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Circular distribution

In probability and statistics, a circular distribution is a probability distribution defined on the unit circle, modeling random variables that represent angles, directions, or other cyclical phenomena where the scale wraps around (e.g., 0 and 360 degrees are equivalent).^[1] These distributions are essential for analyzing data with inherent periodicity, such as compass bearings or times of day, as standard linear distributions fail to account for the modulo 2π topology of the circle.^[2] Circular distributions form the foundation of directional statistics (also known as circular statistics), a subfield that addresses challenges like measuring central tendency (e.g., mean direction via the resultant vector) and dispersion (e.g., circular variance, ranging from 0 for full concentration to 1 for uniformity) differently from linear methods.^[3] Key properties include rotational symmetry in probability density and the use of trigonometric moments to summarize data, enabling tests for uniformity (e.g., Rayleigh's test) and goodness-of-fit assessments.^[4] Prominent examples include the circular uniform distribution, which assigns equal probability to all directions and serves as a null model for randomness; the von Mises distribution, a symmetric, unimodal analog to the Gaussian distribution characterized by a mean direction μ and concentration parameter κ (where higher κ indicates tighter clustering), introduced by Richard von Mises in 1918; and the wrapped normal distribution, derived by folding a normal distribution onto the circle to model moderate concentrations.^[3]^[4] Other families encompass intrinsic distributions (natively defined on the circle, like the cardioid) and wrapped distributions (over 45 continuous variants, including wrapped exponential and Cauchy, obtained by summing linear densities over integer cycles).^[1] The development of circular distributions traces back to 18th-century astronomical inquiries by figures like Bernoulli and Gauss, but the field formalized in the early 20th century with Rayleigh's 1919 test for uniformity and von Mises's foundational work on the "circular normal."^[4] Mid-century advances by Watson, Guttman, and others introduced correlation and regression tools, while 1970s–1980s contributions from Mardia (e.g., comprehensive inference frameworks) and Batschelet (e.g., biological applications in his 1981 book Circular Statistics in Biology) expanded parametric and nonparametric methods.^[4] Recent innovations include weighted and skew variants for asymmetric data, score-driven models for time series, and software like the R package "Circular" for fitting and simulation.^[5] Applications are diverse, including environmental science (e.g., modeling wind or ocean current directions), biology (e.g., animal migration paths or circadian rhythms), astronomy (e.g., star orientations), and even econometrics and machine learning for periodic patterns.^[1]^[2] With growing data volumes in these areas, circular distributions continue to evolve, supporting advanced analyses like mixture models and Bayesian inference on the circle.^[2]

Fundamentals

Definition

A circular random variable is a random variable that represents an angle or direction, typically taking values in the interval [0, 2\pi) or equivalently on the unit circle.^[6] Unlike linear random variables defined on the real line, circular random variables account for the periodic and bounded nature of angular data, where the endpoints 0 and $2\pi are identified as the same point.^[7] The key distinction between circular and linear distributions lies in the failure of standard linear statistics, such as the arithmetic mean and variance, to appropriately summarize circular data due to the wrap-around property. For example, angles of 359° and 1° (or approximately $6.28 radians and $0.02 radians) are spatially adjacent on the circle, yet their linear average of 180° misrepresents their proximity. This periodicity necessitates specialized measures, like circular means and variances, to capture the geometry of the data.^[7] Formally, a circular distribution is a probability measure on the circle, specified by a probability density function f(\theta) where f(\theta) \geq 0 for \theta \in [0, 2\pi) and the normalization condition holds:

\int_0^{2\pi} f(\theta) \, d\theta = 1. $&#36; [](https://faculty.ucr.edu/~ashis/publication/publications/CALMAT04.PDF) The sample space for such distributions is the circle group $S^1$, a one-dimensional manifold topologically equivalent to the unit circle in the plane, parameterized by the angle $\theta$.[](https://arxiv.org/pdf/2104.03194) This framework extends to higher dimensions, such as the torus for multivariate circular data, but the univariate case focuses on $S^1$.[](https://arxiv.org/pdf/2104.03194) Examples of circular data include wind directions in meteorology and animal orientations in biology, which motivate the development of these distributions.[](https://math.montana.edu/grad_students/writing-projects/2002/02scott.pdf) ### Historical Development The origins of circular distribution theory trace back to 18th-century astronomical inquiries, but gained momentum in the 19th century, when astronomers and meteorologists encountered challenges in analyzing angular data, such as celestial positions and wind directions, which did not conform to linear statistical models.[](https://ecommons.luc.edu/cgi/viewcontent.cgi?article=3500&context=luc_diss) Formal statistical treatments began to emerge in the early 20th century, with Lord Rayleigh's 1919 work on random flights in two dimensions providing an early foundation for understanding uniform distributions on the circle, particularly in meteorological contexts like wind direction patterns. This paper introduced probabilistic models for isotropic scattering, laying groundwork for later developments in directional statistics. A pivotal example in the early 20th century was the von Mises distribution, proposed by Richard von Mises in 1918 as a circular analogue to the normal distribution for modeling concentrated angular data. In the mid-20th century, Ronald A. Fisher formalized key inference methods for directional data, particularly in his seminal 1953 paper "Dispersion on a Sphere," which established the Fisher distribution for spherical orientations.[](https://arxiv.org/pdf/2405.17919) These contributions marked a shift toward rigorous statistical frameworks for non-linear data. The 1950s saw advancements in wrapped distributions, with Gumbel, Greenwood, and Durand's 1953 paper introducing the wrapped normal (circular normal) distribution, extending linear extreme value theory to the circle for applications in meteorology and beyond. By the 1960s, Fourier analysis influenced the field through the development of trigonometric moments, enabling efficient characterization of circular densities via harmonic expansions, as explored in early spectral methods for angular data. Modern formalization arrived with K. V. Mardia's 1972 book *Statistics of Directional Data*, which synthesized models, moments, and inference techniques into a comprehensive framework, solidifying directional statistics as a distinct subfield.[](https://www.sciencedirect.com/book/9780124711501/statistics-of-directional-data) Subsequent work in the 1980s, including Peter E. Jupp and Mardia's 1980 paper on correlation coefficients for directional data and their 1989 review of theoretical advances from 1975–1988, advanced inference methods and unified disparate approaches. ## Mathematical Framework ### Probability Density Functions The probability density function (PDF) of a circular distribution, denoted as $f(\theta; \boldsymbol{\eta})$ where $\boldsymbol{\eta}$ represents the parameters, is defined on the interval $[0, 2\pi)$ and satisfies periodic boundary conditions such that $f(0) = f(2\pi)$.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) This periodicity reflects the inherent circular topology, ensuring continuity across the identification of 0 and $2\pi$.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) The normalization condition for such PDFs requires that $\int_0^{2\pi} f(\theta) \, d\theta = 1$.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) A common representation exploits the periodic nature through a Fourier series expansion, expressed in real form as

f(\theta) = \frac{1}{2\pi} + \sum_{k=1}^\infty \left[ a_k \cos(k\theta) + b_k \sin(k\theta) \right],

where the coefficients $a_k = \frac{1}{\pi} \int_0^{2\pi} f(\theta) \cos(k\theta) \, d\theta$ and $b_k = \frac{1}{\pi} \int_0^{2\pi} f(\theta) \sin(k\theta) \, d\theta$ for $k \geq 1$.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) Equivalently, in complex exponential form,

f(\theta) = \frac{1}{2\pi} \sum_{p=-\infty}^\infty \phi_p e^{-i p \theta},

with $\phi_p = \int_0^{2\pi} f(\theta) e^{i p \theta} \, d\theta = \mathbb{E}[e^{i p \theta}]$ denoting the $p$-th trigonometric moment, which inherently preserves normalization since $\phi_0 = 1$.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) The uniform distribution serves as the baseline case, where $f(\theta) = 1/(2\pi)$ and all higher-order coefficients vanish.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) Circular PDFs exhibit varied symmetry properties, ranging from unimodal densities with a single mode to multimodal ones featuring multiple peaks around the circle.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) Reflection symmetry, often termed axial symmetry in this context, occurs when the density satisfies $f(\theta) = f(\pi - \theta)$ (up to a location shift), indicating invariance under reflection across a specific diameter, whereas full rotational symmetry implies uniformity independent of $\theta$.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) To derive circular distributions from linear ones, a common transformation wraps the support onto the circle using the two-argument arctangent function: for a bivariate normal random vector $(X, Y)$ with mean vector $(\mu_x, \mu_y)$ and covariance matrix $\Sigma$, the angle $\theta = \atantwo(Y, X) \mod 2\pi$ yields the projected normal distribution as the induced density on $[0, 2\pi)$.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) This mapping preserves the probabilistic structure while accounting for the modular arithmetic of angles.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) ### Moments and Characteristic Functions In circular statistics, trigonometric moments provide a fundamental way to summarize the location, dispersion, and higher-order features of a distribution on the unit circle. The $k$-th order trigonometric moment is defined in complex form as $\rho_k = E[e^{ik\theta}] = \int_0^{2\pi} e^{ik\theta} f(\theta) \, d\theta$, where $f(\theta)$ is the probability density function and $k$ is an integer; the magnitude satisfies $|\rho_k| \leq 1$, with equality holding only for degenerate distributions concentrated at a single point. These moments fully characterize unimodal circular distributions due to the Fourier series representation of $f(\theta)$. The first-order trigonometric moment $\rho_1$ determines the mean direction $\mu = \arg(\rho_1)$, which serves as the circular analogue of the linear mean. For a sample of $n$ observations $\theta_1, \dots, \theta_n$, the sample mean direction is estimated as $\hat{\mu} = \atantwo\left( \frac{1}{n} \sum_{i=1}^n \sin \theta_i, \frac{1}{n} \sum_{i=1}^n \cos \theta_i \right)$, where $\atantwo(y, x)$ is the two-argument arctangent function that accounts for the correct quadrant. The mean resultant length $R = |\rho_1|$ measures concentration around $\mu$, with $R = 1$ indicating no dispersion and $R = 0$ corresponding to uniformity. The circular variance is then $V = 1 - R$, ranging from 0 (perfect concentration) to 1 (maximum dispersion under uniformity). Higher-order moments $\rho_k$ for $k \geq 2$ capture skewness and kurtosis; for example, they are used in parameter estimation for distributions like the von Mises, where the concentration parameter relates to $R$. The characteristic function for a circular random variable $\theta$ is $\phi(t) = E[e^{it\theta}]$ for real $t$, analogous to the linear case but inherently periodic with $\phi(t + 2\pi k) = \phi(t)$ for integer $k$. Due to the compact support on $[0, 2\pi)$, $\phi(t)$ equals the Fourier coefficients of $f(\theta)$, specifically $\phi(k) = a_k + i b_k$ for integer $k$, linking directly to the trigonometric moments $\rho_k = \phi(k)$. This connection facilitates moment-generating properties and expansions for inference. Cumulants derive from the logarithm of the characteristic function, $\kappa(t) = \log \phi(t)$, providing additive measures for circular skewness and kurtosis. The circular skewness is given by $\gamma_1 = \frac{1 - |\rho_2|^2}{(1 - |\rho_1|^2)^{3/2}}$, which quantifies asymmetry relative to the mean direction and equals zero for symmetric distributions; its absolute value is bounded by 2. Higher cumulants, such as those for kurtosis, follow similar expansions but are less commonly used due to interpretability challenges on the circle. ## Visualization ### Graphical Methods Circular histograms provide a fundamental visualization for circular distributions by dividing the circle into equal angular sectors, or bins, and representing the frequency of observations in each sector as radial bars or segments. The choice of bin width is crucial, as narrower bins offer higher resolution but require larger sample sizes to avoid excessive noise, while wider bins smooth the display at the potential cost of obscuring fine details; typically, the number of bins is selected proportional to the square root of the sample size to balance these trade-offs.[](https://www.cambridge.org/core/books/statistical-analysis-of-circular-data/324A46F3941A5CD641ED0B0910B2C33F) Kernel density estimation plots offer a smoother alternative for visualizing circular distributions, employing a von Mises kernel to account for the periodic nature of the data. The estimated density function is approximated as

\hat{f}(\theta) \approx \frac{1}{n} \sum_{i=1}^n K\left( \frac{\theta - \theta_i}{h} \right),

where $ K $ denotes the von Mises kernel, $ \theta_i $ are the observed angles, $ n $ is the sample size, and $ h $ is the bandwidth parameter controlling smoothness. Bandwidth selection methods, such as plug-in rules, ensure the estimate adapts to the data's concentration while preserving angular topology.[](https://www.sciencedirect.com/science/article/abs/pii/S0167947307004367)[](https://www.cambridge.org/core/books/statistical-analysis-of-circular-data/324A46F3941A5CD641ED0B0910B2C33F) Q-Q plots for circular data facilitate comparison between the empirical distribution and a theoretical circular distribution, such as the von Mises, by plotting ordered angular deviations or transformed quantiles against their expected values under the model. These plots preserve circular structure through angular unwrapping or projection techniques, highlighting deviations in tail behavior or modality via departures from a straight reference line.[](https://www.cambridge.org/core/books/statistical-analysis-of-circular-data/324A46F3941A5CD641ED0B0910B2C33F) Software implementations, such as the R package 'circular', enable the generation of these plots through dedicated functions for histograms, density estimates, and Q-Q comparisons, supporting seamless integration into statistical workflows.[](https://cran.r-project.org/web/packages/circular/circular.pdf) These methods are particularly useful for displaying densities of uniform or von Mises distributions in exploratory analysis. ### Interpretations of Plots In visualizations of circular distributions, such as rose diagrams or circular histograms, the concentration of data is inferred from the spatial arrangement of points or bars. Tight clustering around a mean direction signifies high concentration and low angular variance, corresponding to a mean resultant length $ R $ close to 1, where $ R = \left| \sum_{j=1}^n \exp(i \theta_j) \right| / n $ measures the length of the vector sum relative to sample size $ n $.[](https://books.google.com/books/about/Statistical_Analysis_of_Circular_Data.html?id=wGPj3EoFdJwC) Conversely, a broad spread with points or bars distributed evenly across the circle indicates low concentration approaching uniformity, with $ R $ near 0, reflecting no preferred direction.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) Multimodality is detected through the presence of multiple distinct peaks or clusters in the plot. For example, two or more separated concentrations in a rose diagram suggest a bimodal or multimodal structure, often arising in applications like migration directions or seasonal patterns, whereas a single dominant peak points to a unimodal distribution.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) Such visual cues provide intuitive evidence of underlying subpopulations or modes before quantitative analysis. Interpreting circular plots requires caution against linear projections, which unfold the circle into a line and distort tail behavior, especially for data straddling the 0/2π boundary. This can create misleading gaps or artificial multimodality, as seen in wrapped distributions where endpoints are contiguous.[](https://books.google.com/books/about/Statistical_Analysis_of_Circular_Data.html?id=wGPj3EoFdJwC) Comparative plots overlay empirical distributions with fitted theoretical curves to assess goodness of fit visually. Deviations, such as mismatches in peak heights or locations, highlight poor alignment; these can be summarized by Kuiper's statistic $ V = \sup_{\theta} |F_n(\theta) - F(\theta)| $, where $ F_n(\theta) $ is the empirical cumulative distribution function and $ F(\theta) $ the theoretical one, with smaller $ V $ values indicating closer agreement.[](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Circular_Data_Analysis.pdf) ## Specific Distributions ### Circular Uniform Distribution The circular uniform distribution is the simplest and most fundamental probability distribution on the circle, characterized by equal likelihood of all directions or angles. It arises naturally as the unique distribution that is invariant under arbitrary rotations and reflections of the circle, making it an ideal baseline for modeling random directional data without any preferred orientation. In directional statistics, this distribution represents complete isotropy, where observations are scattered uniformly around the circumference, analogous to a fair randomization of angles. The probability density function (PDF) of the circular uniform distribution is constant over the interval $[0, 2\pi)$, given by

f(\theta) = \frac{1}{2\pi}, \quad \theta \in [0, 2\pi).

This constant density ensures that the total probability integrates to 1, as $\int_0^{2\pi} f(\theta) \, d\theta = 1$. A key property is that every direction is equally probable, reflecting perfect uniformity with no concentration toward any particular angle. The mean resultant length, which measures the concentration of the distribution, is $R = 0$, indicating maximal dispersion. Additionally, all trigonometric moments vanish except the zeroth-order moment: $\rho_k = E[\cos(k\theta)] + i E[\sin(k\theta)] = 0$ for $k \neq 0$, and $\rho_0 = 1$, underscoring the lack of any directional bias. The cumulative distribution function (CDF) is linear and straightforward:

F(\theta) = \frac{\theta}{2\pi}, \quad \theta \in [0, 2\pi).

Sampling from this distribution can be achieved directly through inversion of the CDF or by generating angles from a standard linear uniform distribution. Specifically, if $U \sim \text{Uniform}[0, 1)$, then $\theta = 2\pi U$ yields a sample from the circular uniform distribution. This method leverages the simplicity of uniform random number generators available in most statistical software. The circular uniform distribution is directly related to the linear uniform distribution on $[0, 1)$ (or equivalently $[-\pi, \pi)$) via a modular wrapping transformation, where angles are projected onto the circle by adding multiples of $2\pi$ as needed. This wrapping preserves uniformity, providing a bridge between linear and circular statistical frameworks. As a null model, it serves as a baseline for assessing deviations toward more concentrated distributions, such as the von Mises. ### von Mises Distribution The von Mises distribution serves as the primary unimodal model for circular data exhibiting a preferred direction, analogous to the normal distribution on the linear scale. It is symmetric about its mean direction and widely applied in fields such as biology, geology, and meteorology to analyze angular observations like animal orientations, fault directions, or wind patterns. The distribution's tractability stems from its simple functional form involving trigonometric and Bessel functions, making it suitable for both theoretical analysis and computational implementation. The probability density function is

f(\theta; \mu, \kappa) = \frac{1}{2\pi I_0(\kappa)} \exp\left(\kappa \cos(\theta - \mu)\right),

where $\theta \in [0, 2\pi)$, $\mu \in [0, 2\pi)$ denotes the mean direction (location parameter), $\kappa \geq 0$ is the concentration parameter, and $I_0(\kappa)$ is the modified Bessel function of the first kind and order zero, which acts as a normalizing constant.[](http://www.math.ualberta.ca/~thillen/paper/vonmises.pdf) When $\kappa = 0$, the density simplifies to the circular uniform distribution, indicating no directional preference. As $\kappa$ increases, the distribution concentrates more sharply around $\mu$; for large $\kappa$, it approximates the wrapped normal distribution, providing a Gaussian-like behavior on the circle with effective variance $1/\kappa$.[](https://link.springer.com/article/10.1007/s11222-025-10563-4) The mean resultant length, a key summary statistic equivalent to the magnitude of the first trigonometric moment, is given by $\rho_1 = I_1(\kappa)/I_0(\kappa)$, where $I_1(\kappa)$ is the modified Bessel function of order one. This $\rho_1$ measures linear concentration and ranges from 0 (uniform case) to 1 (degenerate concentration). For large $\kappa$, the approximation $\rho_1 \approx 1 - 1/(2\kappa)$ holds, yielding a circular variance of $1 - \rho_1 \approx 1/\kappa$.[](http://www.math.ualberta.ca/~thillen/paper/vonmises.pdf) Proposed by Richard von Mises in 1918 to model deviations in atomic weights treated circularly, the distribution has since become foundational in directional statistics. ### Wrapped Normal Distribution The wrapped normal distribution arises as the circular analog of the univariate normal distribution, constructed by wrapping a linear normal random variable around the unit circle via the modulo 2π operation. This results in an infinite sum of normal densities shifted by multiples of 2π, providing a flexible model for directional data that is symmetric and unimodal when concentrated. It is particularly useful in applications such as wind direction analysis or animal orientation studies, where data exhibit Gaussian-like spread on the line but periodicity on the circle.[](https://doi.org/10.3390/math12162440) The probability density function of the wrapped normal distribution is given by

f(\theta; \mu, \sigma^2) = \frac{1}{\sigma \sqrt{2\pi}} \sum_{k=-\infty}^{\infty} \exp\left( -\frac{(\theta + 2\pi k - \mu)^2}{2\sigma^2} \right),

where $\theta \in [0, 2\pi)$, $\mu \in \mathbb{R}$ is the location parameter representing the mean direction (modulo $2\pi$), and $\sigma^2 > 0$ is the spread parameter controlling concentration. For small values of $\sigma$, the distribution peaks sharply around $\mu \mod 2\pi$, reflecting low dispersion; as $\sigma$ increases, the density flattens and approaches the uniform distribution on the circle, capturing high uncertainty in directional data. An equivalent representation uses the Fourier series form, where the characteristic coefficients are $\phi_k = \exp(i k \mu - k^2 \sigma^2 / 2)$ for integer $k$, allowing efficient computation via trigonometric expansion for numerical evaluation.[](https://tisl.cs.utoronto.ca/publication/201410-sdf-wrapped_normal_evaluation/201410-sdf-wrapped_normal_evaluation.pdf)[](https://doi.org/10.3390/math12162440) The trigonometric moments of the distribution are $\rho_k = \exp(i k \mu - k^2 \sigma^2 / 2)$, with the first-order moment $\rho_1 = \exp(i \mu - \sigma^2 / 2)$ yielding the mean resultant length $|\rho_1| = \exp(-\sigma^2 / 2)$. The circular variance, a key measure of dispersion on the circle, is then $1 - \exp(-\sigma^2 / 2)$, which ranges from 0 (perfect concentration at $\mu$) to 1 (uniform dispersion). This variance quantifies how much the distribution deviates from a Dirac delta at the mean direction, with higher values indicating greater angular spread.[](https://doi.org/10.3390/math12162440) The wrapped normal distribution maintains a direct connection to the bivariate normal distribution and is closely related to the projected normal distribution through geometric interpretation on the plane, where the angle arises from the direction of a non-centered bivariate Gaussian vector. This linkage highlights its utility in embedding linear Gaussian models into circular settings while approximately preserving normality properties. For moderate values of $\sigma$, the wrapped normal can be approximated by the von Mises distribution for simpler inference. ### Wrapped Cauchy Distribution The wrapped Cauchy distribution arises from wrapping the standard Cauchy distribution on the real line around the unit circle, resulting in a circular probability distribution characterized by heavy tails and symmetry about its location parameter.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) This distribution is particularly useful for modeling directional data where extreme deviations are more probable than in lighter-tailed alternatives, such as the wrapped normal distribution. The probability density function of the wrapped Cauchy distribution is given by

f(\theta; \mu, \gamma) = \frac{1}{\pi} \sum_{k=-\infty}^{\infty} \frac{\gamma}{(\theta - \mu + 2\pi k)^2 + \gamma^2},

where $\theta \in [0, 2\pi)$, $\mu \in [0, 2\pi)$ is the location parameter representing the mean direction, and $\gamma > 0$ is the scale parameter controlling concentration.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) An equivalent closed-form expression exists in terms of the concentration parameter $\rho = e^{-\gamma}$:

f(\theta; \mu, \rho) = \frac{1 - \rho^2}{2\pi (1 + \rho^2 - 2\rho \cos(\theta - \mu))}.

[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) Like its linear counterpart, the wrapped Cauchy distribution exhibits infinite variance, reflecting the heavy-tailed nature of the Cauchy, which leads to no finite second moment even after wrapping.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) The characteristic function is $\phi(t) = \exp(i t \mu) / (1 + |t| \gamma)$, with periodic extension to the circle via evaluation at integer orders.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) The trigonometric moments decay exponentially, with the modulus of the $k$-th moment given by $|\rho_k| = \exp(-|k| \gamma)$, enabling the distribution to capture multimodal patterns through multiple wrappings.[](http://palaeo.spb.ru/pmlibrary/pmbooks/mardia&jupp_2000.pdf) This property makes it suitable for approximating circular stable laws, as the wrapped Cauchy serves as a special case of the wrapped symmetric $\alpha$-stable family when $\alpha = 1$, facilitating models of stable processes on the circle such as Lévy flights in directional data. ### Wrapped Lévy Distribution The wrapped Lévy distribution, also known as the wrapped stable distribution, arises from folding a linear Lévy stable distribution onto the unit circle, providing a versatile model for circular data that captures asymmetry, heavy tails, and multimodal behavior. This distribution is particularly suited for scenarios involving directional extremes and skewness, extending the symmetric cases like the wrapped Cauchy (at α=1, β=0).[](https://doi.org/10.1016/j.csda.2007.11.005) The probability density function is expressed as an infinite Fourier series:

f(\theta; \mu, c, \alpha, \beta) = \frac{1}{2\pi} \sum_{k=-\infty}^{\infty} \exp\left(ik(\theta - \mu)\right) \exp\left(-|ck|^\alpha \left[1 - i\beta \operatorname{sgn}(k) \tan\left(\frac{\pi\alpha}{2}\right)\right]\right),

for θ ∈ [0, 2π) and α ∈ (0, 2), α ≠ 1, where μ is the location parameter (mean direction), c > 0 is the scale parameter, α is the stability index controlling tail heaviness, and β ∈ [-1, 1] is the skewness parameter. For α = 1, the form adjusts to incorporate a logarithmic term: the exponent becomes -|c k| [1 + i β (2/π) sgn(k) log|c k|]. This structure derives from the characteristic function of the underlying linear stable distribution, ensuring the wrapped version inherits properties like infinite divisibility.[](https://doi.org/10.1016/j.csda.2007.11.005) Key properties include heavy right tails when α < 2 and β > 0, making it ideal for modeling asymmetric extremes on the circle, such as directional outliers in angular data. The trigonometric moments are given by the linear characteristic function evaluated at integers, E[exp(ikθ)] = exp(ikμ - |c k|^α [1 - i β sgn(k) tan(πα/2)]), which exist for all integer k but reflect the linear distribution's moment non-existence beyond order α. The mean resultant length is |E[exp(iθ)]|, decreasing with scale c and tail index α. As a generalization of the wrapped Cauchy at α=1, it is obtained by wrapping increments of a linear Lévy stable process, which is central to applications in financial angular risks, such as modeling volatile directional propagations in high-frequency trading or asset orientation extremes.[](https://doi.org/10.1016/j.csda.2007.11.005) ### Projected Normal Distribution The projected normal distribution is a probability distribution on the unit circle derived from the radial projection of a bivariate normal random vector onto the circle. Specifically, if $(X, Y)^\top \sim \mathcal{N}_2(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\boldsymbol{\mu} = (\mu_x, \mu_y)^\top$ is the mean vector and $\boldsymbol{\Sigma}$ is the 2×2 positive definite covariance matrix, then the random direction $\boldsymbol{U} = (X, Y)^\top / \sqrt{X^2 + Y^2}$ follows the projected normal distribution $\mathrm{PN}_2(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, and the corresponding angle is $\theta = \atan2(Y, X)$. This model captures directional data by normalizing the Gaussian vector to lie on the unit circle, allowing for flexible shapes influenced by the location and spread of the underlying bivariate normal.[](https://pmc.ncbi.nlm.nih.gov/articles/PMC3773532/)[](https://projecteuclid.org/journals/bayesian-analysis/volume-12/issue-1/The-General-Projected-Normal-Distribution-of-Arbitrary-Dimension--Modeling/10.1214/15-BA989.full) The probability density function of $\theta$ involves an integral over the radial component and lacks a simple closed form in general. It is given by

f(\theta) = \frac{1}{2\pi} \int_0^\infty r , \phi(r \mathbf{u}(\theta); \boldsymbol{\mu}, \boldsymbol{\Sigma}) , dr, \quad \theta \in [0, 2\pi),

where $\phi(\cdot; \boldsymbol{\mu}, \boldsymbol{\Sigma})$ denotes the bivariate normal density with mean $\boldsymbol{\mu}$ and covariance $\boldsymbol{\Sigma}$, and $\mathbf{u}(\theta) = (\cos \theta, \sin \theta)^\top$. Equivalently, this can be expressed as

f(\theta) = \frac{|\boldsymbol{\Sigma}|^{-1/2} \exp\left( -\frac{1}{2} \boldsymbol{\mu}^\top \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu} \right)}{2\pi} \int_0^\infty r \exp\left( -\frac{1}{2} q(\theta, r) \right) dr,

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continuous random variable having the probability density function f(x; n). ... a circular distribution. Further, the trigonometric moment of order p for a ...
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Statistics of Directional Data - ScienceDirect.com
Statistics of Directional Data. A volume in Probability and Mathematical Statistics: A Series of Monographs and Textbooks. Book • 1972. Author: K.V. MARDIA ...
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[PDF] Directional Statistics - palaeo
May 3, 2016 · ... Peter E. Jupp. University of Leeds. UK. Uradttersity of St Andrews, UK ... Inference on Spheres ......................... 193. 10.1 ...
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Statistical Analysis of Circular Data
Cambridge Core - General and Classical Physics - Statistical Analysis of Circular Data.
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Automatic bandwidth selection for circular density estimation
The goal of this paper is to obtain an equivalent plug-in rule for density estimation on the circle. Specifically, we consider the estimator in which the kernel ...
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[PDF] Circular Statistics - CRAN
Computes circular summary statistics including the sample size, mean direction and mean resultant ... trigonometric.moment. Trigonometric Moments.
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Statistical Analysis of Circular Data - N. I. Fisher - Google Books
Oct 12, 1995 · A unified, up-to-date account of circular data-handling techniques, useful throughout science. Preview this book
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[PDF] moments of von mises and fisher distributions and applications
Nov 22, 2016 · To explain the use of the von Mises and Fisher distribution in population models, we consider a simple random walk model for oriented animal/ ...
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An analysis of the modality and flexibility of the inverse ...
Jan 13, 2025 · For the one dimensional case, it was shown in Kent (1978) that the von Mises distribution closely approximates the wrapped normal distribution ...
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https://doi.org/10.3390/math12162440
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[PDF] Efficient Evaluation of the Probability Density Function of a Wrapped ...
Abstract—The wrapped normal distribution arises when the density of a one-dimensional normal distribution is wrapped around the circle infinitely many times. ...
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https://doi.org/10.1016/j.csda.2007.11.005
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Directional data analysis under the general projected normal ...
The projected normal distribution is an under-utilized model for explaining directional data. In particular, the general version provides flexibility.
[21]
https://projecteuclid.org/journals/bayesian-analysis/volume-12/issue-1/The-General-Projected-Normal-Distribution-of-Arbitrary-Dimension--Modeling/10.1214/15-BA989.full
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Projected Normal Distribution: Moment Approximations and ... - arXiv
Jun 20, 2025 · In this work, we derive analytic approximations to the first and second moments of the projected normal distribution using Taylor expansions.Missing: atan2 integral
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Directional Statistics - Wiley Online Library
Jan 3, 1999 · Presents new and up-dated material on both the underlying theory and the practical methodology of directional statistics.
[25]
On maximum likelihood estimation of the concentration parameter of ...
Maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions involves inverting the ratio Rν=Iν+1/Iν of modified Bessel ...
[26]
[PDF] Bayesian Analysis of Circular Data Using Wrapped Distributions
Summary. Circular data arise in a number of different areas such as geological, meteoro- logical, biological and industrial sciences.<|control11|><|separator|>
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On the estimation of the parameters of the von mises distribution
Jun 27, 2007 · Abstract. The usual maximum likelihood estimators of the parameters of the von Mises distribution are shown to perform badly in small samples.
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Circular data - Lee - 2010 - WIREs Computational Statistics
Jul 15, 2010 · A comprehensive set of methods for circular data analysis have been developed, covering summary statistics, probability distributions, one- and multi-sample ...Univariate Distributions · Regression And Correlation · Circular Correlation
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Circular data in biology: advice for effectively implementing ... - PMC
Jul 11, 2018 · We provide guidance for biologists helping them to make decisions when testing circular data for single or multiple departures from uniformity.Missing: multimodality | Show results with:multimodality
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Bayesian tests for circular uniformity - ScienceDirect.com
... mean resultant length ... The circular uniform distribution is defined by p ( θ ) = ( 2 ...
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Tests concerning random points on a circle - Semantic Scholar
Semantic Scholar extracted view of "Tests concerning random points on a circle" by N. Kuiper.
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A class of goodness-of-fit tests for circular distributions based on ...
We propose a class of goodness-of-fit test procedures for arbitrary parametric families of circular distributions with unknown parameters.