{"id":2990,"date":"2018-09-18T13:11:48","date_gmt":"2018-09-18T17:11:48","guid":{"rendered":"https:\/\/www.med.unc.edu\/bigs2\/?page_id=2990"},"modified":"2020-10-13T18:26:33","modified_gmt":"2020-10-13T22:26:33","slug":"sivc","status":"publish","type":"page","link":"https:\/\/www.med.unc.edu\/bigs2\/sivc\/","title":{"rendered":"LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices"},"content":{"rendered":"<div id=\"content1\">\n<div id=\"main\">\n<div id=\"right\">\n<div id=\"right_text\">\n<div class=\"box\">\n<p>Local polynomial regression has received extensive attention for the nonparametric\u00a0estimation of regression functions when both the response and the covariate are in Euclidean\u00a0space. However, little has been done when the response is in a Riemannian manifold. We develop\u00a0an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite\u00a0(SPD) matrices as responses that lie in a Riemannian manifold with covariate in Euclidean space.\u00a0The primary motivation and application of the proposed methodology is in computer vision and\u00a0medical 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\u00a0bandwidth selection method, derive the asymptotic bias, variance, and normality of the intrinsic\u00a0local constant and local linear estimators, and compare their asymptotic mean square errors. Simulation\u00a0studies are further used to compare the estimators under the two metrics and to examine\u00a0their finite sample performance. We use our method to detect diagnostic differences between\u00a0diffusion tensors along fiber tracts in a study of human immunodeficiency virus.<\/p>\n<p><a href=\"http:\/\/www.nitrc.org\/projects\/fadtts\/\"><strong>Software Download<\/strong> <\/a><\/p>\n<p><strong><a href=\"https:\/\/www.med.unc.edu\/bigs2\/wp-content\/uploads\/sites\/822\/2018\/09\/localSPD_revise_Supplement_sub.pdf\">Supplement Document<\/a><\/strong><\/p>\n<p><strong>Citation<\/strong>: Ying Yuan, Hongtu Zhu, Weili Lin, J. S. Marron (2011). Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices. <em>Journal of the Royal Statistical Society Series B: Statistical Methodology<\/em>. 2013.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"footer\">\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p><!-- footer ends--><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Local polynomial regression has received extensive attention for the nonparametric\u00a0estimation of regression functions when both the response and the covariate are in Euclidean\u00a0space. However, little has been done when the response is in a Riemannian manifold. We develop\u00a0an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite\u00a0(SPD) matrices as responses that lie &hellip; <a href=\"https:\/\/www.med.unc.edu\/bigs2\/sivc\/\" aria-label=\"Read more about LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices\">Read more<\/a><\/p>\n","protected":false},"author":1503,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-2990","page","type-page","status-publish","hentry","odd"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices - BIG-S2<\/title>\n<meta name=\"robots\" content=\"noindex, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices - BIG-S2\" \/>\n<meta property=\"og:description\" content=\"Local polynomial regression has received extensive attention for the nonparametric\u00a0estimation of regression functions when both the response and the covariate are in Euclidean\u00a0space. However, little has been done when the response is in a Riemannian manifold. We develop\u00a0an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite\u00a0(SPD) matrices as responses that lie &hellip; Read more\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.med.unc.edu\/bigs2\/sivc\/\" \/>\n<meta property=\"og:site_name\" content=\"BIG-S2\" \/>\n<meta property=\"article:modified_time\" content=\"2020-10-13T22:26:33+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.med.unc.edu\/bigs2\/sivc\/\",\"url\":\"https:\/\/www.med.unc.edu\/bigs2\/sivc\/\",\"name\":\"LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices - BIG-S2\",\"isPartOf\":{\"@id\":\"https:\/\/www.med.unc.edu\/bigs2\/#website\"},\"datePublished\":\"2018-09-18T17:11:48+00:00\",\"dateModified\":\"2020-10-13T22:26:33+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/www.med.unc.edu\/bigs2\/sivc\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.med.unc.edu\/bigs2\/sivc\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.med.unc.edu\/bigs2\/sivc\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/www.med.unc.edu\/bigs2\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.med.unc.edu\/bigs2\/#website\",\"url\":\"https:\/\/www.med.unc.edu\/bigs2\/\",\"name\":\"BIG-S2\",\"description\":\"Biostatistics and Imaging Genomics analysis lab - Statistics &amp; Signal\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.med.unc.edu\/bigs2\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices - BIG-S2","robots":{"index":"noindex","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"og_locale":"en_US","og_type":"article","og_title":"LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices - BIG-S2","og_description":"Local polynomial regression has received extensive attention for the nonparametric\u00a0estimation of regression functions when both the response and the covariate are in Euclidean\u00a0space. However, little has been done when the response is in a Riemannian manifold. We develop\u00a0an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite\u00a0(SPD) matrices as responses that lie &hellip; Read more","og_url":"https:\/\/www.med.unc.edu\/bigs2\/sivc\/","og_site_name":"BIG-S2","article_modified_time":"2020-10-13T22:26:33+00:00","twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/www.med.unc.edu\/bigs2\/sivc\/","url":"https:\/\/www.med.unc.edu\/bigs2\/sivc\/","name":"LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices - BIG-S2","isPartOf":{"@id":"https:\/\/www.med.unc.edu\/bigs2\/#website"},"datePublished":"2018-09-18T17:11:48+00:00","dateModified":"2020-10-13T22:26:33+00:00","breadcrumb":{"@id":"https:\/\/www.med.unc.edu\/bigs2\/sivc\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.med.unc.edu\/bigs2\/sivc\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/www.med.unc.edu\/bigs2\/sivc\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.med.unc.edu\/bigs2\/"},{"@type":"ListItem","position":2,"name":"LocalSPD: Local Polynomial Regression for Symmetric Positive\u00a0Definite Matrices"}]},{"@type":"WebSite","@id":"https:\/\/www.med.unc.edu\/bigs2\/#website","url":"https:\/\/www.med.unc.edu\/bigs2\/","name":"BIG-S2","description":"Biostatistics and Imaging Genomics analysis lab - Statistics &amp; Signal","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.med.unc.edu\/bigs2\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"}]}},"_links_to":[],"_links_to_target":[],"_links":{"self":[{"href":"https:\/\/www.med.unc.edu\/bigs2\/wp-json\/wp\/v2\/pages\/2990","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.med.unc.edu\/bigs2\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.med.unc.edu\/bigs2\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.med.unc.edu\/bigs2\/wp-json\/wp\/v2\/users\/1503"}],"replies":[{"embeddable":true,"href":"https:\/\/www.med.unc.edu\/bigs2\/wp-json\/wp\/v2\/comments?post=2990"}],"version-history":[{"count":0,"href":"https:\/\/www.med.unc.edu\/bigs2\/wp-json\/wp\/v2\/pages\/2990\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.med.unc.edu\/bigs2\/wp-json\/wp\/v2\/media?parent=2990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}