analyze directional data

Directional data are often constrained to local intervals on the unit circle (e.g., the first quadrant), and existing multivariate models struggle to simultaneously ensure bounded marginal supports and flexible dependence structures. To address this, we propose a copula-based Bayesian multivariate model: marginal variables are defined on subsets of the unit circle, while joint dependence is flexibly modeled via a copula function. We introduce a projected Gamma prior and a two-stage MCMC sampling scheme for posterior inference. This work constitutes the first systematic extension of multivariate modeling frameworks to bounded circular data. Through extensive simulations and real-data applications, we demonstrate the model’s high accuracy in estimating both the joint distribution and underlying parameters. It significantly enhances statistical modeling capability for restricted directional data, offering improved flexibility, interpretability, and inferential precision compared to existing approaches.

Nonparametric inference for directional/axial data is hindered by the intractability of normalizing constants in higher-order exponential families. Method: This paper proposes an empirical distribution framework based on Cartesian coordinates to nonparametrically estimate the mean direction, dispersion, and full distribution of spherical directional and axial data; axial symmetry is uniformly handled via projection matrices, and compact confidence sets are constructed using bootstrap resampling. Contribution/Results: The approach circumvents restrictive assumptions of classical exponential-family models, enabling multi-mean comparisons and trend testing in high dimensions. It yields model-free, geometrically consistent estimates of the distribution function on the sphere. Rigorously grounded in asymptotic theory and computationally feasible, this framework constitutes the first systematic nonparametric inferential toolkit for directional statistics.

Traditional geostatistical methods rely on second-order moments and Gaussian assumptions, which are inadequate for capturing non-Gaussian spatial dependence. This work addresses this limitation by leveraging Sklar’s theorem to introduce a copula-based framework that decouples marginal distributions from the spatial dependence structure. The authors systematically develop spatial copula models applicable at both fixed point sets and process levels, emphasizing Kolmogorov consistency to clarify distinctions between these two modeling paradigms. The framework is further extended to spatio-temporal settings and supports flexible marginal specifications. By integrating spatial statistics, copula theory, and stochastic processes, this study establishes a unified approach for modeling non-Gaussian spatial dependence, elucidating the relationships, strengths, and limitations of existing methodologies, thereby advancing both theoretical understanding and practical applications in the field.

Existing directional statistics tools are seldom adopted in engineering and computer science due to terminological barriers and lack of practical interfaces for modeling orientation data—such as angles, unit vectors, rotation matrices, and quaternions—in applications ranging from robotics to 3D vision. Method: We introduce the first comprehensive, practitioner-oriented reference guide for probability distributions over multi-degree-of-freedom orientation domains (1D–3D), employing a unified, engineering-friendly notation. The guide systematically presents density functions, maximum-likelihood parameter estimation procedures, and inverse-transform or rejection-sampling algorithms for six canonical directional distributions. Contribution/Results: We release an open-source Python library (built on NumPy/SciPy) supporting distribution fitting and random sampling. Empirical validation on robot pose calibration and 3D point cloud normal estimation demonstrates its practical efficacy, substantially bridging the gap between theoretical directional statistics and real-world engineering deployment.

Existing spatial data models struggle to simultaneously capture complex extremal features such as tail dependence, asymptotic independence, and tail asymmetry. This work proposes a novel approach by introducing multivariate Pareto mixture distributions into a spatial copula framework, yielding a flexible model capable of jointly modeling both bulk and tail behaviors. The resulting formulation provides a unified treatment of the three aforementioned extremal dependence structures while preserving permutation asymmetry. Based on copula theory and maximum likelihood estimation, the method is validated through finite-sample simulations that demonstrate its computational feasibility and favorable parameter estimation performance. Empirical analysis of temperature data successfully reveals intricate tail structures, underscoring the model’s theoretical rigor and practical utility.

This study addresses the challenge of distinguishing directional asymmetry from tail-ratio deviations in multivariate distributions by proposing a quantile-based projection diagnostic framework that avoids reliance on higher-order moments. The method integrates directional skewness and tail-ratio measures through one-dimensional projections, sparse rank-one computations, and directional search to robustly classify heavy-tailed multivariate distributions into four categories: symmetric baseline tails, symmetric tail deviations, skewed baseline tails, and skewed tail deviations. Theoretical analysis establishes population-level properties, finite-sample uniform bounds, and classifier consistency, while revealing the complementary roles of coordinate and random directions in high dimensions, thereby offering a reliable foundation for multivariate modeling choices.

This study addresses the inefficiency of variogram estimation in traditional multivariate extreme-value models that condition on a single component. To overcome this limitation, the authors propose a directional extremal variogram framework by conditioning the multivariate generalized Pareto vector on an arbitrary half-space, introducing a direction vector \( v \) to define the \( v \)-variogram and establishing its decomposition structure. Analytical expressions are derived for the Logistic, Dirichlet, and Hüsler–Reiss models, revealing a unique optimal direction \( v_0 \) in the Hüsler–Reiss case—linked to a resistance–curvature structure—that governs the bias–variance trade-off. Empirical estimation combining multiple directions substantially reduces variance and enhances statistical efficiency.

This work addresses the problem of multiple hypothesis testing for multivariate Gaussian means under arbitrary covariance dependence structures. The authors propose a novel approach that integrates maximum residual descent (MRD) with a multi-stage calibration scheme. By introducing a new representation of residual statistics based on a single active precision matrix, the method achieves covariance-adaptive residualization, substantially reducing computational complexity. It replaces model-dependent thresholds with a simple multi-stage calibration rule, combining generalized stepwise critical values and precision matrix reconstruction techniques. The proposed procedure significantly lowers the normalized misclassification risk across diverse dependence structures, achieving error discovery rate control close to the nominal level, extremely low missed detection rates, near-perfect statistical power, and accurate estimation of the number of true signals.

This study addresses the challenges of inaccurate parameter estimation and inadequate goodness-of-fit testing for the Frank copula in small-sample settings. To this end, the authors propose a Bayesian estimation approach based on Jeffreys’ prior and conduct a systematic comparison with maximum likelihood estimation. The results demonstrate that, for sample sizes of 25 or fewer, the Jeffreys-prior Bayesian estimator substantially outperforms existing methods in terms of mean squared error. Furthermore, the work uncovers counterintuitive behaviors of commonly used goodness-of-fit test statistics and provides reliable Monte Carlo–based critical value tables. The proposed methodology is successfully applied to model the dependence between groundwater arsenic concentrations and hydrochemical variables in Đồng Tháp Province, Vietnam, thereby enhancing the practical implementation of copula models in small-sample scenarios.

This study addresses the challenge of effectively estimating spatiotemporal dynamic dependencies in high-dimensional multivariate vector autoregressive (VAR) models, where parameter proliferation impedes reliable inference. The authors propose a novel decomposition of the transition matrix into inter-variable dependency coefficients and a spatial transition matrix, the latter constrained by a predefined spatial graph. Structured estimation is achieved via weighted ℓ₁-regularized least squares, embedding spatial graph information—hitherto unexplored in high-dimensional VAR modeling—directly into the estimation framework. The resulting biconvex optimization problem is efficiently solved using the alternating direction method of multipliers (ADMM), with theoretical guarantees established under stability and restricted eigenvalue conditions. Empirical results demonstrate that the proposed method significantly outperforms existing two-stage ℓ₁ approaches in both support recovery and estimation accuracy, successfully uncovering interpretable variable dependency networks and inter-regional spatial interaction patterns in North American climate data.