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.

We develop nonparametric smoothing for regression when both the predictor and the response variables are defined on a sphere of whatever dimension. A local polynomial fitting approach is pursued, which retains all the advantages in terms of rate optimality, interpretability, and ease of implementation widely observed in the standard setting. Our estimates have a multi-output nature, meaning that each coordinate is separately estimated, within a scheme of a regression with a linear response. The main properties include linearity and rotational equivariance. This research has been motivated by the fact that very few models describe this kind of regression. Such current methods are surely not widely employable since they have a parametric nature, and also require the same dimensionality for prediction and response spaces, along with nonrandom design. Our approach does not suffer these limitations. Real-data case studies and simulation experiments are used to illustrate the effectiveness of the method.

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 L2-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.

Abstract. Regression with a circular response is a topic of current interest. We introduce non‐parametric smoothing for this problem. Simple adaptations of a weight function enable a unified formulation for both real‐line and circular predictors, whereas these cases have been tackled by quite distinct parametric methods. Additionally, we discuss various methodological extensions, obtaining a number of promising techniques – totally new in circular statistics – such as confidence intervals for the value of a circular regression and non‐parametric autoregression in circular time series. The findings are also illustrated through real data examples.

Although most circular datasets are in the form of time series, not much research has been done in the field of circular time series analysis. We propose a nonparametric theory for smoothing and prediction in the time domain for circular time series data. Our model is based on local polynomial fitting which minimizes an angular risk function. Both asymptotic arguments and empirical examples are used to describe the accuracy of our methods.