## Functional Data AnalysisScientists 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. |

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### Contents

2 | |

v | 11 |

Summary statistics for functional data | 22 |

Phaseplane plots of periodic effects | 29 |

om functional data to smooth functions | 37 |

The Fourier basis system for periodic data | 45 |

Other useful basis systems | 53 |

noothing functional data by least squares | 59 |

Computational issues | 235 |

Further reading and notes | 244 |

Longterm and seasonal trends in the nondurable goods | 251 |

Conﬁdence intervals | 257 |

Regularization using restricted basis functions | 264 |

The direct penalty method for computing 3 | 271 |

Further reading and notes | 276 |

Assessing goodness of ﬁt | 290 |

noothing functional data with a roughness penalty | 82 |

Conﬁdence intervals for function values and functional | 100 |

Further reading and notes | 109 |

The performance of spline smoothing revisited | 117 |

Fitting a linear model with estimation of the density | 123 |

Shift registration | 129 |

Using the warping function h to register ac | 137 |

fl P111tlmQ1 rnarlino anrl 1nntFQ | 144 |

Deﬁning functional PCA | 148 |

TfFQ | 150 |

agularized principal components analysis | 173 |

incipal components analysis of mixed data | 187 |

The temperature data reconsidered | 195 |

Principles of classical CCA | 204 |

Algorithmic considerations | 210 |

nctional linear models | 217 |

Force plate data for walking horses | 229 |

rivatives and functional linear models | 297 |

JC0lD | 305 |

Beyond the constant COff1C1Ilt ﬁrst order linear equation | 311 |

Some linear differential equation facts | 319 |

Initial conditions boundary conditions and other con | 323 |

A principal differential analysis of lip movement | 329 |

Techniques for principal differential analysis | 338 |

Further readings and notes | 348 |

Further reading and notes | 357 |

The optimal basis for spline smoothing | 363 |

Some case studies | 370 |

me perspectives on FDA 379 | 378 |

Some algebraic and functional techniques | 385 |

Further aspects of inner product spaces | 391 |

Kronecker Products | 398 |

IlCS | 405 |

420 | |

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### Common terms and phrases

acceleration algorithm applied approach argument values B-spline basis expansion basis functions basis system breakpoints Chapter coefﬁcients coeﬁicients compute consider constraint context correlation corresponding covariance cross-validation curves cycle deﬁned deﬁnition degrees of freedom differential equation eigenanalysis error estimate example Figure ﬁnd ﬁrst ﬁt ﬁtting criterion ﬁxed Fourier Fourier series func function values functional data analysis functional linear model growth inﬁnite inner product integration interval Journal knots landmark least squares linear differential operator linear model matrix mean measure methods minimizing multivariate notation orthogonal panel penalized plot point-wise polynomial precipitation principal components analysis probability density function problem Ramsay reﬁect registration regression reproducing kernel residuals roughness penalty satisﬁes second derivative Section shift Silverman smoothing parameter solution speciﬁc spline spline function spline smoothing statistics sum of squares techniques temperature tive tlrip transformation variability variance-covariance matrix variation vector velocity warping function wavelet weight functions zero