{"id":2964,"date":"2018-09-17T17:49:40","date_gmt":"2018-09-17T21:49:40","guid":{"rendered":"https:\/\/www.med.unc.edu\/bigs2\/?page_id=2964"},"modified":"2019-10-08T18:01:56","modified_gmt":"2019-10-08T22:01:56","slug":"regression-models","status":"publish","type":"page","link":"https:\/\/www.med.unc.edu\/bigs2\/projects\/shape-analysis\/regression-models\/","title":{"rendered":"Regression Models"},"content":{"rendered":"
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Title:\u00a0Regression models on Riemannian symmetric spaces<\/b><\/h4>\n
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 <\/p>\n

Introduction<\/h4>\n

Manifold-valued responses in curved spaces frequently arise in many disciplines including medical imaging,\u00a0computational biology and computer vision, among many others. For instance, in medical and molecular imaging,\u00a0it is interesting to delineate the changes in the shape and anatomy of a molecule.\u00a0Regression anal- ysis is a fundamental statistical tool for relating a response variable to a covariate, such as age and gender.\u00a0In particular, when both the response and the covariate(s) are in Euclidean space, the classical linear regression model\u00a0and its variants have been widely used in various fields. However, when the response is in a Riemannian symmetric space (RSS)\u00a0and the covariates are in Euclidean space, developing regression models for this type is difficult.<\/p>\n

Questions<\/h4>\n

An intriguing question is whether there is a general regression framework for manifold-valued response in an RSS and\u00a0covariates in a multi-dimensional Euclidean space. The aim of this paper is to give an affirmative answer to such a question.<\/p>\n

Methods<\/h4>\n
    \n
  1. We propose an intrinsic regression model solely based on an intrinsic conditional moment for the response in an RSS,\u00a0thus avoiding specifying any parametric distributions in a general RSS\u2014the model can handle multiple covariates in Euclidean space.<\/li>\n
  2. We develop several \u2018efficient\u2019 estimation methods for estimating the regression coefficients in this intrinsic model.<\/li>\n
  3. We develop several test statistics for testing linear hypotheses of the regression coefficients.<\/li>\n
  4. We develop a general asymptotic frame work for the estimates of the regression coefficients and test statistics.<\/li>\n
  5. We systematically investigate the geometrical properties (e.g. chart invariance) of these parameter estimates and test statistics.<\/li>\n<\/ol>\n

    Findings<\/h4>\n

    We are interested in characterizing the change of the CC contour shape as a function of three covariates\u00a0including gender, age and AD diagnosis.\u00a0We assessed whether there is an age\u00d7diagnosis interaction effect on the shape of the CC contour or not.\u00a0The resting result indicates that there is a strong age-dependent diagnosis effect on the shape of the\u00a0CC contours. It can be observed that there is a difference in\u00a0shape along the inner side of the posterior splenium and isthmus subregions in both male and female groups.\u00a0The splenium seems to be less rounded and the isthmus is thinner in subjects with AD than in healthy controls.\u00a0We also assessed whether there is a gender effect on the shape of the CC contour or not. The Wald test was applied with its\u00a0p-value 0.116. Thus, it is not significant at the 0:05 level of significance. It may indicate that there\u00a0is no gender effect on the shape of the CC contours.\u00a0Similar shapes of CC contours can be observed in males and females according to\u00a0the mean age-dependent CC trajectories for the female and male groups within each diagnosis group.<\/p>\n


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    References<\/h4>\n

    Cornea, E., Zhu, H., Kim, P., Ibrahim, J. G. and the Alzheimer’s Disease Neuroimaging Initiative (2016),
    \nRegression models on Riemannian symmetric spaces. J. R. Stat. Soc. B. doi:10.1111\/rssb.12169<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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     <\/p>\n<\/div>\n<\/div>\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

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