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  • 1
    Online Resource
    Online Resource
    An invitation to statistics in Wasserstein space (2020)
    Cham : Springer
    Details
    UID:
    b3kat_BV046652400
    Format: 1 Online-Ressource (xiii, 147 Seiten) , Illustrationen
    ISBN: 9783030384388
    Series Statement: SpringerBriefs in probability and mathematical statistics
    Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-38437-1
    Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-38439-5
    Language: English
    DOI: 10.1007/978-3-030-38438-8
    URL: Volltext  (kostenfrei)
    URL: Volltext  (kostenfrei)
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    EUV Frankfurt
    HTW Berlin
    FH Potsdam
    BTU Cottbus
  • 2
    Online Resource
    Online Resource
    An Invitation to Statistics in Wasserstein Space (2020)
    Cham : Springer Nature
    Details
    UID:
    gbv_1778463061
    Format: 1 Online-Ressource (147 p.)
    ISBN: 9783030384388
    Series Statement: SpringerBriefs in Probability and Mathematical Statistics
    Content: This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph. ; Gives a succinct introduction to necessary mathematical background, focusing on the results useful for statistics from an otherwise vast mathematical literature. Presents an up to date overview of the state of the art, including some original results, and discusses open problems. Suitable for self-study or to be used as a graduate level course text. Open access
    Note: English
    Language: English
    URL: kostenfrei
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    Stabi Berlin
    UB Potsdam
  • 3
    Online Resource
    Online Resource
    An Invitation to Statistics in Wasserstein Space. (2020)
    Cham :Springer International Publishing AG,
    Details
    UID:
    almahu_9949602154802882
    Format: 1 online resource (157 pages)
    Edition: 1st ed.
    ISBN: 9783030384388
    Series Statement: SpringerBriefs in Probability and Mathematical Statistics Series
    Note: 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. , 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.
    Additional Edition: Print version: Panaretos, Victor M. An Invitation to Statistics in Wasserstein Space Cham : Springer International Publishing AG,c2020 ISBN 9783030384371
    Language: English
    Keywords: Electronic books.
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    HU Berlin
    UB Potsdam
  • 4
    Book
    Book
    An invitation to statistics in Wasserstein space (2020)
    Cham : Springer
    Details
    UID:
    b3kat_BV046822977
    Format: xiii, 147 Seiten , Illustrationen
    ISBN: 9783030384371 , 9783030384388
    Series Statement: SpringerBriefs in probability and mathematical statistics
    Additional Edition: Erscheint auch als Online-Ausgabe ISBN 978-3-030-38438-8
    Language: English
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  • 5
    Online Resource
    Online Resource
    An Invitation to Statistics in Wasserstein Space (2020)
    Cham :Springer International Publishing :
    Details
    UID:
    almahu_9948276330302882
    Format: XIII, 147 p. 30 illus., 24 illus. in color. , online resource.
    Edition: 1st ed. 2020.
    ISBN: 9783030384388
    Series Statement: SpringerBriefs in Probability and Mathematical Statistics,
    Content: This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.
    Note: Optimal transportation -- The Wasserstein space -- Fréchet means in the Wasserstein space -- Phase variation and Fréchet means -- Construction of Fréchet means and multicouplings.
    In: Springer eBooks
    Additional Edition: Printed edition: ISBN 9783030384371
    Additional Edition: Printed edition: ISBN 9783030384395
    Language: English
    DOI: 10.1007/978-3-030-38438-8
    URL: https://doi.org/10.1007/978-3-030-38438-8
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    HU Berlin
  • 6
    Online Resource
    Online Resource
    An invitation to statistics in Wasserstein space / (2020)
    Cham :Springer,
    Details
    UID:
    kobvindex_HPB1148226628
    Format: 1 online resource
    ISBN: 9783030384388 , 3030384381
    Series Statement: SpringerBriefs in Probability and Mathematical Statistics,
    Content: This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.
    Note: Optimal transportation -- The Wasserstein space -- Fréchet means in the Wasserstein space -- Phase variation and Fréchet means -- Construction of Fréchet means and multicouplings.
    Additional Edition: Print version: Panaretos, Victor M. Invitation to statistics in Wasserstein space. Cham : Springer, 2020 ISBN 3030384373
    Additional Edition: ISBN 9783030384371
    Language: English
    URL: ProQuest Ebook Central
    URL: SpringerLink
    URL: VLeBooks
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    HPol Brandenburg
  • 7
    Online Resource
    Online Resource
    An Invitation to Statistics in Wasserstein Space (2020)
    Cham : Springer International Publishing | Cham : Imprint: Springer
    Details
    UID:
    gbv_1694052737
    Format: 1 Online-Ressource(XIII, 147 p. 30 illus., 24 illus. in color.)
    Edition: 1st ed. 2020.
    ISBN: 9783030384388
    Series Statement: SpringerBriefs in Probability and Mathematical Statistics
    Content: Optimal transportation -- The Wasserstein space -- Fréchet means in the Wasserstein space -- Phase variation and Fréchet means -- Construction of Fréchet means and multicouplings.
    Content: This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.
    Note: Open Access
    Additional Edition: ISBN 9783030384371
    Additional Edition: ISBN 9783030384395
    Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 9783030384371
    Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 9783030384395
    Language: English
    DOI: 10.1007/978-3-030-38438-8
    URL: kostenfrei
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    Open Access Request
    UB Potsdam
  • 8
    Online Resource
    Online Resource
    An Invitation to Statistics in Wasserstein Space / (2020)
    Cham : Springer Nature | Cham :Springer International Publishing :
    Details
    UID:
    almahu_9948368139102882
    Format: 1 online resource (XIII, 147 p. 30 illus., 24 illus. in color.)
    Edition: 1st ed. 2020.
    ISBN: 3-030-38438-1
    Series Statement: SpringerBriefs in Probability and Mathematical Statistics,
    Content: This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.
    Note: Optimal transportation -- The Wasserstein space -- Fréchet means in the Wasserstein space -- Phase variation and Fréchet means -- Construction of Fréchet means and multicouplings. , English
    Additional Edition: ISBN 3-030-38437-3
    Language: English
    DOI: 10.1007/978-3-030-38438-8
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    HU Berlin
  • 9
    Online Resource
    Online Resource
    An Invitation to Statistics in Wasserstein Space / (2020)
    Cham : Springer Nature | Cham :Springer International Publishing :
    Details
    UID:
    edocfu_9959333555402883
    Format: 1 online resource (XIII, 147 p. 30 illus., 24 illus. in color.)
    Edition: 1st ed. 2020.
    ISBN: 3-030-38438-1
    Series Statement: SpringerBriefs in Probability and Mathematical Statistics,
    Content: This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.
    Note: Optimal transportation -- The Wasserstein space -- Fréchet means in the Wasserstein space -- Phase variation and Fréchet means -- Construction of Fréchet means and multicouplings. , English
    Additional Edition: ISBN 3-030-38437-3
    Language: English
    DOI: 10.1007/978-3-030-38438-8
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    FU Berlin
  • 10
    Online Resource
    Online Resource
    An Invitation to Statistics in Wasserstein Space / (2020)
    Cham : Springer Nature | Cham :Springer International Publishing :
    Details
    UID:
    edoccha_9959333555402883
    Format: 1 online resource (XIII, 147 p. 30 illus., 24 illus. in color.)
    Edition: 1st ed. 2020.
    ISBN: 3-030-38438-1
    Series Statement: SpringerBriefs in Probability and Mathematical Statistics,
    Content: This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.
    Note: Optimal transportation -- The Wasserstein space -- Fréchet means in the Wasserstein space -- Phase variation and Fréchet means -- Construction of Fréchet means and multicouplings. , English
    Additional Edition: ISBN 3-030-38437-3
    Language: English
    DOI: 10.1007/978-3-030-38438-8
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