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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.

We have a growing understanding of the light-sensing organs and light-influenced behaviours of animals with distributed visual systems, but we have yet to learn how these animals convert visual input into behavioural output. It has been suggested they consolidate visual information early in their sensory-motor pathways, resulting in them being able to detect visual cues (spatial resolution) without being able to locate them (spatial vision). To explore how an animal with dozens of eyes processes visual information, we analysed the responses of the bay scallop Argopecten irradians to both static and rotating visual stimuli. We found A. irradians distinguish between static visual stimuli in different locations by directing their sensory tentacles towards them and were more likely to point their extended tentacles towards larger visual stimuli. We also found that scallops track rotating stimuli with individual tentacles and with rotating waves of tentacle extension. Our results show, to our knowledge for the first time that scallops have both spatial resolution and spatial vision, indicating their sensory-motor circuits include neural representations of their visual surroundings. Exploring a wide range of animals with distributed visual systems will help us learn the different ways non-cephalized animals convert sensory input into behavioural output.

There is high confidence that precipitation extremes are projected to become more frequent and severe and, to a lesser extent, that their seasonality may change. However, these precipitation characteristics are dealt with separately, without examining whether magnitude and seasonality are jointly projected to change. Here we assess how the seasonality and magnitude of precipitation extremes are jointly projected to change for different climate scenarios. We perform analyses at the global scale using nine global climate models and four different emission scenarios. We identify large areas of the globe where the magnitude of the extremes is expected to increase as the emissions increase; at the same time, large changes in the seasonality of these extremes are projected to impact regions mainly located in the tropical and sub-tropical areas. These changes could impact our response and mitigation efforts and our resilience against such phenomena in response to climate change.

In recent years there has been growing interest in measuring time-varying functional connectivity between different brain regions using resting-state functional magnetic resonance imaging (rs-fMRI) data. One way to assess the relationship between signals from different brain regions is to measure their phase synchronization (PS) across time. There are several ways to perform such analyses, and we compare methods that utilize a PS metric together with a sliding window, referred to here as windowed phase synchronization (WPS), with those that directly measure the instantaneous phase synchronization (IPS). In particular, IPS has recently gained popularity as it offers single time-point resolution of time-resolved fMRI connectivity. In this paper, we discuss the underlying assumptions required for performing PS analyses and emphasize the importance of band-pass filtering the data to obtain valid results. Further, we contrast this approach with the use of Empirical Mode Decomposition (EMD) to achieve similar goals. We review various methods for evaluating PS and introduce a new approach within the IPS framework denoted the cosine of the relative phase (CRP). We contrast methods through a series of simulations and application to rs-fMRI data. Our results indicate that CRP outperforms other tested methods and overcomes issues related to undetected temporal transitions from positive to negative associations common in IPS analysis. Further, in contrast to phase coherence, CRP unfolds the distribution of PS measures, which benefits subsequent clustering of PS matrices into recurring brain states.

Willig et al. (Methods in Ecology and Evolution, 15, 868–885, 2024) cautioned that unequal sampling effort and pseudoreplication can bias the characterisation of species phenology using circular statistics. Borrowing concepts from rarefaction, they proposed bootstrapping to control for time‐varying marginal totals that arise from unequal sampling effort over time. This study extends their cautionary notes to regressions of phenological time series, where bootstrapping can be replaced by various built‐in functionalities of generalised linear mixed‐effect models. I further take this opportunity to borrow a key innovation in model‐based ordination and joint species distribution modelling—generalised linear latent variable models (GLLVM)—to illustrate its ability to extract more information out of multispecies phenological data beyond circular statistics. Synthesis. With sampling‐bias adjustment, GLLVMs, or regressions in general, are robust predictive and inferential tools that enrich our phenological understandings in conjunction with circular statistics for hypothesis testing.

•Circular tests are differentially sensitive to different coupling modes.•The Watson test is a good all-rounder method.•Phase Opposition Sum is robust to imbalances in relative trial number.•Modulation Index detects more complex phase-behavior relationships. Growing experimental evidence points at relationships between the phase of a cortical or bodily oscillation and behavior, using various circular statistical tests. Here, we systematically compare the performance (sensitivity, False Positive rate) of four circular statistical tests (some commonly used, i.e. Phase Opposition Sum, Circular Logistic Regression, others less common, i.e., Watson test, Modulation Index). We created semi-artificial datasets mimicking real two-alternative forced choice experiments with 30 participants, where we imposed a link between a simulated binary behavioral outcome with the phase of a physiological oscillation. We systematically varied the strength of phase-outcome coupling, the coupling mode (1:1 to 4:1), the overall number of trials and the relative number of trials in the two outcome conditions. We evaluated different strategies to estimate phase-outcome coupling chance level, as well as significance at the individual or group level. The results show that the Watson test, although seldom used in the experimental literature, is an excellent first intention test, with a good sensitivity and low False Positive rate, some sensitivity to 2:1 coupling mode and low computational load. Modulation Index, initially designed for continuous variables but that we find useful to estimate coupling between phase and a binary outcome, should be preferred if coupling mode is higher than 2:1. Phase Opposition Sum, coupled with a resampling procedure, is the only test retaining a good sensitivity in the case of a large unbalance in the number of occurrences of the two behavioral outcomes.

This paper studies circular correlations for the bivariate von Mises sine and cosine distributions. These are two simple and appealing models for bivariate angular data with five parameters each that have interpretations connected to those in the ordinary bivariate normal model. However, the variability and association of the angle pairs cannot be easily deduced from the model parameters unlike the bivariate normal. Thus to compute such summary measures, tools from circular statistics are needed. We derive analytic expressions and study the properties of the Jammalamadaka–Sarma and Fisher–Lee circular correlation coefficients for the von Mises sine and cosine models. Likelihood-based inference of these coefficients from sample data is then presented. The correlation coefficients are illustrated with numerical and visual examples, and the maximum likelihood estimators are assessed on simulated and real data, with comparisons to their non-parametric counterparts. Implementations of these computations for practical use are provided in our R package BAMBI.

Electroencephalography and magnetoencephalography recordings are non-invasive and temporally precise, making them invaluable tools in the investigation of neural responses in humans. However, these recordings are noisy, both because the neuronal electrodynamics involved produces a muffled signal and because the neuronal processes of interest compete with numerous other processes, from blinking to day-dreaming. One fruitful response to this noisiness has been to use stimuli with a specific frequency and to look for the signal of interest in the response at that frequency. Typically this signal involves measuring the coherence of response phase: here, a Bayesian approach to measuring phase coherence is described. This Bayesian approach is illustrated using two examples from neurolinguistics and its properties are explored using simulated data. We suggest that the Bayesian approach is more descriptive than traditional statistical approaches because it provides an explicit, interpretable generative model of how the data arises. It is also more data-efficient: it detects stimulus-related differences for smaller participant numbers than the standard approach. Phase coherence is a measurement of waves, for example, brain waves, which quantifies the similarity of their oscillatory behaviour at a fixed frequency. That is, while the waves may vibrate the same number of times per minute, the relative timing of the waves with respect to each other may be different (incoherent) or similar (coherent). In neuroscience, scientists study phase coherence in brain waves to understand how the brain responds to external stimuli, for example if they occur at a fixed frequency during an experiment. To do this, phase coherence is usually quantified with a statistic known as ‘inter-trial phase coherence’ (ITPC). When ITPC equals one, the waves are perfectly coherent, that is, there is no shift between the two waves and the peaks and troughs occur at exactly the same time. When ITPC equals zero, the waves are shifted from each other in an entirely random way. Phase coherence can also be modelled on phase angles – which describe the shift in each wave relative to a reference angle of zero – and wrapped distributions. Wrapped distributions are probability distributions over phase angles that express their relative likelihood. Wrapped distributions have statistics, including a mean and a variance. The variance of a wrapped distribution can be used to model phase coherence because it explicitly represents the similarity of phase angles relative to the mean: larger variance means less coherence. While the ITPC is a popular method for analysing phase coherence, it is a so-called ‘summary statistic’. Analyses using the ITPC discard useful information in the trial-to-trial-level data, which might not be lost using phase angles. Thus, Dimmock, O’Donnell and Houghton set out to determine whether they could create a model of phase coherence that works directly on phase angles (rather than on the ITPC) and yields better results than existing methods. Dimmock, O’Donnell and Houghton compare their model to the ITPC using both experimental and simulated data. The comparison demonstrates that their model can detect entrainment of the brain to grammatical phrases compared to ungrammatical ones at smaller sample sizes than ITPC, and with fewer false positives. Traditional tools for studying how the brain processes language often yield a lot of noise in the data, which makes it difficult to analyse measurements. Dimmock, O’Donnell and Houghton demonstrates that the brain is not simply responding to the ‘surprise factor’ of words in a phrase, as some have suggested, but also to their grammatical category. These results of this study will benefit scientists who analyse phase coherence. By using the model in addition to other approaches to study phase coherence, researchers can provide a different perspective on their results and potentially identify new features in their data. This will be particularly powerful in studies with small sample sizes, such as pilot studies where maximising the use of data is important.

Circular dichroism (CD) spectroscopy is a widely used technique for the study of protein structure. Numerous algorithms have been developed for the estimation of the secondary structure composition from the CD spectra. These methods often fail to provide acceptable results on α/β-mixed or β-structure–rich proteins. The problem arises from the spectral diversity of β-structures, which has hitherto been considered as an intrinsic limitation of the technique. The predictions are less reliable for proteins of unusual β-structures such as membrane proteins, protein aggregates, and amyloid fibrils. Here, we show that the parallel/antiparallel orientation and the twisting of the β-sheets account for the observed spectral diversity. We have developed a method called β-structure selection (BeStSel) for the secondary structure estimation that takes into account the twist of β-structures. This method can reliably distinguish parallel and antiparallel β-sheets and accurately estimates the secondary structure for a broad range of proteins. Moreover, the secondary structure components applied by the method are characteristic to the protein fold, and thus the fold can be predicted to the level of topology in the CATH classification from a single CD spectrum. By constructing a web server, we offer a general tool for a quick and reliable structure analysis using conventional CD or synchrotron radiation CD (SRCD) spectroscopy for the protein science research community. The method is especially useful when X-ray or NMR techniques fail. Using BeStSel on data collected by SRCD spectroscopy, we investigated the structure of amyloid fibrils of various disease-related proteins and peptides. Significance Circular dichroism (CD) spectroscopy is widely used for protein secondary structure analysis. However, quantitative estimation for β-sheet–containing proteins is problematic due to the huge morphological and spectral diversity of β-structures. We show that parallel/antiparallel orientation and twisting of β-sheets account for the observed spectral diversity. Taking into account the twist of β-structures, our method accurately estimates the secondary structure for a broad range of protein folds, particularly for β-sheet–rich proteins and amyloid fibrils. Moreover, the method can predict the protein fold down to the topology level following the CATH classification. We provide a general tool for a quick and reliable structure analysis using conventional or synchrotron radiation CD spectroscopy, which is especially useful when X-ray or NMR techniques fail.

The hydro-meteorological factors influencing flood timing and magnitude are shifting due to natural and anthropogenic climate change. Regionally, the association between floods and their driving factors/descriptors is complex. This necessitates a deeper understanding of flood generation to enhance forecasting, modeling, and risk analyses—critical aspects of effective flood management. Thus, to better understand flood generation in India, we investigate the dominant flood-generating descriptors and their relationship with watershed characteristics across central and southern peninsular India using circular statistics. We find that flood generation is primarily influenced by soil moisture and precipitation excess, dominating 89% of the analyzed (231) watersheds. In particular, larger watersheds (>70000 km2) are predominantly influenced by soil moisture, while smaller ones (<16000 km2) are influenced by precipitation. Interestingly, watersheds covering similar areas produce higher flood flows if predominantly influenced by soil moisture. The explicit evaluation suggests a positive influence of antecedent soil moisture (ASM) on flood flows across all watersheds. An attempt to relate the morphological characteristics with flood descriptors reveals a positive (negative) influence of the topographic wetness index (TWI) on annual maximum flows for soil moisture-dominated (precipitation-dominated) watersheds. This indicates that ponding/accumulation is a driving (limiting) factor for soil moisture (precipitation) dominated watersheds. The relative importance of the ASM compared to precipitation decreases when the precipitation intensity (PI) increases, implying exchanges of influence at certain levels of PI. Further exploration could reveal insights into the interplay between ASM and precipitation, crucial for flood magnitude and hazard assessments. Given that flood behavior is significantly influenced by dominant descriptors, it is advisable to adopt a segregated approach in analyzing flood escalation under climate change. In addition, incorporation of dominant flood descriptors into cascade flood modeling is essential for enhancing flood hazard and risk modeling.