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Even If Google Sent You Here, What Follows Is NOT the Book RANDOM i Note: Even if Google sent you here, what follows is NOT the book RANDOM FIELDS AND GEOMETRY published with Springer in 2007, but rather a companion volume, still under production, that gives a simpler version of the theory of the first book as well as lots of applications. You can find the original Random Fields and Geometry on the Springer site. Meanwhile, enjoy what is available of the second volume, and keep in mind that we are happy to receive all comments, suggestions and, most of all, cor- rections. ii Applications of RANDOM FIELDS AND GEOMETRY Foundations and Case Studies Robert J. Adler Faculty of Industrial Engineering and Management and Faculty of Electrical Engineering Technion { Israel Institute of Technology Haifa, Israel e-mail: [email protected] Jonathan E. Taylor Department of Statistics Stanford University Stanford, California. e-mail: [email protected] Keith J. Worsley Department of Mathematics and Statistics McGill University Montr´eal,Canada and Department of Statistics University of Chicago Chicago, Illinois. e-mail: [email protected] May 20, 2015 0 Preface Robert J. Adler Jonathan E. Taylor1 Keith Worsley Haifa, Israel Stanford, California Montreal, Canada ie.technion.ac.il/Adler.phtml www-stat.stanford.edu/∼jtaylo www.math.mcgill.ca/keith 1Research supported in part by NSF grant DMS-0405970 Contents 0 Preface .................................................... iii 1 Introduction ............................................... 1 1.1 An Initial Example......................................1 1.2 Gaussian and Related Random Fields......................3 1.3 Shape and the Euler Characteristic........................4 1.4 The Large-Scale Structure of the Universe..................7 1.4.1 Survey Data......................................7 1.4.2 Designing and Testing a Model......................8 1.5 Brain Imaging.......................................... 12 1.5.1 The Data........................................ 12 1.5.2 Statistical Testing................................. 13 1.6 Beyond the Euler Characteristic........................... 17 Part I The Underlying Theory 2 Random Fields ............................................ 23 2.1 Stochastic Processes and Random Fields................... 23 2.2 Gaussian and Gaussian Related Random Fields............. 25 2.2.1 Gaussian Variables................................ 25 2.2.2 Gaussian Fields................................... 27 2.2.3 Gaussian Related Fields............................ 28 2.3 Examples of Random Fields.............................. 30 2.3.1 Stationary and Isotropic Fields...................... 30 2.3.2 Random Polynomials.............................. 31 2.3.3 The Cosine Process................................ 32 2.3.4 The Cosine Field.................................. 34 2.3.5 Orthogonal Expansions............................ 36 2.4 Stationarity and Isotropy................................. 39 2.4.1 Basic Definitions.................................. 39 vi Contents 2.4.2 Spectral Distribution Theorem...................... 41 2.4.3 Spectral Moments and Derivatives of Random Fields... 42 2.4.4 Constant Variance................................. 44 2.4.5 White Noise and Integration........................ 45 2.4.6 Spectral Representation Theorem.................... 47 2.4.7 Isotropy.......................................... 52 2.4.8 Isotropic Fields on the Sphere and Other Homogenous Spaces........................................... 57 2.4.9 Transforming to Stationarity and Isotropy............ 58 2.5 Smoothness of Random Fields............................. 60 2.5.1 The General Gaussian Theory...................... 62 2.5.2 Gaussian Fields on RN ............................. 65 2.5.3 Non-Gaussian Processes............................ 69 2.6 Gaussian Exceedence Probabilities......................... 70 2.6.1 Borell-TIS Inequality.............................. 71 2.6.2 Comparison Inequalities............................ 74 2.6.3 Exceedence Probabilities for Smooth Processes........ 77 2.7 An Expectation Meta-Theorem: The Rice-Kac Formula...... 78 2.8 Exercises............................................... 82 3 Geometry ................................................. 89 3.1 Basic Complexes and the Euler Characteristic............... 89 3.2 Excursion Sets and Morse Theory......................... 93 3.3 Volume of Tubes and Intrinsic Volumes.................... 99 3.3.1 Euclidean Sets.................................... 99 3.3.2 Intrinsic Volumes for Simplicial Complexes........... 104 3.3.3 Volumes of Tubes on Spheres....................... 105 3.3.4 Probabilities of Tubes.............................. 107 3.4 Some Integral Geometric Formulae........................ 109 3.4.1 The Euclidean KFF............................... 109 3.4.2 The KFF on Spheres.............................. 110 3.4.3 Crofton's Formula................................. 111 3.5 Stratified Riemannian Manifolds........................... 112 3.6 Exercises............................................... 115 Part II Quantifiable Properties 4 The Expected Euler Characteristic . 121 4.1 Regularity Conditions.................................... 122 4.2 Stationary Gaussian Fields............................... 122 4.3 Non-Stationary Gaussian Fields........................... 127 4.4 The Gaussian Kinematic Formula, I....................... 131 4.5 Euler Characteristic Densities............................. 134 4.5.1 Gaussian Fields................................... 135 Contents vii 2 4.5.2 χk Fields......................................... 135 2 4.5.3χ ¯K Fields........................................ 138 4.5.4 F Fields......................................... 141 4.5.5 Student T Fields.................................. 146 4.5.6 Hotelling T 2 Fields................................ 147 4.5.7 Roy's Maximum Root Field......................... 149 4.5.8 Conjunctions/Correlated........................... 149 4.5.9 Correlation Fields................................. 149 4.5.10 Another Look at the F Field....................... 149 4.6 Some Lipschitz-Killing Curvatures......................... 149 4.6.1 Time-Space Fields................................. 150 4.6.2 Scale Space Fields................................. 150 4.6.3 Rotation Space Fields.............................. 163 4.7 Mean Lipschitz-Killing Curvatures under Isotropy........... 163 4.8 The Gaussian Kinematic Formula, II....................... 165 4.9 Exercises............................................... 167 5 Exceedence Probabilities ..................................169 5.1 The Mean Number of Maxima............................ 171 5.2 The Mean Number of Maxima and the Mean Euler Characteristic........................................... 174 5.3 Expected Euler Characteristic Heuristic.................... 178 5.3.1 About the proof................................... 180 2 2 5.3.2 χ andχ ¯K Fields.................................. 181 5.3.3 Numerical Examples............................... 182 5.4 Alternative Approaches to Exceedence Probabilities.......... 182 5.4.1 Volume of Tubes and Gaussian Exceedence Probabilities182 5.4.2 The Double Sum Method........................... 185 5.4.3 Double sums of moments........................... 187 5.5 Aproximately Gaussian Random Fields..................... 189 5.6 Exercises............................................... 189 6 The Structure of Excursion Sets . 193 6.1 Palm Distributions and Conditioning...................... 194 6.2 The Local Structure of Gaussian Maxima................... 196 6.2.1 The Exact Slepian Model........................... 197 6.2.2 Excursions Above High Levels...................... 202 6.3 More on the Local Structure of Gaussian Fields............. 203 6.3.1 Low Maxima and High Minima..................... 204 6.3.2 High Excursions................................... 205 6.3.3 Overshoot Distributions............................ 206 6.3.4 Numerical Approaches............................. 208 6.4 The Local Structure of Some Non-Gaussian Fields........... 209 2 6.4.1 χk fields.......................................... 209 6.4.2 Student T fields................................... 214 viii Contents 6.4.3 F fields.......................................... 215 6.4.4 On Computing the Mean Euler Characteristic......... 216 6.5 The Size of an Excursion Set Component................... 217 6.5.1 The Gaussian Case................................ 217 6.5.2 The Non-Gaussian Case............................ 220 6.6 The Poisson Clumping Heuristic and Excursions............. 221 6.6.1 Exceedence Probabilities, Again..................... 222 6.6.2 The Largest Component of an Excursion Set.......... 223 6.6.3 Numerical Experiments............................ 224 6.7 Exercises............................................... 227 Part III Numerics 7 Simulating Random Fields .................................231 7.1 Fields on Euclidean Spaces............................... 231 7.2 Fields on Spheres........................................ 231 7.3 Fields on Manifolds...................................... 231 8 Discrete approximation ....................................233 8.1 Estimating Intrinsic Volumes............................. 233 8.2 Data on a Lattice/Triangulated Surface.................... 233 8.3 Computing the Euler Characteristic....................... 233 8.4 Continuity Correction for Supremum Distribution........... 233 8.5 Improved Bonferroni Inequality........................... 233 Part IV Applications 9 Neuroimaging .............................................237
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