Modeling sparse longitudinal data on Riemannian manifolds

Dai, Xiongtao; Lin, Zhenhua; Müller, Hans‐Georg

doi:10.1111/biom.13385

Cited by 22 publications

(26 citation statements)

References 47 publications

(70 reference statements)

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“…All of these works assume fully observed functional data. The development by Dai et al (2020) targets discretely and noisily recorded Riemannian functional data, but again requires a Euclidean ambient space.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic Riemannian functional data analysis

Lin¹,

Yao²

2019

Ann. Statist.

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In this work we develop a novel and foundational framework for analyzing general Riemannian functional data, in particular a new development of tensor Hilbert spaces along curves on a manifold. Such spaces enable us to derive Karhunen-Loève expansion for Riemannian random processes. This framework also features an approach to compare objects from different tensor Hilbert spaces, which paves the way for asymptotic analysis in Riemannian functional data analysis. Built upon intrinsic geometric concepts such as vector field, Levi-Civita connection and parallel transport on Riemannian manifolds, the developed framework applies to not only Euclidean submanifolds but also manifolds without a natural ambient space. As applications of this framework, we develop intrinsic Riemannian functional principal component analysis (iRFPCA) and intrinsic Riemannian functional linear regression (iRFLR) that are distinct from their traditional and ambient counterparts. We also provide estimation procedures for iRF-PCA and iRFLR, and investigate their asymptotic properties within the intrinsic geometry. Numerical performance is illustrated by simulated and real examples.

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Section: Introductionmentioning

confidence: 99%

Intrinsic Riemannian functional data analysis

Lin¹,

Yao²

2019

Ann. Statist.

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“…While the role of smoothing individual trajectories in FDA is well understood in the Euclidean case (Hall and Van Keilegom 2007;Zhang and Chen 2007), it remains an open problem to investigate its properties in the much more general setting of longitudinal object data. An even bigger challenge that is also left for future research is the case where measurement grids are sparse and irregular, a problem that was recently studied in Lin et al (2020); Dai et al (2020) for data on Riemannian manifolds.…”

Section: Discussionmentioning

confidence: 99%

Modeling Time-Varying Random Objects and Dynamic Networks

Dubey¹,

Müller²

2021

Preprint

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Samples of dynamic or time-varying networks and other random object data such as timevarying probability distributions are increasingly encountered in modern data analysis. Common methods for time-varying data such as functional data analysis are infeasible when observations are time courses of networks or other complex non-Euclidean random objects that are elements of general metric spaces. In such spaces, only pairwise distances between the data objects are available and a strong limitation is that one cannot carry out arithmetic operations due to the lack of an algebraic structure. We combat this complexity by a generalized notion of mean trajectory taking values in the object space. For this, we adopt pointwise Fréchet means and then construct pointwise distance trajectories between the individual time courses and the estimated Fréchet mean trajectory, thus representing the time-varying objects and networks by functional data. Functional principal component analysis of these distance trajectories can reveal interesting features of dynamic networks and object time courses and is useful for downstream analysis. Our approach also makes it possible to study the empirical dynamics of time-varying objects, including dynamic regression to the mean or explosive behavior over time. We demonstrate desirable asymptotic properties of sample based estimators for suitable population targets under mild assumptions. The utility of the proposed methodology is illustrated with dynamic networks, time-varying distribution data and longitudinal growth data.

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“…The proposed metric halfspace depth, therefore, applies to a wide range of non-Euclidean data objects. This includes data lying on smooth Riemannian manifolds such as directional data on a sphere (Mardia and Jupp, 2009); bivariate molecular torsion angles on a flat torus (Eltzner et al, 2018); and constrained matrix-valued data, such as rotations (Bingham et al, 2009) and covariance matrices (Dai et al, 2020). Nonsmooth objects with possible degeneracy lying on a geodesic space, such as phylogenetic trees (Feragen and Nye, 2020), networks (Kolaczyk et al, 2020), and shapes (Dryden and Mardia, 2016) can also be investigated by the proposed depth.…”

Section: Our Contributionsmentioning

confidence: 99%

“…Let M = SPD(k) be the manifold of k × k symmetric positive definite (SPD) matrices. This matrix manifold has seen wide application in modeling brain connectivity matrices (Dai et al, 2020) and diffusion tensors (Pennec et al, 2006). Endowed with the affine-invariant geometry (Pennec et al, 2006), the geodesic distance on M is defined as…”

Section: Examples: Metric Halfspace Depth In Common Spacesmentioning

confidence: 99%

Tukey's Depth for Object Data

Dai,

Lopez-Pintado

2021

Preprint

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We develop a novel exploratory tool for non-Euclidean object data based on data depth, extending the celebrated Tukey's depth for Euclidean data. The proposed metric halfspace depth, applicable to data objects in a general metric space, assigns to data points depth values that characterize the centrality of these points with respect to the distribution and provides an interpretable center-outward ranking. Desirable theoretical properties that generalize standard depth properties postulated for Euclidean data are established for the metric halfspace depth. The depth median, defined as the deepest point, is shown to have high robustness as a location descriptor both in theory and in simulation. We propose an efficient algorithm to approximate the metric halfspace depth and illustrate its ability to adapt to the intrinsic data geometry. The metric halfspace depth was applied to an Alzheimer's disease study, revealing group differences in the brain connectivity, modeled as covariance matrices, for subjects in different stages of dementia. Based on phylogenetic trees of 7 pathogenic parasites, our proposed metric halfspace depth was also used to construct a meaningful consensus estimate of the evolutionary history and to identify potential outlier trees.

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Modeling sparse longitudinal data on Riemannian manifolds

Cited by 22 publications

References 47 publications

Intrinsic Riemannian functional data analysis

Intrinsic Riemannian functional data analysis

Modeling Time-Varying Random Objects and Dynamic Networks

Tukey's Depth for Object Data

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