Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis
CRC Press – 2015 – 536 pages
Requiring only a background in mathematical statistics or analysis, this book provides the first comprehensive look at the statistics of manifolds. It presents location and spread parameters for distribution on manifolds, followed by key large sample theory results. The text also addresses inference of two samples, nonparametric MANOVA, principal component analysis, and density estimation on manifolds. A special section is dedicated to a nonparametric statistical analysis on certain special manifolds arising in statistics. The final part of the text focuses on concrete applications in astronomy, image analysis, medical imaging, bioinformatics, and pattern recognition.
Nonparametric Statistics on Manifolds. Data on Manifolds. Basic NonparametricMultivariate Inference. Geometry and Topology of Manifolds. Consistency of Fréchet Moments on Manifolds. Nonparametric Distributions of Fréchet Means. Inference for two samples on manifolds. Function Estimation on Manifolds. Asymptotic Theory and Nonparametric Bootstrap on Special Manifolds. Statistics on Homogeneous Hadamard Manifolds. Analysis on Stiefel Manifolds. Asymptotic Distributions on Projective Spaces. Nonparametric Statistics on Hilbert Manifolds. Analysis on Spaces of Congruences of k - ads. Similarity Shape Analysis. Statistics on Grassmannians. Applications in Object Data Analysis on Manifolds. DTI Data Analysis. Application of Directional Data Analysis. Direct Similarity Shape Analysis in Medical Imaging. Similarity Shape Analysis of Planar Contours. Estimating Mean Skull Size and Shape from CT scans. Affine Shape and Linear Shape Applications. Projective Shape Analysis of Planar Contours. Two Sample Tests for Mean Projective Shapes. Mean Glaucomatous Shape Change Detection. Application of Density Estimation on Manifolds. Additional Topics. Persistent Homology. Further Directions in Statistics on Manifolds. Bibliography.
Victor Patrangenaru, Florida State University, Tallahassee, USA