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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.

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.

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.

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.

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.

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.

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.

Simple algorithms are given to simulate samples from multimodal circu-lar distributions corresponding to the wrapped Cauchy distribution. These algorithms are used in a acceptance-rejection method to simulate samples from multimodal circular distributions of the von Mises type.

The wrapped Cauchy distribution is an alternative to the Fisher-von Mises distribution for modeling symmetric data on the circle, and its maximum likelihood estimate (m.l.e.) represents a robust alternative to the mean direction for estimating the location for circular data. Surprisingly, there appear to be no previous results on the m.l.e. for the wrapped Cauchy distribution. It is shown that for sample sizes greater than two, the m.l.e. exists, is unique, and can be found by solving the likelihood equations. Also, a simple algorithm is presented which converges to the m.l.e.