Recent advances in directional statistics

A dependent circular-linear model for multivariate biomechanical data: Ilizarov ring fixator study

Biomechanical and orthopaedic studies frequently encounter complex datasets that encompass both circular and linear variables. In most cases (i) the circular and linear variables are considered in isolation with dependency between variables neglected and (ii) the cyclicity of the circular variables is disregarded resulting in erroneous decision making. Given the inherent characteristics of circular variables, it is imperative to adopt methods that integrate directional statistics to achieve precise modelling. This paper is motivated by the modelling of biomechanical data, that is, the fracture displacements, that is used as a measure in external fixator comparisons. We focus on a dataset, based on an Ilizarov ring fixator, comprising of six variables. A modelling framework applicable to the six-dimensional joint distribution of circular-linear data based on vine copulas is proposed. The pair-copula decomposition concept of vine copulas represents the dependence structure as a combination of circular-linear, circular-circular and linear-linear pairs modelled by their respective copulas. This framework allows us to assess the dependencies in the joint distribution as well as account for the cyclicity of the circular variables. Thus, a new approach for accurate modelling of mechanical behaviour for Ilizarov ring fixators and other data of this nature is imparted.

Analyzing physiological signals recorded with a wearable sensor across the menstrual cycle using circular statistics

This study aims to identify the most significant features in physiological signals representing a biphasic pattern in the menstrual cycle using circular statistics which is an appropriate analytic method for the interpretation of data with a periodic nature. The results can be used empirically to determine menstrual phases. A non-uniform pattern was observed in ovulating subjects, with a significant periodicity (p< 0.05) in mean temperature, heart rate (HR), Inter-beat Interval (IBI), mean tonic component of Electrodermal Activity (EDA), and signal magnitude area (SMA) of the EDA phasic component in the frequency domain. In contrast, non-ovulating cycles displayed a more uniform distribution (p> 0.05). There was a significant difference between ovulating and non-ovulating cycles (p< 0.05) in temperature, IBI, and EDA but not in mean HR. Selected features were used in training an Autoregressive Integrated Moving Average (ARIMA) model, using data from at least one cycle of a subject, to predict the behavior of the signal in the last cycle. By iteratively retraining the algorithm on a per-day basis, the mean temperature, HR, IBI and EDA tonic values of the next day were predicted with root mean square error (RMSE) of 0.13 ± 0.07 (C°), 1.31 ± 0.34 (bpm), 0.016 ± 0.005 (s) and 0.17 ± 0.17 (μS), respectively.

Robust Minimum Divergence Estimation for the Multinomial Circular Logistic Regression Model

Circular data are extremely important in many different contexts of natural and social science, from forestry to sociology, among many others. Since the usual inference procedures based on the maximum likelihood principle are known to be extremely non-robust in the presence of possible data contamination, in this paper, we develop robust estimators for the general class of multinomial circular logistic regression models involving multiple circular covariates. Particularly, we extend the popular density-power-divergence-based estimation approach for this particular set-up and study the asymptotic properties of the resulting estimators. The robustness of the proposed estimators is illustrated through extensive simulation studies and few important real data examples from forest science and meteorology.

An experimental platform for stochastic analyses of single serotonergic fibers in the mouse brain

The self-organization of the serotonergic matrix, a massive axon meshwork in all vertebrate brains, is driven by the structural and dynamical properties of its constitutive elements. Each of these elements, a single serotonergic axon (fiber), has a unique trajectory and can be supported by a soma that executes one of the many available transcriptional programs. This "individuality" of serotonergic neurons necessitates the development of specialized methods for single-fiber analyses, both at the experimental and theoretical levels. We developed an integrated platform that facilitates experimental isolation of single serotonergic fibers in brain tissue, including regions with high fiber densities, and demonstrated the potential of their quantitative analyses based on stochastic modeling. Single fibers were visualized using two transgenic mouse models, one of which is the first implementation of the Brainbow toolbox in this system. The trajectories of serotonergic fibers were automatically traced in the three spatial dimensions with a novel algorithm, and their properties were captured with a single parameter associated with the directional von Mises-Fisher probability distribution. The system represents an end-to-end workflow that can be imported into various studies, including those investigating serotonergic dysfunction in brain disorders. It also supports new research directions inspired by single-fiber analyses in the serotonergic matrix, including supercomputing simulations and modeling in physics.

Understanding the influence of news on society decision making: application to economic policy uncertainty

The abundance of digital documents offers a valuable chance to gain insights into public opinion, social structure, and dynamics. However, the scale and volume of these digital collections makes manual analysis approaches extremely costly and not scalable. In this paper, we study the potential of using automated methods from natural language processing and machine learning, in particular weak supervision strategies, to understand how news influence decision making in society. Besides proposing a weak supervision solution for the task, which replaces manual labeling to a certain extent, we propose an improvement of a recently published economic index. This index is known as economic policy uncertainty (EPU) index and has been shown to correlate to indicators such as firm investment, employment, and excess market returns. In summary, in this paper, we present an automated data efficient approach based on weak supervision and deep learning (BERT + WS) for identification of news articles about economical uncertainty and adapt the calculation of EPU to the proposed strategy. Experimental results reveal that our approach (BERT + WS) improves over the baseline method centered in keyword search, which is currently used to construct the EPU index. The improvement is over 20 points in precision, reducing the false positive rate typical to the use of keywords.

Linear convergence of the subspace constrained mean shift algorithm: from Euclidean to directional data

This paper studies the linear convergence of the subspace constrained mean shift (SCMS) algorithm, a well-known algorithm for identifying a density ridge defined by a kernel density estimator. By arguing that the SCMS algorithm is a special variant of a subspace constrained gradient ascent (SCGA) algorithm with an adaptive step size, we derive the linear convergence of such SCGA algorithm. While the existing research focuses mainly on density ridges in the Euclidean space, we generalize density ridges and the SCMS algorithm to directional data. In particular, we establish the stability theorem of density ridges with directional data and prove the linear convergence of our proposed directional SCMS algorithm.

Discrete mixture representations of spherical distributions

We obtain discrete mixture representations for parametric families of probability distributions on Euclidean spheres, such as the von Mises–Fisher, the Watson and the angular Gaussian families. In addition to several special results we present a general approach to isotropic distribution families that is based on density expansions in terms of special surface harmonics. We discuss the connections to stochastic processes on spheres, in particular random walks, discrete mixture representations derived from spherical diffusions, and the use of Markov representations for the mixing base to obtain representations for families of spherical distributions.

From dark to darkness, negative phototropism influences the support-tree location of the massive woody climber Hydrangea serratifolia (Hydrangeaceae) in a Chilean temperate rainforest

Climbing plants rely on suitable support to provide the light conditions they require in the canopy. Negative phototropism is a directional search behavior proposed to detect a support-tree, which indicates growth or movement away from light, based on light attenuation. In a Chilean temperate rainforest, we addressed whether the massive woody climber Hydrangea serratifolia (H. et A.) F. Phil (Hydrangeaceae) presents a support-tree location pattern influenced by light availability. We analyzed direction and light received in two groups of juvenile shoots: searching shoots (SS), with plagiotropic (creeping) growth vs. ascending shoots (AS), with orthotropic growth. We found that, in accordance with light attenuation, SS and AS used directional orientation to search and then ascend host trees. The light available to H. serratifolia searching shoots was less than that of the general forest understory; the directional orientation in both groups showed a significant deviation from a random distribution, with no circular statistical difference between them. Circular-linear regression indicated a relationship between directional orientations and light availability. Negative phototropism encodes the light environment's heterogeneous spatial and temporal information, guiding the shoot apex to the most shaded part of the support-tree base, the climbing start point.

Families of Discrete Circular Distributions with Some Novel Applications

We give a unified treatment of constructing families of circular discrete distributions. Some of these families are deduced from established distributions such as von Mises and wrapped Cauchy. Some others are derived directly such as a flexible family based on trigonometric sums and the circular location family. Results interrelating these families are discussed. These distributions have been motivated by two examples of discrete circular data: casino roulette spins and smart health acrophase monitoring, and these data are analyzed using our proposed models. We discuss how using continuous circular models for circular discrete data can be misleading.

sconce: a cosmic web finder for spherical and conic geometries

The latticework structure known as the cosmic web provides a valuable insight into the assembly history of large-scale structures. Despite the variety of methods to identify the cosmic web structures, they mostly rely on the assumption that galaxies are embedded in a Euclidean geometric space. Here, we present a novel cosmic web identifier called sconce (Spherical and CONic Cosmic wEb finder) that inherently considers the 2D (RA, DEC) spherical or the 3D (RA, DEC, z) conic geometry. The proposed algorithms in sconce generalize the well-known subspace constrained mean shift (scms) method and primarily address the predominant filament detection problem. They are intrinsic to the spherical/conic geometry and invariant to data rotations. We further test the efficacy of our method with an artificial cross-shaped filament example and apply it to the SDSS galaxy catalogue, revealing that the 2D spherical version of our algorithms is robust even in regions of high declination. Finally, using N-body simulations from Illustris, we show that the 3D conic version of our algorithms is more robust in detecting filaments than the standard scms method under the redshift distortions caused by the peculiar velocities of haloes. Our cosmic web finder is packaged in python as sconce-scms and has been made publicly available.

Bayesian Inference on the Bimodality of the Generalized von Mises Distribution

This article introduces Bayesian inference on the bimodality of the generalized von Mises (GvM) distribution for planar directions (Gatto and Jammalamadaka in Stat Methodol 4(3):341–353, 2007). The GvM distribution is a flexible model that can be axial symmetric or asymmetric, unimodal or bimodal. Two inferential approaches are analysed. The first is the test of null hypothesis of bimodality and Bayes factors are obtained. The second approach provides a two-dimensional highest posterior density (HPD) credible set for two parameters relevant to bimodality. Based on the identification of the two-dimensional parametric region associated with bimodality, the inclusion of the HPD credible set in that region allows us to infer on the bimodality of the underlying GvM distribution. A particular implementation of the Metropolis–Hastings algorithm allows for the computation of the Bayes factors and the HPD credible sets. A Monte Carlo study reveals that, whenever the samples are generated under a bimodal GvM, the Bayes factors and the HPD credible sets do clearly confirm the underlying bimodality.

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.

Detecting (non)parallel evolution in multidimensional spaces: angles, correlations and eigenanalysis

Parallelism between evolutionary trajectories in a trait space is often seen as evidence for repeatability of phenotypic evolution, and angles between trajectories play a pivotal role in the analysis of parallelism. However, properties of angles in multidimensional spaces have not been widely appreciated by biologists. To remedy this situation, this study provides a brief overview on geometric and statistical aspects of angles in multidimensional spaces. Under the null hypothesis that trajectory vectors have no preferred directions (i.e. uniform distribution on hypersphere), the angle between two independent vectors is concentrated around the right angle, with a more pronounced peak in a higher-dimensional space. This probability distribution is closely related to t- and beta distributions, which can be used for testing the null hypothesis concerning a pair of trajectories. A recently proposed method with eigenanalysis of a vector correlation matrix can be connected to the test of no correlation or concentration of multiple vectors, for which simple test procedures are available in the statistical literature. Concentration of vectors can also be examined by tools of directional statistics such as the Rayleigh test. These frameworks provide biologists with baselines to make statistically justified inferences for (non)parallel evolution.

The importance of time of day for magnetic body alignment in songbirds

Spontaneous magnetic alignment is the simplest known directional response to the geomagnetic field that animals perform. Magnetic alignment is not a goal directed response and its relevance in the context of orientation and navigation has received little attention. Migratory songbirds, long-standing model organisms for studying magnetosensation, have recently been reported to align their body with the geomagnetic field. To explore whether the magnetic alignment behaviour in songbirds is involved in the underlying mechanism for compass calibration, which have been suggested to occur near to sunset, we studied juvenile Eurasian reed warblers (Acrocephalus scirpaceus) captured at stopover during their first autumn migration. We kept one group of birds in local daylight conditions and an experimental group under a 2 h delayed sunset. We used an ad hoc machine learning algorithm to track the birds' body alignment over a 2-week period. Our results show that magnetic body alignment occurs prior to sunset, but shifts to a more northeast-southwest alignment afterwards. Our findings support the hypothesis that body alignment could be associated with how directional celestial and magnetic cues are integrated in the compass of migratory birds.