bayes-space fpca

This work proposes a dimensionality reduction and structural decomposition approach for multivariate probability density functions characterized by relative and constrained properties, formulated within the Bayesian paradigm. By embedding density functions into a Hilbert space via the centered log-ratio (clr) transformation and applying functional principal component analysis (FPCA), the method achieves effective dimensionality reduction. An orthogonal decomposition is further introduced to disentangle independent and interactive components. The core contribution lies in establishing an optimal variance decomposition framework for multivariate densities in the Bayes space, which endows FPCA outcomes with clear geometric and statistical interpretability. Experimental results on housing and geological datasets demonstrate the method’s efficacy in producing interpretable low-dimensional representations and accurately identifying constituent components.

This study addresses dimensionality reduction and covariance analysis for scalar-valued functional data indexed over a compact d-dimensional Riemannian manifold, where the indexing domain—not the function values—exhibits geometric structure. The work extends functional principal component analysis (FPCA) to manifold-indexed settings by introducing an intrinsic kernel estimation framework that incorporates geodesic distance and Riemannian volume density correction, accommodating heterogeneous sampling frequencies and weighted estimation strategies across individuals. Theoretically, by integrating intrinsic kernel methods, VC-type empirical process theory, and spectral perturbation analysis, the paper establishes uniform convergence rates for the mean, covariance, and eigen-objects under non-Lipschitz kernels. It further reveals that the transition from sparse to dense observation regimes is governed by the intrinsic manifold dimension, reducing to classical results when d = 1. Experiments on S¹, S², and real-world SONICOM head-related transfer function data demonstrate consistent and substantial performance gains over baseline methods that ignore geometric structure.

This paper addresses the problem of functional principal component estimation from noisy, discretely sampled functional data, with the goal of characterizing the statistical impact of denoising preprocessing. Under a double-asymptotic regime where both sample size and number of observation grid points grow, we propose a histogram-projection-based estimator for functional principal components. We establish, for the first time under realistic joint constraints of noise and discrete sampling, the minimax optimal convergence rate for functional principal component estimation and rigorously prove that our method achieves this rate. Theoretically, we show that smoothing preprocessing improves the convergence order but does not alter the fundamental minimax difficulty. Extensive simulations validate the method’s effectiveness, and we demonstrate its practical utility through functional visualization analysis of genomic data.

Traditional functional principal component analysis (FPCA) treats principal component estimates as deterministic quantities, neglecting their sampling variability and thereby yielding inaccurate uncertainty quantification. To address this, we propose Bayes-FPCA—a fast Bayesian FPCA framework that models principal components directly on the Stiefel manifold for the first time. It achieves efficient dimensionality reduction via orthogonal spline basis projection and constructs a uniform prior on the manifold using the polar decomposition. A stable MCMC sampling strategy is designed to respect the structural constraints inherent in FPCA. Evaluated on DASH4D continuous glucose monitoring data, Bayes-FPCA accurately characterizes postprandial glucose dynamics and substantially improves uncertainty estimation accuracy while accelerating computation by an order of magnitude. All code and simulation routines are publicly available.

This paper addresses the challenge of modeling population-level variability in replicated point processes. We propose a novel functional principal component analysis (fPCA) framework grounded in random measures and the cumulative mass function (CMF). Introducing the concept of “principal measures,” we establish the Karhunen–Loève expansion for random measures and derive a Mercer-type theorem for their covariance measures, enabling consistent parameter-rate estimation of eigencomponents. The method integrates fPCA, random measure theory, and nonparametric/semiparametric estimation, yielding closed-form solutions for Poisson and Hawkes processes. Evaluated on seismological, single-cell spatial transcriptomic, and neurophysiological datasets, our approach significantly improves both the accuracy of identifying population-level variation structures in point patterns and their biological interpretability.

This study addresses the limited power of existing two-sample tests for functional data in finite-sample settings by proposing a hybrid Gaussian random projection method that integrates Haar and Fourier Gaussian components to simultaneously capture local discontinuities and global oscillatory differences. The approach innovatively incorporates a label-invariant, data-adaptive covariance operator, enhancing detection sensitivity while preserving the validity of permutation-based inference. Theoretical analysis establishes the consistency of the proposed test, and empirical evaluations demonstrate its strong specificity and robustness in simulations. When applied to the ECG5000 dataset, the method effectively identifies class distinctions primarily driven by local features and significantly outperforms competing approaches, particularly in scenarios involving changes in covariance structure.

This work addresses the challenge of modeling circular density data characterized by periodicity and relative structure by proposing a novel periodic spline approach within the Bayes space framework. By applying the centered log-ratio transformation, densities are mapped into an L² subspace subject to a zero-integral constraint, enabling the construction of spline bases that simultaneously respect periodicity and Hilbert space structure. The method unifies smoothing and penalized spline estimation in a matrix formulation for computational efficiency. It represents the first integration of periodic splines with Bayes space theory, preserving the relative nature and interpretability of densities while facilitating subsequent functional data analysis. Experiments on wind direction data demonstrate that the proposed approach yields smooth, plausible, and interpretable density estimates, offering a new paradigm for modeling complex circular density data.