Bayesian Framework for Simultaneous Registration and Estimation of Noisy, Sparse and Fragmented Functional Data

Mathematical and Physical Sciences: 3rd Place (The Ohio State University Edward F. Hayes Graduate Research Forum)In many applications, smooth processes generate data that is recorded under a variety of observation regimes, such as dense sampling and sparse or fragmented observations that are often contaminated with error. The statistical goal of registering and estimating the individual underlying functions from discrete observations has thus far been mainly approached sequentially without formal uncertainty propagation, or in an application-specific manner by pooling information across subjects. We propose a unified Bayesian framework for simultaneous registration and estimation, which is flexible enough to accommodate inference on individual functions under general observation regimes. Our ability to do this relies on the specification of strongly informative prior models over the amplitude component of function variability. We provide two strategies for this critical choice: a data-driven approach that defines an empirical basis for the amplitude subspace based on available training data, and a shape-restricted approach when the relative location and number of local extrema is well-understood. The proposed methods build on the elastic functional data analysis framework to separately model amplitude and phase variability inherent in functional data. We emphasize the importance of uncertainty quantification and visualization of these two components as they provide complementary information about the estimated functions. We validate the proposed framework using simulation studies, and real applications to estimation of fractional anisotropy profiles based on diffusion tensor imaging measurements, growth velocity functions and bone mineral density curves.No embarg

Robust Persistence Homology Through Amplitude-Phase Separation in Persistence Landscapes

Persistence homology of a point cloud sampled from a manifold embedded in $\mathbb R^d$ using geometric filtered complexes, such as Rips or \v{C}ech complexes, is sensitive to scaling, sampling variability, and geometric configurations of the points in $\mathbb R^d$ relative to the manifold. This can result in markedly different persistence diagrams for point clouds sampled from topologically identical manifolds. In this paper, we propose a practical solution to this problem by determining transformations of points in persistence diagrams, which act as a denoising mechanism by amplifying the topological signal. The transformations are obtained by computing phase functions that align the corresponding persistence landscapes, which are in bijection with the diagrams. We view persistence landscapes as piecewise linear parameterized closed curves, and carry out amplitude-phase separation using an elastic Riemannian metric. We are thus able to compute a mean persistence landscape that better preserves the overall shape of landscapes in the sample. The estimated phase functions are tied to the resolution parameter that determines the filtration of simplicial complexes used to construct persistence diagrams. For a dataset obtained under geometric, scale and sampling variabilities, the phase function prescribes an optimal rate at which to increase the radii of balls to ensure that the simplicial complex adapts to local scale changes that arise due to scale, sampling and geometric variabilities. We demonstrate benefits of our approach through several simulated examples, a semi-synthetic data example concerning cancer cells, and a real data example concerning structure of brain artery trees

Bayesian Functional Principal Component Analysis using Relaxed Mutually Orthogonal Processes

Functional Principal Component Analysis (FPCA) is a prominent tool to characterize variability and reduce dimension of longitudinal and functional datasets. Bayesian implementations of FPCA are advantageous because of their ability to propagate uncertainty in subsequent modeling. To ease computation, many modeling approaches rely on the restrictive assumption that functional principal components can be represented through a pre-specified basis. Under this assumption, inference is sensitive to the basis, and misspecification can lead to erroneous results. Alternatively, we develop a flexible Bayesian FPCA model using Relaxed Mutually Orthogonal (ReMO) processes. We define ReMO processes to enforce mutual orthogonality between principal components to ensure identifiability of model parameters. The joint distribution of ReMO processes is governed by a penalty parameter that determines the degree to which the processes are mutually orthogonal and is related to ease of posterior computation. In comparison to other methods, FPCA using ReMO processes provides a more flexible, computationally convenient approach that facilitates accurate propagation of uncertainty. We demonstrate our proposed model using extensive simulation experiments and in an application to study the effects of breastfeeding status, illness, and demographic factors on weight dynamics in early childhood. Code is available on GitHub at https://github.com/jamesmatuk/ReMO-FPC

Bayesian Framework for Simultaneous Registration and Estimation of Noisy, Sparse and Fragmented Functional Data

In many applications, smooth processes generate data that is recorded under a variety of observational regimes, including dense sampling and sparse or fragmented observations that are often contaminated with error. The statistical goal of registering and estimating the individual underlying functions from discrete observations has thus far been mainly approached sequentially without formal uncertainty propagation, or in an application-specific manner by pooling information across subjects. We propose a unified Bayesian framework for simultaneous registration and estimation, which is flexible enough to accommodate inference on individual functions under general observational regimes. Our ability to do this relies on the specification of strongly informative prior models over the amplitude component of function variability using two strategies: a data-driven approach that defines an empirical basis for the amplitude subspace based on training data, and a shape-restricted approach when the relative location and number of extrema is well-understood. The proposed methods build on the elastic functional data analysis framework to separately model amplitude and phase variability inherent in functional data. We emphasize the importance of uncertainty quantification and visualization of these two components as they provide complementary information about the estimated functions. We validate the proposed framework using multiple simulation studies and real applications

Circadian control of abscisic acid biosynthesis and signalling pathways revealed by genome-wide analysis of LHY binding targets

The LATE ELONGATED HYPOCOTYL (LHY) transcription factor functions as part of the oscillatory mechanism of the Arabidopsis circadian clock. This paper reports the genome‐wide analysis of its binding targets and reveals a role in the control of abscisic acid (ABA) biosynthesis and downstream responses. LHY directly repressed expression of 9‐cis‐epoxycarotenoid dioxygenase enzymes, which catalyse the rate‐limiting step of ABA biosynthesis. This suggested a mechanism for the circadian control of ABA accumulation in wild‐type plants. Consistent with this hypothesis, ABA accumulated rhythmically in wild‐type plants, peaking in the evening. LHY‐overexpressing plants had reduced levels of ABA under drought stress, whereas loss‐of‐function mutants exhibited an altered rhythm of ABA accumulation. LHY also bound the promoter of multiple components of ABA signalling pathways, suggesting that it may also act to regulate responses downstream of the hormone. LHY promoted expression of ABA‐responsive genes responsible for increased tolerance to drought and osmotic stress but alleviated the inhibitory effect of ABA on seed germination and plant growth. This study reveals a complex interaction between the circadian clock and ABA pathways, which is likely to make an important contribution to plant performance under drought and osmotic stress conditions

MYB96 shapes the circadian gating of ABA signaling in Arabidopsis

Circadian clocks regulate the rhythms of biological activities with a period of approximately 24-hours and synchronize plant metabolism and physiology with the environmental cycles. The clock also gates responses to environmental stresses to maximize fitness advantages. Here we report that the MYB96 transcription factor is connected with the clock oscillator to shape the circadian gating of abscisic acid (ABA) responses. MYB96 directly binds to the TIMING OF CAB EXPRESSION 1 (TOC1) promoter to positively regulate its expression. The use of myb96 mutant plants shows that this regulation is essential for the gated induction of TOC1 by ABA. In turn, MYB96 induction by ABA is also altered in toc1-3 mutant plants. The increased tolerance to drought of MYB96 over-expressing plants is decreased in the toc1-3 mutant background, suggesting that MYB96 and TOC1 intersect the circadian clock and ABA signaling. The MYB96-TOC1 function might be also regulated by the clock component CIRCADIAN CLOCK-ASSOCIATED 1 (CCA1), which binds to the MYB96 promoter and alters its circadian expression. Thus, a complex circuitry of CCA1-MYB96-TOC1 regulatory interactions provides the mechanistic basis underlying the connection between circadian and stress signaling to optimize plant fitness to ambient stresses