Rabindra N Bhattacharya
Professor, BIO5 Institute
Professor, Mathematics
Professor, Statistics-GIDP
Primary Department
(520) 626-5647
Research Interest
Rabi N. Bhattacharya is a Professor of Mathematics at the University of Arizona and a Member of BIO5. He is internationally known for his work on asymptotic theories of probability and statistics with diverse applications in many areas of science and engineering. His recent work on statistics on shape spaces has important applications in morphometrics, medical imaging and diagnostics and machine vision, in particular. This original work has inspired a large US and international following. During the 2011-12 academic year he was an invited speaker at international conferences and workshops at the University of Goettingen (Germany), University of Bielefeld (Germany), IMS/APRM 2012 in Tsukuba (Japan), and the Mathematical Biosciences Institute (MBI), Ohio State University. He has published nearly 100 research articles and co-authored four research monographs and two graduate texts. Two of these books have been recently reprinted by SIAM (Society for Industrial and Applied Mathematics) as Classics in Applied Mathematics. He is a Fellow of the IMS and a Member of the AMS, and a recipient of a Humboldt Prize and a Guggenheim Fellowship.Professor Bhattacharya has been intimately associated with the establishment of the Statistics GIDP at the U of A since 2002, and was a member of the Executive Committee of the GIDP for the past six years.

Publications

Bhattacharya, R. N., & Majumdar, M. (1973). Random exchange economies. Journal of Economic Theory, 6(1), 37-67.
Bhattacharya, R., & Patrangenaru, V. (2014). Rejoinder to the discussion. Journal of Statistical Planning and Inference, 145, 42-48.
Sposito, G., Gupta, V. K., & Bhattacharya, R. (1979). Foundation theories of solute transport in porous media: a critical review. Advances in Water Resources, 2(C), 59-68.

Abstract:

The theories that have been employed to derive the macroscopic differential equations that describe solute transport through porous media are reviewed critically. These foundational theories may be grouped into three classes: (1) those based in fluid mechanics, (2) those based in kinematic approaches employing the mathematics of the theory of Markov processes, and (3) those based in a formal analogy between statistical thermodynamics and hydrodynamic dispersion. It is shown that the theories of class 1 have had to employ highly artificial models of a porous medium in order to produce a well-defined velocity field in the pore space that can be analysed rigorously or have had to assume that well-defined solutions of the equations of fluid mechanics exist in the pore space of a natural porous medium and then adopt an ad hoc definition of the solute difusivity tensor. The theories of class 2 do not require the validity of fluid mechanics but they suffer from the absence of a firm dynamical basis, at the molecular level, for the stochastic properties they attribute to the velocity of a solute molecule, or they ignore dynamics altogether and make kinematic assumptions directly on the position process of a solute molecule. The theories of class 3 have been purely formal in nature, with an unclear physical content, or have been no different in content from empirically based theories that make use of the analogy between heat and matter flow at the macroscopic level. It is concluded that none of the existing foundational theories has yet achieved the objectives of: (1) deriving, in a physically meaningful and mathematically rigorous fashion, the macroscopic differential equations of solute transport theory, and (2) elucidating the structure of the empirical coefficients appearing in these equations. © 1979.

Bandulasiri, A., Bhattacharya, R. N., & Patrangenaru, V. (2009). Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging. Journal of Multivariate Analysis, 100(9), 1867-1882.

Abstract:

For all p > 2, k > p, a size-and-reflection-shape space S R Σp, 0k of k-ads in general position in Rp, invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold R Σp, 0k, a space of orbits of scaled k-ads in general position under the group of isometries of Rp, providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p > 2. The Veronese embedding of the planar Kendall shape manifold Σ2k is extended to an equivariant embedding of the size-and-shape manifold S Σ2k, which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory. © 2009 Elsevier Inc. All rights reserved.

Bhattacharya, R., & Majumdar, M. (2010). Random iterates of monotone maps. Review of Economic Design, 14(1-2), 185-192.

Abstract:

In this paper we prove the existence, uniqueness and stability of the invariant distribution of a random dynamical system in which the admissible family of laws of motion consists of monotone maps from a closed subset of a finite dimensional Euclidean space into itself. © 2008 Springer-Verlag.