9780387400808

Functional Data Analysis

by ;
  • ISBN13:

    9780387400808

  • ISBN10:

    038740080X

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 2005-06-30
  • Publisher: Springer Verlag
  • Purchase Benefits
  • Free Shipping On Orders Over $35!
    Your order must be $35 or more to qualify for free economy shipping. Bulk sales, PO's, Marketplace items, eBooks and apparel do not qualify for this offer.
  • Get Rewarded for Ordering Your Textbooks! Enroll Now
List Price: $199.00 Save up to $5.97
  • Buy New
    $193.03
    Add to Cart Free Shipping

    PRINT ON DEMAND: 2-4 WEEKS. THIS ITEM CANNOT BE CANCELLED OR RETURNED.

Supplemental Materials

What is included with this book?

  • The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

Summary

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.

Author Biography

Bernard Silverman is Master of St. Peter's College and Professor of Statistics at Oxford University. Jim Ramsay is Professor of Psychology at McGill University.

Table of Contents

Preface to the Second Edition v
1 Introduction
1(18)
1.1 What are functional data?
1(4)
1.2 Functional models for nonfunctional data
5(1)
1.3 Some functional data analyses
5(4)
1.4 The goals of functional data analysis
9(2)
1.5 The first steps in a functional data analysis
11(4)
1.5.1 Data representation: smoothing and interpolation
11(1)
1.5.2 Data registration or feature alignment
12(1)
1.5.3 Data display
13(1)
1.5.4 Plotting pairs of derivatives
13(2)
1.6 Exploring variability in functional data
15(1)
1.6.1 Functional descriptive statistics
15(1)
1.6.2 Functional principal components analysis
15(1)
1.6.3 Functional canonical correlation
16(1)
1.7 Functional linear models
16(1)
1.8 Using derivatives in functional data analysis
17(1)
1.9 Concluding remarks
18(1)
2 Tools for exploring functional data
19(18)
2.1 Introduction
19(1)
2.2 Some notation
20(2)
2.2.1 Scalars, vectors, functions and matrices
20(1)
2.2.2 Derivatives and integrals
20(1)
2.2.3 Inner products
21(1)
2.2.4 Functions of functions
21(1)
2.3 Summary statistics for functional data
22(4)
2.3.1 Functional means and variances
22(1)
2.3.2 Covariance and correlation functions
22(2)
2.3.3 Cross-covariance and cross-correlation functions
24(2)
2.4 The anatomy of a function
26(3)
2.4.1 Functional features
26(1)
2.4.2 Data resolution and functional dimensionality
27(1)
2.4.3 The size of a function
28(1)
2.5 Phase-plane plots of periodic effects
29(5)
2.5.1 The log nondurable goods index
29(1)
2.5.2 Phase-plane plots show energy transfer
30(3)
2.5.3 The nondurable goods cycles
33(1)
2.6 Further reading and notes
34(3)
3 From functional data to smooth functions
37(22)
3.1 Introduction
37(1)
3.2 Some properties of functional data
38(5)
3.2.1 What makes discrete data functional?
38(1)
3.2.2 Samples of functional data
39(1)
3.2.3 The interplay between smooth and noisy variation
39(1)
3.2.4 The standard model for error and its limitations
40(1)
3.2.5 The resolving power of data
41(1)
3.2.6 Data resolution and derivative estimation
41(2)
3.3 Representing functions by basis functions
43(2)
3.4 The Fourier basis system for periodic data
45(1)
3.5 The spline basis system for open-ended data
46(7)
3.5.1 Spline functions and degrees of freedom
47(2)
3.5.2 The B-spline basis for spline functions
49(4)
3.6 Other useful basis systems
53(3)
3.6.1 Wavelets
53(1)
3.6.2 Exponential and power bases
54(1)
3.6.3 Polynomial bases
54(1)
3.6.4 The polygonal basis
55(1)
3.6.5 The step-function basis
55(1)
3.6.6 The constant basis
55(1)
3.6.7 Empirical and designer bases
56(1)
3.7 Choosing a scale for t
56(1)
3.8 Further reading and notes
57(2)
4 Smoothing functional data by least squares
59(22)
4.1 Introduction
59(1)
4.2 Fitting data using a basis system by least squares
59(3)
4.2.1 Ordinary or unweighted least squares fits
60(1)
4.2.2 Weighted least squares fits
61(1)
4.3 A performance assessment of least squares smoothing
62(1)
4.4 Least squares fits as linear transformations of the data
63(4)
4.4.1 How linear smoothers work
64(2)
4.4.2 The degrees of freedom of a linear smooth
66(1)
4.5 Choosing the number K of basis functions
67(3)
4.5.1 The bias/variance trade-off
67(2)
4.5.2 Algorithms for choosing K
69(1)
4.6 Computing sampling variances and confidence limits
70(3)
4.6.1 Sampling variance estimates
70(1)
4.6.2 Estimating Σe
71(1)
4.6.3 Confidence limits
72(1)
4.7 Fitting data by localized least squares
73(6)
4.7.1 Kernel smoothing
74(2)
4.7.2 Localized basis function estimators
76(1)
4.7.3 Local polynomial smoothing
77(1)
4.7.4 Choosing the bandwidth h
78(1)
4.7.5 Summary of localized basis methods
78(1)
4.8 Further reading and notes
79(2)
5 Smoothing functional data with a roughness penalty
81(30)
5.1 Introduction
81(1)
5.2 Spline smoothing
82(9)
5.2.1 Two competing objectives in function estimation
83(1)
5.2.2 Quantifying roughness
84(1)
5.2.3 The penalized sum of squared errors fitting criterion
84(1)
5.2.4 The structure of a smoothing spline
85(1)
5.2.5 How spline smooths are computed
86(1)
5.2.6 Spline smoothing as a linear operation
87(2)
5.2.7 Spline smoothing as an augmented least squares problem
89(1)
5.2.8 Estimating derivatives by spline smoothing
90(1)
5.3 Some extensions
91(3)
5.3.1 Roughness penalties with fewer basis functions
91(1)
5.3.2 More general measures of data fit
92(1)
5.3.3 More general roughness penalties
92(1)
5.3.4 Computing the roughness penalty matrix
93(1)
5.4 Choosing the smoothing parameter
94(6)
5.4.1 Some limits imposed by computational issues
94(2)
5.4.2 The cross-validation or CV method
96(1)
5.4.3 The generalized cross-validation or GCV method
97(2)
5.4.4 Spline smoothing the simulated growth data
99(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

Rewards Program

Write a Review