PrefacetotheSecondEdition.
1Introduction
1.1Whatarefunctionaldata?
1.2Functionalmodelsfornonfunctionaldata
1.3Somefunctionaldataanalyses
1.4Thegoalsoffunctionaldataanalysis
1.5Thefirststepsinafunctionaldataanalysis
1.5.1Datarepresentation:smoothingandinterpolation
1.5.2Dataregistrationorfeaturealignment
1.5.3Datadisplay
1.5.4Plottingpairsofderivatives
1.6Exploringvariabilityinfunctionaldata
1.6.1Functionaldescriptivestatistics
1.6.2Functionalprincipalcomponentsanalysis
1.6.3Functionalcanonicalcorrelation
1.7Functionallinearmodels
1.8Usingderivativesinfunctionaldataanalysis
1.9Concludingremarks
2Toolsforexploringfunctionaldata
2.1Introduction
2.2Somenotation
2.2.1Scalars,vectors,functionsandmatrices
2.2.2Derivativesandintegrals
2.2.3Innerproducts
2.2.4Functionsoffunctions
2.3Summarystatisticsforfunctionaldata
2.3.1Functionalmeansandvariances
2.3.2Covarianceandcorrelationfunctions
2.3.3Cross-covarianceandcross-correlationfunctions
2.4Theanatomyofafunction
2.4.1Functionalfeatures
2.4.2Dataresolutionandfunctionaldimensionality
2.4.3Thesizeofafunction
2.5Phase-planeplotsofperiodiceffects
2.5.1Thelognondurablegoodsindex
2.5.2Phase-planeplotsshowenergytransfer
2.5.3Thenondurablegoodscycles
2.6Furtherreadingandnotes
3Fromfunctionaldatatosmoothfunctions
3.1Introduction
3.2Somepropertiesoffunctionaldata
3.2.1Whatmakesdiscretedatafunctional?
3.2.2Samplesoffunctionaldata.
3.2.3Theinterplaybetweensmoothandnoisyvariation
3.2.4Thestandardmodelforerroranditslimitations
3.2.5Theresolvingpowerofdata
3.2.6Dataresolutionandderivativeestimation
3.3Representingfunctionsbybasisfunctions
3.4TheFourierbasissystemforperiodicdata
3.5Thesplinebasissystemforopen-endeddata
3.5.1Splinefunctionsanddegreesoffreedom
3.5.2TheB,splinebasisforsplinefunctions
3.6Otherusefulbasissystems
3.6.1Wavelets
3.6.2Exponentialandpowerbases
3.6.3Polynomialbases
3.6.4Thepolygonalbasis
3.6.5Thestep-functionbasis
3.6.6Theconstantbasis
3.6.7Empiricalanddesignerbases
3.7Choosingascalefort
3.8Furtherreadingandnotes
4Smoothingfunctionaldatabyleastsquares
4.1Introduction
4.2Fittingdatausingabasissystembyleastsquares
4.2.1Ordinaryorunweightedleastsquaresfits
4.2.2Weightedleastsquaresfits
4.3Aperformanceassessmentofleastsquaressmoothing
4.4Leastsquaresfitsaslineartransformationsofthedata
4.4.1Howlinearsmootherswork
4.4.2Thedegreesoffreedomofalinearsmooth
4.5ChoosingthenumberKofbasisfunctions
4.5.1Thebias/variancetrade-off
4.5.2AlgorithmsforchoosingK
4.6Computingsamplingvariancesandconfidencelimits
4.6.1Samplingvarianceestimates
4.6.2Estimatinge
4.6.3Confidencelimits
4.7Fittingdatabylocalizedleastsquares
4.7.1Kernelsmoothing
4.7.2Localizedbasisfunctionestimators
4.7.3Localpolynomialsmoothing
4.7.4Choosingthebandwidthh
4.7.5Summaryoflocalizedbasismethods
4.8Furtherreadingandnotes
5Smoothingfunctionaldatawitharoughnesspenalty
5.1Introduction
5.2Splinesmoothing
5.2.1Twocompetingobjectivesinfunctionestimation
5.2.2Quantifyingroughness
5.2.3Thepenalizedsumofsquarederrorsfittingcriterion
5.2.4Thestructureofasmoothingspline
5.2.5Howsplinesmoothsarecomputed
5.2.6Splinesmoothingasalinearoperation
5.2.7Splinesmoothingasanaugmentedleastsquares problem
5.2.8Estimatingderivativesbysplinesmoothing
5.3Someextensions
5.3.1Roughnesspenaltieswithfewerbasisfunctions
5.3.2Moregeneralmeasuresofdatafit
5.3.3Moregeneralroughnesspenalties
5.3.4Computingtheroughnesspenaltymatrix
5.4Choosingthesmoothingparameter
5.4.1Somelimitsimposedbycomputationalissues
5.4.2Thecross-validationorCVmethod
5.4.3Thegeneralizedcross-validationorGCVmethod
5.4.4Splinesmoothingthesimulatedgrowthdata
5.5Confidenceintervalsforfunctionvaluesandfunctional probes
5.5.1Linearfunctionalprobes
5.5.2Twolinearmappingsdefiningaprobevalue
5.5.3Computingconfidencelimitsforfunctionvalues
5.5.4Confidencelimitsforgrowthacceleration
5.6Abi-resolutionanalysiswithsmoothingsplines
5.6.1Complementarybases
5.6.2Specifyingtheroughnesspenalty
5.6.3Somepropertiesoftheestimates
5.6.4Relationshiptotheroughnesspenaltyapproach
5.7Furtherreadingandnotes
6Constrainedfunctions
6.1Introduction
6.2Fittingpositivefunctions
6.2.1Apositivesmoothingspline
6.2.2Representingapositivefunctionbyadifferential equation
6.3Fittingstrictlymonotonefunctions
6.3.1Fittingthegrowthofababy'stibia
6.3.2Expressingastrictlymonotonefunctionexplicitly
6.3.3Expressingastrictlymonotonefunctionasadiffer entialequation
6.4Theperformanceofsplinesmoothingrevisited
6.5Fittingprobabilityfunctions.
6.6Estimatingprobabilitydensityfunctions
6.7Functionaldataanalysisofpointprocesses
6.8Fittingalinearmodelwithestimationofthedensityofresiduals
6.9Furthernotesandreadings
7Theregistrationanddisplayoffunctionaldata
7.1Introduction
7.2Shiftregistration
7.2.1Theleastsquarescriterionforshiftalignment
7.3Featureorlandmarkregistration
7.4Usingthewarpingfunctionhtoregisterx
7.5Amoregeneralwarpingfunctionh
7.6Acontinuousfittingcriterionforregistration
7.7Registeringtheheightaccelerationcurves
7.8Somepracticaladvice
7.9Computationaldetails
7.9.1ShiftregistrationbytheNewton-Raphsonalgorithm
7.10Furtherreadingandnotes
Principalcomponentsanalysisforfunctionaldata
8.1Introduction
8.2DefiningfunctionalPCA
8.2.1PCAformultivariatedata
8.2.2DefiningPCAforfunctionaldata
8.2.3Defininganoptimalempiricalorthonormalbasis
8.2.4PCAandeigenanalysis
8.3Visualizingtheresults
8.3.1Plottingcomponentsasperturbationsofthemean
8.3.2Plottingprincipalcomponentscores
8.3.3Rotatingprincipalcomponents
8.4ComputationalmethodsforfunctionalPCA
8.4.1Discretizingthefunctions
8.4.2Basisfunctionexpansionofthefunctions
8.4.3Moregeneralnumericalquadrature
8.5BivariateandmultivariatePCA
8.5.1DefiningmultivariatefunctionalPCA
8.5.2Visualizingtheresults
8.5.3Innerproductnotation:Concludingremarks
8.6Furtherreadingsandnotes
9Regularizedprincipalcomponentsanalysis
9.1Introduction
9.2TheresultsofsmoothingthePCA
9.3Thesmoothingapproach
9.3.1Estimatingtheleadingprincipalcomponent
9.3.2Estimatingsubsequentprincipalcomponents
9.3.3ChoosingthesmoothingparameterbyCV
9.4FindingtheregularizedPCAinpractice
9.4.1Theperiodiccase
9.4.2Thenonperiodiccase
9.5Alternativeapproaches
9.5.1SmoothingthedataratherthanthePCA
9.5.2Astepwiseroughnesspenaltyprocedure
9.5.3Afurtherapproach
10Principalcomponentsanalysisofmixeddata
10.1Introduction
10.2Generalapproachestomixeddata
10.3ThePCAofhybriddata
10.3.1Combiningfunctionandvectorspaces
10.3.2Findingtheprincipalcomponentsinpractice
10.3.3Incorporatingsmoothing..
10.3.4Balancebetweenfunctionalandvectorvariation
10.4CombiningregistrationandPCA
10.4.1Expressingtheobservationsasmixeddata.
10.4.2Balancingtemperatureandtimeshifteffects
10.5Thetemperaturedatareconsidered
10.5.1Takingaccountofeffectsbeyondphaseshift
10.5.2Separatingoutthevectorcomponent
11Canonicalcorrelationanddiscriminantanalysis
11.1Introduction
11.1.1Thebasicproblem
11.2PrinciplesofclassicalCCA
11.3Functionalcanonicalcorrelationanalysis
11.3.1Notationandassumptions
11.3.2Thenaiveapproachdoesnotgivemeaningfulresults
11.3.3Choiceofthesmoothingparameter
11.3.4Thevaluesofthecorrelations
11.4Applicationtothestudyoflupusnephritis
11.5Whyisregularizationnecessary?
11.6Algorithmicconsiderations
11.6.1Discretizationandbasisapproaches
11.6.2Theroughnessofthecanonicalvariates
11.7Penalizedoptimalscoringanddiscriminantanalysis
11.7.1Theoptimalscoringproblem
11.7.2Thediscriminantproblem
11.7.3TherelationshipwithCCA
11.7.4Applications
11.8Furtherreadingsandnotes
12Functionallinearmodels
12.1Introduction
12.2Afunctionalresponseandacategoricalindependentvari able
12.3Ascalarresponseandafunctionalindependentvariable
12.4Afunctionalresponseandafunctionalindependentvariable
12.4.1Concurrent
12.4.2Annualortotal
12.4.3Short-termfeed-forward
12.4.4Localinfluence
12.5Whataboutpredictingderivatives?
12.6Overview
13Modellingfunctionalresponseswithmultivariatecovari ates
13.1Introduction
13.2Predictingtemperaturecurvesfromclimatezones
13.2.1Fittingthemodel
13.2.2Assessingthefit
13.3Forceplatedataforwalkinghorses
13.3.1Structureofthedata
13.3.2Afunctionallinearmodelforthehorsedata
13.3.3Effectsandcontrasts
13.4Computationalissues
13.4.1Thegeneralmodel
13.4.2Pointwiseminimization
13.4.3Functionallinearmodellingwithregularizedbasis expansions
13.4.4UsingtheKroneckerproducttoexpressB
13.4.5Fittingtherawdata
13.5Confidenceintervalsforregressionfunctions
13.5.1Howtocomputeconfidenceintervals
13.5.2Confidenceintervalsforclimatezoneeffects
13.5.3Somecautionsoninterpretingconfidenceintervals
13.6Furtherreadingandnotes
14Functionalresponses,functionalcovariatesandthecon currentmodel
14.1Introduction
14.2Predictingprecipitationprofilesfromtemperaturecurves
14.2.1Themodelforthedailylogarithmofrainfall
14.2.2Preliminarysteps
14.2.3Fittingthemodelandassessingfit
14.3Long-termandseasonaltrendsinthenondurablegoods index
14.4Computationalissues
14.5Confidenceintervals
14.6Furtherreadingandnotes
15Functionallinearmodelsforscalarresponses
15.1Introduction
15.2Anaiveapproach:Discretizingthecovariatefunction
15.3Regularizationusingrestrictedbasisfunctions
15.4Regularizationwithroughnesspenalties
15.5Computationalissues
15.5.1Computingtheregularizedsolution
15.5.2Computingconfidencelimits
15.6Cross-validationandregressiondiagnostics
15.7Thedirectpenaltymethodforcomputing
15.7.1Functionalinterpolation
15.7.2Thetwo-stageminimizationprocess
15.7.3Functionalinterpolationrevisited
15.8Functionalregressionandintegralequations
15.9Furtherreadingandnotes
16Functionallinearmodelsforfunctionalresponses
16.1Introduction:Predictinglogprecipitationfromtempera ture
16.1.1Fittingthemodelwithoutregulaxization
16.2Regularizingthefitbyrestrictingthebases
16.2.1Restrictingthebasisη(s)
16.2.2Restrictingthebasisθ0(t)
16.2.3Restrictingbothbases
16.3Assessinggoodnessoffit
16.4Computationaldetails
16.4.1Fittingthemodelwithoutregularization
16.4.2Fittingthemodelwithregulaxization
16.5Thegeneralcase
16.6Furtherreadingandnotes
17Derivativesandfunctionallinearmodels
17.1Introduction
17.2Theoilrefinerydata
17.3Themelanomadata
17.4Somecomparisonsoftherefineryandmelanomaanalyses
18Differentialequationsandoperators
18.1Introduction
18.2Exploringasimplelineardifferentialequation
18.3Beyondtheconstantcoefficientfirst-orderlinearequation
18.3.1Nonconstantcoefficients
18.3.2Higherorderequations
18.3.3Systemsofequations
18.3.4Beyondlineaxity
18.4Someapplicationsoflineardifferentialequationsand operators
18.4.1Differentialoperatorstoproducenewfunctional observations
18.4.2Thegrossdomesticproductdata
18.4.3Differentialoperatorstoregularizeorsmoothmod els
18.4.4Differentialoperatorstopartitionvariation
18.4.5Operatorstodefinesolutionstoproblems.
18.5Somelineardifferentialequationfacts
18.5.1Derivativesaxerougher
18.5.2Findingalineardifferentialoperatorthatannihi latesknown functions
18.5.3FindingthefunctionsξjsatisfyingLξj=0
18.6Initialconditions,boundaryconditionsandothercon straints
18.6.1Whyadditionalconstraintsaxeneededtodefinea solution
18.6.2HowLandBpartitionfunctions
18.6.3TheinnerproductdefinedbyoperatorsLandB
18.7Furtherreadingandnotes
19Principaldifferentialanalysis
19.1Introduction
19.2Definingtheproblem
19.3Aprincipaldifferentialanalysisoflipmovement
19.3.1Thebiomechanicsoflipmovement
19.3.2VisualizingthePDAresults
19.4PDAofthepinchforcedata.
19.5Techniquesforprincipaldifferentialanalysis
19.5.1PDAbypoint-wiseminimization
19.5.2PDAusingtheconcurrentfunctionallinearmodel
19.5.3PDAbyiteratingtheconcurrentlinearmodel
19.5.4AssessingfitinPDA
19.6ComparingPDAandPCA
19.6.1PDAandPCAbothminimizesumsofsquared errors
19.6.2PDAandPCAbothinvolvefindinglinearoperators
19.6.3Differencesbetweendifferentialoperators(PDA)andprojectionoperators(PCA).
19.7Furtherreadingsandnotes..
20Green'sfunctionsandreproducingkernels
20.1Introduction
20.2TheGreen'sfunctionforsolvingalineardifferentiale quation
20.2.1ThedefinitionoftheGreen'sfunction
20.2.2AmatrixanalogueoftheGreen'sfunction
20.2.3ArecipefortheGreen'sfunction
20.3ReproducingkernelsandGreen'sfunctions
20.3.1Whatisareproducingkernel?
20.3.2ThereproducingkernelforkerB
20.3.3ThereproducingkernelforkerL
20.4Furtherreadingandnotes
21Moregeneralroughnesspenalties
21.1Introduction
21.1.1Thelipmovementdata
21.1.2Theweatherdata
21.2Theoptimalbasisforsplinesmoothing
21.3AnO(n)algorithmforL-splinesmoothing
21.3.1Theneedforagoodalgorithm
21.3.2Settingupthesmoothingprocedure
21.3.3Thesmoothingphase
21.3.4Theperformanceassessmentphase
21.3.5OtherO(n)algorithms
21.4AcompactsupportbasisforL-splines
21.5Somecasestudies
21.5.1Thegrossdomesticproductdata
21.5.2Themelanomadata
21.5.3TheGDPdatawithseasonaleffects
21.5.4Smoothingsimulatedhumangrowthdata
22SomeperspectivesonFDA
22.1Thecontextoffunctionaldataanalysis
22.1.1Replicationandregularity
22.1.2Somefunctionalaspectselsewhereinstatistics
22.1.3Functionalanalytictreatments
22.2Challengesforthefuture
22.2.1Probabilityandinference
22.2.2Asymptoticresults
22.2.3Multidimensionalarguments
22.2.4Practicalmethodologyandapplications
22.2.5Backtothedata!
Appendix:Somealgebraicandfunctionaltechniques
A.1Innerproducts(x,y)
A.1.1Somespecificexamples
A.1.2Generalproperties
A.1.3Descriptivestatisticsininnerproductnotation
A.1.4Someextendedusesofinnerproductnotation
A.2Furtheraspectsofinnerproductspaces
A.2.1Projections
A.2.2Quadraticoptimization
A.3Matrixdecompositionsandgeneralizedinverses
A.3.1Singularvaluedecompositions
A.3.2Generalizedinverses
A.3.3TheQRdecomposition
A.4Projections
A.4.1Projectionmatrices
A.4.2Findinganappropriateprojectionmatrix
A.4.3Projectionsinmoregeneralinnerproductspaces
A.5Constrainedmaximizationofaquadraticfunction
A.5.1Thefinite-dimensionalcase
A.5.2Theprobleminamoregeneralspace
A.5.3Generalizedeigenproblems
A.6KroneckerProducts
A.7Themultivariatelinearmodel
A.7.1Linearmodelsfromatransformationperspective
A.7.2TheleastsquaressolutionforB
A.8Regularizingthemultivariatelinearmodel
A.8.1Definitionofregularization
A.8.2Hard-edgedconstraints
A.8.3Soft-edgedconstraints
References
Index...
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