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

Authors: Yogendra P. Chaubey
Journal: Journal of Probability and Statistics
 (2018)

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## Abstract

The circular kernel density estimator, with the wrapped Cauchy kernel, is derived from the empirical version of Carathéodory function that is used in the literature on orthogonal polynomials on the unit circle. An equivalence between the resulting circular kernel density estimator, to Fourier series density estimator, has also been established. This adds further weight to the considerable role of the wrapped Cauchy distribution in circular statistics.


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## Body

## 1. Introduction

Consider an absolutely continuous (with respect to the Lebesgue measure) circular densityfθ, θ∈-π,π; that is, fθ is 2π-periodic,(1)fθ≥0forθ∈R,∫-ππfθdθ=1.In the literature on modeling circular data, starting from the classical text of Mardia [1], the standard texts such as Fisher [2], Jammalamadaka and SenGupta [3], and Mardia and Jupp [4] cover parametric models along with many inference problems. More recently various alternatives to these classical parametric models, exhibiting asymmetry and multimodality, have been investigated with respect to their mathematical properties and goodness of fit to some real data; see Abe and Pewsey [5], Jones and Pewsey [6], Kato and Jones [7], Kato and Jones [8], Minh and Farnum [9], and Shimizu and Lida [10].As elaborated in Nuñez-Antonio et al. [11], circular data possess characteristics such as high skewness or kurtosis and multimodality in many situations, for example, the data on directions of clinical vectorcardiogram (see Downs [12]), wind directions (see Fisher and Lee [13]), and animal orientation (see Oliveira et al. [14]). Such data are not well fitted by standard parametric models, and in such cases, semiparametric or nonparametric modeling may be considered more appropriate.Fernändez-Durän [15] and Mooney et al. [16] considered modeling of circular data by semiparametric models based on mixture of circular normal and von Mises distributions whereas Hall et al. [17] and Bai et al. [18] considered nonparametric kernel-based density estimation for data on sphere and Fisher [19] and Taylor [20] considered kernel density estimation for circular data. 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 d-dimensional torus; the special case of d=1 provides circular kernel density estimator that is described below.Given a random sample(θ1,…,θN) from the density (1), the circular kernel density estimator is given by (2)f^θ;κ=1N∑j=1NKκθ-θj,where Kκθ-ϕ is a circular kernel where κ is a concentration parameter and ϕ 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 κ 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.In this note we consider the circular kernel density estimator using the wrapped Cauchy kernel that is given by(3)f^WCθ;ρ=1N∑j=1NKWCθ-θj;0,ρ,where(4)KWCθ;μ,ρ=12π1-ρ21+ρ2-2ρcos⁡θ-μ,-π≤θ<π,and we show that it is derived from the empirical version of Carathéodory function, used in the literature on orthogonal polynomials on the unit circle. We also show that this approach leads to Fourier series density estimation; however no truncation of the series is required.In Section2, some basic results from the literature on orthogonal polynomials on the unit circle are presented first and then the strategy of estimating fθ is introduced. This in turn produces the nonparametric circular kernel density estimator given in (3). The details are in Section 3. The next section describes the Fourier series estimator of fθ that is shown to be equivalent to the circular kernel density estimator (2) in a limiting sense when wrapped Cauchy kernel is employed.


## 2. Some Preliminary Results on Orthogonal Polynomials on Unit Circle

LetD be the open unit disk, z∣z<1, in the complex plane, and let μ be a continuous measure defined on the boundary ∂D, that is, the circle C=z∣z=1. The point z∈D will be represented by z=reiθ for r∈0,1, θ∈0,2π and i=-1. The closure of D will be denoted by D¯.Definition 1.
A sequence of polynomialsϕnz defined on C are orthogonal with respect to a Borel measure μ, if they satisfy(5)∫Cϕrzϕsz¯dμz=δrs,where δrs>0 for r=s and it equals 0, otherwise.The importance of these polynomials is in approximation of bounded functionsgz defined on the unit circle by the representation (see Cantero and Iserles [23])(6)gz=∑n=1∞g^nϕnz,whereg^n=g,ϕnϕnμ2,where (7)g,h=∫z=1gzhz¯dμ,gμ=g,g1/2.Here hz¯ refers to the complex conjugate of hz. The reader is referred to the excellent book on the subject of orthogonal polynomials on the unit circle by Simon [24]. We will use this result for approximating Fz defined below,(8)Fz=∫-ππeiθ+zeiθ-zfθdθ.The above function is a “special function,” known as Carathéodory function, that plays an important role in the study of orthogonal polynomials defined on the unit circle (see Simon [24], pp. 25). The orthogonal expansion of Fz with respect to the basis 1,z,z2,… is given by (see 4.15 and 4.16 of Simon [25]) (9)Fz=1+2∑n=1∞cnzn,where(10)cn=∫-ππe-inθfθdθis the jth trigonometric moment of the circular distribution with density f. The integrand in (8) involves the function(11)Cz,ω=ω+zω-zthat is known as the complex Poisson kernel in the theory of complex analysis. Its relation with the wrapped Cauchy kernel lies in the fact that(12)Re⁡Creiθ,eiφ=Prθ,φ=1-r21+r2-2rcos⁡θ-φfor θ,φ∈-π,π and r∈0,1, where “Re” denotes the real part. The above function is known as the real Poisson kernel and it is clearly related to the wrapped Cauchy kernel since(13)Pρθ,φ=2πKWCθ-φ;0,ρ.A standard result in complex analysis, known as thePoisson representation (see ([24], p. 27)), says that if g is analytic in a neighborhood of D¯, with g0 real, then for z∈D, (14)gz=∫-ππeiθ+zeiθ-zRe⁡geiθdθ2π.This representation leads to the result (see (ii) in Section 5 of Simon [24]) that for Lebesgue a.e. θ(15)limr↑1⁡Freiθ≡Feiθexists and that (16)fθ=12πReFeiθ=12πlimr↑1⁡Re⁡Freiθ.


## 3. Smooth Circular Density Estimator Derived from an Estimator ofFz

Recognizing the expression forFz as the expectation of eiθ+z/eiθ-z, its empirical version is given by (17)FNreiθ=1N∑j=1Neiθj+reiθeiθj-reiθ.Thus we may define an estimator of fθ motivated by FNz, the identities (15) and (16), as(18)f^rθ=12πRe⁡FNreiθ.Now we show that the above estimator is of the same form as the circular density estimator given in (3). Recognize that (19)FNreiθ=1N∑j=1NCreiθ,ωj,where ωj=eiθj; then using (12), we have (20)Re⁡FNreiθ=1N∑j=1NPrθ,θj,and therefore(21)f^rθ=12πN∑j=1NPrθ,θj=1N∑j=1NKWCθ-θj;0,r,which is of the same form as in (3).


## 4. A Connection of the Circular Density Estimator to Fourier Series Estimator

The orthogonal series forFz given in (9) may be directly used to define an estimator of fθ. This method also provides the same estimator as derived in the previous section that is demonstrated as follows. Estimating the coefficients cn, n=1,2,…, by (22)c^n=1N∑j=1Ne-inθj,an estimator of Fz is given by (23)F^z=1+∑n=1∞c^nzn.Substituting the expression for c^n from (22), we can write (24)F^z=1+2N∑j=1N∑n=1∞e-inθjzn=1+2N∑j=1N∑n=1∞ω¯jzn;ωj=einθj=1+2N∑j=1Nω¯jz1-ω¯jz=2N∑j=1N12+ω¯jz1-ω¯jz=1N∑j=1N1+ω¯jz1-ω¯jz=1N∑j=1NCz,ωj,which is the same as FNz given in (19).Equation (9) may be used to derive the Fourier series estimator. The reader may be referred to Efromovich [26] for the details about Fourier series density estimator. Truncating the infinite sum in (9) at some large index n∗, we have an estimator of fθ(25)f^Sθ=12π+1πN∑j=1N∑n=1n∗cos⁡nθ-θj.Here n∗ is chosen according to some criteria, for example, to minimize the integrated squared error. The reader may be referred to Efromovich [26]. Thus we have two contrasting situations; in one we have to choose n∗ and in the other case we have to choose a suitable concentration parameter ρ. Numerically as well as technically, the second choice, that is, the circular kernel density estimator, may be considered more suitable.



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*Source: 5372803-2018-04-01.xml*