A Note on Density Estimation for Circular Data

A Note on Density Estimation for Circular Data Marco Di Marzio, Agnese Panzera, and Charles C. Taylor

Abstract We discuss kernel density estimation for data lying on the d -dimensional torus (d  1). We consider a specific class of product kernels, and formulate exact and asymptotic L2 properties for the estimators equipped with these kernels. We also obtain the optimal smoothing for the case when the kernel is defined by the product of von Mises densities. A brief simulation study illustrates the main findings.

1 Introduction A circular observation can be regarded as a point on the unit circle, and may be represented by an angle  2 Œ0; 2/. Typical examples include flight direction of birds from a point of release, wind and ocean current direction. A circular observation is periodic, i.e.  D  C 2m for m 2 Z. This periodicity sets apart circular statistical analysis from standard real-line methods. Recent accounts are given by Jammalamadaka and SenGupta (2001) and Mardia and Jupp (1999). Concerning circular density estimation we observe that almost all of the related methods appear to have a parametric nature, with the exception of a few contributions on kernel density estimation for data lying on the circle or on the sphere (Bai et al. 1988; Beran 1979; Hall et al. 1987; Klemel¨a 2000; Taylor 2008). In this paper we consider kernel density estimation when a support point  is a point on the d -dimensional torus Td WD Œ; d , d  1. To this end, we define estimators equipped with kernels belonging to a suitable class, and derive their exact and asymptotic L2 properties.

M. Di Marzio ()  A. Panzera DMQTE, G. d’Annunzio University, Viale Pindaro 42, 65127 Pescara, Italy e-mail: [email protected]; [email protected] C. C. Taylor Department of Statistics, University of Leeds, Leeds West Yorkshire LS2 9JT, UK e-mail: [email protected] A. Di Ciaccio et al. (eds.), Advanced Statistical Methods for the Analysis of Large Data-Sets, Studies in Theoretical and Applied Statistics, DOI 10.1007/978-3-642-21037-2 27, © Springer-Verlag Berlin Heidelberg 2012

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We also obtain the optimal smoothing degree for the case in which the kernel is a d -fold product of von Mises densities. In particular, in Sect. 2 we discuss the class of toroidal kernels, and in Sect. 3 we obtain the exact and asymptotic integrated mean squared error along with the optimal smoothing for our estimators. Finally, in Sect. 4, we give some numerical evidence which confirms our asymptotic results.

2 Toroidal Kernels By toroidal density we mean a continuous probability density function whose support is Td and such that f ./  f ./ where  D .;    ; /. For estimating a smooth toroidal density, we consider the following class of kernels introduced by Di Marzio et al. (2011). Definition 1 (Toroidal kernels). A d -dimensional toroidal kernel with concentration (smoothing) parameters C WD .s 2 RC ; s D 1;    ; d /, is the d -fold product Q KC WD dsD1 Ks , where K W T ! R is such that P (i) It admits an uniformly convergent Fourier series f1C2 1 j D1 j ./ cos.j/g= .2/,  2 T, where  ./ is a strictly monotonic function of . j R (ii) T K D 1, and, if K takes negative values, there exists 0 < M < 1 such that, for all  > 0 Z jK ./j d  M: T

(iii) For all 0 < ı < , Z lim

!1 ıj j

jK ./j d D 0:

These kernels are continuous and symmetric about the origin, so the d -fold product of von Mises, wrapped normal and the wrapped Cauchy distributions are included. As more general examples, we now list families of densities whose d -fold products are candidates as toroidal kernels: 1. Wrapped symmetric stable family of Mardia (1972, p. 72). 2. The extensions of the von Mises distribution due to Batschelet (1981, p. 288, equation (15.7.3)). 3. The unimodal symmetric distributions in the family of Kato and Jones (2009). 4. The family of unimodal symmetric distributions of Jones and Pewsey (2005). 5. The wrapped t family of Pewsey et al. (2007). Definition R2 (Sin-order). Given the one-dimensional toroidal kernel K , let j .K / D T sinj ./K ./d. We say that K has sin-order q if and only if j .K / D 0; for 0 < j < q; and q .K / ¤ 0:

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Q Note that KC WD dsD1 Ks has sin-order q if and only if Ks has sin-order q for s 2 f1;    ; d g. Moreover, observe that the quantity j .K / plays a similar rˆole as the j th moment of a kernel in the euclidean theory and its expression is given by the following Lemma 1. Given a positive even q, if K has sin-order q, then 9 8 ! ! q=2 = < X 1 q q1 .1/qCs q .K / D q1 2s ./ : C ; : q=2 2 q=2 C s sD1 Proof. See Appendix. Remark 1. Notice that if K has sin-order q, then  j ./ D 1 for each j < q, and since lim!1 q ./ D 1, it results q .K / D O f1  q ./g21q .

3 Toroidal Density Estimation Definition 3 (Kernel estimator of toroidal density). Let f ` ; ` D 1;    ; ng, with  ` WD .`1 ;    ; `d / 2 Td , be a random sample from a continuous toroidal density f . The kernel estimator of f at  2 Td is defined as 1X fO.I C/ WD KC .   i /: n i D1 n

(1)

From now on we always assume that s D  for each s 2 f1;    ; d g, i.e. we assume that C is a multiset with element  and multiplicity d . Letting gO be a nonparametric estimator of a square-integrable curveRg, the mean integrated squared error (MISE) for gO isR defined by MISEŒg O WD EŒfg./ O  R g./g2 d  D fEŒg./ O  g./g2 d  C VarŒg./d O . In what follows we will derive a Fourier expansion of the exact MISE for the estimator (1). Before stating the main result, we need to introduce a little notation. Given j D .j1 ;    ; jd / 2 Zd , for a function f defined on Td we have f ./ D

1 X i j cj e ; .2/d d

(2)

j2Z

R where i 2 D 1, cj WD Td f ./e i j d , and j   is the inner product of j and . Q Given the d -dimensional toroidal kernel KC ./ D dsD1 K .s /, and letting R Q d j .C/ WD Td KC ./e i j d  D sD1 js ./, the estimator in (1), being the convolution between the empirical version of f and KC , can be expressed as

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fO.I C/ D

1 X cQj j .C/e i j ; .2/d d

(3)

j2Z

where cQj WD n1

Pn `D1

e i j ` .

Theorem 1. Suppose that both f and KC belong to L2 .Td /, then h i MISE fO.I C/ D

X  2  2 1 cj   .C/ 1  j n.2/d d j2Z

C

2  2 1 X˚ cj  : 1   .C/ j .2/d d j2Z

Proof. See Appendix. Now we derive the asymptotic MISE (AMISE) for the estimator in (1) equipped with a kernel given by the d - fold product of univariate second sin-order toroidal kernels. Theorem 2. Given the random sample f ` ; ` D 1;    ; ng, consider the estimator Q fO.I C/ having as the kernel KC WD dsD1 K , with K being a univariate second sin-order toroidal kernel. If: (i) (ii) (iii) (iv)

K is such that lim!1 .1  2 .//=.1  j .// D j 2 =4: lim!1 2 .K / D 0: P 2 d limn!1 n1 .2/d f1 C 2 1 i D1 j ./g D 0: the hessian matrix of f at , Hf ./, has entries piecewise continuous and square integrable.

then 1 f1  2 ./g2 AMISEŒfO.I C/ D 16

Z

1 tr fHf ./gd  C n 2

(

1C2

P1

2 i D1 j ./

)d

2

Proof. See Appendix. Remark 2. For circular kernels the expansion of the convolution behaves differently from its euclidean counterpart. In fact, here higher order terms do not necessarily vanish faster for whatever kernel, assumption .i / being required to this end. Such an assumption is satisfied by many symmetric, unimodal densities, even though, surely, not by the wrapped Cauchy. Important cases are given by the wrapped normal and von Mises. But also the class introduced by Batschelet (1981, p. 288, equation (15.7.3)) matches the condition, along with the unimodal symmetric densities in the family introduced by Kato and Jones (2009). The above result can be easily extended to estimators equipped with higher sinorder toroidal kernels. These latter can be constructed from second-sin-order ones as

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a direct consequence of the result in Lemma 1. For a discussion on higher sin-order and smaller bias, see Di Marzio et al. (2011). Now we derive the AMISE-optimal Qd concentration parameter for the estimator having as the kernel VC ./ WD sD1 V .s /, the d -fold product of von Mises density V ./ WD f2I0 ./g1 e  cos. / , with Ij ./ denoting the modified Bessel function of the first kind and order j . Theorem 3. Given the random sample f ` ; ` D 1;    ; ng, consider the estimator fO.I C/ having as the kernel VC ./. Assume that condition (iv) of Theorem 2 holds, and: (i) limn!1  1 D 0: (ii) limn!1 n1  d=2 D 0: then the AMISE optimal concentration parameter for fO.I C/ is "

#2=.4Cd / R 2d  d=2 n tr2 fHf ./gd  : d

Proof. See Appendix.

4 Numerical Evidence In order to illustrate the methodology and results a small simulation was carried out. Taking f to be a bivariate von Mises distribution (with independent components) with concentration parameters 2 and 4, we simulated 50 datasets for each size of n D 100; 500; 2; 500. For each dataset a variety of smoothing parameters () were used in the toroidal density estimate, with kernel VC , and for each value of  we compute the average integrated squared error (ISE) (over the 50 simulations) and MISE using (4) in the appendix. In addition, for each value of n we compute the AMISE optimal concentration parameter for fO.I C/ given by Theorem 3. The results are shown in Fig. 1. It can be seen that ISE decreases with n, and that the approximation of MISE improves as n increases. Finally we note that the optimal smoothing parameter given by Theorem 3 also improves with n.

Appendix Proof of Lemma 1. First observe that for odd j we have that sinj ./ is orthogonal in L1 .T/ to each function in the set f1=2; cos./; cos.2/;    g, which implies that j .K / D 0. When j is even, sinj ./ is not orthogonal

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0.03 0.02

n = 100

0.01

integrated squared error

0.04

302

n = 500 n = 2500 10

20

30

40

50

κ

Fig. 1 Asymptotic mean integrated squared error (dashed lines) and average integrated squared error (continuous lines) over 50 simulations of size n from a bivariate von Mises distribution. The minimum values are shown by symbols, and the vertical lines represent the AMISE optimal values given by Theorem 3

in L1 .T/ to 1=2 and to the set fcos.2s/; 0 < s  j=2g, and in particular one has ! Z j 1  sinj ./ d D and 2 j=2 2j 1 T ! Z j .1/j Cs  j sin ./ cos.2s/d D ; 2j 1 j=2 C s T t u

which gives the result.

R Proof ofPTheorem 1. First observe that by Parseval’s identity Td ff ./g2 d  D .2/d j2Zd jjcj jj2 , where jjgjj stands for the L2 norm of g. Then use the results in (2) and in (3), the identities EŒcQj  D cj , EŒjjcQj  cj jj2  D n1 .1  jjcj jj2 /, and some algebraic manipulations, to get Z E Td

2 o2 n fO.I C/  f ./ d  D E 4

3 ˇˇ ˇˇ2 ˇˇ 1 X ˇˇˇˇ ˇˇ 5 ˇˇcQj j .C/  cj ˇˇ .2/d d j2Z

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i 1 X  h cQj  cj 2  2 .C/ E j .2/d d j2Z

˚

C 1  j .C/ D

2  2  cj 

X  2  2 1 cj   .C/ 1  j n.2/d d j2Z

C

2  2 1 X˚ 1  j .C/ cj  : d .2/ d j2Z

t u Proof of Theorem 2. Put Su WD fsin.u1 /;    ; sin.ud /gT , and use Df ./ to denote the first-order partial derivatives vector of the function f at . Using the expansion f .uC/ D f ./CSTu Df ./C 12 STu Hf ./Su CO.STu Su /, and recalling assumptions (i) and (ii), a change of variables leads to EŒfO.I C/ D

Z Td

KC .

 /f . /d

1 D f ./ C f1  2 ./gtrfHf ./g C o.1/: 4 Now, recalling assumption (iii), we obtain 1 VarŒfO.I C/ D n D

1 n

Z Z

Td

Td

f ./ D n

fKC .

 /g2 f . /d



o2 1n O EŒf .I C/ n

fKC .u/g2 ff ./ C o.1/gd u  (

1C2

P1

2 j D1 j ./

2

)d

1 ff ./ C o.1/g2 n

C o.1/ : t u

Proof of Theorem 3. First observe that for the von Mises kernel j ./ D Ij ./=I0 ./, then follow the proof of Theorem 2 to get MISEŒfO.I C/ D

 Z I1 ./ 2 tr2 fHf ./gd  I0 ./  d  2  I0 .2/ 1 1 d=2  : C o C n  C p n 2fI0 ./g2 1 4



(4)

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Now, replace I1 ./=I0 ./ by 1 with an error of magnitude O. 1 /, use  lim

!1

I0 .2/ 2fI0 ./g2

then minimize the leading term of (4).

d

D

  d=2 ; 4 t u

References Z. D. Bai, R. C. Rao, and L. C. Zhao. Kernel estimators of density function of directional data. Journal of Multivariate Analysis, 27: 24–39, 1988. E. Batschelet. Circular Statistics in Biology. Academic Press, London, 1981. R. Beran. Exponential models for directional data. The Annals of Statistics, 7: 1162–1178, 1979. M. Di Marzio, A. Panzera, and C.C. Taylor. Density estimation on the torus. Journal of Statistical Planning & Inference, 141: 2156–2173, 2011. P. Hall, G.S. Watson, and J. Cabrera. Kernel density estimation with spherical data. Biometrika, 74: 751–762, 1987. S. R. Jammalamadaka and A SenGupta. Topics in Circular Statistics. World Scientific, Singapore, 2001. M.C. Jones and A. Pewsey. A family of symmetric distributions on the circle. Journal of the American Statistical Association, 100: 1422–1428, 2005. S. Kato and M.C. Jones. A family of distributions on the circle with links to, and applications arising from, m¨obius transformation. Journal of the American Statistical Association, to appear, 2009. J. Klemel¨a. Estimation of densities and derivatives of densities with directional data. Journal of Multivariate Analysis, 73: 18–40, 2000. K. V. Mardia. Statistics of Directional Data. Academic Press, London, 1972. K. V. Mardia and P. E. Jupp. Directional Statistics. John Wiley, New York, NY, 1999. A. Pewsey, T. Lewis, and M. C. Jones. The wrapped t family of circular distributions. Australian & New Zealand Journal of Statistics, 49: 79–91, 2007. C. C. Taylor. Automatic bandwidth selection for circular density estimation. Computational Statistics & Data Analysis, 52: 3493–3500, 2008.