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|a 9783030384388
|q (electronic bk.)
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|z 9783030384371
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|a (MiAaPQ)EBC6135409
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|a (Au-PeEL)EBL6135409
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|a (OCoLC)1148226628
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|a QA273.A1-274.9
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|a Panaretos, Victor M.
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|a An Invitation to Statistics in Wasserstein Space.
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|a 1st ed.
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|a Cham :
|b Springer International Publishing AG,
|c 2020.
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|c ©2020.
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| 300 |
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|a 1 online resource (157 pages)
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| 336 |
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|a text
|b txt
|2 rdacontent
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| 337 |
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|a computer
|b c
|2 rdamedia
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| 338 |
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|a online resource
|b cr
|2 rdacarrier
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| 490 |
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|a SpringerBriefs in Probability and Mathematical Statistics Series
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|a Intro -- Preface -- Contents -- 1 Optimal Transport -- 1.1 The Monge and the Kantorovich Problems -- 1.2 Probabilistic Interpretation -- 1.3 The Discrete Uniform Case -- 1.4 Kantorovich Duality -- 1.4.1 Duality in the Discrete Uniform Case -- 1.4.2 Duality in the General Case -- 1.5 The One-Dimensional Case -- 1.6 Quadratic Cost -- 1.6.1 The Absolutely Continuous Case -- 1.6.2 Separable Hilbert Spaces -- 1.6.3 The Gaussian Case -- 1.6.4 Regularity of the Transport Maps -- 1.7 Stability of Solutions Under Weak Convergence -- 1.7.1 Stability of Transference Plans and CyclicalMonotonicity -- 1.7.2 Stability of Transport Maps -- 1.8 Complementary Slackness and More General Cost Functions -- 1.8.1 Unconstrained Dual Kantorovich Problem -- 1.8.2 The Kantorovich-Rubinstein Theorem -- 1.8.3 Strictly Convex Cost Functions on Euclidean Spaces -- 1.9 Bibliographical Notes -- 2 The Wasserstein Space -- 2.1 Definition, Notation, and Basic Properties -- 2.2 Topological Properties -- 2.2.1 Convergence, Compact Subsets -- 2.2.2 Dense Subsets and Completeness -- 2.2.3 Negative Topological Properties -- 2.2.4 Covering Numbers -- 2.3 The Tangent Bundle -- 2.3.1 Geodesics, the Log Map and the Exponential Mapin W2(X) -- 2.3.2 Curvature and Compatibility of Measures -- 2.4 Random Measures in Wasserstein Space -- 2.4.1 Measurability of Measures and of Optimal Maps -- 2.4.2 Random Optimal Maps and Fubini's Theorem -- 2.5 Bibliographical Notes -- 3 Fréchet Means in the Wasserstein Space W2 -- 3.1 Empirical Fréchet Means in W2 -- 3.1.1 The Fréchet Functional -- 3.1.2 Multimarginal Formulation, Existence, and Continuity -- 3.1.3 Uniqueness and Regularity -- 3.1.4 The One-Dimensional and the Compatible Case -- 3.1.5 The Agueh-Carlier Characterisation -- 3.1.6 Differentiability of the Fréchet Functional and Karcher Means -- 3.2 Population Fréchet Means.
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|a 3.2.1 Existence, Uniqueness, and Continuity -- 3.2.2 The One-Dimensional Case -- 3.2.3 Differentiability of the Population Fréchet Functional -- 3.3 Bibliographical Notes -- 4 Phase Variation and Fréchet Means -- 4.1 Amplitude and Phase Variation -- 4.1.1 The Functional Case -- 4.1.2 The Point Process Case -- 4.2 Wasserstein Geometry and Phase Variation -- 4.2.1 Equivariance Properties of the Wasserstein Distance -- 4.2.2 Canonicity of Wasserstein Distance in Measuring Phase Variation -- 4.3 Estimation of Fréchet Means -- 4.3.1 Oracle Case -- 4.3.2 Discretely Observed Measures -- 4.3.3 Smoothing -- 4.3.4 Estimation of Warpings and Registration Maps -- 4.3.5 Unbiased Estimation When X=R -- 4.4 Consistency -- 4.4.1 Consistent Estimation of Fréchet Means -- 4.4.2 Consistency of Warp Functions and Inverses -- 4.5 Illustrative Examples -- 4.5.1 Explicit Classes of Warp Maps -- 4.5.2 Bimodal Cox Processes -- 4.5.3 Effect of the Smoothing Parameter -- 4.6 Convergence Rates and a Central Limit Theoremon the Real Line -- 4.7 Convergence of the Empirical Measure and Optimality -- 4.8 Bibliographical Notes -- 5 Construction of Fréchet Means and Multicouplings -- 5.1 A Steepest Descent Algorithm for the Computation of FréchetMeans -- 5.2 Analogy with Procrustes Analysis -- 5.3 Convergence of Algorithm 1 -- 5.4 Illustrative Examples -- 5.4.1 Gaussian Measures -- 5.4.2 Compatible Measures -- 5.4.2.1 The One-Dimensional Case -- 5.4.2.2 Independence -- 5.4.2.3 Common Copula -- 5.4.3 Partially Gaussian Trivariate Measures -- 5.5 Population Version of Algorithm 1 -- 5.6 Bibliographical Notes -- References.
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|a Description based on publisher supplied metadata and other sources.
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|a Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2023. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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| 655 |
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|a Electronic books.
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| 700 |
1 |
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|a Zemel, Yoav.
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| 776 |
0 |
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|i Print version:
|a Panaretos, Victor M.
|t An Invitation to Statistics in Wasserstein Space
|d Cham : Springer International Publishing AG,c2020
|z 9783030384371
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| 797 |
2 |
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|a ProQuest (Firm)
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| 830 |
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0 |
|a SpringerBriefs in Probability and Mathematical Statistics Series
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| 856 |
4 |
0 |
|u https://ebookcentral.proquest.com/lib/matrademy/detail.action?docID=6135409
|z Click to View
|
