Thursday, Aug 7: 10:30 AM - 12:20 PM
4221
Contributed Papers
Music City Center
Room: CC-202A
Main Sponsor
Section on Nonparametric Statistics
Presentations
In this work, we propose an adaptive bivariate penalized spline regression framework for spatially varying coefficient models in the context of streaming data. The method accommodates the dynamic insertion of new spatial locations by sequentially updating summary statistics, enabling real-time modeling as data arrives. We establish the consistency of the estimators for the functional coefficients under this framework. Simulation studies and a real-world application are presented to illustrate the effectiveness and adaptability of the proposed approach.
Keywords
bivariate splines
semi-parametric
spatial
triangulation
First Author
Jingru Mu, Kansas State University
Presenting Author
Jingru Mu, Kansas State University
Shape-constrained functional data encompass a wide array of application fields, such as activity profiling, growth curves, healthcare and mortality. Most existing methods for general functional data analysis often ignore that such data are subject to inherent shape constraints, while some specialized techniques rely on strict distributional assumptions. We propose an approach for modeling such data that harnesses the intrinsic geometry of functional trajectories by decomposing them into size and shape components. We focus on the two most prevalent shape constraints, positivity and monotonicity, and develop individual-level estimators for the size and shape components. Furthermore, we demonstrate the applicability of our approach by conducting subsequent analyses involving Fréchet mean and Fréchet regression and establish rates of convergence for the empirical estimators. Illustrative examples include simulations and data applications for activity profiles for Mediterranean fruit flies during their entire lifespan and for data from the Zürich longitudinal growth study.
Keywords
Fréchet regression
Functional data analysis
Longitudinal studies
Monotonicity
Positivity
Size-shape decomposition
In this talk, we introduce and develop a projective shape analysis for the study of cognitive abilities evaluated based on learning behaviour in the DSNT (Dresden Spatial Navigation Task) virtual navigational experiment ([1]). DSNT adapts the classical water maze test for humans, and was developed at DZNE (The Research Institute for Neurodegenerative Diseases from Dresden, Germany). This new mathematical modelling of the spatial orientation and learning is based on recent concepts in object-oriented data analysis like extrinsic covariance and extrinsic cross-covariance as well as novel statistical testing methods for random objects on manifolds ([2]). Additionally, new numerical algorithms will be developed, studied and finally implemented in an open-source mathematical software like R and will be used to evaluate our conclusions and to present the data visually.
Bibliography
1. Garthe A., Kempermann G., An old test for new neurons: refining the Morris water maze to study the functional relevance of adult hippocampal neurogenesis, Front. in Neuro., 7 (2013).
2. Wong K.C., Patrangenaru V., Paige R.L.,Pricop Jeckstadt M., Extrinsic Principal Component Analysis, arXiv (2024).
Keywords
object-oriented data analysis
projective shape,
nonparametric statistics,
spatial learning, virtual reality
This work introduces a novel method for exploring object data viewed as random elements in a metric space. The approach evaluates the Fréchet mean of elements within a ball centered at a specific point in the space. We investigate the behaviors of these ball Fréchet means as the ball radius increases, treating them as functional objects dependent on the radius and ball centers. We reduce them into real-valued functions by taking their distance from the global Fréchet mean, thereby enabling the application of traditional functional data analysis techniques. Theoretical results are provided, and the method is illustrated with simulations and applications to human mortality data and U.S. electricity generation data, demonstrating its potential for clustering and outlier detection.
Keywords
Data exploration
Fréchet mean
Functional data analysis
Object data analysis
Gaussian processes (GPs) are popular models for random functions in statistics, machine learning and data science. This talk focuses on the estimation of covariance and precision operator of GPs. The first part of the talk will give minimax rates for structured covariance operator estimation, including banded operators with kernels that decay rapidly off-the-diagonal and $L^q$-sparse operators with unordered sparsity pattern. We identify the fundamental dimension-free quantities that determine the sample complexity and show that tapering and thresholding estimators attain the optimal rate. In the second part of the talk, I will discuss the estimation of large ill-conditioned precision matrices and Cholesky factors obtained by observing a GP at many locations. Under general assumptions, we show that the sample complexity scales poly-logarithmically with the size of the matrices. For precision estimation, our theory hinges on a local regression technique on the lattice graph exploiting the sparsity implied by the screening effect. For Cholesky factor estimation, we leverage a block-Cholesky decomposition recently used to establish complexity bounds for sparse Cholesky factorization.
Keywords
Gaussian processes
Minimax theory
Dimension-free analysis
Covariance estimation
Precision estimation
Cholesky factor estimation
High-dimensional medical imaging data are rapidly expanding, yet their complex structure and measurement errors pose significant challenges for reliable scientific discovery. We propose a robust distributed image-on-scalar Regression (R-DISR) framework that integrates spatially varying coefficient models with triangulated spherical spline smoothing via domain decomposition. This approach is designed to handle heavy-tailed noise and measurement errors while achieving near-linear computational speedup and minimizing communication overhead in distributed computing environments. We rigorously establish that the R-DISR estimators attain the same convergence rate as full-sample global estimators and derive their asymptotic distributions. Moreover, a weighted bootstrap procedure is developed to construct simultaneous confidence corridors for the spatially varying coefficient functions. Extensive simulation studies demonstrate the method's finite-sample performance, and its application to cortical surface-based functional magnetic resonance imaging data from the Human Connectome Project illustrates its effectiveness and scalability for analyzing large-scale imaging datasets.
Keywords
Nonparametric Smoothing
Penalized Splines
Robust estimation
Robust Inference
Simultaneous Confidence Corridors
Triangulation