Extrinsic Local Regression on Manifold-Valued Data

Lin, Lizhen; Thomas, B.S.; Zhu, Hongtu; Dunson, David B.

doi:10.1080/01621459.2016.1208615

Cited by 54 publications

(38 citation statements)

References 37 publications

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“…Sometimes functional data X(t) are observed only at densely spaced time points and observations might be contaminated with measurement errors. In these situations one can presmooth the observations using smoothers that are adapted to a Riemannian manifold (Jupp and Kent 1987;Lin et al 2016), treating the presmoothed curves as fully observed underlying curves.…”

Section: Estimationmentioning

confidence: 99%

“…The intrinsic dimension d is only reflected in the rate constant but not the speed of convergence. Our situation is analogous to that of estimating the mean of Euclidean-valued random functions (Bosq 2000), or more generally, Fréchet regression with Euclidean responses , where the speed of convergence does not depend on the dimension of the Euclidean space, in contrast to common nonparametric regression settings (Lin et al 2016;Lin and Yao 2017). The root-n rate is not improvable in general since it is the optimal rate for mean estimates in the special Euclidean case.…”

Section: Component Analysismentioning

confidence: 99%

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Principal component analysis for functional data on Riemannian manifolds and spheres

Dai¹,

Müller²

2018

Ann. Statist.

105

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Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fréchet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. Specifically, we derive a central limit theorem for the mean function, as well as root-n uniform convergence rates for other model components, including the covariance function, eigenfunctions, and functional principal component scores. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. Riemannian functional principal component analysis is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications and simulations and is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores.

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Section: Estimationmentioning

confidence: 99%

Section: Component Analysismentioning

confidence: 99%

Principal component analysis for functional data on Riemannian manifolds and spheres

Dai¹,

Müller²

2018

Ann. Statist.

105

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“…Regression models for this special case have been well studied (Fisher, Lewis and Embleton 1987;Chang 1989;Prentice 1989;Fisher 1995), including intrinsic models for geodesic regression (Fletcher 2013;Niethammer, Huang and Vialard 2011;Cornea et al 2016), semiparametric regression (Shi et al 2009) and local kernel regression as a generalization of the classical Nadaraya-Watson smoother (Pelletier 2006;Davis et al 2007;Hinkle et al 2012;Yuan et al 2012). Recently, the extrinsic regression model in Lin et al (2015) extends the notion of extrinsic means (see, e.g., Ch. 11 and 18 of Patrangenaru and Ellingson 2015), where extrinsic approaches have been reported to have computational advantages (Bhattacharya et al 2012).…”

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confidence: 99%

Fréchet regression for random objects with Euclidean predictors

Petersen¹,

Müller²

2019

Ann. Statist.

113

180

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Increasingly, statisticians are faced with the task of analyzing complex data that are non-Euclidean and specifically do not lie in a vector space. To address the need for statistical methods for such data, we introduce the concept of Fréchet regression. This is a general approach to regression when responses are complex random objects in a metric space and predictors are in R p , achieved by extending the classical concept of a Fréchet mean to the notion of a conditional Fréchet mean. We develop generalized versions of both global least squares regression and local weighted least squares smoothing. The target quantities are appropriately defined population versions of global and local regression for response objects in a metric space. We derive asymptotic rates of convergence for the corresponding fitted regressions using observed data to the population targets under suitable regularity conditions by applying empirical process methods. For the special case of random objects that reside in a Hilbert space, such as regression models with vector predictors and functional data as responses, we obtain a limit distribution. The proposed methods have broad applicability. Illustrative examples include responses that consist of probability distributions and correlation matrices, and we demonstrate both global and local Fréchet regression for demographic and brain imaging data. Local Fréchet regression is also illustrated via a simulation with response data which lie on the sphere.

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“…In St. Thomas et al (2014), for example, the manifold of the parameters of a statistical model is embedded into a big sphere, while Lin et al (2016) embeds the response manifold of a regression model into a Euclidean space for inference. In section 3.1, a simulation study is carried out to compare the performances of an eGP model with that of an intrinsic one in a regression model with predictors on a sphere.…”

Section: Examplesmentioning

confidence: 99%

Extrinsic Gaussian Processes for Regression and Classification on Manifolds

Lin¹,

Niu²,

Cheung³

et al. 2019

Bayesian Anal.

Self Cite

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Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Although there is an increasingly vast literature on applications, methods, theory and algorithms related to GPs, the overwhelming majority of this literature focuses on the case in which the input domain corresponds to a Euclidean space. However, particularly in recent years with the increasing collection of complex data, it is commonly the case that the input domain does not have such a simple form. For example, it is common for the inputs to be restricted to a non-Euclidean manifold, a case which forms the motivation for this article. In particular, we propose a general extrinsic framework for GP modeling on manifolds, which relies on embedding of the manifold into a Euclidean space and then constructing extrinsic kernels for GPs on their images. These extrinsic Gaussian processes (eGPs) are used as prior distributions for unknown functions in Bayesian inferences. Our approach is simple and general, and we show that the eGPs inherit fine theoretical properties from GP models in Euclidean spaces. We consider applications of our models to regression and classification problems with predictors lying in a large class of manifolds, including spheres, planar shape spaces, a space of positive definite matrices, and Grassmannians.Our models can be readily used by practitioners in biological sciences for various regression and classification problems, such as disease diagnosis or detection. Our work is also likely to have impact in spatial statistics when spatial locations are on the sphere or other geometric spaces.

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Extrinsic Local Regression on Manifold-Valued Data

Cited by 54 publications

References 37 publications

Principal component analysis for functional data on Riemannian manifolds and spheres

Principal component analysis for functional data on Riemannian manifolds and spheres

Fréchet regression for random objects with Euclidean predictors

Extrinsic Gaussian Processes for Regression and Classification on Manifolds

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