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Inferring the most appropriate distribution (or distributions) to describe observed directional data is important in many applications of circular statistics. In particular, animal movement paths are typically analysed and modelled by considering the distribution of step lengths and turning (or absolute) angles. Here we demonstrate that a single-wrapped Cauchy distribution can appear to fit directional data mixed from two different underlying wrapped normal distributions. We derive mathematical expressions to calculate the parameter space for which this occurs and illustrate the result by analysing an example data set of the movements of African bull elephants (Loxodonta Africana). We conclude that the presence of a wrapped Cauchy distribution in observed directional data can, in certain cases, be explained by data coming from two distinct underlying distributions. We discuss how this may relate to the presence of multiple movement modes within an observed path when analysing animal movement data.

A general method of constructing a circular model is to apply transformation to its argument. In this paper, we propose a new transformation in order to construct a large class of new skew-symmetric circular models. We consider a symmetric unimodal distribution as our base model and provide general results for the modality, skewness and shape properties of the resulting circular distributions. The new class enfolds two well-known and common classes of unimodal symmetric and asymmetric circular distributions. The skewness parameter in the new class greatly improves the flexibility of our model to cover symmetric, asymmetric, unimodal and multimodal data. Further, we consider Wrapped Cauchy distribution as a special case for our base distribution and introduce the Skew-Symmetric Wrapped Cauchy (SSWC) distribution, discuss its sub-models and estimate its parameters by the maximum likelihood method. Finally, an application of SSWC distribution and its inferential methods are illustrated using two real data examples.

Circular data can arise from many sources, such as image processing, protein structure, and geological data, just to name a few. Wrapped stable family of distributions constitute one of the most widely used class of distributions for the analysis of such data. Wrapped Cauchy distribution is a member of this family and it is the only one known to have a single term explicit pdf compared to the infinite series representations for all the others. We develop in this paper reference priors and probability matching priors for this distribution, which turn out to be quite elegant and provide good frequentist performance as well.

Circular data jointly observed with linear data are common in various disciplines. Since circular data require different techniques than linear data, it is often misleading to use usual dependence measures for joint data of circular and linear observations. Moreover, although a mutual information measure between circular variables exists, the measure has drawbacks in that it is defined only for a bivariate extension of the wrapped Cauchy distribution and has to be approximated using numerical methods. In this paper, we introduce two measures of dependence, namely, (i) circular-linear mutual information as a measure of dependence between circular and linear variables and (ii) circular-circular mutual information as a measure of dependence between two circular variables. It is shown that the expression for the proposed circular-linear mutual information can be greatly simplified for a subfamily of Johnson–Wehrly distributions. We apply these two dependence measures to learn a circular-linear tree-structured Bayesian network that combines circular and linear variables. To illustrate and evaluate our proposal, we perform experiments with simulated data. We also use a real meteorological data set from different European stations to create a circular-linear tree-structured Bayesian network model .

A cylindrical distribution whose linear part models heavy-tailedness is proposed. The conditional distribution of the linear variable given the circular variable is a generalized Pareto-type distribution. Therefore, it may not have any conditional moments; however, the mode and median have closed-form expressions. The circular marginal distribution is a wrapped Cauchy distribution, and the conditional distribution of the circular variable given the linear variable belongs to a family of symmetric distributions. These properties allow its application to cylindrical data, whose linear observations may take large values and whose circular observations are symmetric. As illustrative examples, the proposed distribution is fitted to two data sets, and the results are compared with those by other cylindrical distributions that cannot model heavy-tailedness for the linear parts.

We discuss the asymptotic properties with respect to nonparametric regression for circular data. We reveal theoretical properties for circular nonparametric regression by applying von Mises (VM) and wrapped Cauchy (WC) kernels. We derive the asymptotic normalities and the convergence rate of the weighted conditional mean integrated squared errors regarding VM and WC kernels. The numerical experiment shows that WC kernel outperforms VM kernel in the small samples, and the theoretical properties are supported in the large samples.

Spatial direction vector data can be represented cylindrically by linear magnitudes and circular angles. We analyze such data by using a hierarchical Markov random field model with latent discrete classes and conditionally independent cylindrical data given the classes. The structure of a Potts model segments the spatial domain, and each class defines a cylindrical density that represents a specific structure. We consider two types of cylindrical distributions; the Weibull sine-skewed von Mises distribution, which is skewed in the circular part, and the generalized Pareto-type wrapped Cauchy distribution, which is heavy-tailed in the linear part. In this setting, we develop a statistically efficient block composite likelihood method for parameter estimation. The method is shown to provide much faster convergence than an expectation–maximization approach. However, the convergence is less stable for the block composite likelihood method, and we suggest a hybrid estimation approach for practical use. We apply the approach to study ocean surface currents in the Norwegian Sea. The models are able to describe the currents in terms of interpretable local regimes of two–three classes. Scoring rules are used to measure predictive performance of the two cylindrical densities. Results indicate that there is clearly skew angular components, and possibly also some heavy tails in magnitude. This is an open access article distributed under the terms of the Creative Commons CC-BY license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Accurate modeling of wind speed and direction plays a significant role in wind energy research. This paper proposes a probabilistic approach to model the randomness and correlation of wind direction at two nearby wind sites. Such an approach introduces mixture Wrapped Cauchy distribution to fit the marginal wind direction probability distribution at each location. The circular correlation coefficient is applied to measure the correlation between wind directions at two nearby sites. The analytical relationship between the circular correlation coefficient and the copula parameter is derived, which facilitates the determination of the appropriate copula parameter using the bisection method. The joint wind direction probability distribution with the appropriate copula parameter could preserve the circular correlation coefficient at a desirable level. A sampling procedure that produces random and correlated wind directions has also been developed. The collected wind direction data from two wind sites in the U.S are used to verify the proposed approach. The results show that the proposed method can appropriately model the randomness and correlation of collected wind direction data. Furthermore, the produced wind direction samples share similar circular mean value, circular standard deviation, and the circular correlation coefficient with those of collected wind direction samples.

We present some properties of a Cauchy family of distributions on the sphere, which is a spherical extension of the wrapped Cauchy family on the circle. The spherical Cauchy family is closed under the Möbius transformations on the sphere and the parameter of the transformed family is expressed using extended Möbius transformations on the compactified Euclidean space. Stereographic projection transforms the spherical Cauchy family into a multivariate $t$-family with a certain degree of freedom on Euclidean space. The Möbius transformations and stereographic projection enable us to obtain some results related to the spherical Cauchy family such as an efficient algorithm for random variate generation, a simple form of pivotal statistic and straightforward calculation of probabilities of a region. A method of moments estimator and an asymptotically efficient estimator are expressed in closed form. Maximum likelihood estimation is also straightforward.

In this article, we consider the circular–circular regression model using Möbius transformation. We first consider the model provided by Kato et al. (2008) for only one circular regressor and prove the identifiability of the model. After that, a methodology is discussed to reduce the prediction error of this model. We then introduce the two multiple circular–circular regression models with multiple circular regressors. We prove the identifiability of the models and discuss their geometry. We then discuss the parameter estimation procedure followed by simulation study. The methodologies are illustrated by some real datasets.

Capturing the dependences among circular variables within supervised classification models is a challenging task. In this paper, we propose four different supervised Bayesian classification algorithms where the predictor variables follow all circular wrapped Cauchy distributions. For this purpose, we introduce four wrapped Cauchy classifiers. The bivariate wrapped Cauchy distribution is the only bivariate circular distribution whose marginals and conditionals are also wrapped Cauchy distributions, a property that makes it possible to define these models easily. Furthermore, the wrapped Cauchy tree-augmented naive Bayes classifier requires the definition of a conditional circular mutual information measure between variables that follow wrapped Cauchy distributions. Synthetic data is used to illustrate, compare and evaluate the classification algorithms (including a comparison with the Gaussian TAN classifier, decision tree, random forest, multinomial logistic regression, support vector machine and simple neural network), leading to satisfactory predictive results. We also use a real neuromorphological dataset obtained from juvenile rat somatosensory cortex cells, where we measure the bifurcation angles of the dendritic basal arbors.

A new Reduced-Reference (RR) image quality metric based on statistical models in the complex wavelet transform domain is proposed. The magnitude and the relative phase information of the complex wavelet coefficients is modeled by using probability density function, and a strategy based on the information criterion is proposed to optimally approximate the distribution. To further improve the accuracy of the metric, a comparison of the candidate models is studied, and the inverse Gaussian distribution and the wrapped Cauchy distribution are selected to model the magnitude and the relative phase distributions, respectively. The Kullback–Leibler divergence between the distributions of the reference image and the distorted one serves as the RR feature to measure the distortion. Finally, a generalized regression neural network is employed to map the RR feature into an objective score. Experimental studies confirmed that the proposed RR image quality metric is quality-aware and highly correlated with the human visual system.

The von Mises distribution is widely used for modeling angular data. When such data are seen in a quality control setting, there may be interest in checking whether the values are in statistical control or have gone out of control. A cumulative sum (cusum) control chart has desirable properties for checking whether the distribution has changed from an in-control to an out-of-control setting. This paper develops cusums for a change in the mean direction and concentration of angular data and illustrates some of their properties.

The way of investigating a distribution knowing its interesting properties might be often inadequate when the shapes of two distributions are almost similar. In each of these circumstances, the accurate decision about the genesis of a random sample from any of the two parent distributions will be very much ambiguous even with the availability of the existing testing procedure of the circular data. A sequential discrimination procedure has been suggested which is also invariant to the sample size. The performance of the proposed discrimination procedure has been evaluated by checking its capability of detecting the genesis of the known samples from the two identically shaped wrapped distributions.

We propose a five-parameter bivariate wrapped Cauchy distribution as a unimodal model for toroidal data. It is highly tractable, displays numerous desirable properties, including marginal and conditional distributions that are all wrapped Cauchy, and arises as an appealing submodel of a six-parameter distribution obtained by applying Mobius transformation to a pre-existing bivariate circular model. Method of moments and maximum likelihood estimation of its parameters are fast, and tests for independence and goodness-of-fit are available. An analysis involving dihedral angles of the proteinogenic amino acid Tyrosine illustrates the distribution’s application. A Markov process for circular data is also explored.

Local phase is now known to carry information about image features or object motions. But it is harder to use directly compared with amplitude, so far. In this paper, we propose that the relative local phase, which is a function of scale, position and time, really matters in representing the information of image structures or movements. A unified description of relative phase is given in this paper based on a bilinear representation of natural image series via multi-scale orientated dual tree complex wavelets. Then, the behaviors of nontrivial relative phase, especially for their distribution on multi-scale and multi-subband, are investigated. We propose a new generalized model, which is derived from Mobius transform, to describe various relative phases. Numerical experiments for a large amount of test images show that the new model performs best compared with the von Mises or wrapped Cauchy distribution. Especially for those with asymmetric pdf, our function fits with the histogram quite well while the other two may fail. We thus lay a groundwork for relative phase-based image processing methods, such as classification, deblurring and motion perception. Copyright © 2014 John Wiley & Sons, Ltd.

This article is concerned with the analogue of copulas for circular distributions, which we call ‘circulas’. We concentrate on one particular class of circulas, which is pre-existing but not studied in such explicit form or detail before. This class is appealing in many ways but does not necessarily result in especially attractive bivariate circular models for arbitrary non-uniform marginals. A major exception to this is an elegant bivariate wrapped Cauchy distribution previously proposed and developed by two of the current authors. We look both at properties of the circulas themselves, including their density behaviour, distribution function, and dependence measures, and at properties of various distributions based on these circulas by transformation to non-uniform marginal distributions. We consider inference for the latter distributions and present two applications of them to modelling data. We concentrate mostly on the bivariate case, but also briefly consider extension to the multivariate case.

We establish the kinetics of ballistic-to-diffusive (BD) transition observed in two-dimensional random walk using directional statistics. Directional correlation is parameterized using the walker's turning angle distribution, which follows the commonly adopted wrapped Cauchy distribution (WCD) function. During the BD transition, the concentration factor (ρ) governing the WCD shape is observed to decrease from its initial value. We next analytically derive the relationship between effective ρ and time, which essentially quantifies the BD transition rate. The prediction of our kinetic expression agrees well with the empirical datasets obtained from correlated random walk simulation. We further connect our formulation with the conventionally used scaling relationship between the walker's mean-square displacement and time.

In this note we provide a simple approximation theory motivation for the circular kernel density estimation and further explore the usefulness of the wrapped Cauchy kernel in this context. It is seen that the wrapped Cauchy kernel appears as a natural candidate in connection to orthogonal series density estimation on a unit circle. This adds further weight to the considerable role of the wrapped Cauchy in circular statistics., 9 pages

Modelling the relationship between directional variables is a nearly unexplored field. The bivariate wrapped Cauchy distribution has recently emerged as the first closed family of bivariate directional distributions (marginals and conditionals belong to the same family). In this paper, we introduce a tree-structured Bayesian network suitable for modelling directional data with bivariate wrapped Cauchy distributions. We describe the structure learning algorithm used to learn the Bayesian network. We also report some simulation studies to illustrate the algorithms including a comparison with the Gaussian structure learning algorithm and an empirical experiment on real morphological data from juvenile rat somatosensory cortex cells.

We present a retrospective method for studying forest disturbance regimes, and especially the role of windthrows, based on circular statistical models of directions of fallen logs. This approach was applied to fallen log data from three areas of pristine Picea abies -dominated boreal forests in northern Europe. The data consisted of 5 plots from each of the three areas, totaling 15 plots and covering an area of 24 ha. The disturbance history of the plots, which varied from area to area, was known from previous detailed studies. We selected and fitted the most suitable circular model for each plot, based on goodness-of-fit and the Akaike information criterion. For uneven-aged forests, the symmetric von Mises distribution , was best fitted, while for the even-aged forest the sine-skewed wrapped Cauchy distribution was selected. The degree of concentration around the mean direction of fallen trees was strongest for the late-successional even-aged forest most exposed to windthrow, while an uneven-aged forest with drought-driven mortality had the lowest concentration and the greatest variance over the mean directions. For the third area, characterized by an uneven age structure and tree mortality driven by heart-rot fungi in old trees in interaction with wind, an intermediate between these two were derived. Our results suggested the utility of circular distributions of fallen logs and their statistical models for retrospective assessments of forest disturbance regimes.

In this paper, a method for omnidirectional sound source tracking using a circular microphone array is proposed. The sequential updating histogram estimated every two microphones are integrated for the sound source tracking. The histogram is estimated by weighting those reliability to results obtained every adjacent microphone pair. In addition, the wrapped Cauchy distribution is used to detect the omnidirectional DOA. As a result, the accurate omnidirectional sound source tracking can be achieved. Several experimental results are shown to present the effectiveness of the proposed method.

Summary 1. Large data sets containing precise movement data from free-roaming animals are now becoming commonplace. One means of analysing individual movement data is through discrete, random walk–based models. 2. Random walk models are easily modified to incorporate common features of animal movement, and the ways that these modifications affect the scaling of net displacement are well studied. Recently, ecologists have begun to explore more complex statistical models with multiple latent states, each of which are characterized by a distribution of step lengths and have their own unimodal distribution of turning angles centred on one type of turn (e.g. reversals). 3. Here, we introduce the compound wrapped Cauchy distribution, which allows for multimodal distributions of turning angles within a single state. When used as a single state model, the parameters provide a straightforward summary of the relative contributions of different turn types. The compound wrapped Cauchy distribution can also be used to build multiple state models. 4. We hypothesize that a multiple state model with unimodal distributions of turning angles will best describe movement at finer resolutions, while a multiple state model using our multimodal distribution will better describe movement at intermediate temporal resolutions. At coarser temporal resolutions, a single state model using our multimodal distribution should be sufficient. We parameterize and compare the performance of these models at four different temporal resolutions (1, 4, 12 and 24 h) using data from eight individuals of Loxodonta cyclotis and find support for our hypotheses. 5. We assess the efficacy of the different models in extrapolating to coarser temporal resolution by comparing properties of data simulated from the different models to the properties of the observed data. At coarser resolutions, simulated data sets recreate many aspects of the observed data; however, only one of the models accurately predicts step length, and all models underestimate the frequency of reversals. 6. The single state model we introduce may be adequate to describe movement data at many resolutions and can be interpreted easily. Multiscalar analyses of movement such as the ones presented here are a useful means of identifying inconsistencies in our understanding of movement.

We propose a family of four-parameter distributions on the circle that contains the von Mises and wrapped Cauchy distributions as special cases. The family is derived by transforming the von Mises distribution via a Mobius transformation, which maps the unit circle onto itself. The densities in the family have a symmetric or asymmetric, unimodal or bimodal shape, depending on the values of the parameters. Conditions for unimodality are explored. Further properties of the proposed model are obtained, many by applying the theory of Mobius transformation. Properties of a three-parameter symmetric submodel are investigated as well; these include maximum likelihood estimation, its asymptotics, and a reparameterization that proves useful quite generally. A three-parameter asymmetric subfamily, which often proves to be an adequate model, is also discussed, with emphasis on its mean direction and circular skewness. The proposed family and subfamilies are used to model an asymmetrically distributed data set and ar...

In this paper, we develop a new approach which exploits the probabilistic properties from the phase information of 2-D complex wavelet coefficients for image modeling. Instead of directly using phases of complex wavelet coefficients, we demonstrate why relative phases should be used. The definition, properties and statistics of relative phases of complex coefficients are studied in detail. We proposed von Mises and wrapped Cauchy for the probability density function (pdf) of relative phases in the complex wavelet domain. The maximum-likelihood method is used to estimate two parameters of von Mises and wrapped Cauchy. We demonstrate that the von Mises and wrapped Cauchy fit well with real data obtained from various real images including texture images as well as standard images. The von Mises and wrapped Cauchy models are compared, and the simulation results show that the wrapped Cauchy fits well with the peaky and heavy-tailed pdf of relative phases and the von Mises fits well with the pdf which is in Gaussian shape. For most of the test images, the wrapped Cauchy model is more accurate than the von Mises model, when images are decomposed by different complex wavelet transforms including dual-tree complex wavelet (DTCWT), pyramidal dual-tree directional filter bank (PDTDFB) and uniform discrete curvelet transform (UDCT). Moreover, the relative phase is applied to obtain new features for texture image retrieval and segmentation applications. Instead of using only real or magnitude coefficients, the new approach uses a feature in which phase information is incorporated, yielding a higher accuracy in texture image retrieval as well as in segmentation. The relative phase information which is complementary to the magnitude is a promising approach in image processing.

An extended version of t -distribution on the unit circle is generated by conditioning a normal mixture distribution, which is broadened to include not only unimodality and symmetry, but also bimodality and asymmetry, depending on the values of parameters. After reparametrization, the distribution contains four circular distributions as special cases: symmetric Jones-Pewsey, generalized von Mises, generalized cardioid and generalized wrapped Cauchy distributions. As an illustrative example, the proposed model is fitted to the number of occurrences of the thunder in a day.

We shall discuss a problem of how to construct angular distributions on the circle considering the generalization of the well-known von Mises distributions.We introduce three methods of the transformations to polar co-ordinates from a bivariate normal distribution instead of the radial projection of the distribution. Furthermore, the scale mixture of the obtained distributions with the inverse Gamma distribution for the mixing distribution is considered.

This paper considers the three-parameter family of symmetric unimodal distributions obtained by wrapping the location-scale extension of Student's t distribution onto the unit circle. The family contains the wrapped normal and wrapped Cauchy distributions as special cases, and can be used to closely approximate the von Mises distribution. In general, the density of the family can only be represented in terms of an infinite summation, but its trigonometric moments are relatively simple expressions involving modified Bessel functions. Point estimation of the parameters is considered, and likelihood-based methods are used to fit the family of distributions in an illustrative analysis of cross-bed measurements. The use of the family as a means of approximating the von Mises distribution is investigated in detail, and new efficient algorithms are proposed for the generation of approximate pseudo-random von Mises variates.

We propose a new family of symmetric unimodal distributions on the circle that contains the uniform, von Mises, cardioid, and wrapped Cauchy distributions, among others, as special cases. The basic form of the densities of this family is very simple, although its normalization constant involves an associated Legendre function. The family of distributions can also be derived by conditioning and projecting certain bivariate spherically and elliptically symmetric distributions on to the circle. Trigonometric moments are available, and a measure of variation is discussed. Aspects of maximum likelihood estimation are considered, and likelihood is used to fit the family of distributions to an example set of data. Finally, extension to a family of rotationally symmetric distributions on the sphere is briefly made.