Möbius Transformation-Induced Distributions Provide Better Modelling for Protein Architecture

Proteins are found in all living organisms and constitute a large group of macromolecules with many functions. Proteins achieve their operations by adopting distinct three-dimensional structures encoded within the sequence of the constituent amino acids in one or more polypeptides. New, more flexible distributions are proposed for the MCMC sampling method for predicting protein 3D structures by applying a Möbius transformation to the bivariate von Mises distribution. In addition to this, sine-skewed versions of the proposed models are introduced to meet the increasing demand for modelling asymmetric toroidal data. Interestingly, the marginals of the new models lead to new multimodal circular distributions. We analysed three big datasets consisting of bivariate information about protein domains to illustrate the efficiency and behaviour of the proposed models. These newly proposed models outperformed mixtures of well-known models for modelling toroidal data. A simulation study was carried out to find the best method for generating samples from the proposed models. Our results shed new light on proposal distributions in the MCMC sampling method for predicting the protein structure environment.

Towards extracting cosmic magnetic field structures from cosmic-ray arrival directions

We present a novel method to search for structures of coherently aligned patterns in ultra-high energy cosmic-ray arrival directions simultaneously across the entire sky. This method can be used to obtain information on the Galactic magnetic field, in particular the integrated component perpendicular to the line of sight, from cosmic-ray data only. Using a likelihood-ratio approach, neighboring cosmic rays are related by rotatable, elliptically shaped density distributions and the significance of their alignment with respect to circular distributions is evaluated. In this way, a vector field tangential to the celestial sphere is fitted which approximates the local deflections in cosmic magnetic fields if significant deflection structures are detected. The sensitivity of the method is evaluated on the basis of astrophysical simulations of the ultra-high energy cosmic-ray sky, where a discriminative power between isotropic and signal-induced scenarios is found.

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.

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

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.