estimate participation ratio and anisotropy

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.

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.

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.

Existing spatial spectral analysis methods are constrained by data types (e.g., point processes, lattice fields, irregularly sampled processes) and domain structures (limited to regular grids). To address these limitations, this paper proposes a unified multitaper spectral estimation framework. Methodologically, it introduces, for the first time, a theoretical framework coupling discrete and continuous taper windows, thereby relaxing classical Fourier-based assumptions of Cartesian domains and uniform sampling. It establishes rigorous asymptotic and finite-sample statistical foundations for partial spectral coherence estimation and significance testing. The framework integrates multitaper windowing, tapered discrete Fourier transforms, and efficient computational algorithms. Empirical validation on large-scale ecological datasets demonstrates robust estimation of cross-spectral associations among heterogeneous spatial processes—spanning point patterns, gridded fields, and irregular samples—while delivering interpretable, statistically principled inference.

This work addresses local structure modeling of point clouds in product spaces endowed with mixed Euclidean and directional metrics. We propose the first subspace-constrained mean shift algorithm tailored to such hybrid metric spaces, enabling joint estimation of density modes and density ridges. Theoretically, we establish convergence guarantees on product manifolds and provide practical implementation criteria. By integrating manifold gradient analysis with a customized product-space metric design, the method enhances interpretability and fidelity in capturing heterogeneous multi-source structures. Experiments on synthetic and real-world data—including 3D human poses and motion trajectories—demonstrate substantial improvements over state-of-the-art approaches in mode and ridge localization accuracy, robustness to noise, and structural interpretability. Our framework establishes a new paradigm for density-based topological modeling in complex geometric domains.

This study addresses the problem of testing statistical hypotheses concerning the independence or homogeneity between marks and spatial locations in marked point processes at local scales. It proposes a chi-square-type test statistic based on a local inhomogeneous mark-weighted K-function—a novel application of such K-functions for constructing test statistics. The proposed method simultaneously detects both global and local departures from the null hypothesis and maintains high sensitivity even when mark structures are weak or sample sizes are limited. Empirical validation using real-world datasets, including forest ecology and seismic event data, demonstrates its effectiveness in identifying spatially dependent mark structures in complex scenarios, substantially improving the accuracy and applicability of local pattern inference.

Existing evaluation metrics for generative spatial audio lack systematic investigation into their response characteristics under variations in spatial parameters such as azimuth and elevation. This work proposes the first sensitivity analysis framework that evaluates multiple metrics along continuous spatial trajectories, introducing three key criteria: responsiveness, smoothness, and symmetry. The framework is empirically applied to metrics including Fréchet Audio Distance (FAD), intensity vectors, and acoustic maps within controlled scenes of varying complexity. Results demonstrate that FAD based on directional embeddings and acoustic maps consistently excel across all three criteria, whereas intensity vectors exhibit significant performance degradation as scene complexity increases. These findings reveal substantial differences in the sensitivity of existing metrics to spatial variations, offering critical insights for the design and selection of evaluation methods in spatial audio generation.

Traditional marked point process analyses rely on global statistics and struggle to capture local spatial heterogeneity. This study proposes the Local Indicator of Mark Association (LIMA) framework, which for the first time incorporates compositional marks into local spatial analysis. By leveraging the centered log-ratio (clr) transformation and Aitchison geometry, LIMA maps compositional data into Euclidean space, enabling a pointwise decomposition of mark structure. The method effectively uncovers local clustering and “siphoning” effects that are obscured by global approaches. Simulation experiments demonstrate that LIMA substantially outperforms existing global methods in detecting local clusters. When applied to economic data from Castilla–La Mancha, Spain, LIMA successfully reveals latent regional economic agglomeration patterns.

Existing nonparametric statistical testing methods based on surrogate data are primarily designed for undirected graphs and are ill-suited for directed graph structures. This work extends such approaches to the directed graph setting for the first time by defining wide-sense stationary signals through the eigendecomposition of graph shift operators and constructing a surrogate signal generation framework that preserves the covariance structure. Evaluated on real-world data, the proposed method significantly outperforms conventional undirected-graph approaches and naive permutation strategies, offering enhanced statistical power while maintaining test validity and practical feasibility.

This study addresses the susceptibility of the conventional Moran’s I statistic to distributional outliers, which can introduce bias in both global and local spatial autocorrelation estimates. The authors systematically evaluate and compare three classes of robust estimation methods—plug-in robust estimators, trimmed least squares (TLS), and Theil–Sen–type estimators—for constructing robust Moran indices and LISA maps, accompanied by tailored visualization strategies. This work presents the first comprehensive comparison of multiple robust spatial association measures and proposes the Theil–Sen Moran estimator as a preferred default for exploratory spatial data analysis. Empirical results demonstrate its superior performance across diverse scenarios, while plug-in methods also exhibit strong scalability and reliability in large datasets, collectively enhancing the robustness of spatial outlier detection.