Practical Smoothing: The Joys of P-splines

213 Pages·2021·9.241 MB·English

by  Paul H.C. Eilers, Brian D. Marx

PracticalSmoothing
ThisisapracticalguidetoP-splines,asimple,flexible,andpowerfultoolfor
smoothing.P-splinescombineregressiononB-splineswithsimple,discrete,roughness
penalties.Theywereintroducedbytheauthorsin1996andhavebeenusedinmany
diverseapplications.Theregressionbasismakesitstraightforwardtohandle
non-normaldata,likeingeneralizedlinearmodels.Theauthorsdemonstrateoptimal
smoothing,usingmixedmodeltechnologyandBayesianestimation,inadditionto
classicaltoolslikecross-validationandAIC,coveringtheoryandapplicationswith
codeinR.Goingfarbeyondsimplesmoothing,theyalsoshowhowtouseP-splines
forregressiononsignals,varying-coefficientmodels,quantileandexpectile
smoothing,andcompositelinksforgroupeddata.Penaltiesarethecrucialelementsof
P-splines;withpropermodificationstheycanhandleperiodicandcirculardataaswell
asshapeconstraints.CombiningpenaltieswithtensorproductsofB-splinesextends
theseattractivepropertiestomultipledimensions.Theappendicesofferasystematic
comparisontoothersmoothers.
paul h. c. eilers isProfessorEmeritusofGeneticalStatisticsattheErasmus
UniversityMedicalCenter,Rotterdam,TheNetherlands.HereceivedhisPhDin
biostatistics.Hisresearchinterestsincludehigh-throughputgenomicdataanalysis,
chemometrics,smoothing,longitudinaldataanalysis,survivalanalysis,andstatistical
computing.Hehaspublishedextensivelyonthesesubjects.
brian d. marx isProfessorintheDepartmentofExperimentalStatisticsat
LouisianaStateUniversity.HereceivedhisPhDinstatistics.Hismainresearch
interestsincludesmoothing,ill-conditionedregressionproblems,andhigh-dimensional
chemometricapplications,andhehasnumerouspublicationsonthesetopics.Heis
currentlyservingascoordinatingeditorforthejournalStatisticalModelling.Heis
coauthoroftwobooksandisaFellowoftheAmericanStatisticalAssociation.

Practical Smoothing
The Joys of P-splines
Paul H. C. Eilers
ErasmusUniversityMedicalCenter
Brian D. Marx
LouisianaStateUniversity
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DOI:10.1017/9781108610247
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Names:Eilers,PaulH.C.,1948–author.|Marx,BrianD.,1960–author.
Title:Practicalsmoothing:thejoysofP-splines/PaulH.C.Eilers,
ErasmusUniversityMedicalCenter,BrianD.Marx,LouisianaStateUniversity.
Description:Cambridge,UK;NewYork,NY:CambridgeUniversityPress,
2021.|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2020016638(print)|LCCN2020016639(ebook)|
ISBN9781108482950(hardback)|ISBN9781108610247(epub)
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Contents
Preface pagexi
1 Introduction 1
2 Bases,Penalties,andLikelihoods 6
2.1 LinearandPolynomialRegression 6
2.2 B-splines 9
2.3 PenalizedLeastSquares 14
2.4 InterpolationandExtrapolation 18
2.5 Derivatives 20
2.6 TheEffectiveDimension 21
2.7 StandardErrors 23
2.8 HeavySmoothingandPolynomialLimits 24
2.9 P-splinesasaParametricModel 24
2.10 Whittaker:P-splineswithoutB-splines 26
2.11 EquivalentKernels 26
2.12 SmoothingofaNon-normalResponse 28
2.12.1 PoissonSmoothing 28
2.12.2 BinomialSmoothing 31
2.12.3 GLMEffectiveDimensionandStandardErrors 32
2.13 NotesandDetails 34
3 OptimalSmoothinginAction 36
3.1 Cross-Validation 37
3.2 Akaike’sInformationCriterion 38
3.3 DensityEstimation 40
3.4 MixedModels 41
3.5 BayesianP-splines 45
3.6 DangersofAutomaticSmoothing 51
vii
viii Contents
3.7 L-andV-curves 54
3.8 TransformationoftheIndependentVariable 56
3.9 NotesandDetails 58
4 MultidimensionalSmoothing 59
4.1 GeneralizedAdditiveModels 60
4.2 VaryingCoefficientModels 63
4.3 TensorProductModels 67
4.4 TensorProductBases 69
4.5 Two-DimensionalPenalties 71
4.6 InterpolationandExtrapolation 73
4.7 SmoothingonLargeGrids 74
4.8 GeneralizedTwo-DimensionalSmoothing 75
4.9 OptimalTwo-DimensionalSmoothing 77
4.10 IssueswithIsotropicSmoothing 79
4.11 HigherDimensions 79
4.12 NestedBasesandPS-ANOVA 79
4.13 NotesandDetails 83
5 SmoothingofScaleandShape 84
5.1 QuantileSmoothing 85
5.2 ExpectileSmoothing 91
5.3 ModelsforShapeandScaleParameters 97
5.4 BaselineEstimation 101
5.5 NotesandDetails 102
6 ComplexCountsandCompositeLinks 103
6.1 HistogramswithWideBins 104
6.2 HistogramsandScaleTransformation 107
6.3 IndividualCensoring 109
6.4 LatentMixtures 110
6.5 NotesandDetails 112
7 SignalRegression 114
7.1 AChemicalCalibrationProblem 115
7.2 ExtensionstotheGeneralizedLinearModel 120
7.3 MultidimensionalSignalRegression 122
7.4 FurtherExtensions 126
7.5 NotesandDetails 129
8 SpecialSubjects 131
8.1 TheProperB-splineBasis 132
8.2 HarmonicSmoothing 132
Contents ix
8.3 CircularSmoothing 135
8.4 SignalSeparationwithPenalties 138
8.5 DoublePenalties 141
8.6 PiecewiseConstantSmoothing 143
8.7 ShapeConstraints 146
8.8 VariableandAdaptivePenalties 152
8.9 SurvivalAnalysisandMortalityModeling 155
8.10 NotesandDetails 158
AppendixA P-splinesfortheImpatient 159
AppendixB P-splinesandCompetitors 161
AppendixC ComputationalDetails 168
AppendixD ArrayAlgorithms 174
AppendixE MixedModelEquations 176
AppendixF StandardErrorsinDetail 182
AppendixG TheWebsite 184
References 188
Index 196