Topological complexity of smooth random functions : École d'Été de Probabilités de Saint-Flour XXXIX - 2009 / Robert J. Adler, Jonathan E. Taylor.

Format:

Book

Conference/Event

Author/Creator:

Adler, Robert J.

Contributor:

Taylor, Jonathan E.

Conference Name:

Ecole d'été de probabilités de Saint-Flour.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 2019.

Lecture notes in mathematics, 0075-8434 ; 2019

Language:

English

Subjects (All):

Gaussian processes.

Random fields.

Genre:

Conference papers and proceedings.

Physical Description:

viii, 122 pages : illustrations (some color) ; 24 cm.

Place of Publication:

Heidelberg ; New York : Springer, [2011]

Summary:

These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors' 2007 Springer monograph "Random Fields and Geometry." While not as exhaustive as the full monograph, they are also less exhausting, while still covering the basic material, typically at a more intuitive and less technical level. They also cover some more recent material relating to random algebraic topology and statistical applications. The notes include an introduction to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness. This is followed by a quick review of geometry, both integral and Riemannian, with an emphasis on tube formulae, to provide the reader with the material needed to understand and use the Gaussian kinematic formula, the main result of the notes. This is followed by chapters on topological inference and random algebraic topology, both of which provide applications of the main results. Book jacket.

Contents:

1 Introduction 1

1.1 Random Fields on Stratified Manifolds 1

1.2 opological Complexity 3

1.3 Random Fields and Complexity 7

1.3.1 Statistical Implications 10

1.3.2 Connections with Sample Path Behaviour 10

1.3.3 Geometry 11

2 Gaussian Processes 13

2.1 The Cosine Process 14

2.2 The Cosine Field 16

2.3 Constructing Gaussian Processes 18

2.4 The Canonical Process on S(R¹) 20

2.4.1 The Canonical Processes and Exceedence Probabilities 22

2.4.2 The Canonical Process and Geometry 24

2.5 The Basic Theory of Gaussian Fields 25

2.5.1 Regularity for Gaussian Process 26

2.5.2 Gaussian Fields on R^N 28

2.5.3 Differentiability 29

2.6 Stationarity, Isotropy, and Constant Variance 31

2.6.1 Spectral Moments and Derivatives of Random Fields 31

2.6.2 Local Isotropy and the Induced Metric 33

2.7 Three Gaussian Facts 34

3 Some Geometry and Some Topology 37

3.1 Some Notation for Riemannian Manifolds 37

3.2 Coarea Formula 39

3.3 Stratified Manifolds 40

3.4 Tube Formulae and Lipschitz-Killing Curvatures 42

3.4.1 Describing Tubes 43

3.4.2 Computing Volumes 46

3.4.3 Intrinsic Volumes 50

3.5 Probabilities of Tubes: Gaussian Minkowski Functionals 51

3.6 Kinematic Formulae 54

3.7 Crofton's Formula 55

3.8 Morse's Theorem 56

4 The Gaussian Kinematic Formula 59

4.1 The Kac-Rice Metatheorem 60

4.2 Real Isotropic Fields on Rectangles: Euler Characteristic 65

4.3 Real Isotropic Fields: Lipschitz-Killing Curvatures 70

4.4 Real Stationary Fields on Rectangles: Euler Characteristic 71

4.5 The Induced Metric and the Need for Riemannian Geometry 73

4.6 The Canonical Isotropic Process on the Sphere 77

4.6.1 A Model Process 77

4.6.2 The GKF for the Model Process 79

4.6.3 Back to the Canonical Process 81

4.7 Fields with Finite Expansions 82

4.8 The GKF in the General Case 83

4.9 Not Just Excursion Sets 84

4.10 Infinite Dimensions 84

5 On Applications: Topological Inference 87

5.1 Local Structure of Extrema and the Euler Characteristic Heuristic 88

5.2 Gaussian Related Random Fields 91

5.2.1 χ² Fields 93

5.3 Brain Imaging 94

5.4 Estimating Lipschitz-Killing Curvatures 98

5.5 Cosmology 102

6 Algebraic Topology of Excursion Sets: A New Challenge 107

6.1 Persistent Homology and Barcodes 108

6.1.1 Barcodes of Excursion Sets 108

6.2 Barcode Distributions 110

6.3 The Mean Euler Characteristic of the Barcodes of Gaussian Excursion Sets 112.

Notes:

Based on lectures delivered in Saint Flour.

Includes bibliographical references and index.

ISBN:

9783642195792

3642195792

OCLC:

733988174