circular statistics

This study addresses the lack of effective visualization methods for grouped periodic angular data in fields such as psychology, genomics, and meteorology. The authors propose concentric circular boxplots and circular quartile plots to characterize grouped angular distributions, and further extend the approach to a three-dimensional toroidal visualization for multiple groups to reveal periodic patterns. A novel scaling strategy is introduced, wherein box width is inversely proportional to the square root of the distance from the center, enhancing visual perception. This work is the first to integrate concentric circular boxplots with toroidal 3D visualization specifically for angular data. The effectiveness and practical utility of the proposed methods are demonstrated through applications to real-world datasets, including motor resonance phases, circadian clock gene expression phases, and periodic wind direction patterns.

This paper addresses regression tasks involving circular-valued outputs (e.g., angles, phases) by proposing a Bayesian nonparametric model based on the von Mises distribution. Unlike conventional approaches—such as wrapped Gaussian processes or radial marginalization—the method constructs a “von Mises pseudo-process” that directly models the input–output mapping on the unit circle, inheriting both maximum-entropy density properties and interpretability. Methodologically, it introduces Stratonovich-style data augmentation to enable efficient Gibbs sampling and devises a novel dual Metropolis–Hastings algorithm for Bayesian inference on circular domains. Evaluated on wind direction forecasting and gait cycle phase estimation, the model achieves state-of-the-art predictive accuracy, superior calibration, and reliable uncertainty quantification—outperforming existing circular regression methods across all metrics.

Existing methods struggle to effectively model the joint dependence between discrete circular variables—such as finitely many equally spaced directional observations—and continuous linear variables. This work proposes the first analytically tractable joint model, employing a wrapped symmetric geometric distribution for the discrete circular component and a Weibull distribution for the linear component, with their dependence structure induced via a trigonometric linking function. The model yields closed-form marginal and conditional distributions along with conditional moments, offers clear theoretical interpretations of its parameters, and establishes monotonicity of the conditional mean and variance with respect to the dependence parameter. Moreover, it enables direct sampling through the inverse transform method. Simulation studies demonstrate excellent parameter estimation performance, and applications to two real-world environmental datasets underscore the model’s effectiveness and practical utility.

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 paper addresses the challenge of detecting joint changepoints in circular time series—such as wind direction—where both the mean direction and concentration undergo simultaneous abrupt shifts. We propose the first nonparametric changepoint test grounded in toroidal differential geometry. By defining an intrinsic “squared angle” and a novel “curvature variance” measure, we construct a unified framework capable of detecting changes in direction, concentration, and their co-variation. Theoretically, we derive the asymptotic distribution of the test statistic; empirically, we validate its efficacy via Monte Carlo simulations and real-world meteorological data. Simulation results demonstrate substantial superiority over existing linear or single-parameter methods. Applied to cyclone “Amphan” wind data, our method successfully identifies joint changepoints aligned with landfall and intensity surges. Further analyses across three meteorological datasets yield scientifically interpretable insights. Our core innovation lies in pioneering the integration of toroidal geometry into circular-data changepoint analysis, enabling highly sensitive, geometrically consistent detection of joint structural breaks.

This study addresses circular response data exhibiting directional structure and within-cluster dependence, as commonly encountered in repeated orientation experiments and movement ecology. The authors propose a mixed-effects model within the generalized angular regression framework, defining the mean direction via a two-dimensional consensus vector and incorporating both a von Mises-distributed circular random intercept and a Gaussian scalar random slope. By deriving an analytical marginalization, the marginal likelihood is reduced to a one-dimensional integral, circumventing computationally intensive high-dimensional numerical integration. The work establishes conditions for model identifiability and develops associated asymptotic theory. Simulation studies demonstrate excellent numerical stability and favorable finite-sample performance. Application to sandhopper orientation data confirms the validity and practical utility of the proposed approach for variance component inference.

This work addresses the geometric misalignment inherent in traditional regression approaches when modeling multimodal, skewed, or asymmetric circular data, as they only predict the conditional mean. To overcome this limitation, the authors propose ANGLE, a lightweight, nonparametric framework for circular distributional regression. ANGLE leverages deep generative models to learn the full conditional distribution of angular responses, incorporating pre- and post-additive noise modeling and optimizing via a generalized circular energy score (GCES) loss. The method is theoretically grounded, ensuring strict propriety and rotational equivariance, and enables distributional extrapolation, effective dimensionality reduction, and equivalence testing of conditional distributions. Empirical evaluations on object pose estimation and wind direction forecasting demonstrate that ANGLE substantially outperforms existing methods, delivering both high predictive accuracy and reliable uncertainty quantification.

Many datasets are observed on a finite set of equally spaced directions instead of the exact angles, such as the wind direction data. However, in the statistical literature, bivariate models are only available for continuous circular random variables. This article presents two bivariate circular distributions, namely bivariate wrapped geometric (BWG) and bivariate generalized wrapped geometric (BGWG), for analyzing bivariate discrete circular data. We consider wrapped geometric distributions and a trigonometric function to construct the models. The models are analytically tractable due to the exact closed-form expressions for the trigonometric moments. We thoroughly discuss the distributional properties of the models, including the interpretation of parameters and dependence structure. The estimation methodology based on maximizing the likelihood functions is illustrated for simulated datasets. Finally, the proposed distributions are utilized to analyze pairwise wind direction measurements obtained at different stations in India, and the interpretations for the fitted models are briefly discussed.

This study addresses the limitations of existing circular logistic regression methods, which are confined to symmetric link functions and struggle to effectively model the relationship between circular predictors and binary or binomial responses. The authors propose a generalized linear framework that incorporates circular covariates into the linear predictor via sine and cosine transformations and, for the first time, systematically evaluates the performance of both symmetric and asymmetric link functions. Through Monte Carlo simulations based on the von Mises distribution—assessed using AIC, deviance, and empirical analyses of meteorological and seismic data—the study reveals that link function choice significantly impacts model performance when circular data are dispersed and responses are imbalanced. Under high concentration, symmetric links demonstrate greater robustness, whereas asymmetric links tend to be unstable. This work extends the theoretical boundaries of circular–binary response modeling and offers practical guidance for applied researchers.

This study addresses the challenge of accurately modeling how sensory conditions influence directional errors in spatial orientation, particularly when covariates comprise a mixture of continuous and categorical variables. To this end, the authors propose a novel nonparametric circular regression framework that integrates product kernel estimation to handle mixed-type covariates and introduces a bootstrap-based bandwidth selection criterion tailored to the cosine loss function inherent to circular responses. This approach extends nonparametric circular regression to mixed-covariate settings for the first time and constructs simultaneous confidence bands to quantify estimation uncertainty. Evaluations on both simulated and real-world data—including participants with blindness, low vision, and normal vision—demonstrate the method’s superior bias-variance trade-off, its robustness in uncovering nonlinear patterns of directional error across sensory conditions, and its reliability for statistical inference.