Functional Data Analysis (Springer Series in Statistics)

326 Pages·1997·3.098 MB·English

by  J. Ramsay, B. W. Silverman

Preface
This book is a snapshot of a highly social and, therefore, decidedly
unpredictableprocess.Thecombinedpersonalviewoffunctionaldata
analysis that it presents has emerged over a number of years of
research and contact, and has been greatly nourished by delightful
collaborations withmany friends.We hope thatreaderswill enjoy the
book as much as we have enjoyed writing it, whether they are our
colleagues as researchers or applied data analysts reading the book
asaresearchmonograph,orstudentsusingitasacoursetext.
After some introductory material in Chapters 1 and 2, we discuss
aspects of smoothing methods in Chapters 3 and 4. Chapter 5 deals
with curve registration, the alignment of common characteristics of
a sample of curves. In the next three chapters, we turn to functional
principalcomponentsanalysis,oneofourmainexploratorytechniques.
In Chapters 9 to 11, problems that involve covariate information are
considered in various functional versions of the linear model. The
functional analogue of canonical correlation analysis in Chapter 12
explores the relationship between two functional variables treating
themsymmetrically.Chapters13to 15developspecificallyfunctional
methods that exploit derivative information and the use of linear
differential operators. Our final chapter provides historical remarks
andsomepointerstopossiblefuturedevelopments.
Data arising in real applications are used throughout for both
motivationandillustration,showinghow functionalapproachesallow
vi Preface
us to see new things, especially by exploiting the smoothness of the
processesgeneratingthedata.Thedatasetsexemplifythewidescope
of functional data analysis; they are drawn from growth analysis,
meteorology, biomechanics, equine science, economics, andmedicine.
Some of these data sets, together with the software we have used to
analysethem,areavailablefromaworld-widewebsitedescribedatthe
endofChapter1.
Many people have been of great help to us in our work on this
book. In particular we would like to thank Michal Abrahamowicz,
Philippe Besse, Darrell Bock, Jeremy Burn, Catherine Dalzell, Shelly
Feran, Randy Flanagan, Rowena Fowler, Theo Gasser, Mary Gauthier,
Vince Gracco, David Green, Nancy Heckman, Anouk Hoedeman, Steve
Hunka, Iain Johnstone, Alois Kneip, Wojtek Krzanowski, Xiaochun Li,
Duncan Murdoch, Kevin Munhall, Guy Nason, Richard Olshen, David
Ostry, Tim Ramsay, John Rice, Xiaohui Wang and Alan Wilson. We
gratefully acknowledge financial support from the Natural Science
and Engineering Research Council of Canada, the National Science
Foundation and the National Institute of Health of the USA, and the
BritishEngineeringand Physical Sciences ResearchCouncil. Above all,
ourthanksareduetotheRoyalStatisticalSocietyResearchSection;our
firstcontactwasatadiscussionmeetingwhereoneofusreadapaper
andtheotherproposedthevoteofthanks,notalwaysanoccasionthat
leadstoameetingofminds!
November1996 JimRamsay
BernardSilverman
Contents
1 Introduction 1
1.1 Whatarefunctionaldata? . . . . . . . . . . . . . . . . . . . 1
1.2 Somefunctionaldataanalyses . . . . . . . . . . . . . . . . 3
1.3 Thegoalsoffunctionaldataanalysis . . . . . . . . . . . . 8
1.4 Firststepsinafunctionaldataanalysis . . . . . . . . . . 9
1.4.1 Datarepresentation:smoothingandinterpolation 9
1.4.2 Dataregistrationorfeaturealignment. . . . . . . 9
1.4.3 Datadisplay . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Summarystatisticsforfunctionaldata. . . . . . . . . . . 11
1.5.1 Meanandvariancefunctions . . . . . . . . . . . . 11
1.5.2 Covarianceandcorrelationfunctions . . . . . . . 11
1.5.3 Cross-covarianceandcross-correlationfunctions 13
1.6 Exploringvariabilityinmoredetail . . . . . . . . . . . . . 16
1.6.1 Functional principalcomponentsanalysis . . . . 16
1.6.2 Functional linearmodelling . . . . . . . . . . . . . 16
1.6.3 Functional canonicalcorrelation . . . . . . . . . . 17
1.7 Usingderivativesinfunctionaldataanalysis . . . . . . . 18
1.7.1 Correlationplotsforderivatives . . . . . . . . . . 18
1.7.2 Plottingpairsofderivatives . . . . . . . . . . . . . 19
1.7.3 Principaldifferentialanalysis . . . . . . . . . . . . 20
1.8 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . 21
viii Contents
2 Notationandtechniques 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Generalnotation . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Scalars,vectors,functions andmatrices. . . . . . 24
2.2.2 Combiningquantities . . . . . . . . . . . . . . . . . 25
2.3 Innerproducts(cid:1)x,y(cid:2) . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Somespecificexamples . . . . . . . . . . . . . . . . 25
2.3.2 Generalproperties . . . . . . . . . . . . . . . . . . . 27
2.3.3 Descriptivestatisticsininnerproductnotation . 29
2.3.4 Someextendedusesofinnerproductnotation . 30
2.4 Furtheraspectsofinnerproductspaces . . . . . . . . . . 31
2.4.1 Projections . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.2 Quadraticoptimization . . . . . . . . . . . . . . . . 32
2.5 Themultivariatelinearmodel . . . . . . . . . . . . . . . . 32
2.5.1 Atransformationperspective . . . . . . . . . . . . 33
2.5.2 Theleastsquaressolution . . . . . . . . . . . . . . 33
2.6 Regularizingthemultivariatelinearmodel . . . . . . . . 34
2.6.1 Definitionofregularization . . . . . . . . . . . . . 34
2.6.2 Hard-edgedconstraints . . . . . . . . . . . . . . . . 35
2.6.3 Soft-edgedconstraints . . . . . . . . . . . . . . . . 35
3 Representingfunctionaldataassmoothfunctions 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Observedfunctionaldata . . . . . . . . . . . . . . . 37
3.1.2 Samplingandobservationalerror . . . . . . . . . 38
3.1.3 Linearsmoothing. . . . . . . . . . . . . . . . . . . . 43
3.2 Basisfunctionmethods . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Leastsquaresfittingofbasisexpansions . . . . . 44
3.2.2 Choosingagoodbasis . . . . . . . . . . . . . . . . 46
3.2.3 Fourierseries . . . . . . . . . . . . . . . . . . . . . . 47
3.2.4 Polynomial bases . . . . . . . . . . . . . . . . . . . . 48
3.2.5 Regressionsplinebases . . . . . . . . . . . . . . . . 48
3.2.6 Waveletbases . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Smoothingbylocalweighting . . . . . . . . . . . . . . . . 51
3.3.1 Kernelfunctions . . . . . . . . . . . . . . . . . . . . 51
3.3.2 Kernelsmoothing . . . . . . . . . . . . . . . . . . . 52
3.3.3 Localizedbasisfunctionestimators . . . . . . . . 54
3.3.4 Localpolynomialsmoothing . . . . . . . . . . . . . 55
3.3.5 Choosingthebandwidth . . . . . . . . . . . . . . . 55
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Contents ix
4 Theroughnesspenaltyapproach 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Splinesmoothing . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.1 Twocompetingobjectives infunctionestimation 58
4.2.2 Estimatingderivativesbysplinesmoothing . . . 60
4.2.3 Moregeneralmeasuresoffit. . . . . . . . . . . . . 60
4.2.4 Moregeneralroughnesspenalties . . . . . . . . . 61
4.3 Theregularizedbasisapproach . . . . . . . . . . . . . . . 62
4.3.1 Complementarybases . . . . . . . . . . . . . . . . . 62
4.3.2 Specifyingtheroughnesspenalty . . . . . . . . . . 63
4.3.3 Somepropertiesoftheestimates . . . . . . . . . . 65
4.3.4 Relationshiptotheroughnesspenaltyapproach 65
5 Theregistrationanddisplay offunctionaldata 67
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Shiftregistration . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.1 Typesofalignment . . . . . . . . . . . . . . . . . . 68
5.2.2 Estimatingthealignment . . . . . . . . . . . . . . . 69
5.2.3 UsingtheNewton–Raphsonalgorithm . . . . . . . 71
5.3 Featureorlandmarkregistration . . . . . . . . . . . . . . 73
5.4 Moregeneraltransformations . . . . . . . . . . . . . . . . 79
5.4.1 Familiesoftransformations . . . . . . . . . . . . . 79
5.4.2 Fittingtransformationsusingregularization . . . 80
5.4.3 Aregularizedbasisapproach . . . . . . . . . . . . 81
5.4.4 Registeringtheheightaccelerationcurves . . . . 82
5.4.5 Somecautionaryremarks. . . . . . . . . . . . . . . 83
6 Principalcomponentsanalysisforfunctionaldata 85
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 DefiningfunctionalPCA . . . . . . . . . . . . . . . . . . . . 86
6.2.1 PCAformultivariatedata. . . . . . . . . . . . . . . 86
6.2.2 DefiningPCAforfunctionaldata . . . . . . . . . . 87
6.2.3 Defininganoptimalempiricalorthonormalbasis 90
6.2.4 PCAandeigenanalysis . . . . . . . . . . . . . . . . 90
6.3 Visualizingtheresults . . . . . . . . . . . . . . . . . . . . . 92
6.3.1 Plottingperturbationsofthemean . . . . . . . . . 92
6.3.2 Plottingprincipalcomponentscores . . . . . . . . 94
6.3.3 Rotatingprincipalcomponents . . . . . . . . . . . 95
6.4 ComputationalmethodsforfunctionalPCA . . . . . . . 99
6.4.1 Discretizingthefunctions . . . . . . . . . . . . . . 99
6.4.2 Basisfunctionexpansionofthefunctions . . . . 100
6.4.3 Moregeneralnumericalquadrature . . . . . . . . 103