Contents
Preface v
1 Introduction 1
1.1 Why consider functional data at all? . . . . . . . . . . . 1
1.2 TheWeb site . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The case studies . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 How is functional data analysis distinctive? . . . . . . . . 14
1.5 Conclusion and bibliography . . . . . . . . . . . . . . . . 15
2 Life Course Data in Criminology 17
2.1 Criminology life course studies . . . . . . . . . . . . . . . 17
2.1.1 Background . . . . . . . . . . . . . . . . . . . . . 17
2.1.2 The life course data . . . . . . . . . . . . . . . . . 18
2.2 First steps in a functional approach . . . . . . . . . . . . 19
2.2.1 Turning discrete values into a functional datum . 19
2.2.2 Estimating the mean . . . . . . . . . . . . . . . . 21
2.3 Functional principal component analyses . . . . . . . . . 23
2.3.1 The basic methodology . . . . . . . . . . . . . . . 23
2.3.2 Smoothing the PCA . . . . . . . . . . . . . . . . 26
2.3.3 Smoothed PCA of the criminology data . . . . . 26
2.3.4 Detailed examination of the scores . . . . . . . . 28
2.4 What have we seen? . . . . . . . . . . . . . . . . . . . . . 31

2.5 How are functions stored and processed? . . . . . . . . . 33
2.5.1 Basis expansions . . . . . . . . . . . . . . . . . . 33
2.5.2 Fitting basis coefficients to the observed data . . 35
2.5.3 Smoothing the sample mean function . . . . . . . 36
2.5.4 Calculations for smoothed functional PCA . . . . 37
2.6 Cross-validation for estimating the mean . . . . . . . . . 38
2.7 Notes and bibliography . . . . . . . . . . . . . . . . . . . 40
3 The Nondurable Goods Index 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Transformation and smoothing . . . . . . . . . . . . . . . 43
3.3 Phase-plane plots . . . . . . . . . . . . . . . . . . . . . . 44
3.4 The nondurable goods cycles . . . . . . . . . . . . . . . . 47
3.5 What have we seen? . . . . . . . . . . . . . . . . . . . . . 54
3.6 Smoothing data for phase-plane plots . . . . . . . . . . . 55
3.6.1 Fourth derivative roughness penalties . . . . . . . 55
3.6.2 Choosing the smoothing parameter . . . . . . . . 55
4 Bone Shapes from a Paleopathology Study 57
4.1 Archaeology and arthritis . . . . . . . . . . . . . . . . . . 57
4.2 Data capture . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 How are the shapes parameterized? . . . . . . . . . . . . 59
4.4 A functional principal components analysis . . . . . . . . 61
4.4.1 Procrustes rotation and PCA calculation . . . . . 61
4.4.2 Visualizing the components of shape variability . 61
4.5 Varimax rotation of the principal components . . . . . . 63
4.6 Bone shapes and arthritis: Clinical relationship? . . . . . 65
4.7 What have we seen? . . . . . . . . . . . . . . . . . . . . . 66
4.8 Notes and bibliography . . . . . . . . . . . . . . . . . . . 66
5 Modeling Reaction-Time Distributions 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Nonparametric modeling of density functions . . . . . . . 71
5.3 Estimating density and individual differences . . . . . . . 73
5.4 Exploring variation across subjects with PCA . . . . . . 76
5.5 What have we seen? . . . . . . . . . . . . . . . . . . . . . 79
5.6 Technical details . . . . . . . . . . . . . . . . . . . . . . . 80
6 Zooming in on Human Growth 83
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Height measurements at three scales . . . . . . . . . . . 84
6.3 Velocity and acceleration . . . . . . . . . . . . . . . . . . 86
6.4 An equation for growth . . . . . . . . . . . . . . . . . . . 89
6.5 Timing or phase variation in growth . . . . . . . . . . . . 91
6.6 Amplitude and phase variation in growth . . . . . . . . . 93
6.7 What we have seen? . . . . . . . . . . . . . . . . . . . . . 96
6.8 Notes and further issues . . . . . . . . . . . . . . . . . . 97
6.8.1 Bibliography . . . . . . . . . . . . . . . . . . . . . 97
6.8.2 The growth data . . . . . . . . . . . . . . . . . . 98
6.8.3 Estimating a smooth monotone curve to fit data . 98
7 Time Warping Handwriting and Weather Records 101
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Formulating the registration problem . . . . . . . . . . . 102
7.3 Registering the printing data . . . . . . . . . . . . . . . . 104
7.4 Registering the weather data . . . . . . . . . . . . . . . . 105
7.5 What have we seen? . . . . . . . . . . . . . . . . . . . . . 110
7.6 Notes and references . . . . . . . . . . . . . . . . . . . . 110
7.6.1 Continuous registration . . . . . . . . . . . . . . . 110
7.6.2 Estimation of the warping function . . . . . . . . 113
8 How Do Bone Shapes Indicate Arthritis? 115
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.2 Analyzing shapes without landmarks . . . . . . . . . . . 116
8.3 Investigating shape variation . . . . . . . . . . . . . . . . 120
8.3.1 Looking at means alone . . . . . . . . . . . . . . 120
8.3.2 Principal components analysis . . . . . . . . . . . 120
8.4 The shape of arthritic bones . . . . . . . . . . . . . . . . 123
8.4.1 Linear discriminant analysis . . . . . . . . . . . . 123
8.4.2 Regularizing the discriminant analysis . . . . . . 125
8.4.3 Why not just look at the group means? . . . . . . 127
8.5 What have we seen? . . . . . . . . . . . . . . . . . . . . . 128
8.6 Notes and further issues . . . . . . . . . . . . . . . . . . 128
8.6.1 Bibliography . . . . . . . . . . . . . . . . . . . . . 128
8.6.2 Why is regularization necessary? . . . . . . . . . 129
8.6.3 Cross-validation in classification problems . . . . 130
9 Functional Models for Test Items 131
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.2 The ability space curve . . . . . . . . . . . . . . . . . . . 132
9.3 Estimating item response functions . . . . . . . . . . . . 135
9.4 PCA of log odds-ratio functions . . . . . . . . . . . . . . 136
9.5 Do women and men perform differently on this test? . . 138
9.6 A nonlatent trait: Arc length . . . . . . . . . . . . . . . . 140
9.7 What have we seen? . . . . . . . . . . . . . . . . . . . . . 143
9.8 Notes and bibliography . . . . . . . . . . . . . . . . . . . 143
10 Predicting Lip Acceleration from Electromyography 145
10.1 The neural control of speech . . . . . . . . . . . . . . . . 145
10.2 The lip and EMG curves . . . . . . . . . . . . . . . . . . 147
10.3 The linear model for the data . . . . . . . . . . . . . . . 148
10.4 The estimated regression function . . . . . . . . . . . . . 150
10.5 How far back should the historical model go? . . . . . . 152
10.6 What have we seen? . . . . . . . . . . . . . . . . . . . . . 155
10.7 Notes and bibliography . . . . . . . . . . . . . . . . . . . 155
11 The Dynamics of Handwriting Printed Characters 157
11.1 Recording handwriting in real time . . . . . . . . . . . . 157
11.2 An introduction to dynamic models . . . . . . . . . . . . 158
11.3 One subject’s printing data . . . . . . . . . . . . . . . . . 160
11.4 A differential equation for handwriting . . . . . . . . . . 162
11.5 Assessing the fit of the equation . . . . . . . . . . . . . . 165
11.6 Classifying writers by using their dynamic equations . . 166
11.7 What have we seen? . . . . . . . . . . . . . . . . . . . . . 170
12 A Differential Equation for Juggling 171
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 171
12.2 The data and preliminary analyses . . . . . . . . . . . . 172
12.3 Features in the average cycle . . . . . . . . . . . . . . . . 173
12.4 The linear differential equation . . . . . . . . . . . . . . 176
12.5 What have we seen? . . . . . . . . . . . . . . . . . . . . . 180
12.6 Notes and references . . . . . . . . . . . . . . . . . . . . 181
References 183
Index 187