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Circular data – data whose values lie in the interval [0,2π) – are important in a number of application areas. In some, there is a suspicion that a sequence of circular readings may contain two or more segments following different models. An analysis may then seek to decide whether there are multiple segments, and if so, to estimate the changepoints separating them. This paper presents an optimal method for segmenting sequences of data following the von Mises distribution. It is shown by example that the method is also successful in data following a distribution with much heavier tails. [ABSTRACT FROM AUTHOR]

Directional data appears in several branches of research. In some cases, those directional variables are only defined in subsets of the K-dimensional unit sphere. For example, in some applications, angles as measured responses are limited on the positive orthant. Analysis on subsets of the K-dimensional unit sphere is challenging and nowadays there are not many proposals that discuss this topic. Thus, from a methodological point of view, it is important to have probability distributions defined on bounded subsets of the K-dimensional unit sphere. Specifically, in this paper, we introduce a nonparametric Bayesian model to describe directional variables restricted to the first orthant. This model is based on a Dirichlet process mixture model with multivariate projected Gamma densities as kernel distributions. We show how to carry out inference for the proposed model based on a slice sampling scheme. The proposed methodology is illustrated using simulated data sets as well as a real data set., Competing Interests: No potential conflict of interest was reported by the authors., (© 2022 Informa UK Limited, trading as Taylor & Francis Group.)

There is a very short and beautiful proof that the number of distinct non-empty palindromes in a word of length n is at most n . In this paper we show, with a very complicated proof, that the number of distinct non-empty palindromes with length at most n in a circular word of length n is less than 5 n / 3 . For n divisible by 3 we present circular words of length n containing 5 n / 3 − 2 distinct palindromes, so the bound is almost sharp. The paper finishes with some open problems. [ABSTRACT FROM AUTHOR]

Researchers in psychology often encounter data measured in angles (e.g., directions, or measurements on circular scales such as the circumplex model of affect). Due to periodicity, the evaluation of these circular data requires special statistical methods. This article introduces new tests for the analysis of order constrained hypotheses for circular data. Through these tests, researchers can evaluate their expectations regarding the outcome of an experiment directly by representing their ideas in the form of a hypothesis containing inequality constraints. The resulting data analysis is generally more powerful than one using null hypothesis testing. An example of circular data from psychology is presented to illustrate the use of the tests. Results from a simulation study show that the tests perform well in terms of type I error and power. [ABSTRACT FROM AUTHOR]

Circular data are commonly encountered in the earth sciences and statistical descriptions and inferences about such data are necessary in structural geology. In this paper we compare two statistical distributions appropriate for complex circular data sets: the mixture of von Mises and the projected normal distribution. We show how the number of components in a mixture of von Mises distribution may be chosen, and how one may choose between the projected normal distribution and the mixture of von Mises for a particular data set. We illustrate these methods with a few structural geological data, showing how the fitted models can complement geological interpretation and permit statistical inference. One of our data sets suggests a special case of the projected normal distribution which we discuss briefly. [ABSTRACT FROM AUTHOR]

Uniform scores test is a rank-based method that tests the homogeneity ofk-populations in circular data problems. The influence of ties on the uniform scores test has been emphasized by several authors in several articles and books. Moreover, it is suggested that the uniform scores test should be used with caution if ties are present in the data. This paper investigates the influence of ties on the uniform scores test by computing the power of the test using average, randomization, permutation, minimum, and maximum methods to break ties. Monte Carlo simulation is performed to compute the power of the test under several scenarios such as having 5% or 10% of ties and tie group structures in the data. The simulation study shows no significant difference among the methods under the existence of ties but the test loses its power when there are many ties or complicated group structures. Thus, randomization or average methods are equally powerful to break ties when applying uniform scores test. Also, it can be concluded thatk-sample uniform scores test can be used safely without sacrificing the power if there are only less than 5% of ties or at most two groups of a few ties. [ABSTRACT FROM AUTHOR]

Von Mises distribution, also known as the natural analog of the normal distribution on real line, has two parameters namely the mean direction and concentration parameter. This paper considers the problem related to finding the best confidence intervals (CI) for a large concentration parameter. Various CI are derived and their performances are evaluated via simulation studies. We found that the CI based on asymptotic distribution of κ and CI based on Bootstrap-t Method are good in terms of coverage probability even though the CI based on distribution of mean direction and resultant length gives shorter expected length. [ABSTRACT FROM AUTHOR]

Smoothing methods and SiZer (SIgnificant ZERo crossing of the derivatives) are useful tools for exploring significant underlying structures in data samples. An extension of SiZer to circular data, namely CircSiZer, is introduced. Based on scale-space ideas, CircSiZer presents a graphical device to assess which observed features are statistically significant, both for density and regression analysis with circular data. The method is intended for analyzing the behavior of wind direction in the atlantic coast of Galicia (NW Spain) and how it has an influence over wind speed. The performance of CircSiZer is also checked with some simulated examples. [ABSTRACT FROM AUTHOR]

Abstract: It is crucial to check validation of any statistical model after fitting it for a given set of data. In Bayesian statistics, a researcher can check the fit of the model using a variety of strategies. In this paper we consider two major aspects, first checking that the posterior inferences are reasonable, given the substantive context of the model; and then examining the sensitivity of inferences to reasonable changes in the prior distribution and the likelihood. Here we consider functional Bregman divergence between posterior distributions for model diagnostics, which produce methods for outlier detection as well as for prior sensitivity analysis. The methodology is exemplified through a logistic regression and a circular data model. [Copyright &y& Elsevier]

A novel augmented complex-valued common spatial pattern (CSP) algorithm is introduced in order to cater for general complex signals with noncircular probability distributions. This is a typical case in multichannel electroencephalogram (EEG), due to the power difference or correlation between the data channels, yet current methods only cater for a very restrictive class of circular data. The proposed complex-valued CSP algorithms account for the generality of complex noncircular data, by virtue of the use of augmented complex statistics and the strong-uncorrelating transform (SUT). Depending on the degree of power difference of complex signals, the analysis and simulations show that the SUT based algorithm maximizes the inter-class difference between two motor imagery tasks. Simulations on both synthetic noncircular sources and motor imagery experiments using real-world EEG support the approach. [ABSTRACT FROM AUTHOR]

This study examined the temporal dynamics of the inter-limb angles of skilled and less skilled ice climbers to determine how they explored ice fall properties to adapt their coordination patterns during performance. We observed two circular time series corresponding to the upper- and lower-limbs of seven expert and eight inexperienced ice climbers. We analyzed these data through a multiple change-point analysis of the geodesic (or Fréchet) mean on the circle. Guided by the nature of the geodesic mean obtained by an optimization procedure, we extended the filtered derivative method, known to be computationally very cheap and fast, to circular data. Local estimation of the variability was assessed through the number of change-points computed via the filtered derivatives withp-value method for the time series and integrated squared error (ISE). Results of this change-point analysis did not reveal significant differences of the number of change-points between groups but indicated higher ISE that supported the existence of plateaux for beginners. These results emphasized higher local variability of limb angles for experts than for beginners suggesting greater dependence on the properties of the performance environment and adaptive behaviors in the former. Conversely, the lower local variance of limb angles assessed in beginners may reflect their independence of the environmental constraints, as they focused mainly on controlling body equilibrium. [ABSTRACT FROM AUTHOR]

This paper deals with optimality aspects of complete block designs with interference effects arising from the neighbouring units up to distance 2 (first and second order) from one side. Conditions have been obtained for the block design to be universally optimal for estimating direct and interference effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct effects, first order neighbour effects and second order neighbour effects. [ABSTRACT FROM AUTHOR]

We introduce a new test of isotropy or uniformity on the circle, based on the Gini mean difference of the sample arc-lengths and obtain both the exact and asymptotic distributions under the null hypothesis of circular uniformity. We also provide a table of upper percentile values of the exact distribution for small to moderate sample sizes. Illustrative examples in circular data analysis are also given. It is shown that a “generalized” Gini mean difference test has better asymptotic efficiency than a corresponding “generalized” Rao's test in the sense of Pitman asymptotic relative efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we study an unsupervised clustering problem. The originality of this problem lies in the data, which consist of the positions of five separate X-ray beams on a circle. Radiation therapists positioned the five X-ray beam 'projectors' around each patient on a predefined circle. However, similarities exist in positioning for certain groups of patients, and we aim to describe these similarities with the goal of creating pre-adjustment settings that could help save time during X-ray positioning. We therefore performed unsupervised clustering of observed X-ray positions. Because the data for each patient consist of five angle measurements, Euclidean distances are not appropriated. Furthermore, we cannot perform k-means algorithm, usually used for minimizing corresponding distortion because we cannot calculate centers of clusters. We present here solutions to these problems. First, we define a suitable distance on the circle. Then, we adapt an algorithm based on simulated annealing to minimize distortion. This algorithm is shown to be theoretically convergent. Finally, we present simulations on simulated and real data. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Because of its role in many ecological processes, movement of animals in response to landscape features is an important subject in ecology and conservation biology. In this paper, we develop models of animal movement in relation to objects or fields in a landscape. We took a finite mixture modeling approach in which the component densities are conceptually related to different choices for movement in response to a landscape feature, and the mixing proportions are related to the probability of selecting each response as a function of one or more covariates. We combined particle swarm optimization and an expectation‐maximization (EM) algorithm to obtain maximum‐likelihood estimates of the model parameters. We used this approach to analyze data for movement of three bobcats in relation to urban areas in southern California, USA. A behavioral interpretation of the models revealed similarities and differences in bobcat movement response to urbanization. All three bobcats avoided urbanization by moving either parallel to urban boundaries or toward less urban areas as the proportion of urban land cover in the surrounding area increased. However, one bobcat, a male with a dispersal‐like large‐scale movement pattern, avoided urbanization at lower densities and responded strictly by moving parallel to the urban edge. The other two bobcats, which were both residents and occupied similar geographic areas, avoided urban areas using a combination of movements parallel to the urban edge and movement toward areas of less urbanization. However, the resident female appeared to exhibit greater repulsion at lower levels of urbanization than the resident male, consistent with empirical observations of bobcats in southern California. Using the parameterized finite mixture models, we mapped behavioral states to geographic space, creating a representation of a behavioral landscape. This approach can provide guidance for conservation planning based on analysis of animal movement data using statistical models, thereby linking connectivity evaluations to empirical data. [ABSTRACT FROM AUTHOR]

The goal of this work is to introduce nonparametric kernel methods for density and regression estimation for circular data, and illustrate their use by a brief simulation study and real data application. Apart from supplying practitioners with a license free and easy to run code for the use of these methods, our aim is also to provide solutions to practical problems that may be encountered in their application. The real data examples belong to the International Polar Year project, concerned with the environmental change in the polar regions. [ABSTRACT FROM AUTHOR]

In this article, we study the SB-robustness of various estimators of the mean direction (μ) and the concentration parameter (ρ) of the wrapped normal distribution. The functional corresponding to the sample mean direction is seen to be not SB-robust as an estimator of μ at the family of wrapped normal distributions with varying ρ, whereas the γ-trimmed mean direction is SB-robust at the same family of distributions for the different dispersion measures considered in this article. We also study the SB-robustness of the moment estimator of ρ and also that for a newly introduced trimmed estimator of ρ. [ABSTRACT FROM PUBLISHER]

The identification of sea regimes from environmental multivariate times series is complicated by the mixed linear-circular support of the data, by the occurrence of missing values, by the skewness of some variables, and by the temporal autocorrelation of the measurements. We address these issues simultaneously by a hidden Markov approach, and segment the data into pairs of toroidal and skew-elliptical clusters by means of the inferred sequence of latent states. Toroidal clusters are defined by a class of bivariate von Mises densities, while skew-elliptical clusters are defined by mixed linear models with positive random effects. The core of the classification procedure is an EM algorithm accounting for missing measurements, unknown cluster membership, and random effects as different sources of incomplete information. Moreover, standard simulation routines allow for the efficient computation of bootstrap standard errors. The proposed procedure is illustrated for a multivariate marine time series, and identifies a number of wintertime regimes in the Adriatic Sea. [ABSTRACT FROM AUTHOR]

We study conditions under which the characteristic vector of a normal lcQS-manifold is a torsion-forming or even a concircular vector field. We prove that the following assertions are equivalent: [ABSTRACT FROM AUTHOR]

This article considers the derivation of approximate distributions for two types of statistics that can be used in developing new tests of discordance in circular samples from the von Mises distribution. An alternative test of discordance is proposed based on the circular distance between sample points. The advantage of the test is that it allows users to detect possible outliers in both univariate and bivariate circular data. For illustration, the test is applied to two real circular data sets. [ABSTRACT FROM AUTHOR]

The EM algorithm is the standard method for estimating the parameters in finite mixture models. Yang and Pan [25] proposed a generalized classification maximum likelihood procedure, called the fuzzy c-directions (FCD) clustering algorithm, for estimating the parameters in mixtures of von Mises distributions. Two main drawbacks of the EM algorithm are its slow convergence and the dependence of the solution on the initial value used. The choice of initial values is of great importance in the algorithm-based literature as it can heavily influence the speed of convergence of the algorithm and its ability to locate the global maximum. On the other hand, the algorithmic frameworks of EM and FCD are closely related. Therefore, the drawbacks of FCD are the same as those of the EM algorithm. To resolve these problems, this paper proposes another clustering algorithm, which can self-organize local optimal cluster numbers without using cluster validity functions. These numerical results clearly indicate that the proposed algorithm is superior in performance of EM and FCD algorithms. Finally, we apply the proposed algorithm to two real data sets. [ABSTRACT FROM PUBLISHER]

Researchers studying the movements of the human body often encounter data measured in angles (e.g., angular displacements of joints). The evaluation of these circular data requires special statistical methods. The authors introduce a new test for the analysis of order-constrained hypotheses for circular data. Through this test, researchers can evaluate their expectations regarding the outcome of an experiment directly by representing their ideas in the form of a hypothesis containing inequality constraints. The resulting data analysis is generally more powerful than one using standard null hypothesis testing. Two examples of circular data from human movement science are presented to illustrate the use of the test. Results from a simulation study show that the test performs well. [ABSTRACT FROM AUTHOR]

We give a brief history of the results which led to the introduction of asymmetry into symmetric circular distributions. This is followed by the presentation of another method of introducing asymmetry. Some properties of the induced distributions are studied. Finally, this new distribution is shown to be a reasonable fit to the Jander ant data as presented in Fisher (1993). [ABSTRACT FROM AUTHOR]

Cluster analysis is a useful tool for data analysis. Clustering methods are used to partition a data set into clusters such that the data points in the same cluster are the most similar to each other and the data points in the different clusters are the most dissimilar. The mean shift was originally used as a kernel-type weighted mean procedure that had been proposed as a clustering algorithm. However, most mean shift-based clustering (MSBC) algorithms are used for numeric data. The circular data that are the directional data on the plane have been widely used in data analysis. In this paper, we propose a MSBC algorithm for circular data. Three types of mean shift implementation procedures with nonblurring, blurring and general methods are furthermore compared in which the blurring mean shift procedure is the best and recommended. The proposed MSBC for circular data is not necessary to give the number of cluster. It can automatically find a final cluster number with good clustering centers. Several numerical examples and comparisons with some existing clustering methods are used to demonstrate its effectiveness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Psychological researchers in different fields sometimes encounter circular or directional data. Circular data are data measured in the form of angles or two-dimensional orientations. As an example, experiments investigating the development of spatial memory and the influence of visual experience on haptic orientation perception are presented. Three permutation tests are proposed for the evaluation of ordered hypotheses. The quality of the permutation tests is investigated by means of several simulation studies. The results of these studies show the expected increase in power when the permutation tests for ordered hypotheses are compared to a common non-directional test for circular data. The differences in power between the three tests for ordered alternatives are small. [ABSTRACT FROM AUTHOR]

Motivated by classification issues that arise in marine studies, we propose a latent-class mixture model for the unsupervised classification of incomplete quadrivariate data with two linear and two circular components. The model integrates bivariate circular densities and bivariate skew normal densities to capture the association between toroidal clusters of bivariate circular observations and planar clusters of bivariate linear observations. Maximum-likelihood estimation of the model is facilitated by an expectation maximization (EM) algorithm that treats unknown class membership and missing values as different sources of incomplete information. The model is exploited on hourly observations of wind speed and direction and wave height and direction to identify a number of sea regimes, which represent specific distributional shapes that the data take under environmental latent conditions. [ABSTRACT FROM PUBLISHER]

In this paper we consider the problem of outliers for the functional relationship model of circular variables by transforming the circular data to continuous or real line data set via complex form. The COVRATIO statistic is extended from the linear regression models to the proposed model to detect any possible outliers. The cut-off points are obtained and the power of performance is examined by simulation studies. The model is illustrated with an application to the analysis of wind direction data recorded by two different techniques and the detection procedure of outliers is implied. [ABSTRACT FROM AUTHOR]

Abstract: Stone pavements are widespread landforms of arid environments. They typically develop by accretionary rise above a thickening aeolian mantle which usually contains a surficial vesicular unit. Hence, stone pavements are regarded as stable landforms and, thus, are often used as surface-age indicators. We studied orientation patterns of pavement clasts and depict their geometrical properties. We introduce a new statistical approach for quantitative description of circular data. From four study regions in a wide variety of environmental settings we show that preferred, non-random orientation of clast length axes is a recurrent feature of modern stone pavements. Orientation is primarily determined by no other relief parameter than slope aspect. In 40 out of 52 measured plots an angle of 72±9° separates two angle maxima. The same patterns are present in stone lines buried under a cover of aeolian dust. We interpret these features as fossil stone pavements, accordingly. Furthermore, on artificially cleared plots disturbed stone pavements begin to recover within few months by a process which creates essentially the same orientation pattern. We suggest that lateral surficial processes such as unconcentrated overland flow and creep contribute to the formation and maintenance of stone pavements as well as to this preferred clast orientation. They arrange clasts symmetrical to slope aspect with an angle determined by the axes ratio of the transported object. This results in the frequent bimodal pattern of stone length axes. This special type of transport requires previous accumulation of aeolian material forming a fine grained, even vesicular layer. Thus, there is close relationship between stone pavement and vesicular layer. We conclude that lateral processes take part in stone pavement formation. The commonly accepted model of dust accretion, which is not rendered obsolete by our findings, has to be complemented by this lateral component. This has implications for the stability of stone pavement-covered surfaces and for their applicability as age indicators. Quickly resealing disturbances as well as fossilised stone pavements imply this prominent desert surface feature may become fragile and transient. [Copyright &y& Elsevier]

When ecological traits differ seasonally among biological groups, environmental adaptation is expected. However, current analytical methods for such seasonal data ignore the circular nature of data and therefore are likely to be flawed. We propose a simple bootstrap hypothesis test which statistically quantifies the presence/absence of differences in peak months for multiple groups taking the circular nature of the data into account. The test is based on a robust distribution-free method. Simulations showed that the test gives a satisfactory performance. The test is illustrated using data for anchovy and sardine egg abundances in the western North Pacific. [ABSTRACT FROM AUTHOR]

We give a linear-time recognition algorithm for circular-arc graphs based on the algorithm of Eschen and Spinrad (Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 128-137, ) and Eschen (PhD thesis, ). Our algorithm both improves the time bound of Eschen and Spinrad, and fixes some flaws in it. Our algorithm is simpler than the earlier linear-time recognition algorithm of McConnell (Algorithmica 37(2):93-147, ), which is the only linear time recognition algorithm previously known. [ABSTRACT FROM AUTHOR]

In this paper, a class of row-column design incorporating neighbour effects has been studied. A row-column design is said to be neighbour balanced if every treatment has all other treatments appearing as neighbour(s) constant number of times. We have considered here row-column designs with non-directional neighbour effects and method of constructing these designs has been developed. The characterization properties of the designs have also been studied and the designs are found to be variance balanced for the estimation of direct and neighbour effects of treatments. [ABSTRACT FROM AUTHOR]

This paper proposes a maximum clustering similarity (MCS) method for determining the number of clusters in a data set by studying the behavior of similarity indices comparing two (of several) clustering methods. The similarity between the two clusterings is calculated at the same number of clusters, using the indices of Rand (R), Fowlkes and Mallows (FM), and Kulczynski (K) each corrected for chance agreement. The number of clusters at which the index attains its maximum is a candidate for the optimal number of clusters. The proposed method is applied to simulated bivariate normal data, and further extended for use in circular data. Its performance is compared to the criteria discussed in Tibshirani, Walther, and Hastie (2001). The proposed method is not based on any distributional or data assumption which makes it widely applicable to any type of data that can be clustered using at least two clustering algorithms. [ABSTRACT FROM AUTHOR]

Inferences based on regression models for a directional response are usually problematic. This paper presents a Bayesian analysis of a regression model for circular data using the projected normal distribution. Inferences about the model are based on samples from the posterior densities which are obtained using the Gibbs sampler after the introduction of suitable latent variables. The problem of missing data in the response variable is also addressed in this context as is the use of a predictive criterion for model selection. The procedures are illustrated using two simulated datasets a dataset previously analysed in the literature and a real dataset concerning wind directions. [ABSTRACT FROM AUTHOR]

This paper considers the three-parameter family of symmetric unimodal circular distributions proposed by Batschelet in [1], an extension of the von Mises distribution containing distributional forms ranging from the highly leptokurtic to the very platykurtic. The family's fundamental properties are given, and likelihood-based techniques described which can be used to perform estimation and hypothesis testing. Analyses are presented of two data sets which illustrate how the family and three of its most direct competitors can be applied in the search for parsimonious models for circular data. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the robustness of the directional mean (a.k.a. circular mean) for different families of circular distributions. We show that the directional mean is robust in the sense of finite standardized gross error sensitivity (SB-robust) for the following families: (1) mixture of two circular normal distributions, (2) mixture of wrapped normal and circular normal distributions and (3) mixture of two wrapped normal distributions. We also show that the directional mean is not SB-robust for the family of all circular normal distributions with varying concentration parameter. We define the circular trimmed mean and prove that it is SB-robust for this family. In general the property of SB-robustness of an estimator at a family of probability distributions is dependent on the choice of the dispersion measure. We introduce the concept of equivalent dispersion measures and prove that if an estimator is SB-robust for one dispersion measure then it is SB-robust for all equivalent dispersion measures. Three different dispersion measures for circular distributions are considered and their equivalence studied. [ABSTRACT FROM AUTHOR]

There are many well-known methods applied in classification problem for linear data with both known and unknown distribution. Here, we deal with classification involving data on torus and cylinder. A new method involving a generalized likelihood ratio test is developed for classifying in two populations using directional data. The approach assumes that one of the probabilities of misclassification is known. The procedure is constructed by applying Gibbs sampler on the conditionally specified distribution. A parametric bootstrap approach is also presented. An application to data involving linear and circular measurements on human skull from two tribal populations is given. [ABSTRACT FROM AUTHOR]

The proposed method in this paper uses circular statistics for the analysis of errors in the positional accuracy of geometric corrections satellite images using Independent Check Lines (ICL) instead of Independent Check Points (ICP). Circular statistics has been preferred because of the vectorial nature of the spatial error. A study case has been presented and discussed in detail. From the TERRA-ASTER images of Extremadura area (Spain), the Ground Control Point (GCP), ICP, and ICL data were acquired using differential GPS through field survey, and the planimetric positional accuracy was analyzed by both the conventional method (using ICP) and the proposed method (using ICL). Comparing conventional and proposed methods, the results indicated that modulus statistics are similar (e.g., RMSE of Geometric Correction 1 were 17.5 for the conventional method and 17.2 m for proposed method). But as additional results, azimuthal component statistics was calculated (e.g., mean direction: 247.2° in Geometric Correction 1), and several tests were made which showed the error distribution are not uniform and normal. [ABSTRACT FROM AUTHOR]

We propose a discrete time Markov process which takes values on the unit circle. Some properties of the process, including the limiting behaviour and ergodicity, are investigated. Many computations associated with this process are shown to be greatly simplified if the variables and parameters of the model are represented in terms of complex numbers. A further discussion is given on some submodels, in particular on the stationary process. The proposed model is compared with some existing Markov processes for circular data. Some statistical issues of the model, such as statistical inference, model selection and diagnostic checks, are considered. Finally, an application of the model to wind direction data is provided. [ABSTRACT FROM AUTHOR]