.1 Inner products (x, y)

385

(6)

A.1.1 Some specific examples

386

(1)

A.1.2 General properties

387

(2)

A.1.3 Descriptive statistics in inner product notation

389

(1)

A.1.4 Some extended uses of inner product notation

390

(1)

A.2 Further aspects of inner product spaces

391

(1)

A.2.1 Projections

391

(1)

A.2.2 Quadratic optimization

392

(1)

A.3 Matrix decompositions and generalized inverses

392

(2)

A.3.1 Singular value decompositions

392

(1)

A.3.2 Generalized inverses

393

(1)

A.3.3 The QR decomposition

393

(1)

A.4 Projections

394

(2)

A.4.1 Projection matrices

394

(1)

A.4.2 Finding an appropriate projection matrix

395

(1)

A.4.3 Projections in more general inner product spaces

395

(1)

A.5 Constrained maximization of a quadratic function

396

(2)

A.5.1 The finite-dimensional case

396

(1)

A.5.2 The problem in a more general space

396

(1)

A.5.3 Generalized eigenproblems

397

(1)

A.6 Kronecker Products

398

(1)

A.7 The multivariate linear model

399

(2)

A.7.1 Linear models from a transformation perspective

399

(1)

A.7.2 The least squares solution for B

400

(1)

A.8 Regularizing the multivariate linear model

401

(4)

A.8.1 Definition of regularization

401

(1)

A.8.2 Hard-edged constraints

401

(1)

A.8.3 Soft-edged constraints

402

(3)

References

405

(14)

Index

419

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Functional Data Analysis

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