Analysis Of Circular Data Research Articles

A new measure of preferred direction for circular data using angular wrapping

The statistical techniques which are developed for the analysis of data in the linear number system cannot be applied to directional data directly. Circular data may be discontinuous in some principal interval. These discontinuities cause failure results in the circular statistics. Because of that the proposed wrapping operator must be used for data, which are defined in the discontinuous range. However, in both continuity and discontinuity, the wrapping operator works correctly. The most common preferred directions for circular data are circular mean and variance summarizing and comparing them. Although circular data has a very important role in statistics, the literature is weak in terms of statistical analysis of circular data. It creates a gap in this field. This study examines the preferred direction of circular data to fill this gap and presents a new measure of preferred direction for circular data using angular wrapping. Four different artificial and three real datasets are employed to evaluate the performance of the proposed methods. The results demonstrate the superiority of the proposed methods in terms of the absolute error and absolute percentage error. Consequently, it has been seen that the proposed methods giv e more consistent and more accurate results than thevectorial methods.

Estimating the concentration parameter of a von Mises distribution: a systematic simulation benchmark

In directional statistics, the von Mises distribution is a key element in the analysis of circular data. While there is a general agreement regarding the estimation of its location parameter μ, several methods have been proposed to estimate the concentration parameter κ. We here provide a thorough evaluation of the behavior of 12 such estimators for datasets of size N ranging from 2 to 8192 generated with a κ ranging from 0 to 100. We provide detailed results as well as a global analysis of the results, showing that (1) for a given κ, most estimators have behaviors that are very similar for large datasets () and more variable for small datasets, and (2) for a given estimator, results are very similar if we consider the mean absolute error for and the mean relative absolute error for

Generalized Cardioid Distributions for Circular Data Analysis

The Cardioid (C) distribution is one of the most important models for modeling circular data. Although some of its structural properties have been derived, this distribution is not appropriate for asymmetry and multimodal phenomena in the circle, and then extensions are required. There are various general methods that can be used to produce circular distributions. This paper proposes four extensions of the C distribution based on the beta, Kumaraswamy, gamma, and Marshall–Olkin generators. We obtain a unique linear representation of their densities and some mathematical properties. Inference procedures for the parameters are also investigated. We perform two applications on real data, where the new models are compared to the C distribution and one of its extensions.

Examining Periodic Differences of Suicide Cases with Circular Data Analysis

Examining Periodic Differences of Suicide Cases with Circular Data Analysis

Descriptive analysis of circular data with outliers using Python programming language

Descriptive statistics are commonly used in data analysis to describe the basic features of raw data. Descriptive summaries enable us to present the data in a more simple and meaningful way so that the interpretation will be easier to understand. The descriptive analysis of circular data with outliers is discussed in this study. Circular data is different from linear data in many aspects such as statistical modeling, descriptive statistics and etc. Hence, unlike linear data, the availability of statistical software specialises in analysing circular data is very limited. Python is a programming language which frequently used by data analysts nowadays. However, the package for circular statistics is not fully developed and it is not ready to use like in Splus or R programming language. In this study, the descriptive analysis of circular data is performed using the in-demand programming language, Python. Descriptive statistics of the circular data especially with the existence of outliers are discussed and the proposed Python code is available to use.

Exploring Anderson localization in one-dimensional single-electron lattice systems via randomness of eigenvalue spectra

Connecting electron localization properties with energy spectra statistics is a fundamental issue in condensed matter physics. Inspired by the circular data analysis and the Kuramoto order parameter developed in classical systems, the sequence of the ratio of consecutive level spacing {rm} is mapped to a set of unit vectors {eˆm} distributed on a unit circle. A quantity Ir=Z2M is proposed to measure the randomness of eigenvalue spectra of quantum systems, where M is the sequence length and Z=1M|∑m=1Meˆm|. For the other sequence {Rm} related to {rm}, a similar quantity IR is also defined. It is found that 〈Ir〉>〈IR〉 for localized phases and 〈Ir〉<〈IR〉 for delocalized ones, which can serve as a fruitful index to monitor Anderson transition.

A smoothing spline model for multimodal and skewed circular responses: Applications in meteorology and oceanography

AbstractThe analysis of circular data is the main subject in many disciplines, such as meteorology and oceanography. In this article, we introduce a new multimodal skew‐circular model as an extension of the circular beta distribution. We propose a truncated power smoothing spline for modeling the skewness parameter and identifying significant factors of the asymmetry. A Markov chain Monte Carlo scheme is provided to perform statistical inference from a Bayesian perspective. Then, the performance of our modeling methodology to analyze specific circular responses is assessed through several simulation studies. To illustrate the usefulness of the new model in practical applications, we analyze measurements on the wind and wave directions in Norway. We also fit various regression models to show that the cubic smoothing spline approach performs better than competitive models in practical applications. Findings, based on prediction values, confirm that the proposed model can reasonably fit multimodal skewed‐circular responses.

TORUS GRAPHS FOR MULTIVARIATE PHASE COUPLING ANALYSIS.

Angular measurements are often modeled as circular random variables, where there are natural circular analogues of moments, including correlation. Because a product of circles is a torus, a d-dimensional vector of circular random variables lies on a d-dimensional torus. For such vectors we present here a class of graphical models, which we call torus graphs, based on the full exponential family with pairwise interactions. The topological distinction between a torus and Euclidean space has several important consequences. Our development was motivated by the problem of identifying phase coupling among oscillatory signals recorded from multiple electrodes in the brain: oscillatory phases across electrodes might tend to advance or recede together, indicating coordination across brain areas. The data analyzed here consisted of 24 phase angles measured repeatedly across 840 experimental trials (replications) during a memory task, where the electrodes were in 4 distinct brain regions, all known to be active while memories are being stored or retrieved. In realistic numerical simulations, we found that a standard pairwise assessment, known as phase locking value, is unable to describe multivariate phase interactions, but that torus graphs can accurately identify conditional associations. Torus graphs generalize several more restrictive approaches that have appeared in various scientific literatures, and produced intuitive results in the data we analyzed. Torus graphs thus unify multivariate analysis of circular data and present fertile territory for future research.

Estimation of finite mixture models of skew-symmetric circular distributions

Analysis of circular data is challenging, since the usual statistical methods are unsuitable and it is necessary to use circular periodic probabilistic models. Because some actual circular datasets exhibit asymmetry and/or multimodality, finite mixtures of symmetric circular distributions to model and fit these data have been investigated. However, it is necessary to question the predominant assumption that each component in the finite mixture model is symmetric. In this study, we consider a finite mixture model of possibly skewed circular distributions and discuss the expectation-maximization (EM) algorithm for the maximum likelihood estimate. It is shown that the maximum likelihood estimator is strongly consistent under some suitable conditions in a finite mixture of skew-symmetric circular distributions. A modified M-step in the EM algorithm is proposed in order to estimate the unknown parameter vectors effectively. To investigate the performance of our proposed model with its estimation procedure, we provide a numerical example as well as data analysis using the records of the time of day of fatal traffic accidents.

Model‐based clustering for noisy longitudinal circular data, with application to animal movement

AbstractIn this work, we introduce a model for circular data analysis to robustly estimate parameters, under a longitudinal clustering setting. A hidden Markov model for longitudinal circular data combined with a uniform conditional density on the circle to capture noise observations is proposed. A set of exogenous covariates is available; they are assumed to affect the evolution of clustering over time. Parameter estimation is carried out through a hybrid expectation–maximization algorithm, using recursions widely adopted in the hidden Markov model literature. Examples of application of the proposal on real and simulated data are performed to show the effectiveness of the proposal.

Application of circular statistics to life science data

Objective: The aim of the study was to explain circular statistics and hypothesis tests with life science data. The accuracy of the statistical method used in scientific research is related to the data structure and scale type. Therefore, scale types and data structures should be well defined. It is possible to frequently come across circular data in many different scientific fields such as medicine, biology and physics. These data are usually obtained by compass or clock. Compass; the flight direction of any animal that is released, the direction of the wind or the direction of current in the ocean, clock; the birth time of infants, the time of crisis, circadian rhythms or biological rhythms can be shown as examples. Apart from the clock, such data may also be obtained by a scale that expresses a time such as day, month and year. Material and Methods: The data related to 179 normal deliveries that took place in Yuzuncu Yil University Medical Faculty Hospital in 2008 were used. Circular data analysis was performed using the NCSS2007 statistical package program. Results: The times of birth of infants show a uniform distribution. No significant difference at a significance level of 5% was found between the times of birth according to gender. Conclusion: It has been stated that circular data cannot be analyzed by the analysis methods developed for linear data due to several reasons. If circular data are analyzed by linear statistical methods, inaccurate or nonsense results usually emerge. Therefore, it was emphasized that appropriate statistical methods should be used.

Bayesian nonparametric inference for the overlap of daily animal activity patterns

The study of the interaction among species is an active area of research in Ecology. In particular, it is of interest to evaluate the overlap of their ecological niches. Temporal activity is one of the niche’s axes most commonly used to explore ecological segregation among animal species, and many contributions focus on the overlap of this variable. Once the information of the temporal activity is obtained in the wild, the data is treated as a random sample. There exist different methods to estimate the overlap. Specifically, in the case of two species, one possibility is to estimate the density of the temporal activity of each species and then evaluate the overlap between these density functions. This leads naturally to the analysis of circular data. Most of the procedures currently in use impose some rather restrictive assumptions on the probabilistic models used to describe the phenomena, and only provide approximate measures of the uncertainty involved in the process. In this article, we propose a Bayesian nonparametric approach which incorporates a well-defined noninformative prior. We take advantage of the data structure to define such a prior in terms of the predictive distribution. To the best of our knowledge, this is a novel approach. Our procedure is compared with a well-known method using simulated data, and applied to the analysis of real camera-trap data concerning two mammalian species from the El Triunfo biosphere reserve (Chiapas, Mexico).

One Direction? A Tutorial for Circular Data Analysis Using R With Examples in Cognitive Psychology.

Circular data is data that is measured on a circle in degrees or radians. It is fundamentally different from linear data due to its periodic nature (0° = 360°). Circular data arises in a large variety of research fields. Among others in ecology, the medical sciences, personality measurement, educational science, sociology, and political science circular data is collected. The most direct examples of circular data within the social sciences arise in cognitive and experimental psychology. However, despite numerous examples of circular data being collected in different areas of cognitive and experimental psychology, the knowledge of this type of data is not well-spread and literature in which these types of data are analyzed using methods for circular data is relatively scarce. This paper therefore aims to give a tutorial in working with and analyzing circular data to researchers in cognitive psychology and the social sciences in general. It will do so by focusing on data inspection, model fit, estimation and hypothesis testing for two specific models for circular data using packages from the statistical programming language R.

Bringing the analysis of animal orientation data full circle: model-based approaches with maximum likelihood.

In studies of animal orientation, data are often represented as directions that can be analyzed using circular statistical methods. Although several circular statistical tests exist to detect the presence of a mean direction, likelihood-based approaches may offer advantages in hypothesis testing - especially when data are multimodal. Unfortunately, likelihood-based inference in animal orientation remains rare. Here, we discuss some of the assumptions and limitations of common circular tests and report a new R package called CircMLE to implement the maximum likelihood analysis of circular data. We illustrate the use of this package on both simulated datasets and an empirical example dataset in Chinook salmon (Oncorhynchus tshawytscha). Our software provides a convenient interface that facilitates the use of model-based approaches in animal orientation studies.

A comparative simulation of multiple testing procedures in circular data problems

ABSTRACTBonferroni correction procedures are commonly used for performing multiple hypothesis tests in linear data problems. Moreover, several improved Bonferroni type procedures have been proposed and shown that they attain a type-I error rate better than the classical Bonferroni approach, but all of the studies are considered for linear data problems. The circular data analysis is a developing field of statistics that lacks similar studies, and also lacks computer programs to implement Bonferroni procedures. The aim of this study is to perform a comparative study of improved Bonferroni procedures and also to provide a computer program in R that performs classical and improved Bonferroni procedures for circular data problems.

Influence Diagnostics in Possibly Asymmetric Circular-Linear Multivariate Regression Models

Distributional studies and regression models have played important roles in statistical analysis of circular data. Asymmetric circular-linear multivariate regression models (SenGupta and Ugwuowo Environ. Ecol. Stat.13(3), 299–309 2006) are motivated by and applied to predict some environmental characteristics based on both circular and linear predictors. In this paper, we consider a likelihood approach (Cook J. R. Stat. Soc. Ser. B Stat Methodol.48(2), 133–169 1986) to study influence diagnostic analysis for these models, using the maximum likelihood estimation and influence diagnostics methods. The observed information matrices and normal curvatures are derived. Simulated and real data examples are then provided to illustrate our approach and establish the utility of our results.

Analyzing longitudinal circular data by projected normal models: a semi-parametric approach based on finite mixture models

The analysis of circular data has been recently the focus of a wide range of literature, with the general objective of providing reliable parameter estimates in the presence of heterogeneity and/or dependence among observations under a longitudinal setting. In this paper, we extend the variance component model approach to the analysis of longitudinal circular data, defining a mixed effects model for radial projections onto the circle and introducing dependence between projections through a set of correlated random coefficients. Estimation is carried out by numerical integration through an expectation-maximization algorithm without parametric assumptions upon the random coefficients distribution. The resulting model is a finite mixture of projected normal distributions. A simulation study has been carried out to investigate the behavior of the proposed model in a series of empirical situations. The proposed model is computationally parsimonious and, when applied to a real dataset on animal orientation, produces novel results.

On the Gini Mean Difference Test for Circular Data

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 distributionsunder 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.

Bayesian Analysis of Circular Data Using Wrapped Distributions

Circular data arise in a number of different areas such as geological, meteorological, biological and industrial sciences. Standard statistical techniques can not be used to model circular data due to the circular geometry of the sample space. One of the common methods to analyze circular data is known as the wrapping approach. This approach is based on a simple fact that a probability distribution on a circle can be obtained by wrapping a probability distribution defined on the real line. A large class of probability distributions that are flexible to account for different features of circular data can be obtained by the aforementioned approach. However, the likelihood-based inference for wrapped distributions can be very complicated and computationally intensive. The EM algorithm to compute the MLE is feasible, but is computationally unsatisfactory. A data augmentation method using slice sampling is proposed to overcome such computational difficulties. The proposed method turns out to be flexible and computationally efficient to fit a wide class of wrapped distributions. In addition, a new model selection criteria for circular data is developed. Results from an extensive simulation study are presented to validate the performance of the proposed estimation method and the model selection criteria. Application to a real data set is also presented and parameter estimates are compared to those that are available in the literature.

DATA‐DRIVEN SMOOTH TESTS AND A DIAGNOSTIC TOOL FOR LACK‐OF‐FIT FOR CIRCULAR DATA

SummaryTwo contributions to the statistical analysis of circular data are given. First we construct data‐driven smooth goodness‐of‐fit tests for the circular von Mises assumption. As a second method, we propose a new graphical diagnostic tool for the detection of lack‐of‐fit for circular distributions. We illustrate our methods on two real datasets.