Nonparametric matrix regression function estimation over symmetric positive definite matrices
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 suitab...
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| Published in | Journal of the Korean Statistical Society Vol. 50; no. 3; pp. 795 - 817 |
|---|---|
| Main Authors | , , , , , |
| Format | Journal Article |
| Language | English |
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Singapore
Springer Singapore
01.09.2021
한국통계학회 |
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| Online Access | Get full text |
| ISSN | 1226-3192 2005-2863 |
| DOI | 10.1007/s42952-020-00082-5 |
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| Abstract | 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. |
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| AbstractList | 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. KCI Citation Count: 0 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. |
| Author | Bak, Kwan-Young Koo, Ja-Yong Kim, Kwang-Rae Kim, Peter T. Zhu, Hongtu Park, Changyi |
| Author_xml | – sequence: 1 givenname: Kwan-Young surname: Bak fullname: Bak, Kwan-Young organization: Department of Statistics, Korea University – sequence: 2 givenname: Kwang-Rae surname: Kim fullname: Kim, Kwang-Rae organization: SAS Software Korea Ltd – sequence: 3 givenname: Peter T. surname: Kim fullname: Kim, Peter T. organization: Department of Mathematics and Statistics, University of Guelph – sequence: 4 givenname: Ja-Yong orcidid: 0000-0002-1035-6102 surname: Koo fullname: Koo, Ja-Yong email: jykoo@korea.ac.kr organization: Department of Statistics, Korea University – sequence: 5 givenname: Changyi surname: Park fullname: Park, Changyi organization: Department of Statistics, University of Seoul – sequence: 6 givenname: Hongtu surname: Zhu fullname: Zhu, Hongtu organization: Gillings School of Global Public Health, University of North Carolina at Chapel Hill |
| BackLink | https://www.kci.go.kr/kciportal/ci/sereArticleSearch/ciSereArtiView.kci?sereArticleSearchBean.artiId=ART002762882$$DAccess content in National Research Foundation of Korea (NRF) |
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| Keywords | Matrix regression function Stein loss Minimaxity Cholesky factorization Wishart distribution Diffusion tensor imaging |
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In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability. Contributions to the theory of statistics (Vol. 1, pp. 361–379). Berkeley, Calif.: University of California Press. KooJ-YKimW-CWavelet density estimation by approximation of log-densitiesStatistics & Probability Letters199626271278139490310.1016/0167-7152(95)00020-8 PourahmadiMJoint mean-covariance models with applications to longitudinal data: unconstrained parameterisationBiometrika199986677690172378610.1093/biomet/86.3.677 GolubGHLoanCFMatrix computations19892BaltimoreThe Johns Hopkins University Press0733.65016 SchwartzmanAMascarenhasWFTaylorJEInference for eigenvalues and eigenvectors of Gaussian symmetric matricesAnnals of Statistics20083614231431248501610.1214/08-AOS628 CaiTTZhouHHOptimal rates of convergence for sparse covariance matrix estimationAnnals of Statistics201240523892420309760710.1214/12-AOS998 PourahmadiMMaximum likelihood estimation of generalized linear models for multivariate normal covariance matrixBiometrika200087425435178248810.1093/biomet/87.2.425 StoneCJThe dimensionality reduction principle for generalized additive modelsAnnals of Statistics19861459060684051610.1214/aos/1176349940 BasserPJJonesDKDiffusion-tensor MRI: theory, experimental design and data analysis—a technical reviewNMR in Biomedicine20021545646710.1002/nbm.783 ZhuHTChenYSIbrahimJGLiYMLinWLIntrinsic regression models for positive-definite matrices with applications to diffusion tensor imagingJournal of the American Statistical Association200910412031212275024510.1198/jasa.2009.tm08096 HasanKMNarayanaPAComputation of the fractional anisotropy and mean diffusivity maps without tensor decoding and diagonalization: theoretical analysis and validationMagnetic Resonance in Medicine20035058959810.1002/mrm.10552 Jaquier, N., & Calinon, S. (2017). Gaussian mixture regression on symmetric positive definite matrices manifolds: Application to wrist motion estimation with sEMG. In 2017 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 59–64), Vancouver, BC. HärdleWKerkyacharianGPicardDTsybakovAWavelets, approximation and statistical applications. Lecture notes in statistics1998New YorkSpringer10.1007/978-1-4612-2222-4 SaidSBombrunLBerthoumieuYMantonJHRiemannian gaussian distributions on the space of symmetric positive definite matricesIEEE Transactions on Information Theory201763421532170362686210.1109/TIT.2017.2653803 DonohoDLJohnstoneIMKerkyacharianGPicardDDensity estimation by wavelet thresholdingAnnals of Statistics199624508539139497410.1214/aos/1032894451 Davis, B. C., Bullitt, E., Fletcher, P. T., & Joshi, S. (2007). Population shape regression from random design data. In IEEE 11th international conference on computer vision (pp 1–7). Rio de Janeiro Kim, P. T., & Richards, D. S. P. 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| References_xml | – reference: GolubGHLoanCFMatrix computations19892BaltimoreThe Johns Hopkins University Press0733.65016 – reference: PourahmadiMJoint mean-covariance models with applications to longitudinal data: unconstrained parameterisationBiometrika199986677690172378610.1093/biomet/86.3.677 – reference: Davis, B. C., Bullitt, E., Fletcher, P. T., & Joshi, S. (2007). Population shape regression from random design data. In IEEE 11th international conference on computer vision (pp 1–7). Rio de Janeiro – reference: de BoorCA practical guide to splines2001New YorkSpringer0987.65015 – reference: TsybakovABIntroduction to nonparametric estimation2009New YorkSpringer10.1007/b13794 – reference: SchottJRMatrix analysis for statistics2005HobokenWiley1076.15002 – reference: Kim, P. T., & Richards, D. S. P. (2011). Deconvolution density estimation on the space of positive definite symmetric matrices. In Nonparametric statistics and mixture models: A Festschrift in honor of Thomas P Hettmansperger (pp 147-168). – reference: BasserPJJonesDKDiffusion-tensor MRI: theory, experimental design and data analysis—a technical reviewNMR in Biomedicine20021545646710.1002/nbm.783 – reference: ZhuHTKongLLiRStynerMGerigGLinWGilmoreJHFadtts: Functional analysis of diffusion tensor tract statisticsNeuroImage2011561412142510.1016/j.neuroimage.2011.01.075 – reference: AndersonTWAn introduction to multivariate statistical analysis20033HobokenWiley1039.62044 – reference: HasanKMBasserPJParkerDLAlexanderALAnalytical computation of the eigenvalues and eigenvectors in DT-MRIJournal of Magnetic Resonance2001152414710.1006/jmre.2001.2400 – reference: SchwartzmanAMascarenhasWFTaylorJEInference for eigenvalues and eigenvectors of Gaussian symmetric matricesAnnals of Statistics20083614231431248501610.1214/08-AOS628 – reference: YuanYZhuHLinWMarronJSLocal polynomial regression for symmetric positive definite matricesJournal of the Royal Statistical Society: Series B201274697719296595610.1111/j.1467-9868.2011.01022.x – reference: ChauJvon SachsRIntrinsic wavelet regression for curves of Hermitian positive definite matricesJournal of the American Statistical Association202010.1080/01621459.2019.17001291464.62377 – reference: ZhuHTChenYSIbrahimJGLiYMLinWLIntrinsic regression models for positive-definite matrices with applications to diffusion tensor imagingJournal of the American Statistical Association200910412031212275024510.1198/jasa.2009.tm08096 – reference: SaidSBombrunLBerthoumieuYMantonJHRiemannian gaussian distributions on the space of symmetric positive definite matricesIEEE Transactions on Information Theory201763421532170362686210.1109/TIT.2017.2653803 – reference: StoneCJThe use of polynomial splines and their tensor products in multivariate function estimationAnnals of Statistics19942211818312720790827.62038 – reference: GrenanderUMillerMIPattern theory from representation to inference2007OxfordOxford University Press1259.62089 – reference: BasserPPierpaoliCMicrostructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRIJournal of Magnetic Resonance199611120921910.1006/jmrb.1996.0086 – reference: PietrzykowskiTA generalization of the potential method for conditional maxima on Banach, reflexive spacesNumerische Mathematik19721836737230518110.1007/BF01404687 – reference: KooJ-YKimW-CWavelet density estimation by approximation of log-densitiesStatistics & Probability Letters199626271278139490310.1016/0167-7152(95)00020-8 – reference: ZhuHXuDRazAHaoXZhangHKangarluABansalRPetersonBSA statistical framework for the classification of tensor morphologies in diffusion tensor imagesMagnetic Resonance Imaging20062456958210.1016/j.mri.2006.01.004 – reference: DeVoreRAPopovVAInterpolation of Besov spacesTransactions of the American Mathematical Society198830539741492016610.1090/S0002-9947-1988-0920166-3 – reference: SchwarzGEstimating the dimension of a modelAnnals of Statistics1978646146446801410.1214/aos/1176344136 – reference: BihanDLLooking into the functional architecture of the brain with diffusion MRINature Reviews Neuroscience2003446948010.1038/nrn1119 – reference: HärdleWKerkyacharianGPicardDTsybakovAWavelets, approximation and statistical applications. Lecture notes in statistics1998New YorkSpringer10.1007/978-1-4612-2222-4 – reference: Jaquier, N., & Calinon, S. (2017). Gaussian mixture regression on symmetric positive definite matrices manifolds: Application to wrist motion estimation with sEMG. In 2017 IEEE/RSJ international conference on intelligent robots and systems (IROS) (pp. 59–64), Vancouver, BC. – reference: DonohoDLJohnstoneIMKerkyacharianGPicardDDensity estimation by wavelet thresholdingAnnals of Statistics199624508539139497410.1214/aos/1032894451 – reference: OsborneDPatrangenaruVEllingsonLGroisserDSchwartzmanANonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and diffusion tensor image analysisJournal of Multivariate Analysis2013119163175306142110.1016/j.jmva.2013.04.006 – reference: James, W., & Stein, C. (1961). Estimation with quadratic loss. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability. Contributions to the theory of statistics (Vol. 1, pp. 361–379). Berkeley, Calif.: University of California Press. – reference: PourahmadiMMaximum likelihood estimation of generalized linear models for multivariate normal covariance matrixBiometrika200087425435178248810.1093/biomet/87.2.425 – reference: BarmpoutisAVemuriBCShepherdTMForderJRTensor splines for interpolation and approximation of DT-MRI with applications to segmentation of isolated rat hippocampiIEEE Transactions on Medical Imaging2007261537154610.1109/TMI.2007.903195 – reference: CaiTTZhouHHOptimal rates of convergence for sparse covariance matrix estimationAnnals of Statistics201240523892420309760710.1214/12-AOS998 – reference: DrydenILKoloydenkoAZhouDNon-Euclidean statistics for covariance matrices with applications to diffusion tensor imagingAnnals of Applied Statistics2009311021123275038810.1214/09-AOAS249 – reference: PierpaoliCBasserPToward a quantitative assessment of diffusion anisotropyMagnetic Resonance in Medicine19963689390610.1002/mrm.1910360612 – reference: StoneCJThe dimensionality reduction principle for generalized additive modelsAnnals of Statistics19861459060684051610.1214/aos/1176349940 – reference: HasanKMNarayanaPAComputation of the fractional anisotropy and mean diffusivity maps without tensor decoding and diagonalization: theoretical analysis and validationMagnetic Resonance in Medicine20035058959810.1002/mrm.10552 – reference: YatracosYGA lower bound on the error in nonparametric regression type problemsAnnals of Statistics1988161180118795919510.1214/aos/1176350954 – ident: 82_CR18 – volume: 16 start-page: 1180 year: 1988 ident: 82_CR34 publication-title: Annals of Statistics doi: 10.1214/aos/1176350954 – volume-title: Pattern theory from representation to inference year: 2007 ident: 82_CR14 – volume: 63 start-page: 2153 issue: 4 year: 2017 ident: 82_CR27 publication-title: IEEE Transactions on Information Theory doi: 10.1109/TIT.2017.2653803 – volume: 104 start-page: 1203 year: 2009 ident: 82_CR37 publication-title: Journal of the 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| Snippet | 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... |
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| Title | Nonparametric matrix regression function estimation over symmetric positive definite matrices |
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