Models for Circular-Linear and Circular-Circular Data Constructed from Circular Distributions Based on Nonnegative Trigonometric Sums

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.

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Comparing two circular distributions: advice for effective implementation of statistical procedures in biology

10.1101/2021.03.25.436932 ◽

2021 ◽

Author(s):

Lukas Landler ◽

Graeme D Ruxton ◽

Erich Pascal Malkemper

Keyword(s):

Statistical Power ◽

Type I Error ◽

Statistical Treatment ◽

Good Control ◽

Time Of Day ◽

Circular Data ◽

Type I ◽

Biological Variables ◽

Circular Distributions ◽

Two Samples

Many biological variables, often involving timings of events or directions, are recorded on a circular rather than linear scale, and need different statistical treatment for that reason. A common question that is asked of such circular data involves comparison between two groups or treatments: Are the populations from which the two samples drawn differently distributed around the circle? For example, we might ask whether the distribution of directions from which a stalking predator approaches its prey differs between sunny and cloudy conditions; or whether the time of day of mating attempts differs between lab mice subject to one of two hormone treatments. An array of statistical approaches to these questions have been developed. We compared 18 of these (by simulation) in terms of both abilities to control type I error rate near the nominal value, and statistical power. We found that only eight tests offered good control of type I error in all our test situations. Of these eight, we are able to identify Watsons U^2 test and MANOVA based on trigonometric functions of the data as offering the best power in the overwhelming majority of our test circumstances. There was often little to choose between these tests in terms of power, and no situation where either of the remaining six tests offered substantially better power than either of these. Hence, we recommend the routine use of either Watsons U^2 test or MANOVA when comparing two samples of circular data.

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Testing for seasonality using circular distributions based on non-negative trigonometric sums as alternative hypotheses

Statistical Methods in Medical Research ◽

10.1177/0962280211411531 ◽

2011 ◽

Vol 23 (3) ◽

pp. 279-292 ◽

Cited By ~ 1

Author(s):

J J Fernández-Durán ◽

M M Gregorio-Domínguez

Keyword(s):

Trigonometric Sums ◽

Circular Distributions ◽

Alternative Hypotheses

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On Bimodal Offset Cauchy Distribution

Journal of Applied Mathematics Statistics and Informatics ◽

10.2478/jamsi-2013-0006 ◽

2013 ◽

Vol 9 (1) ◽

pp. 61-67 ◽

Cited By ~ 1

Author(s):

S.V.S. Girija ◽

A.J.V. Radhika ◽

A.V.Dattatreya Rao

Keyword(s):

Clinical Trials ◽

Queuing Theory ◽

Cauchy Distribution ◽

Circular Data ◽

Circular Model ◽

Circular Distributions ◽

Directional Component ◽

Decreasing Failure Rate ◽

Cauchy Model ◽

Labour Turnover

Abstract The bivariate Cauchy distribution has received applications in many areas, including biological analyses, clinical trials, stochastic modeling of decreasing failure rate life components, study of labour turnover, queuing theory and reliability (Nayak (1987) and Lee and Gross (1991)). In the study of biological analyses, clinical trials and reliability circular distributions will yield suitable results. Circular data arises in a number of different areas such as geological, meteorological, biological and industrial sciences. It is not suggestive to use standard statistical techniques to model circular data, due to the circular geometry of the sample space (p.2 Jammalamadaka and Sen Gupta (2001). It is possible to construct a circular model by transforming a bivariate linear random variate to just its directional component and the resultant model is called ‘offset distribution’. In the literature most of the available circular models were constructed by wrapping a linear model. In recent years some wrapped models were constructed by Dattatreya Rao et al (2007). Here an attempt is made to exploit method of offsetting on Bivariate Cauchy distribution to construct a circular model named by us “OFFSET CAUCHY DISTRIBUTION (OC)”. The characteristic function of the Offset Cauchy model is derived and its characteristics are discussed.

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Circular Distributions Based on Nonnegative Trigonometric Sums

Biometrics ◽

10.1111/j.0006-341x.2004.00195.x ◽

2004 ◽

Vol 60 (2) ◽

pp. 499-503 ◽

Cited By ~ 33

Author(s):

J. J. Fernández-Durán

Keyword(s):

Trigonometric Sums ◽

Circular Distributions

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An IMM Filter Defined in the Linear-Circular Domain, Application to Maneuver Detection with Heading Only

Mathematical Problems in Engineering ◽

10.1155/2018/3531075 ◽

2018 ◽

Vol 2018 ◽

pp. 1-14

Author(s):

Karim El mokhtari ◽

Serge Reboul ◽

Georges Stienne ◽

Jean Bernard Choquel ◽

Benaissa Amami ◽

...

Keyword(s):

A Priori ◽

Synthetic Data ◽

Real Data ◽

Circular Domain ◽

Circular Data ◽

Von Mises Distribution ◽

A Posteriori ◽

Circular Distributions ◽

Vehicle Maneuver ◽

Von Mises

In this article, we propose a multimodel filter for circular data. The so-called Circular Interacting Multimodel filter is derived in a Bayesian framework with the circular normal von Mises distribution. The aim of the proposed filter is to obtain the same performance in the circular domain as the classical IMM filter in the linear domain. In our approach, the mixing and fusion stages of the Circular Interacting Multimodel filter are, respectively, defined from the a priori and from the a posteriori circular distributions of the state angle knowing the measurements and according to a set of models. We propose in this article a set of circular models that will be used in order to detect the vehicle maneuvers from heading measurements. The Circular Interacting Multimodel filter performances are assessed on synthetic data and we show on real data a vehicle maneuver detection application.

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