Nonparametric matrix regression function estimation over symmetric positive definite matrices

• Published in: Journal of the Korean Statistical Society Vol. 50; no. 3; pp. 795 - 817
• Main Authors: Bak, Kwan-Young, Kim, Kwang-Rae, Kim, Peter T., Koo, Ja-Yong, Park, Changyi, Zhu, Hongtu
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.09.2021; 한국통계학회
• Subjects: Applied Statistics; Bayesian Inference; Mathematics and Statistics; Research Article; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; 통계학; Matrix regression function; Stein loss; Minimaxity; Cholesky factorization; Wishart distribution; Diffusion tensor imaging
• ISSN: 1226-3192; 2005-2863
• DOI: 10.1007/s42952-020-00082-5

Symmetric positive definite matrix data commonly appear in computer vision and medical imaging, such as diffusion tensor imaging. The aim of this paper is to develop a nonparametric estimation method for a symmetric positive definite matrix regression function given covariates. By obtaining a suitable parametrization based on the Cholesky decomposition, we make it possible to apply univariate smoothing methods to the matrix regression problem. The parametrization also guarantees that the proposed estimator is symmetric positive definite over the entire domain. We adopt the Wishart log-likelihood and a smoothing technique using the basis methodology to define our estimator. The rate of convergence of the proposed estimator is obtained under some regularity conditions. Simulations are performed to investigate the finite sample properties of our proposed method using natural splines. Moreover, we present the results of an analysis of real diffusion tensor imaging data where the estimated fractional anisotropy is provided using 3 × 3 symmetric positive definite matrices measured at consecutive positions along a fiber tract in the brain of subjects.