## Applied Functional Data Analysis: Methods and Case StudiesAlmost as soon as we had completed our previous book Functional Data Analysis in 1997, it became clear that potential interest in the ?eld was far wider than the audience for the thematic presentation we had given there. At the same time, both of us rapidly became involved in relevant new research involving many colleagues in ?elds outside statistics. This book treats the ?eld in a di?erent way, by considering case st- ies arising from our own collaborative research to illustrate how functional data analysis ideas work out in practice in a diverse range of subject areas. These include criminology, economics, archaeology, rheumatology, psych- ogy, neurophysiology, auxology (the study of human growth), meteorology, biomechanics, and education—and also a study of a juggling statistician. Obviously such an approach will not cover the ?eld exhaustively, and in any case functional data analysis is not a hard-edged closed system of thought. Nevertheless we have tried to give a ?avor of the range of meth- ology we ourselves have considered. We hope that our personal experience, including the fun we had working on these projects, will inspire others to extend “functional” thinking to many other statistical contexts. Of course, manyofourcasestudiesrequireddevelopmentofexistingmethodology,and readersshouldgaintheabilitytoadaptmethodstotheirownproblemstoo. |

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### Contents

1 | |

2 | |

14 How is functional data analysis distinctive? | 14 |

15 Conclusion and bibliography | 15 |

Life Course Data in Criminology | 17 |

212 The life course data | 18 |

22 First steps in a functional approach | 19 |

222 Estimating the mean | 21 |

682 The growth data | 98 |

Time Warping Handwriting and Weather Records | 101 |

72 Formulating the registration problem | 102 |

73 Registering the printing data | 104 |

74 Registering the weather data | 105 |

75 What have we seen? | 110 |

762 Estimation of the warping function | 113 |

How Do Bone Shapes Indicate Arthritis? | 115 |

23 Functional principal component analyses | 23 |

232 Smoothing the PCA | 26 |

234 Detailed examination of the scores | 28 |

24 What have we seen? | 31 |

25 How are functions stored and processed? | 33 |

252 Fitting basis coefficients to the observed data | 35 |

253 Smoothing the sample mean function | 36 |

254 Calculations for smoothed functional PC A | 37 |

26 Crossvalidation for estimating the mean | 38 |

27 Notes and bibliography | 40 |

The Nondurable Goods Index | 41 |

32 Transformation and smoothing | 43 |

33 Phaseplane plots | 44 |

34 The nondurable goods cycles | 47 |

35 What have we seen? | 54 |

36 Smoothing data for phaseplane plots | 55 |

Bone Shapes from a Paleopathology Study | 57 |

42 Data capture | 58 |

43 How are the shapes parameterized? | 59 |

44 A functional principal components analysis | 61 |

45 Varimax rotation of the principal components | 63 |

Clinical relationship? | 65 |

47 What have we seen? | 66 |

Modeling ReactionTime Distributions | 69 |

52 Nonparametric modeling of density functions | 71 |

53 Estimating density and individual differences | 73 |

54 Exploring variation across subjects with PCA | 76 |

55 What have we seen? | 79 |

56 Technical details | 80 |

Zooming in on Human Growth | 82 |

62 Height measurements at three scales | 84 |

63 Velocity and acceleration | 86 |

64 An equation for growth | 89 |

65 Timing or phase variation in growth | 91 |

66 Amplitude and phase variation in growth | 93 |

67 What we have seen? | 96 |

68 Notes and further issues | 97 |

82 Analyzing shapes without landmarks | 116 |

83 Investigating shape variation | 120 |

84 The shape of arthritic bones | 123 |

842 Regularizing the discriminant analysis | 125 |

843 Why not just look at the group means? | 127 |

85 What have we seen? | 128 |

862 Why is regularization necessary? | 129 |

863 Crossvalidation in classification problems | 130 |

Functional Models for Test Items | 131 |

92 The ability space curve | 132 |

93 Estimating item response functions | 135 |

94 PCA of log oddsratio functions | 136 |

95 Do women and men perform differently on this test? | 138 |

Arc length | 140 |

97 What have we seen? | 143 |

Predicting Lip Acceleration from Electromyography | 144 |

102 The lip and EMG curves | 147 |

103 The linear model for the data | 148 |

104 The estimated regression function | 150 |

105 How far back should the historical model go? | 152 |

106 What have we seen? | 155 |

The Dynamics of Handwriting Printed Characters | 157 |

112 An introduction to dynamic models | 158 |

113 One subjects printing data | 160 |

114 A differential equation for handwriting | 162 |

115 Assessing the ﬁt of the equation | 165 |

116 Classifying writers by using their dynamic equations | 166 |

117 What have we seen? | 170 |

A Differential Equation for Juggling | 171 |

122 The data and preliminary analyses | 172 |

123 Features in the average cycle | 173 |

124 The linear differential equation | 176 |

125 What have we seen? | 180 |

126 Notes and references | 181 |

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187 | |

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Applied Functional Data Analysis: Methods and Case Studies J.O. Ramsay,B.W. Silverman No preview available - 2002 |

### Common terms and phrases

acceleration curves ADHD arc length arthritic arthritic bones average B-spline B-spline basis basis functions bone shape Chapter coefficients consider coordinate corresponding criminology cross-validation dashed line datum defined density function discriminant analysis distribution dynamic eburnated eigenvalue electromyography estimated examinees F F F F M F func function h(t functional data analysis functional principal components functions Pi handwriting harmonic indicate individual intercondylar notch item response functions juggling cycle landmarks linear differential equation linear discriminant Linear discriminant analysis lip acceleration mean curve measure method mode of variability msec observed osteoarthritis period phase variation phase-plane plot plotted in Figure points positive principal component weight principal components analysis Ramsay and Silverman reaction record registered regression right panel roughness penalty sample shown in Figure shows Silverman 1997 smoothing parameter space curve square standard values varimax vector velocity warping function weight function X-coordinate zero

### Popular passages

Page 185 - Growth, Maturation and Body Composition: The Fels Longitudinal Study 1929-1991. Cambridge: Cambridge University Press, 1992.

Page 185 - Ramsay, JO (2000). Functional components of variation in handwriting. Journal of the American Statistical Association, 95, 9-15. Ramsay, JO and Bock, RD (2002).