Kernel density estimation on the torus

Abstract

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.

NPCirc: AnRPackage for Nonparametric Circular Methods

“…Finally, Marzio et al (2011) introduced a bootstrap method for choosing the smoothing parameter. The idea is to take the value that minimizes the bootstrap version of MISE, which has a closed expression when the von Mises kernel is used.…”

“…Function bw.boot implements the bootstrap procedure proposed by Marzio et al (2011). The minimum of the bootstrap MISE is obtained by using the optimize function from package stats, which searches the minimum in the interval specified by arguments lower and upper (default values are 0 and 50, respectively) and with accuracy specified by tol (default: tol = 0.1).…”

“…Despite being a simple procedure, the performance of this rule is extremely poor in some distribution settings involving multimodality, peakedness or skewness, as shown by Oliveira, Crujeiras, and Rodríguez-Casal (2012). Marzio, Panzera, and Taylor (2011) introduced a bootstrap method with the same purpose, but with unsatisfactory results for small data samples. Recently, Oliveira et al (2012) devised a new procedure for selecting the smoothing parameter in circular kernel density estimation that performs well in distributional scenarios beyond the von Mises case, and presents better or at least competitive results compared with the previously proposed selectors.…”

Smooth Kernel Estimation of a Circular Density Function: A Connection to Orthogonal Polynomials on the Unit Circle

“…Whereas Fisher [19] adapted the kernels used for linear data to the context of circular data, Taylor [20] used von Mises circular distribution replacing linear kernel used in the classical kernel density estimator that naturally maintains the periodicity in the resulting density estimator. More recently, Di Marzio et al [21,22] provided a theoretical basis for circular kernel density estimator by considering the general setting of nonparametric kernel density estimation on a -dimensional torus; the special case of = 1 provides circular kernel density estimator that is described below.…”

“…, ) from the density (1), the circular kernel density estimator is given bŷ Journal of Probability and Statistics where ( − ) is a circular kernel where is a concentration parameter and is the mean direction. Note that a circular kernel is usually chosen to be a circular density, unimodal and symmetric around its mean direction which is zero, and it is characterized by a concentration parameter which governs the amount of the smoothing (see, e.g., Di Marzio et al [21,22] for details). Classical examples of circular kernels include the von Mises density and the densities of wrapped normal and wrapped Cauchy distributions.…”