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This paper presents a goodness-of-fit test for parametric regression models with scalar response and directional predictor, that is, a vector on a sphere of arbitrary dimension. The testing procedure is based on the weighted squared distance between a smooth and a parametric regression estimator, where the smooth regression estimator is obtained by a projected local approach. Asymptotic behaviour of the test statistic under the null hypothesis and local alternatives is provided, jointly with a consistent bootstrap algorithm for application in practice. A simulation study illustrates the performance of the test in finite samples. The procedure is applied to test a linear model in text mining.

Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of implementation and the corresponding theory are given for predictors taking values in Hilbert spaces and/or Riemannian manifolds. The existence of the estimators, convergence of a backfitting algorithm, rates of convergence and asymptotic distributions of the estimators are discussed. The finite sample performance of the estimators is investigated by means of two simulation studies. Finally, three data sets covering several types of non-Euclidean data are analyzed to illustrate the usefulness of the proposed general approach.

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.

Sustainable Intensification (SI) practices offer adopters exploiting improvement potentials in environmental performance of farming, i.e. enhance ecosystem functionality, while maintaining productivity. This paper proposes a directional meta-frontier approach for measuring farms’ eco-efficiency and respective improvement potentials in the direction of farms’ ecological output for SI evaluation. We account for farms’ selection processes into SI using a behavioural model and rely on a matched sample for adopters and non-adopters of agronomic SI practices from the northern German Plain. We conclude that the SI adopters determined the sample’s system frontier and showed higher mean eco-efficiency, but that most farms in our sample did not fully exploit the improvement potentials in biodiversity as ecological outcome.

Geochemical surveys collect sediment or rock samples, measure the concentration of chemical elements, and report these typically either in weight percent or in parts per million (ppm). There are usually a large number of elements measured and the distributions are often skewed, containing many potential outliers. We present a new robust principal component analysis (PCA) method for geochemical survey data, that involves first transforming the compositional data onto a manifold using a relative power transformation. A flexible set of moment assumptions are made which take the special geometry of the manifold into account. The Kent distribution moment structure arises as a special case when the chosen manifold is the hypersphere. We derive simple moment and robust estimators (RO) of the parameters which are also applicable in high-dimensional settings. The resulting PCA based on these estimators is done in the tangent space and is related to the power transformation method used in correspondence analysis. To illustrate, we analyze major oxide data from the National Geochemical Survey of Australia. When compared with the traditional approach in the literature based on the centered log-ratio transformation, the new PCA method is shown to be more successful at dimension reduction and gives interpretable results.

Background A broad range of scientific studies involve taking measurements on a circular, rather than linear, scale (often variables related to times or orientations). For linear measures there is a well-established statistical toolkit based on linear modelling to explore the associations between this focal variable and potentially several explanatory factors and covariates. In contrast, statistical testing of circular data is much simpler, often involving either testing whether variation in the focal measurements departs from circular uniformity, or whether a single explanatory factor with two levels is supported. Methods We use simulations and example data sets to investigate the usefulness of a MANOVA approach for circular data in comparison to commonly used statistical tests. Results Here we demonstrate that a MANOVA approach based on the sines and cosines of the circular data is as powerful as the most-commonly used tests when testing deviation from a uniform distribution, while additionally offering extension to multi-factorial modelling that these conventional circular statistical tests do not. Conclusions The herein presented MANOVA approach offers a substantial broadening of the scientific questions that can be addressed statistically using circular data.

In this work, we study a modified Watson test for the one sample spherical location problem. Our test is based on a modification of the classical Watson test. As is well-known, the Watson test is asymptotically valid under rotational symmetry and locally and asymptotically optimal in the von Mises case. We show that our modified Watson test enjoys several nice features: (i) it remains asymptotically valid under a large class of distributions including the rotational symmetric ones and (ii) it enjoys some local and asymptotic optimality properties in the vicinity of the von Mises case. Our results are supported by Monte Carlo simulations.

Depth functions offer an array of tools that enable the introduction of quantile- and ranking-like approaches to multivariate and non-Euclidean datasets. We investigate the potential of using depths in the problem of nonparametric supervised classification of directional data, that is classification of data that naturally live on the unit sphere of a Euclidean space. In this paper, we address the problem mainly from a theoretical side, with the final goal of offering guidelines on which angular depth function should be adopted in classifying directional data. A set of desirable properties of an angular depth is put forward. With respect to these properties, we compare and contrast the most widely used angular depth functions. Simulated and real data are eventually exploited to showcase the main implications of the discussed theoretical results, with an emphasis on potentials and limits of the often disregarded angular halfspace depth.

Sky surveys represent the fundamental data basis for detecting and locating as yet undiscovered celestial objects. Since 2008, the Fermi LAT Collaboration has catalogued thousands of γ -ray sources with the aim of extending our knowledge of the highly energetic physical mechanisms and processes that lie at the core of our Universe. In this article, we present a nonparametric clustering algorithm which identifies high-energy astronomical sources using the spatial information of the γ -ray photons detected by the large area telescope onboard the Fermi spacecraft. In particular, the sources are identified using a von Mises–Fisher kernel estimate of the photon count density on the unit sphere via an adjustment of the mean-shift algorithm which accounts for the directional nature of the collected data and the need of local smoothing. This choice entails a number of desirable benefits. It allows us to bypass the difficulties inherent on the borders of any projection of the photon directions onto a 2-dimensional plane, while guaranteeing high flexibility. The smoothing parameter is chosen adaptively, by combining scientific input with optimal selection guidelines, as known from the literature. Using statistical tools from hypothesis testing and classification, we furthermore present an automatic way to skim off sound candidate sources from the γ -ray emitting diffuse background and to quantify their significance. We calibrate and test our algorithm on simulated count maps provided by the Fermi LAT Collaboration.

We propose novel parametric concentric multi-unimodal small-subsphere families of densities for p − 1 ≥ 2-dimensional spherical data. Their parameters describe a common axis for K small hypersubspheres, an array of K directional modes, one mode for each subsphere, and K pairs of concentrations parameters, each pair governing horizontal (within the subsphere) and vertical (orthogonal to the subsphere) concentrations. We introduce two kinds of distributions. In its one-subsphere version, the first kind coincides with a special case of the Fisher–Bingham distribution, and the second kind is a novel adaption that models independent horizontal and vertical variations. In its multisubsphere version, the second kind allows for a correlation of horizontal variation over different subspheres. In medical imaging, the situation of p − 1 = 2 occurs precisely in modeling the variation of a skeletally represented organ shape due to rotation, twisting, and bending. For both kinds,we provide new computationally feasible algorithms for simulation and estimation and propose several tests. To the best knowledge of the authors, our proposedmodels are the first to treat the variation of directional data along several concentric small hypersubspheres, concentrated near modes on each subsphere, let alone horizontal dependence. Using several simulations, we show that our methods are more powerful than a recent nonparametric method and ad hoc methods. Using data from medical imaging, we demonstrate the advantage of our method and infer on the dominating axis of rotation of the human knee joint at different walking phases.