This paper introduces a new non-parametric approach to the modeling of circular data, based
on the use of Bernstein polynomial densities which generalizes the standard Bernstein
polynomial model to account for the specific characteristics of circular data. It is shown that the
trigonometric moments of the proposed circular Bernstein polynomial distribution can all be
derived in closed form. We comment on how to fit the Bernstein polynomial density
approximation to a sample of data and illustrate our approach with a real data example

Ausín Olivera, María Concepción

Carnicero, José Antonio

Wiper, Michael Peter

e-Archivo (Univ. Carlos III de Madrid e-Archivo)

  Working Paper  10-25 Statistics and Econometrics Series 11 May 2010  Departamento de Estadística  Universidad Carlos III de Madrid Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624-98-49 Circular Bernstein polynomial distributions  Jose Antonio Carnicero Carreño, Michael Peter Wiper and Concepción Ausin     Abstract This paper introduces a new non-parametric approach to the modeling of circular data, based on the use of Bernstein polynomial densities which generalizes the standard Bernstein polynomial model to account for the specific characteristics of circular data. It is shown that the trigonometric moments of the proposed circular Bernstein polynomial distribution can all be derived in closed form. We comment on how to fit the Bernstein polynomial density approximation to a sample of data and illustrate our approach with a real data example.  Keywords: circular data, non-parametric modeling, Bernstein polynomials.   Universidad Carlos III de Madrid, Department of Statistics, Facultad de Ciencias Sociales y Jurídicas, Madrid, Spain. e-mail addresses: jcarnice@est-econ.uc3m.es (Jose Antonio Carnicero Carreño), mwiper@est-econ.uc3m.es (Michael Peter Wiper) and causin@est-econ.uc3m.es (Concepción Ausin)     Circular Bernstein polynomial distributionsJ. A. Carnicero, M. P. Wiper, M. C. AusinUniversidad Carlos III de Madrid, Getafe, Madrid, SpainAbstractThis paper introduces a new non-parametric approach to the modeling of circu-lar data, based on the use of Bernstein polynomial densities which generalizesthe standard Bernstein polynomial model to account for the specific characteris-tics of circular data. It is shown that the trigonometric moments of the proposedcircular Bernstein polynomial distribution can all be derived in closed form. Wecomment on how to fit the Bernstein polynomial density approximation to asample of data and illustrate our approach with a real data example.Keywords: circular data, non-parametric modeling, Bernstein polynomials.1. IntroductionProblems where the data are angular directions occur in many differentscientific fields such as biology (direction of movement of migrating animals),meteorology (wind directions) and geology (directions of joints and faults). Also,phenomena that are periodic in time such as times of hospital admittance forbirths or the times when crimes are committed may also be converted to angulardata via a simple transformation modulo some period. Data of this type arecommonly known as circular data and are usually represented as points on thecircumference of an unit circle or as angles, 휃, where 0 ≤ 휃 < 2휋 radians, whichrepresent the positive angle of rotation from some arbitrarily chosen origin,휃 = 0.Email addresses: jcarnice@est-econ.uc3m.es (J. A. Carnicero),mwiper@est-econ.uc3m.es (M. P. Wiper), causin@est-econ.uc3m.es (M. C. Ausin)Preprint submitted to Elsevier April 16, 2010A number of parametric models for circular data have been developed using avariety of techniques, see e.g. Mardia and Jupp (1999) for a full review. However,most parametric models developed for circular data are unimodal and symmetricbut in many cases,both multimodal and asymmetric data may be encountered.In such cases, semi-parametric or non-parametric approaches might be preferred.Semi-parametric approaches based on trigonometric sums and mixtures of vonMises or circular normal distributions have been introduced in Ferna´ndez-Dura´n(2004) and Mooney et al. (2003) respectively, but with the exception of Fisher(1989), where a kernel based approach is considered, there has been little workon non-parametric modeling of circular data.In this paper, we introduce an alternative non-parametric approach based onthe use of Bernstein polynomials. It is well known that the Bernstein polynomialis a useful tool for interpolating functions defined on a closed interval. Bern-stein polynomials have been proposed as density estimators for variables withfinite support in a number of articles, see e.g. (Vitale, 1975; Petrone, 1999a,b;Petrone and Wassermann, 2002; Babu et al., 2002; Kakizawa, 2004). However,a problem with generalizing standard Bernstein polynomial density approachesto circular data is that these can lead to fitted densities with discontinuitieswhich is generally unreasonable for continuous circular data. Here, we showthat this problem is easily solved.The article is organized as follows. In Section 2 we define the Bernsteinpolynomial density approximation and show how this can be extended to circularvariables. In Section 3 we demonstrate how to calculate the circular moments ofa Bernstein polynomial density and in Section 4 we comment on how the modelcan be fitted to a sample of data. Finally, we illustrate our results with a realdata set in Section 5.2. The circular Bernstein polynomial distributionLet 푋 be a random variable with support [0, 1] and continuous distributionfunction 퐹푋(⋅). Then the Bernstein polynomial distribution function of order 푘2is defined to be퐵푘(푥) =푘∑푗=0퐹푋(푗푘)⎛⎝ 푘푗⎞⎠ 푥푗(1− 푥)푘−푗 for 0 ≤ 푥 ≤ 1 and 푘 ∈ N (1)It is well known that 퐵푘(푥) converges uniformly to 퐹푋(푥) as 푘 goes to infinity,see e.g. Vitale, (1975). The associated Bernstein density function is given by푏푘(푥) =푘∑푗=1(퐹푋(푗푘)− 퐹푋(푗 − 1푘))훽(푥 ∣ 푗, 푘 − 푗 + 1), (2)where 훽(⋅ ∣ 푎, 푏) is a beta density function:훽(푥 ∣ 푎, 푏) = 1퐵(푎, 푏)푥푎−1 (1− 푥)푏−1 (3)and 퐵(푎, 푏) = (푎+ 푏− 1)!/(푎− 1)!(푏− 1)!, for 푎, 푏 ∈ N, is the beta function.Clearly it is straightforward to extend the use of Bernstein polynomials todensities defined on any closed interval, e.g. [0, 2휋]. However, in order to definea distribution on the circle, it is first necessary to formally define the densityfunction of a circular random variable.The density function, 푓Θ(휃) of a continuous, circular random variable, Θ, isa non-negative, continuous function such that푓Θ(휃 + 2휋) = 푓Θ(휃) for 휃 ∈ Rand∫ 2휋0푓Θ(휃) 푑휃 = 1. Then, in order to define a cumulative distributionfunction of a circular random variable, it is necessary to establish an origin,0 ≤ 휈 < 1. Given this origin, the cumulative distribution function is퐹 휈Θ(휃) =∫ 휈+휃휈푓Θ(푢) 푑푢 for 0 ≤ 휃 < 2휋.However, if we wish to consider the Bernstein polynomial density approximationof order 푘, with respect to the origin 휈, that is푓휈푘 (휃) =12휋푘∑푗=1(퐹 휈Θ(2휋푗푘)− 퐹 휈Θ(2휋(푗 − 1)푘))훽(휃2휋∣∣∣∣ 푗, 푘 − 푗 + 1)(4)for this to be a strictly continuous, circular density then it is necessary that,퐹 휈Θ(2휋푘)= 1− 퐹 휈Θ(2휋(푘 − 1)푘), (5)3where 퐹 휈Θ(휃). The following theorem guarantees the existence of at least oneorigin satisfying (5).Theorem 1. Let 푓 be a density function for a continuous, circular random vari-able. Then there exists at least one point 휈 ∈ [0, 2휋) such that for any 푘 ∈ N,∫ 휈+ 2휋푘휈푓(휃) 푑휃 =∫ 휈휈− 2휋푘푓(휃) 푑휃.Proof. Define 퐺(휈) =∫ 휈+ 2휋푘휈푓(휃) 푑휃−∫ 휈휈− 2휋푘푓(휃) 푑휃. If there exist two points,0 ≤ 휈1 ∕= 휈2 < 2휋 such that 퐺(휈1) ≤ 0 and 퐺(휈2) ≥ 0, then by Bolzano’stheorem, there exists at least one point, 0 ≤ 휈0 < 2휋 such that 퐺(휈0) = 0.Otherwise, suppose that 퐺(휈) is always positive. Then, we have∫ 휈휈− 2휋푘푓(휃) 푑휃 <∫ 휈+ 2휋푘휈푓(휃) 푑휃 <∫ 휈+2휋휈+2휋− 2휋푘푓(휃) 푑휃which is impossible, as, due to the periodicity of 푓 , we have that∫ 휈휈− 2휋푘푓(휃) 푑휃 =∫ 휈+2휋휈+2휋− 2휋푘푓(휃) 푑휃.Similarly, 퐺 cannot always be negative and so the theorem is proved. □3. Trigonometric moments of the circular Bernstein polynomial dis-tributionFor a circular random variable, Θ, the 푝’th trigonometric moment is definedto be흁′푝 = 퐸[푒푖푝Θ]= 퐸[cos 푝Θ] + 푖퐸[sin 푝Θ] where 푖 =√−1def= 휌푝(cos휇′푝 + 푖 sin휇′푝)where휌푝 =√퐸[cos 푝Θ]2 + 퐸[sin 푝Θ]2 and (6)휇′푝 =⎧⎨⎩tan−1 퐸[sin 푝Θ]퐸[cos 푝Θ] if 퐸[sin 푝Θ] > 0 and 퐸[cos 푝Θ] > 0tan−1 퐸[sin 푝Θ]퐸[cos 푝Θ] + 휋 if 퐸[cos 푝Θ] < 0tan−1 퐸[sin 푝Θ]퐸[cos 푝Θ] + 2휋 if 퐸[sin 푝Θ] < 0 and 퐸[cos 푝Θ] > 0(7)4for 푝 = 1, 2, . . .. In particular, when 푝 = 1, we write 휇 for 휇′1 and 휌 for 휌1. Then,휇 is the mean direction and 휌 is the mean resultant length, see e.g. Mardia andJupp (1999) for more details.The trigonometric moments of the circular Bernstein polynomial distributioncan be derived using the following theorem.Theorem 2. The p’th trigonometric moments of a circular Bernstein polynomialdistribution are given by흁′푝 =푘∑푗=1푤푗 (퐸[cos 2휋푝ℬ푗] + 푖퐸[sin 2휋푝ℬ푗])where휔푗 = 퐹휈Θ(푗푘)− 퐹 휈Θ(푗 − 1푘)and where ℬ푗 is a beta random variable with density function 훽(⋅ ∣ 푗, 푘 − 푗 + 1)as defined in (3) such that퐸[cos(2휋푝ℬ푗)] = 1퐵(푗,푘−푗+1)∑푘−푗푟=0(−1)푟⎛⎝ 푘 − 푗푟⎞⎠ 퐼푝(푗 + 푟 − 1) (8)= 1퐵(푗,푘−푗+1)∑푗−1푟=0(−1)푟⎛⎝ 푗 − 1푟⎞⎠ 퐼푝(푘 − 푗 + 푟) (9)퐸[sin(2휋푝ℬ푗)] = 1퐵(푗,푘−푗+1)∑푘−푗푟=0(−1)푟⎛⎝ 푘 − 푗푟⎞⎠퐽푝(푗 + 푟 − 1) (10)= 1퐵(푗,푘−푗+1)∑푗−1푟=0(−1)푟+1⎛⎝ 푗 − 1푟⎞⎠퐽푝(푘 − 푗 + 푟) (11)where퐼푝(푗) =∫ 10cos(2휋푥)푥푗 푑푥 퐽푝(푗) =∫ 10sin(2휋푥)푥푗 푑푥 for 푗 = 0, 1, 2, . . .and 퐼푝(0) = 퐽푝(0) = 퐼푝(1) = 0, 퐽푝(1) = − 12휋푝 and for 퐶 = 2, 3, 4, . . .,퐼푝(퐶) =∑⌊퐶2⌋푐=1 (−1)푐−1 퐶!(퐶−2푐+1)! 1(2휋푝)2푐 (12)퐽푝(퐶) =∑⌊퐶+12⌋푐=1 (−1)푐 퐶!(퐶−2푐+2)! 1(2휋푝)2푐−1 . (13)5Proof. First note that퐸[cos(2휋푝ℬ푗)] =∫ 10cos(2휋푝푥)1퐵(푗, 푘 − 푗 + 1)푥푗−1(1− 푥)푘−푗 푑푥=1퐵(푗, 푘 − 푗 + 1)푘−푗∑푟=0(−1)푟⎛⎝ 푘 − 푗푟⎞⎠∫ 10cos(2휋푝푥)푥푗−1+푟 푑푥=∫ 10cos(2휋푝푦)1퐵(푗, 푘 − 푗 + 1)(1− 푦)푗−1푦푘−푗 푑푦=1퐵(푗, 푘 − 푗 + 1)푗−1∑푟=0(−1)푟⎛⎝ 푗 − 1푟⎞⎠∫ 10cos(2휋푝푦)푦푘−푗+푟 푑푦which gives the expressions for (8) and (9). In a similar way, the expressions(10) and (11) can be derived, recalling that sin(2휋− 휃) = − sin(휃). Now we candemonstrate formulas (12) and (13) by induction.Now observe that퐼푝(0) =∫ 10cos(2휋푝푥) 푑푥 = 0퐽푝(0) =∫ 10sin(2휋푝푥) 푑푥 = 0퐼푝(1) =∫ 10푥 cos(2휋푝푥) 푑푥 =12휋푝[푥 sin(2휋푝푥)]10 −12휋푝∫ 10sin(2휋푝푥) 푑푥 = 0퐽푝(1) =∫ 10푥 sin(2휋푝푥) 푑푥 = − 12휋푝[푥 cos(2휋푝푥)]10 +12휋푝∫ 10cos(2휋푝푥) 푑푥 = − 12휋푝Now consider 퐼푝(퐶). For 퐶 ≥ 2,퐼푝(퐶) =∫ 10푥퐶 cos(2휋푝푥) 푑푥=12휋푝[푥퐶 sin(2휋푝푥)]10− 퐶2휋푝∫ 10푥퐶−1 sin(2휋푝푥) 푑푥= − 퐶2휋푝∫ 10푥퐶−1 sin(2휋푝푥) 푑푥=퐶(2휋푝)2[푥퐶−1 cos(2휋푝푥)]10− 퐶(퐶 − 1)(2휋푝)2∫ 10푥퐶−2 cos(2휋푝푥) 푑푥=퐶(2휋푝)2− 퐶(퐶 − 1)(2휋푝)2퐼푝(퐶 − 2) (14)and therefore 퐼푝(2) =2(2휋푝)2 and 퐼푝(3) =3(2휋푝)2 which satisfy (12). Assume now6that the formula is valid for 푐 = 2, . . . , 퐶. Then퐼푝(퐶 + 2) =퐶 + 2(2휋푝)2− (퐶 + 2)(퐶 + 1)(2휋푝)2퐼푝(퐶) from (14)=퐶 + 2(2휋푝)2−⌊퐶2⌋∑푐=1(−1)푐−1 퐶!(퐶 − 2푐+ 1)!1(2휋푝)2푐from the induction assumption=퐶 + 2(2휋푝)2+⌊퐶2⌋∑푐=1(−1)푐+1−1 (퐶 + 2)!(퐶 + 2− 2(푐+ 1) + 1)!1(2휋푝)2(푐+1)=퐶 + 2(2휋푝)2+⌊퐶+22⌋∑푐=2(−1)푐−1 (퐶 + 2)!(퐶 + 2− 2푐+ 1)!1(2휋푝)2푐=⌊퐶+22⌋∑푐=1(−1)푐−1 (퐶 + 2)!(퐶 + 2− 2푐+ 1)!1(2휋푝)2푐which demonstrates (12).Equally, we have the recurrence relation퐽푝(퐶) = − 12휋푝− 퐶(퐶 − 1)(2휋푝)2퐽푝(퐶 − 2) (15)which implies that 퐽푝(2) = − 12휋푝 and 퐽푝(3) = − 12휋푝 + 3!(2휋푝)3 which satisfy (13).Assuming the formula is valid for 푐 = 2, . . . , 퐶 then퐽푝(퐶 + 2) = − 12휋푝− (퐶 + 2)(퐶 + 1)(2휋푝)2퐽푝(퐶) from (15)= − 12휋푝− (퐶 + 2)(퐶 + 1)(2휋푝)2⌊퐶+12⌋∑푐=1(−1)푐 퐶!(퐶 − 2푐+ 2)!1(2휋푝)2푐−1from the induction assumption= − 12휋푝−⌊퐶+12⌋∑푐=1(퐶 + 2)!(퐶 − 2푐+ 2)!1(2휋푝)2푐+1=⌊퐶+2+12⌋∑푐=1(−1)푐 (퐶 + 2)!(퐶 + 2− 2푐+ 2)!1(2휋푝)2푐−1which demonstrates (13) and proves the theorem. □4. Estimation for the Circular Bernstein PolynomialGiven a sample of 푛 data generated from a linear variable, 푋 , with support[0, 1], then from (2), the natural Bernstein polynomial estimator of order 푘 for7the density of 푋 is given by,푏ˆ푘(푥) =푘∑푗=1(퐹ˆ푋(푗푘)− 퐹ˆ푋(푗 − 1푘))훽(푥 ∣ 푗, 푘 − 푗 + 1),where 퐹ˆ푋(⋅) is the empirical distribution function, see e.g. Vitale (1975). Ob-serve that this estimator can be seen as a smoothed histogram because a his-togram is simply a functionℎ (푥) =푘∑푗=1(퐹ˆ푋(푗푘)− 퐹ˆ푋(푗 − 1푘))퐼[ 푗−1푘 ,푗푘 ](푥) ,where 퐼퐴 (푥) is the indicator function in the set 퐴.Suppose that we have a sample of 푛 circular data. Then the Bernsteinpolynomial density approximation can be fitted in various steps. Firstly, for agiven order, 푘, we need to select an origin. Analogous to (5), it may be thatthere exist multiple origins, 휈, such that the empirical distribution function,퐹ˆ 휈(⋅), satisfies the condition퐹ˆ 휈(2휋푘)= 1− 퐹ˆ 휈(2휋(푘 − 1)푘)(16)when the fitted Bernstein polynomial density is from (4)푓ˆ 휈ˆ푘 (휃) =12휋푘∑푗=1(퐹ˆ 휈ˆ(2휋푗푘)− 퐹ˆ 휈ˆ(2휋(푗 − 1)푘))훽(휃2휋∣∣∣∣ 푗, 푘 − 푗 + 1)and 휈ˆ is chosen to maximize the log-likelihood estimate푛∑푟=1log푘∑푗=1(퐹ˆ 휈(2휋푗푘)− 퐹ˆ 휈(2휋(푗 − 1)푘))훽(휃푟2휋∣∣∣∣ 푗, 푘 − 푗 + 1).However, as the observed data are discrete, the existence of an origin whichsatisfies (16) is not guaranteed and if no such origin exists, an origin 휈ˆ is chosensuch that the distance,∣∣∣∣퐹ˆ 휈(2휋푘)+ 퐹ˆ 휈(2휋(푘 − 1)푘)− 1∣∣∣∣is minimized. Then the Bernstein polynomial density estimate is given by푓ˆ 휈ˆ푘 (휃) =12휋[12{퐹ˆ 휈ˆ(2휋푘)+ 1− 퐹ˆ 휈ˆ(2휋(푘 − 1)푘)}훽(휃2휋∣∣∣∣ 1, 푘)+8푘−1∑푗=2{퐹ˆ 휈ˆ(2휋푗푘)− 퐹ˆ 휈ˆ(2휋(푗 − 1)푘)}훽(휃2휋∣∣∣∣ 푗, 푘 − 푗 + 1)+12{퐹ˆ 휈ˆ(2휋푘)+ 1− 퐹ˆ 휈ˆ(2휋(푘 − 1)푘)}훽(휃2휋∣∣∣∣ 푘, 1)]so that the weights of the first and last beta densities are equal and the fit-ted density is circular. This procedure is this analogous to that of minimizingenclosure in operational research problems, see e.g. Lutterkort et al. (2001).The problem of choosing the order, 푘, of the Bernstein polynomial can beviewed as similar to the problem of choosing the number of bars for the his-togram. Small values of 푘 will lead to a very poor approximation and highvalues will lead to overfitting. One possibility is to choose 푘 = ⌈√푛⌉. Thegoodness of fit of the fitted Bernstein polynomial density can then be testedusing standard tests such as those of Watson (1961) or Kuiper (1960) and ifthe fitted model is rejected at a given significance level (e.g. 10%), 푘 can beincreased until the fitted model is accepted.5. ExampleHere we consider data which correspond to the twenty four hour clocktimes of 1297 crimes perpetrated in Chicago on May 11th, 2007, obtained fromwww.chicagocrime.org. A Bernstein polynomial density approximation of or-der 푘 = 37 = ⌈√1297⌉ was fitted to these data. Figure 5 shows a histogramof the data and the fitted density. The data are plotted so that the origin isset to the fitted origin time, 휈ˆ = 06 : 20 hours. The fitted and empirical meandirection and 17:39 hours and 17:42 hours and the fitted and empirical meancircular resultant lengths are 0.1998 and 0.1880 respectively. Thus, there is goodagreement between the fitted and empirical moments.Both the Kuiper and Watson tests were used to test the goodness of fit ofthe Bernstein polynomial density and in both cases, the model was not rejectedat a 10% level.906:20 12:20 18:20 00:20 06:2000.010.020.030.040.050.060.070.08θf(θ)Figure 1: Histogram of the Chicago crime data with fitted Bernstein polynomial densityapproximationReferences[1] Babu, G.J., Canty, A.J. and Chaubey, Y.P. (2002). Application of Bernsteinpolynomials for smooth estimation of a distribution and density function.Journal of Statistical Planning and Inference, 105, 377–392.[2] Ferna´ndez-Dura´n, J.J. (2004). Circular distributions based on nonnegativetrigonometric sums. Biometrics, 60, 499–503.[3] Kakizawa, Y. (2004). Bernstein polynomial probability density estimation.Journal of Nonparametric Statistics, 16, 709–729.[4] Kuiper, N. H. (1960). Test concerning random points on a circle. Ned. Akad.Wet. Proc, 63, 38–47.[5] Fisher, N.I. (1989). Smoothing a sample of circular data. Journal of Struc-tural Geology, 11, 775–778.[6] Lutterkort, D. and Peters J. (2001). Optimized refinable enclosures of mul-tivariate polynomial pieces. Computer Aided Geometric Design, 18, 851–863.[7] Mardia, K.V. and Jupp, P. E. (1999). Directional Statistics, Wiley: Chich-ester.10[8] Mooney A., Helms P.J. and Jolliffe I.T. (2003). Fitting mixtures of vonMises distributions: a case study involving sudden infant death syndrome.Computational Statistics and Data Analysis, 41, 505–513.[9] Petrone, S. (1999a). Random Bernstein polynomials. Scandinavian Journalof Statistics, 26, 373–393.[10] Petrone, S. (1999b). Bayesian density estimation using random Bernsteinpolynomials. Canadian Journal of Statistics, 27, 105–126.[11] Petrone, S. and Wassermann, L. (2002). Consistency of Bernstein poly-nomial posteriors. Journal of the Royal Statistical Society Series B, 64,79–100.[12] Vitale, R.A. (1975). A Bernstein polynomial approach to density estima-tion. In Statistical inference and related topics (ed. Madan Lal Puri), Vol2, 87–100. Academic Press, New York.[13] Watson, G. S. (1961). Goodness-of-fit tests on the circle. Biometrika, 48,109–114.11

Circular Bernstein polynomial distributions

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Circular Bernstein polynomial distributions

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