On Nonparametric Density Estimation for Circular Data: An Overview

Abstract

This paper provides a short review of modern smoothing methods for density and distribution functions dealing with circular data. We highlight the usefulness of circular kernels for smooth density estimation in this context and contrast it with smooth density estimation based on orthogonal series. It is seen that the wrapped Cauchy kernel as a choice of circular kernel appears as a natural candidate as it has a close connection to orthogonal series density estimation on a unit circle. In the literature, the use of von Mises circular kernel is investigated, which requires the numerical computation of the Bessel function. On the other hand, the wrapped Cauchy kernel is much simpler to use. This adds further weight to the considerable role of the wrapped Cauchy distribution in circular statistics. We also investigate some transformation-based methods in adapting the linear kernel density estimator to the circular data. This is very useful in practice as widely available software on linear kernel density estimation can easily be adapted to the circular case. These simpler methods are illustrated using some real data.

Appendix: Some Preliminary Results from Complex Analysis

Let $\mathbb {D}$ be the open unit disk, $\{z~| ~|z|<1\},$ in $\mathbb {Z}$ and let $\mu$ be a continuous measure defined on the boundary $\partial \mathbb {D},$ i.e., the circle $\{z~|~ |z|=1\}.$ The point $z\in \mathbb {D}$ will be represented by $z=r\mathrm{e}^{i\theta }$ for $r\in [0,1), \theta \in [0,2\pi )$ and $i=\sqrt{-1}.$

A standard result in complex analysis involves the Poisson representation that involves the real and complex Poisson kernels that are defined as

$$\begin{aligned} P_r(\theta ,\varphi )=\frac{1-r^2}{1+r^2-2r\cos (\theta -\varphi )} \end{aligned}$$

(33)

for $\theta ,\varphi \in [0,2\pi )$ and $r\in [0,1)$ and by

$$\begin{aligned} C(z,\omega )=\frac{\omega +z}{\omega -z} \end{aligned}$$

(34)

for $\omega \in \partial \mathbb {D}$ and $z\in \mathbb {D}.$ The connection between the kernels (33) and (34) is given by the fact that

$$\begin{aligned} P_r(\theta ,\varphi )=\mathrm{Re} ~C(r\mathrm{e}^{i\theta },\mathrm{e}^{i\varphi })=(2\pi ) f_{WC}(\theta ;\varphi ,\rho ). \end{aligned}$$

(35)

The Poisson representation says that if g is analytic in a neighborhood of $\bar{\mathbb {D}}$ with g(0) real, then for $z\in \mathbb {D},$

$$ g(z)=\int \left( \frac{{\mathrm{e}}^{i\theta }+z}{\mathrm{e}^{i\theta }-z} \right) \mathrm{Re}(g(\mathrm{e}^{i\theta }))\frac{d\theta }{2\pi } $$

(see [42, p. 27]). This representation leads to the result (see (ii) in §5 of [42]) that for Lebesgue a.e. $\theta ,$

$$ \lim _{r\uparrow 1} W(r{\mathrm{e}^{i\theta }})\equiv W({\mathrm{e}^{i\theta }}) $$

exists and if $d\mu =w(\theta )\frac{d\theta }{2\pi }+d\mu _s$ with $d\mu _s$ singular, then

$$\begin{aligned} w(\theta )={\mathrm{Re}}W({\mathrm{e}^{i\theta }}), \end{aligned}$$

(36)

where

$$ W(z)= \int \left( \frac{{\mathrm{e}}^{i\theta }+z}{{\mathrm{e}}^{i\theta }-z} \right) d\mu (\theta ). $$

Our strategy for smooth estimation is the fact that for $d\mu _s=0$ we have

$$\begin{aligned} f(\theta )=\frac{1}{2\pi }\lim _{r\uparrow 1} \mathrm{Re}~ W(r\mathrm{e}^{i\theta }), \end{aligned}$$

(37)

where now W(z) is defined as

$$ W(z)= \int \left( \frac{{\mathrm{e}}^{i\theta }+z}{{\mathrm{e}}^{i\theta }-z} \right) f(\theta )d\theta . $$

We define the estimator of $f(\theta )$ motivated by considering an estimator of W(z),  the identity (36) and (37), i.e.

$$\begin{aligned} \hat{f}_r( \theta )=\frac{1}{2\pi }\mathrm{Re}~ W_n(r{\mathrm{e}}^{i\theta }) \end{aligned}$$

(38)

where

$$\begin{aligned} W_n(r{\mathrm{e}^{i\theta }})=\frac{1}{n}\sum _{j=1}^{n}\left( \frac{\mathrm{e}^{i\theta _j}+r{\mathrm{e}}^{i\theta }}{\mathrm{e}^{i\theta _j}-r{\mathrm{e}}^{i\theta }}\right) , \end{aligned}$$

(39)

and r has to be chosen appropriately. Recognize that (39) has the representation

$$ W_n(r{\mathrm{e}^{i\theta }})=\frac{1}{n}\sum _{j=1}^{n}~C(z,\omega _j), $$

where $\omega _j={\mathrm{e}^{i\theta _j}},$ then using (35), we have

$$ \mathrm{Re}~W_n(r{\mathrm{e}^{i\theta }})=\frac{1}{n}\sum _{j=1}^{n}~P_r(\theta ,\theta _j), $$

and therefore (38) becomes

$$\begin{aligned} \hat{f}_r( \theta )= & {} \frac{1}{(2\pi )n}\sum _{j=1}^{n}~P_r(\theta ,\theta _j)\nonumber \\= & {} \frac{1}{n}\sum _{j=1}^{n}~f_{WC}(\theta ;\theta _j,r),\nonumber \end{aligned}$$

that is of the same form as in (9).