Sine-skewed circular distributions

Von Mises-Fisher Elliptical Distribution

Modern probabilistic learning systems mainly assume symmetric distributions, however, real-world data typically obey skewed distributions and are thus not adequately modeled through symmetric distributions. To address this issue, a generalization of symmetric distributions called elliptical distributions are increasingly used, together with further improvements based on skewed elliptical distributions. However, existing approaches are either hard to estimate or have complicated and abstract representations. To this end, we propose a novel approach based on the von-Mises-Fisher (vMF) distribution to obtain an explicit and simple probability representation of skewed elliptical distributions. The analysis shows that this not only allows us to design and implement nonsymmetric learning systems but also provides a physically meaningful and intuitive way of generalizing skewed distributions. For rigor, the proposed framework is proven to share important and desirable properties with its symmetric counterpart. The proposed vMF distribution is demonstrated to be easy to generate and stable to estimate, both theoretically and through examples.

Tractable circula densities from Fourier series

This article proposes an approach, based on infinite Fourier series, to constructing tractable densities for the bivariate circular analogues of copulas recently coined ‘circulas’. As examples of the general approach, we consider circula densities generated by various patterns of nonzero Fourier coefficients. The shape and sparsity of such arrangements are found to play a key role in determining the properties of the resultant models. The special cases of the circula densities we consider all have simple closed-form expressions involving no computationally demanding normalizing constants and display wide-ranging distributional shapes. A highly successful model identification tool and methods for parameter estimation and goodness-of-fit testing are provided for the circula densities themselves and the bivariate circular densities obtained from them using a marginal specification construction. The modelling capabilities of such bivariate circular densities are compared with those of five existing models in a numerical experiment, and their application illustrated in an analysis of wind directions.

Enhancing wind direction prediction of South Africa wind energy hotspots with Bayesian mixture modeling

Wind energy production depends not only on wind speed but also on wind direction. Thus, predicting and estimating the wind direction for sites accurately will enhance measuring the wind energy potential. The uncertain nature of wind direction can be presented through probability distributions and Bayesian analysis can improve the modeling of the wind direction using the contribution of the prior knowledge to update the empirical shreds of evidence. This must align with the nature of the empirical evidence as to whether the data are skew or multimodal or not. So far mixtures of von Mises within the directional statistics domain, are used for modeling wind direction to capture the multimodality nature present in the data. In this paper, due to the skewed and multimodal patterns of wind direction on different sites of the locations understudy, a mixture of multimodal skewed von Mises is proposed for wind direction. Furthermore, a Bayesian analysis is presented to take into account the uncertainty inherent in the proposed wind direction model. A simulation study is conducted to evaluate the performance of the proposed Bayesian model. This proposed model is fitted to datasets of wind direction of Marion island and two wind farms in South Africa and show the superiority of the approach. The posterior predictive distribution is applied to forecast the wind direction on a wind farm. It is concluded that the proposed model offers an accurate prediction by means of credible intervals. The mean wind direction of Marion island in 2017 obtained from 1079 observations was 5.0242 (in radian) while using our proposed method the predicted mean wind direction and its corresponding [Formula: see text] credible interval based on 100 generated samples from the posterior predictive distribution are obtained 5.0171 and (4.7442, 5.2900). Therefore, our results open a new approach for accurate prediction of wind direction implementing a Bayesian approach via mixture of skew circular distributions.

Coming Together of Bayesian Inference and Skew Spherical Data

This paper presents Bayesian directional data modeling via the skew-rotationally-symmetric Fisher-von Mises-Langevin (FvML) distribution. The prior distributions for the parameters are a pivotal building block in Bayesian analysis, therefore, the impact of the proposed priors will be quantified using the Wasserstein Impact Measure (WIM) to guide the practitioner in the implementation process. For the computation of the posterior, modifications of Gibbs and slice samplings are applied for generating samples. We demonstrate the applicability of our contribution via synthetic and real data analyses. Our investigation paves the way for Bayesian analysis of skew circular and spherical data.

Theoretical Relationship Between Two Measures of Spike Synchrony: Correlation Index and Vector Strength

Information processing in the nervous system critically relies on temporally precise spiking activity. In the auditory system, various degrees of phase-locking can be observed from the auditory nerve to cortical neurons. The classical metric for quantifying phase-locking is the vector strength (VS), which captures the periodicity in neuronal spiking. More recently, another metric, called the correlation index (CI), was proposed to quantify the temporally reproducible response characteristics of a neuron. The CI is defined as the peak value of a normalized shuffled autocorrelogram (SAC). Both VS and CI have been used to investigate how temporal information is processed and propagated along the auditory pathways. While previous analyses of physiological data in cats suggested covariation of these two metrics, general characterization of their connection has never been performed. In the present study, we derive a rigorous relationship between VS and CI. To model phase-locking, we assume Poissonian spike trains with a temporally changing intensity function following a von Mises distribution. We demonstrate that VS and CI are mutually related via the so-called concentration parameter that determines the degree of phase-locking. We confirm that these theoretical results are largely consistent with physiological data recorded in the auditory brainstem of various animals. In addition, we generate artificial phase-locked spike sequences, for which recording and analysis parameters can be systematically manipulated. Our analysis results suggest that mismatches between empirical data and the theoretical prediction can often be explained with deviations from the von Mises distribution, including skewed or multimodal period histograms. Furthermore, temporal relations of spike trains across trials can contribute to higher CI values than predicted mathematically based on the VS. We find that, for most applications, a SAC bin width of 50 ms seems to be a favorable choice, leading to an estimated error below 2.5% for physiologically plausible conditions. Overall, our results provide general relations between the two measures of phase-locking and will aid future analyses of different physiological datasets that are characterized with these metrics.

Sine-skewed toroidal distributions and their application in protein bioinformatics

In the bioinformatics field, there has been a growing interest in modeling dihedral angles of amino acids by viewing them as data on the torus. This has motivated, over the past years, new proposals of distributions on the torus. The main drawback of most of these models is that the related densities are (pointwise) symmetric, despite the fact that the data usually present asymmetric patterns. This motivates the need to find a new way of constructing asymmetric toroidal distributions starting from a symmetric distribution. We tackle this problem in this article by introducing the sine-skewed toroidal distributions. The general properties of the new models are derived. Based on the initial symmetric model, explicit expressions for the shape and dependence measures are obtained, a simple algorithm for generating random numbers is provided, and asymptotic results for the maximum likelihood estimators are established. An important feature of our construction is that no extra normalizing constant needs to be calculated, leading to more flexible distributions without increasing the complexity of the models. The benefit of employing these new sine-skewed toroidal distributions is shown on the basis of protein data, where, in general, the new models outperform their symmetric antecedents.

A boxplot for circular data

The box-and-whiskers plot is an extraordinary graphical tool that provides a quick visual summary of an observed distribution. In spite of its many extensions, a really suitable boxplot to display circular data is not yet available. Thanks to its simplicity and strong visual impact, such a tool would be especially useful in all fields where circular measures arise: biometrics, astronomy, environmetrics, Earth sciences, to cite just a few. For this reason, in line with Tukey's original idea, a Tukey-like circular boxplot is introduced. Several simulated and real datasets arising in biology are used to illustrate the proposed graphical tool.

Analysis of CO2 daily cycle in the low atmosphere at a rural site

Directional analysis procedures were extended to study the CO(2) daily cycle in the low atmosphere at a rural site. Weighted average, asymmetric circular functions, and conditional probability curves were applied to concentrations recorded at three levels near surface. Wind speed and potential temperature gradient were the meteorological variables used to establish a link to CO(2). The highest weighted concentration was obtained for wind speed below 4 m s(-1) and for potential temperature gradient above 0.05 °C m(-1). Asymmetric circular functions were fitted to selected percentiles, the generalized von Mises function providing the lowest RMSE. Slight differences among levels were observed in the transition between day and night, and higher gradients were observed during the night. Four intervals were proposed when concentrations were binned by intervals of meteorological variables. Wind speed was only relevant to separate intermediate concentrations. Finally, conditional probability curves provided differences between the highest and lowest levels in up to 20% of cases at 20-22 GMT.

Inverse Batschelet distributions for circular data

We provide four-parameter families of distributions on the circle which are unimodal and display the widest ranges of both skewness and peakedness yet available. Our approach is to transform the scale of a generating distribution, such as the von Mises, using various nontrivial extensions of an approach first used in Batschelet's (1981, Circular Statistics in Biology) book. The key is to employ inverses of Batschelet-type transformations in certain ways; these exhibit considerable advantages over direct Batschelet transformations. The skewness transformation is especially appealing as it has no effect on the normalizing constant. As well as a variety of interesting theoretical properties, when likelihood inference is explored these distributions display orthogonality between elements of a pairing of parameters into (location, skewness) and (concentration, peakedness). Further, the location parameter can sometimes be made approximately orthogonal to all the other parameters. Profile likelihoods come to the fore in practice. Two illustrative applications, one concerning the locomotion of a Drosophila fly larva, the other analyzing a large set of sudden infant death syndrome data, are investigated.