Statistical analysis of discontinuous phenomena with potts functionals [Elektronische Ressource] / vorgelegt von Angela Kempe

Kempe, Angela

Sujets

• Mathematics

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2004
• Nombre de lectures: 14
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Statistical Analysis of Discontinuous
Phenomena with Potts Functionals
DISSERTATION
zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften
des Fachbereichs Mathematik
der Ludwig-Maximilians-Universit˜at Munc˜ hen
vorgelegt von
Angela Kempe
30. Januar 20041. Berichtertstatter: Prof. Dr. G. Winkler
2. Berichterstatter: Prof. Dr. P. L. Davies
Tag der mundlic˜ hen Prufung:˜ 25. Juni 2004Fur˜ meine OmaPreface
This thesis was carried out under the supervision of Prof. Dr. Gerhard Win-
kler. From 01.01.2000 to 31.12.2002 it was funded by the GSF-National Re-
search Center for Environment and Health, Neuherberg, and from 01.01.03
to 31.01.2004 by the Graduate Program Applied Algorithmic Mathematics
of the Deutsche Forschungsgemeinschaft at the Munich University of Tech-
nology.
ThesimulationswererealizedonthesoftwarepackageAntsInFieldsdevel-
oped byF. Friedrich (2003). It was also used for most of the illustrations.
I thank the members of the GSF-Institute of Biomathematics and Biome-
try for their support. I am particularly indebted to Gerhard Winkler, Olaf
Wittich, Volkmar Liebscher, and Felix Friedrich for their sedulous help and
encouragement. Many thanks to my friends and my mother.
Munich, January 2004 Angela Kempe
iContents
Preface i
Zusammenfassung vii
Introduction 1
I Analysis of Potts Functionals
and their Minimizers 5
1 Segmentations and Potts Functionals 9
1.1 Potts Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Segmentations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Relation to Bayesian Approaches . . . . . . . . . . . . . . . . 15
2 MAP Estimators 19
2.1 Splitting the Minimization . . . . . . . . . . . . . . . . . . . . 19
2.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Dependence on Hyperparameters . . . . . . . . . . . . . . . . 28
2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Measurable Section . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Exact Optimization 41
3.1 Minimization for Fixed Hyperparameter . . . . . . . . . . . . 42
3.2 Simultaneous Minimization in ? . . . . . . . . . . . . . . . . . 45
II Choice of Hyperparameters 49
4 Equivariance and Hyp 53
4.1 Equivariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
iiiiv CONTENTS
4.2 Normalization of Data . . . . . . . . . . . . . . . . . . . . . . 58
5 Interval Criteria 63
5.1 Invariant Attributes . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 F-Longest Interval Criteria . . . . . . . . . . . . . . . . . . . . 64
5.3 Longest Interval Criterion . . . . . . . . . . . . . . . . . . . . 72
6 Stopping Criteria 81
6.1 Longest Run Criterion . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Multiresolution . . . . . . . . . . . . . . . . . . . . . 83
7 Model Selection Criteria 85
7.1 The Akaike Information Criterion . . . . . . . . . . . . . . . . 86
7.2 The Schwarz . . . . . . . . . . . . . . . 87
7.3 Equivariant Versions . . . . . . . . . . . . . . . . . . . . . . . 87
8 Further Ideas 89
8.1 Iterative Procedures . . . . . . . . . . . . . . . . . . . . . . . 89
8.2 Constant Estimates . . . . . . . . . . . . . . . . . . . . . . . . 91
8.3 Morphological Criteria . . . . . . . . . . . . . . . . . . . . . . 91
III Application to Data 93
9 Data Sets from Life Sciences 97
9.1 Functional Magnetic Resonance Imaging . . . . . . . . . . . . 97
9.2 Fractionation Experiments . . . . . . . . . . . . . . . . . . . . 103
10 Simulations 111
10.1 Constant Signal . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.2 One Jump Signal . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.3 Boxcar Shaped Signal . . . . . . . . . . . . . . . . . . . . . . . 118
IV Consistency 127
11 Fixed Data Length 131
12 Increasing Data Length 133
212.1 Potts Functionals on L ([0;1)) . . . . . . . . . . . . . . . . . . 135
12.2 Epi-Convergence and Relative Compactness . . . . . . . . . . 138
12.3 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 150CONTENTS v
Discussion and Outlook 153
V Appendix 155
A ?-Scanning 159
B Model Selection Criteria 165
B.1 A Family of Regression Models . . . . . . . . . . . . . . . . . 165
B.2 The Akaike Information Criterion . . . . . . . . . . . . . . . . 168
B.3 The Schwarz . . . . . . . . . . . . . . . 171
C Calculations Model Selection Criteria 177
C.1 Proofs for AIC . . . . . . . . . . . . . . . . . . . . . . . . . . 177
C.2 Regularity Conditions Likelihood Function . . . . . . . . . . . 179
C.3 The Hessian Matrix of the Log Likelihood Function . . . . . . 182
Symbols 185