Nonparametric Inference On Manifolds: An Statistical Analysis Of Shapes Of Images

Guest Author: Abhishek Bhattacharya

Analyses of shapes of digital images play a vital role today in science and technology, and also in many aspects of our daily life. Consider for example the problem in morphometrics of determining the gender of an extinct species based on the shape of fossilized bones of a few specimens. An application in medical diagnostics can be the detection of presence or absence of a disease based on any deformation in the shape of a particular organ. Then in robotics one would need shape analysis tools to help the robot understand a particular scene. There are many such applications in diverse fields of biology, image analysis, machine vision, and many more. This book seeks to advance the analysis of images, through the statistical analysis of shapes of images.

The statistical analysis of shapes involves developing statistical inference tools on the so called spaces of shapes. These shape spaces are special geometric objects called manifolds which are very different from the real line or the higher dimensional planes. Various dimensional planes form a special category of manifolds, the so called linear or Euclidean manifolds. Most statistical literature is dedicated to the analysis of data on Euclidean manifolds and very little work exists when the data is non-linear. This book presents in a systematic manner a general theory of statistics on general manifolds with emphasis on manifolds of shapes, and with applications to diverse fields of science and engineering.

Apart from the shape spaces, another simpler manifold of interest is the sphere of one (circle), two (ball) or higher dimension. Statistical inference on spheres finds applications in directional data analysis, an application being the study of shifts in the Earth’s magnetic poles over geological time, which have an important bearing on the subject of tectonics. This book takes a fresh look in analyzing existing data pertaining to a number of such applications. The goal is to lay the framework for other future applications of this exciting emerging field of statistics.

Most of the past statistical literature on manifolds, especially that on shape spaces, has focused on parametric models which means that the data is assumed to come from a fixed class of distributions and the inference holds only if that assumption is true, which is hardly ever the case except in simulated examples. This book mainly deals with nonparametric theory of statistics on manifolds, which makes no model assumptions on the given data. The inference is shown to be asymptotically consistent irrespective of the underlying model. In all the data examples, the developed methodology seems to provide sharper inferences than do their parametric counterparts.

This book develops both nonparametric frequentist and nonparametric Bayesian inference on manifolds. For frequentist inference concepts of center and spread are defined on manifolds and their properties studied like uniqueness, consistency and asymptotic distribution. Confidence regions are constructed and hypothesis testing performed based on these properties. Their bootstrap counterparts are also used.

For Bayesian inference, nonparametric Bayes procedures are developed for functional inference on manifolds including nonparametric density estimation, regression, classification and hypothesis testing.

The developed methods are applied to manifolds of special interest such as spheres, projective spaces and spaces of various notions of shapes such as similarity shapes, reflection similarity shapes, affine shapes and projective shapes.

This book is suitable for graduate students in mathematics, statistics, engineering and computer science who have taken a graduate course in asymptotic statistics and differential geometry. For such students special topic courses may be based on it. This book is also meant to serve as a reference for researchers in the areas mentioned. For the benefit of the readers two appendices are provided on differential geometry and one each on nonparametric Bayes inference and some commonly used parametric models on spheres and shape spaces. These models play a significant role in constructing nonparametric Bayes priors for density estimation and shape classification in the book chapters.

The book “Nonparametric Inference on Manifolds is the result of the joint efforts by Prof. Abhishek Bhattacharya (Indian Statistical Institute) and Prof. Rabi Bhattacharya (University of Arizona). It is published in 2012 as an Institute of Mathematical Statistics (IMS) monograph by Cambridge University Press.

These IMS monographs are research monographs of high quality on any branch of statistics or probability of sufficient interest to warrant publication as books. Some concern relatively traditional topics in need of up-to-date assessment while others are on emerging themes. In all cases the objective is to provide a balanced view of the field.

Our Guest author, Dr. Abhishek Bhattacharya is Assistant Professor in the Theoretical Statistics and Mathematics Unit (SMU) at the Indian Statistical Institute. He has written several articles for leading journals like IMS collections, AMS proceedings, Sankhya, Bayesian Statistics, Biometrika, JMVA, AISM and JASA including this Cambridge University Press book. Details of his work can be accessed from his webpage http://www.isical.ac.in/~abhishek/.

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