Nonparametric Inference on Manifolds: With Applications to Shape Spaces

252 Pages·2012·1.342 MB·English

by  Abhishek Bhattacharya, Rabi Bhattacharya

NonparametricInferenceonManifolds
Thisbookintroducesinasystematicmannerageneralnonparametrictheoryof
statisticsonmanifolds,withemphasisonmanifoldsofshapes.Thetheoryhas
importantandvariedapplicationsinmedicaldiagnostics,imageanalysis,andmachine
vision.Anearlychapterofexamplesestablishestheeffectivenessofthenewmethods
anddemonstrateshowtheyoutperformtheirparametriccounterparts.
InferenceisdevelopedforbothintrinsicandextrinsicFre´chetmeansofprobability
distributionsonmanifolds,andthenappliedtospacesofshapesdefinedasorbitsof
landmarksunderaLiegroupoftransformations–inparticular,similarity,reflection
similarity,affine,andprojectivetransformations.Inaddition,nonparametricBayesian
theoryisadaptedandextendedtomanifoldsforthepurposesofdensityestimation,
regression,andclassification.Idealforstatisticianswhoanalyzemanifolddataand
wishtodeveloptheirownmethodology,thisbookisalsoofinteresttoprobabilists,
mathematicians,computerscientists,andmorphometricianswithmathematical
training.
abhishek bhattacharya
is Assistant Professor in the Theoretical Statistics
andMathematicsUnit(SMU)attheIndianStatisticalInstitute,Kolkata.
rabi bhattacharya
is Professor in the Department of Mathematics at The
UniversityofArizona,Tucson.
INSTITUTE OF MATHEMATICAL STATISTICS
MONOGRAPHS
EditorialBoard
D.R.Cox(UniversityofOxford)
B.Hambly(UniversityofOxford)
S.Holmes(StanfordUniversity)
X.-L.Meng(HarvardUniversity)
IMSMonographsareconciseresearchmonographsofhighqualityonanybranchof
statistics or probability of sufficient interest to warrant publication as books. Some
concern relatively traditional topics in need of up-to-date assessment. Others are on
emergingthemes.Inallcasestheobjectiveistoprovideabalancedviewofthefield.
Nonparametric Inference
on Manifolds
With Applications to Shape Spaces
ABHISHEK BHATTACHARYA
IndianStatisticalInstitute,Kolkata
RABI BHATTACHARYA
UniversityofArizona
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107019584
(cid:2)c A.BhattacharyaandR.Bhattacharya2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
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ISBN978-1-107-01958-4Hardback
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accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
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Contents
Commonlyusednotation pageix
Preface xi
1 Introduction 1
2 Examples 8
2.1 DataexampleonS1:windandozone 8
2.2 DataexamplesonS2:paleomagnetism 8
2.3 DataexampleonΣk:shapesofgorillaskulls 12
2
2.4 DataexampleonΣk:brainscanshapesofschizophrenicand
2
normalpatients 15
2.5 Data example on affine shape space AΣk: application to
2
handwrittendigitrecognition 17
2.6 Data example on reflection similarity shape space RΣk:
3
glaucomadetection 18
2.7 References 20
3 Locationandspreadonmetricspaces 21
3.1 Introduction 21
3.2 Locationonmetricspaces 22
3.3 Variationonmetricspaces 27
3.4 Asymptoticdistributionofthesamplemean 28
3.5 Asymptoticdistributionofthesamplevariation 30
3.6 Anexample:theunitcircle 31
3.7 DataexampleonS1 34
3.8 References 35
4 Extrinsicanalysisonmanifolds 36
4.1 Extrinsicmeanandvariation 36
4.2 Asymptoticdistributionofthesampleextrinsicmean 37
4.3 Asymptoticdistributionofthesampleextrinsicvariation 39
4.4 Asymptotic joint distribution of the sample extrinsic
meanandvariation 41
v
vi Contents
4.5 Two-sampleextrinsictests 42
4.6 Hypothesistestingusingextrinsicmeanandvariation 46
4.7 Equivariantembedding 48
4.8 ExtrinsicanalysisontheunitsphereSd 49
4.9 Applicationsonthesphere 51
4.10 References 55
5 Intrinsicanalysisonmanifolds 57
5.1 Intrinsicmeanandvariation 57
5.2 Asymptoticdistributionofthesampleintrinsicmean 59
5.3 IntrinsicanalysisonSd 64
5.4 Two-sampleintrinsictests 65
5.5 DataexampleonS2 69
5.6 Someremarks 71
5.7 References 75
6 Landmark-basedshapespaces 77
6.1 Introduction 77
6.2 Geometryofshapemanifolds 78
6.3 References 80
7 Kendall’ssimilarityshapespacesΣk 82
m
7.1 Introduction 82
7.2 Geometryofsimilarityshapespaces 83
7.3 References 86
8 TheplanarshapespaceΣk 87
2
8.1 Introduction 87
8.2 Geometryoftheplanarshapespace 88
8.3 Examples 89
8.4 Intrinsicanalysisontheplanarshapespace 90
8.5 OtherFre´chetfunctions 96
8.6 Extrinsicanalysisontheplanarshapespace 97
8.7 Extrinsicmeanandvariation 98
8.8 Asymptoticdistributionofthesampleextrinsicmean 99
8.9 Two-sampleextrinsictestsontheplanarshapespace 101
8.10 Planarsize-and-shapemanifold 103
8.11 Applications 105
8.12 References 109
9 ReflectionsimilarityshapespacesRΣk 110
m
9.1 Introduction 110
9.2 Extrinsicanalysisonthereflectionshapespace 111
9.3 Asymptoticdistributionofthesampleextrinsicmean 117
Contents vii
9.4 Two-sampletestsonthereflectionshapespaces 122
9.5 Otherdistancesonthereflectionshapespaces 123
9.6 Application:glaucomadetection 125
9.7 References 128
10 StiefelmanifoldsVk,m 130
10.1 Introduction 130
10.2 ExtrinsicanalysisonVk,m 130
10.3 References 134
11 AffineshapespacesAΣk 135
m
11.1 Introduction 135
11.2 Geometryofaffineshapespaces 137
11.3 Extrinsicanalysisonaffineshapespaces 139
11.4 Asymptoticdistributionofthesampleextrinsicmean 141
11.5 Applicationtohandwrittendigitrecognition 144
11.6 References 146
12 Realprojectivespacesandprojectiveshapespaces 147
12.1 Introduction 147
12.2 GeometryoftherealprojectivespaceRPm 148
12.3 GeometryoftheprojectiveshapespaceP Σk 149
0 m
12.4 IntrinsicanalysisonRPm 150
12.5 ExtrinsicanalysisonRPm 151
12.6 Asymptoticdistributionofthesampleextrinsicmean 153
12.7 References 155
13 NonparametricBayesinferenceonmanifolds 156
13.1 Introduction 156
13.2 Densityestimationonmetricspaces 157
13.3 Fullsupportandposteriorconsistency 158
13.4 Posteriorcomputations 163
13.5 ApplicationtounitsphereSd 165
13.6 ApplicationtotheplanarshapespaceΣk 166
2
13.7 Applicationtomorphometrics:classification
ofgorillaskulls 168
13.8 Proofsoftheorems 170
13.9 References 181
14 NonparametricBayesregression,classificationand
hypothesistestingonmanifolds 182
14.1 Introduction 182
14.2 Regressionusingmixturesofproductkernels 183
14.3 Classification 185
viii Contents
14.4 NonparametricBayestesting 192
14.5 Examples 196
14.6 Proofs 202
AppendixA Differentiablemanifolds 209
AppendixB Riemannianmanifolds 214
AppendixC Dirichletprocesses 218
AppendixD ParametricmodelsonSd andΣk 225
2
References 229
Index 235
Commonly used notation
Manifolds
Sd: d-dimensionalsphere
RPk,CPk: k-dimensionalreal,resp.complex,projectivespace;alllines
inRk+1,resp.Ck+1,passingthroughtheorigin
Vk,m: StiefelmanifoldStk(m);allk-frames(kmutuallyorthogonal
directions)inRm;see§10.1
d ,d : extrinsicdistance,geodesicdistance;see§4.1,§5.1
E g
d ,d : Procrustesdistance:full,partial;see§9.5
F P
j: M → E: embeddingofthemanifold M intotheEuclideanspaceE;
M˜ = j(M);see§4.1
T (M)orT M: tangentspaceofthemanifold M atu∈ M;see§4.2
u u
d P: T E →T M˜: differentialoftheprojectionmapP: U → M˜,for
x x P(x)
U aneighborhoodofnonfocalpointμ;see§4.2
D ,D: partialderivativewithrespecttorthcoordinate,resp.vectorof
r
partialderivatives;see§3.4
L= Lμ : orthogonallinearprojectionofvectorsinTμ E ≡ E onto
E E
Tμ M˜;see§4.5.1
E
r∗: exponentialmapat pisinjectiveon{v∈Tp(M):|v| <r∗};see§5.1
Shapespaces
Σk: similarityshapespaceofk-adsinmdimensions;see§7.1
m
Σk : similarityshapespaceof“nonsingular”k-adsinm>2
0m
dimensions;see§7.2
Sk: preshapesphereofk-adsinmdimensions;see§7.1
m
Sk/G: preshapesphereofk-adsinmdimensionsmodulotransformation
m
groupG;see§7.1
AΣk: affineshapespaceofk-adsinmdimensions;see§11.1
m
PΣk: projectiveshapespaceofk-adsinmdimensions;see§12.3
m
RΣk: reflectionshapespaceofk-adsinmdimensions;see§9.1
m
SΣk: planarsize-and-shapespaceofk-ads;see§8.10
2
SAΣk: specialaffineshapespaceofk-adsinmdimensions;see§11.2
2
ix