Dataset Information

Local Polynomial Regression for Symmetric Positive Definite Matrices.

ABSTRACT: Local polynomial regression has received extensive attention for the nonparametric estimation of regression functions when both the response and the covariate are in Euclidean space. However, little has been done when the response is in a Riemannian manifold. We develop an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite (SPD) matrices as responses that lie in a Riemannian manifold with covariate in Euclidean space. The primary motivation and application of the proposed methodology is in computer vision and medical imaging. We examine two commonly used metrics, including the trace metric and the Log-Euclidean metric on the space of SPD matrices. For each metric, we develop a cross-validation bandwidth selection method, derive the asymptotic bias, variance, and normality of the intrinsic local constant and local linear estimators, and compare their asymptotic mean square errors. Simulation studies are further used to compare the estimators under the two metrics and to examine their finite sample performance. We use our method to detect diagnostic differences between diffusion tensors along fiber tracts in a study of human immunodeficiency virus.

SUBMITTER: Yuan Y

PROVIDER: S-EPMC3448376 | biostudies-literature | 2012 Sep

REPOSITORIES: biostudies-literature

ACCESS DATA

Publications

Local Polynomial Regression for Symmetric Positive Definite Matrices.

Yuan Ying Y Zhu Hongtu H Lin Weili W Marron J S JS

Journal of the Royal Statistical Society. Series B, Statistical methodology 20120316 4

Local polynomial regression has received extensive attention for the nonparametric estimation of regression functions when both the response and the covariate are in Euclidean space. However, little has been done when the response is in a Riemannian manifold. We develop an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite (SPD) matrices as responses that lie in a Riemannian manifold with covariate in Euclidean space. The primary motivation and applica ...[more]

PMID: 23008683

Similar Datasets

Project description:BackgroundR package mbend was developed for bending symmetric non-positive-definite matrices to positive-definite (PD). Bending is a procedure of transforming non-PD matrices to PD. The covariance matrices used in multi-trait best linear unbiased prediction (BLUP) should be PD. Two bending methods are implemented in mbend. The first is an unweighted bending with small positive values in a descending order replacing negative eigenvalues (LRS14), and the second method is a weighted (precision-based) bending with a custom small positive value (ϵ) replacing smaller eigenvalues (HJ03). Weighted bending is beneficial, as it relaxes low precision elements to change and it reduces or prohibits the change in high precision elements. Therefore, a weighted version of LRS14 was developed in mbend. In cases where the precision of matrix elements is unknown, the package provides an unweighted version of HJ03. Another unweighted bending method (DB88) was tested, by which all eigenvalues are changed (eigenvalues less than ϵ replaced with 100 × ϵ), and it is originally designed for correlation matrices.ResultsDifferent bending procedures were conducted on a 5 × 5 covariance matrix (V), V converted to a correlation matrix (C) and an ill-conditioned 1000 × 1000 genomic relationship matrix (G). Considering weighted distance statistics between matrix elements before and after bending, weighting considerably improved the bending quality. For weighted and unweighted bending of V and C, HJ03-4 (HJ03, ϵ = 10-4) performed the best. HJ03-2 (HJ03, ϵ = 10-2) ranked better than LRS14 for V, but not for C. Though the differences were marginal, LRS14 performed the best for G. DB88-4 (DB88, ϵ = 10-4) was used for unweighted bending and it ranked the last. This method could perform considerably better with a lower ϵ.ConclusionsR package mbend provides necessary tools for transforming symmetric non-PD matrices to PD, using different methods and parameters. There were benefits in both weighted bending and small positive values in a descending order replacing negative eigenvalues. Thus, weighted LRS14 was implemented in mbend. Different bending methods might be preferable for different matrices, depending on the matrix type (covariance vs. correlation), number and the magnitude of negative eigenvalues, and the matrix size.

Project description:The ability of accurate predictions of biological response (biological activity/property/toxicity) of a given chemical makes the quantitative structure-activity/property/toxicity relationship (QSAR/QSPR/QSTR) models unique among the in silico tools. In addition, experimental data of selected species can also be used as an independent variable along with other structural as well as physicochemical variables to predict the response for different species formulating quantitative activity-activity relationship (QAAR)/quantitative structure-activity-activity relationship (QSAAR) approach. Irrespective of the models' type, the developed model's quality, and reliability need to be checked through multiple classical stringent validation metrics. Among the validation metrics, error-based metrics are more significant as the basic idea of a good predictive model is to improve the predictions' quality by lowering the predicted residuals for new query compounds. Following the concept, we have checked the predictive quality of the QSAR and QSAAR models employing kernel-weighted local polynomial regression (KwLPR) approach over the traditional linear and non-linear regression-based approaches tools such as multiple linear regression (MLR) and k nearest neighbors (kNN). Five datasets which were previously modeled using linear and non-linear regression method were considered to implement the KwPLR approach, followed by comparison of their validation metrics outcomes. For all five cases, the KwLPR based models reported better results over the traditional approaches. The present study's focus is not to develop a better or improved QSAR/QSAAR model over the previous ones, but to demonstrate the advantage, prediction power, and reliability of the KwLPR algorithm and establishing it as a novel, powerful cheminformatic tool. To facilitate the use of the KwLPR algorithm for QSAR/QSPR/QSTR/QSAAR modeling, the authors provide an in-house developed KwLPR.RMD script under the open-source R programming language.

Dataset Information

Local Polynomial Regression for Symmetric Positive Definite Matrices.

Publications

Local Polynomial Regression for Symmetric Positive Definite Matrices.

Similar Datasets

OmicsDI is part of the ELIXIR infrastructure

Tweets