Local polynomial regression for circular predictors

We consider local smoothing of datasets where the design space is the d-dimensional (d >= 1) torus and the response variable is real-valued. Our purpose is to extend least squares local polynomial fitting to this situation. We give both theoretical and empirical results

Nonparametric Regression Estimation for Circular Data

[Abstract] Non-parametric regression with a circular response variable and a unidimensional linear regressor is a topic which was discussed in the literature. In this work, we extend the results to the case of multivariate linear explanatory variables. Nonparametric procedures to estimate the circular regression function are formulated. A simulation study is carried out to study the sample performance of the proposed estimators.Ministerio de Economía y Competitividad; MTM2016-76969-PMinisterio de Economía y Competitividad; MTM2017-82724-RXunta de Galicia; ED481A-2017/361Grupos de Referencia Competitiva; ED431C-2016-015Centro Singular de Investigación de Galicia; ED431G/0

Nonparametric multiple regression estimation for circular response

Versión final aceptada de: https://doi.org/10.1007/s11749-020-00736-wThis version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect postacceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s11749-020-00736-wNonparametric estimators of a regression function with circular response and -valued predictor are considered in this work. Local polynomial estimators are proposed and studied. Expressions for the asymptotic conditional bias and variance of these estimators are derived, and some guidelines to select asymptotically optimal local bandwidth matrices are also provided. The finite sample behavior of the proposed estimators is assessed through simulations, and their performance is also illustrated with a real data set.The authors acknowledge the support from the Xunta de Galicia Grant ED481A-2017/361 and the European Union (European Social Fund—ESF). This research has been partially supported by MINECO Grants MTM2016-76969-P and MTM2017-82724-R, and by the Xunta de Galicia (Grupo de Referencia Competitiva ED431C-2017-38, and Centro de Investigación de Galicia “CITIC” ED431G 2019/01), all of them through the ERDF. The authors thank Prof. Felicita Scapini and his research team who kindly provided the sand hoppers data that are used in this work. Data were collected within the Project ERB ICI8-CT98-0270 from the European Commission, Directorate General XII Science. The authors also thank two anonymous referees for numerous useful comments that significantly improved this article.Xunta de Galicia; ED481A-2017/361Xunta de Galicia; ED431C-2017-38Xunta de Galicia; ED431G 2019/0

Nonparametric circular quantile regression

We discuss nonparametric estimation of conditional quantiles of a circular distribution when the conditioning variable is either linear or circular. Two different approaches are pursued: inversion of a conditional distribution function estimator, and minimization of a smoothed check function. Local constant and local linear versions of both estimators are discussed. Simulation experiments and a real data case study are used to illustrate the usefulness of the methods

Kernel density estimation on the torus

Kernel density estimation for multivariate, circular data has been formulated only when the sample space is the sphere, but theory for the torus would also be useful. For data lying on a d-dimensional torus (d >= 1), we discuss kernel estimation of a density, its mixed partial derivatives, and their squared functionals. We introduce a specific class of product kernels whose order is suitably defined in such a way to obtain L-2-risk formulas whose structure can be compared to their Euclidean counterparts. Our kernels are based on circular densities; however, we also discuss smaller bias estimation involving negative kernels which are functions of circular densities. Practical rules for selecting the smoothing degree, based on cross-validation, bootstrap and plug-in ideas are derived. Moreover, we provide specific results on the use of kernels based on the von Mises density. Finally, real-data examples and simulation studies illustrate the findings

Nonparametric Rotations for Sphere-Sphere Regression

Regression of data represented as points on a hypersphere has traditionally been treated using parametric families of transformations that include the simple rigid rotation as an important, special case. On the other hand, nonparametric methods have generally focused on modeling a scalar response through a spherical predictor by representing the regression function as a polynomial, leading to component-wise estimation of a spherical response. We propose a very flexible, simple regression model where for each location of the manifold a specific rotation matrix is to be estimated. To make this approach tractable, we assume continuity of the regression function that, in turn, allows for approximations of rotation matrices based on a series expansion. It is seen that the nonrigidity of our technique motivates an iterative estimation within a Newton–Raphson learning scheme, which exhibits bias reduction properties. Extensions to general shape matching are also outlined. Both simulations and real data are used to illustrate the results. Supplementary materials for this article are available online