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.Circular data, Non-parametric modeling, Bernstein polynomials

Concepción Ausín

José Antonio Carnicero

Michael P. Wiper

English

Research Papers in Economics

 
 
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 distributions
J. A. Carnicero, M. P. Wiper, M. C. Ausin
Universidad Carlos III de Madrid, Getafe, Madrid, Spain
Abstract
This paper introduces a new non-parametric approach to the modeling of circu-
lar data, based on the use of Bernstein polynomial densities which generalizes
the standard Bernstein polynomial model to account for the speciﬁc characteris-
tics 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.
1. Introduction
Problems where the data are angular directions occur in many diﬀerent
scientiﬁc 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 for
births or the times when crimes are committed may also be converted to angular
data via a simple transformation modulo some period. Data of this type are
commonly known as circular data and are usually represented as points on the
circumference of an unit circle or as angles,  , where 0 ≤   < 2  radians, which
represent the positive angle of rotation from some arbitrarily chosen origin,
  = 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 a
variety of techniques, see e.g. Mardia and Jupp (1999) for a full review. However,
most parametric models developed for circular data are unimodal and symmetric
but in many cases,both multimodal and asymmetric data may be encountered.
In such cases, semi-parametric or non-parametric approachesmight be preferred.
Semi-parametric approaches based on trigonometric sums and mixtures of von
Mises or circular normal distributions have been introduced in Fern´ andez-Dur´ an
(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 work
on non-parametric modeling of circular data.
In this paper, we introduce an alternative non-parametric approach based on
the use of Bernstein polynomials. It is well known that the Bernstein polynomial
is a useful tool for interpolating functions deﬁned on a closed interval. Bern-
stein polynomials have been proposed as density estimators for variables with
ﬁnite 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 approaches
to circular data is that these can lead to ﬁtted densities with discontinuities
which is generally unreasonable for continuous circular data. Here, we show
that this problem is easily solved.
The article is organized as follows. In Section 2 we deﬁne the Bernstein
polynomial density approximation and show how this can be extended to circular
variables. In Section 3 we demonstrate how to calculate the circular moments of
a Bernstein polynomial density and in Section 4 we comment on how the model
can be ﬁtted to a sample of data. Finally, we illustrate our results with a real
data set in Section 5.
2. The circular Bernstein polynomial distribution
Let   be a random variable with support [0,1] and continuous distribution
function   (⋅). Then the Bernstein polynomial distribution function of order  
2is deﬁned to be
  ( ) =
   
 =0
  
 
 
 
 
⎛
⎝  
 
⎞
⎠  (1 −  ) −  for 0 ≤   ≤ 1 and   ∈ N (1)
It is well known that   ( ) converges uniformly to   ( ) as   goes to inﬁnity,
see e.g. Vitale, (1975). The associated Bernstein density function is given by
  ( ) =
   
 =1
 
  
 
 
 
 
−   
 
  − 1
 
  
 (  ∣  ,  −   + 1), (2)
where  (⋅ ∣  , ) is a beta density function:
 (  ∣  , ) =
1
 ( , )
  −1 (1 −  )
 −1 (3)
and  ( , ) = (  +   − 1)!/(  − 1)!(  − 1)!, for  ,  ∈ N, is the beta function.
Clearly it is straightforward to extend the use of Bernstein polynomials to
densities deﬁned on any closed interval, e.g. [0,2 ]. However, in order to deﬁne
a distribution on the circle, it is ﬁrst necessary to formally deﬁne the density
function of a circular random variable.
The density function,  Θ( ) of a continuous, circular random variable, Θ, is
a non-negative, continuous function such that
 Θ(  + 2 ) =  Θ( ) for   ∈ R
and
  2 
0  Θ( )   = 1. Then, in order to deﬁne a cumulative distribution
function of a circular random variable, it is necessary to establish an origin,
0 ≤   < 1. Given this origin, the cumulative distribution function is
 
 
Θ( ) =
   + 
 
 Θ( )   for 0 ≤   < 2 .
However, if we wish to consider the Bernstein polynomial density approximation
of order  , with respect to the origin  , that is
  
 ( ) =
1
2 
   
 =1
 
  
Θ
 
2  
 
 
−   
Θ
 
2 (  − 1)
 
  
 
 
 
2 
 
   
  ,  −   + 1
 
(4)
for this to be a strictly continuous, circular density then it is necessary that,
  
Θ
 
2 
 
 
= 1 −   
Θ
 
2 (  − 1)
 
 
, (5)
3where   
Θ( ). The following theorem guarantees the existence of at least one
origin satisfying (5).
Theorem 1. Let   be a density function for a continuous, circular random vari-
able. Then there exists at least one point   ∈ [0,2 ) such that for any   ∈ N,
   + 2￿
k
 
 ( )   =
   
 − 2￿
k
 ( )  .
Proof. Deﬁne  ( ) =
   + 2￿
k
   ( )  −
   
 − 2￿
k  ( )  . If there exist two points,
0 ≤  1 ∕=  2 < 2  such that  ( 1) ≤ 0 and  ( 2) ≥ 0, then by Bolzano’s
theorem, there exists at least one point, 0 ≤  0 < 2  such that  ( 0) = 0.
Otherwise, suppose that  ( ) is always positive. Then, we have
   
 − 2￿
k
 ( )   <
   + 2￿
k
 
 ( )   <
   +2 
 +2 − 2￿
k
 ( )  
which is impossible, as, due to the periodicity of  , we have that
   
 − 2￿
k
 ( )   =
   +2 
 +2 − 2￿
k
 ( )  .
Similarly,   cannot always be negative and so the theorem is proved. □
3. Trigonometric moments of the circular Bernstein polynomial dis-
tribution
For a circular random variable, Θ, the  ’th trigonometric moment is deﬁned
to be
￿
′
  =  
 
 
  Θ 
=  [cos Θ] +   [sin Θ] where   =
√
−1
def =   
 
cos ′
  +  sin ′
 
 
where
   =
 
 [cos Θ]2 +  [sin Θ]2 and (6)
 
′
  =
⎧
    ⎨
    ⎩
tan−1  [sin Θ]
 [cos Θ] if  [sin Θ] > 0 and  [cos Θ] > 0
tan−1  [sin Θ]
 [cos Θ] +   if  [cos Θ] < 0
tan−1  [sin Θ]
 [cos Θ] + 2  if  [sin Θ] < 0 and  [cos Θ] > 0
(7)
4for   = 1,2,.... In particular, when   = 1, we write   for  ′
1 and   for  1. Then,
  is the mean direction and   is the mean resultant length, see e.g. Mardia and
Jupp (1999) for more details.
The trigonometric moments of the circular Bernstein polynomial distribution
can be derived using the following theorem.
Theorem 2. The p’th trigonometric moments of a circular Bernstein polynomial
distribution are given by
￿′
  =
   
 =1
   ( [cos2  ℬ ] +   [sin2  ℬ ])
where
   =  
 
Θ
 
 
 
 
−  
 
Θ
 
  − 1
 
 
and where ℬ  is a beta random variable with density function  (⋅ ∣  ,  −  +1)
as deﬁned in (3) such that
 [cos(2  ℬ )] = 1
 ( , − +1)
  − 
 =0(−1) 
⎛
⎝   −  
 
⎞
⎠  (  +   − 1) (8)
= 1
 ( , − +1)
  −1
 =0(−1) 
⎛
⎝   − 1
 
⎞
⎠  (  −   +  ) (9)
 [sin(2  ℬ )] = 1
 ( , − +1)
  − 
 =0(−1) 
⎛
⎝   −  
 
⎞
⎠  (  +   − 1) (10)
= 1
 ( , − +1)
  −1
 =0(−1) +1
⎛
⎝   − 1
 
⎞
⎠  (  −   +  ) (11)
where
  ( ) =
  1
0
cos(2  )        ( ) =
  1
0
sin(2  )      for   = 0,1,2,...
and   (0) =   (0) =   (1) = 0,   (1) = − 1
2   and for   = 2,3,4,...,
  ( ) =
 ⌊ C
2 ⌋
 =1(−1) −1  !
( −2 +1)!
1
(2  )2c (12)
  ( ) =
 ⌊
C+1
2 ⌋
 =1 (−1)   !
( −2 +2)!
1
(2  )2c−1. (13)
5Proof. First note that
 [cos(2  ℬ )] =
  1
0
cos(2   )
1
 ( ,  −   + 1)
 
 −1(1 −  )
 −    
=
1
 ( ,  −   + 1)
 −   
 =0
(−1)
 
⎛
⎝   −  
 
⎞
⎠
  1
0
cos(2   ) 
 −1+    
=
  1
0
cos(2   )
1
 ( ,  −   + 1)
(1 −  ) −1  −    
=
1
 ( ,  −   + 1)
 −1  
 =0
(−1)
 
⎛
⎝   − 1
 
⎞
⎠
  1
0
cos(2   ) 
 − +    
which gives the expressions for (8) and (9). In a similar way, the expressions
(10) and (11) can be derived, recalling that sin(2  − ) = −sin( ). Now we can
demonstrate formulas (12) and (13) by induction.
Now observe that
  (0) =
  1
0
cos(2   )   = 0
  (0) =
  1
0
sin(2   )   = 0
  (1) =
  1
0
 cos(2   )   =
1
2  
[ sin(2   )]
1
0 −
1
2  
  1
0
sin(2   )   = 0
  (1) =
  1
0
 sin(2   )   = −
1
2  
[ cos(2   )]
1
0 +
1
2  
  1
0
cos(2   )   = −
1
2  
Now consider   ( ). For   ≥ 2,
  ( ) =
  1
0
   cos(2   )  
=
1
2  
 
   sin(2   )
 1
0 −
 
2  
  1
0
  −1 sin(2   )  
= −
 
2  
  1
0
  −1 sin(2   )  
=
 
(2  )2
 
 
 −1 cos(2   )
 1
0 −
 (  − 1)
(2  )2
  1
0
 
 −2 cos(2   )  
=
 
(2  )2 −
 (  − 1)
(2  )2   (  − 2) (14)
and therefore   (2) = 2
(2  )2 and   (3) = 3
(2  )2 which satisfy (12). Assume now
6that the formula is valid for   = 2,..., . Then
  (  + 2) =
  + 2
(2  )2 −
(  + 2)(  + 1)
(2  )2   ( ) from (14)
=
  + 2
(2  )2 −
⌊ C
2 ⌋  
 =1
(−1) −1  !
(  − 2  + 1)!
1
(2  )2  from the induction assumption
=
  + 2
(2  )2 +
⌊ C
2 ⌋  
 =1
(−1) +1−1 (  + 2)!
(  + 2 − 2(  + 1) + 1)!
1
(2  )2( +1)
=
  + 2
(2  )2 +
⌊
C+2
2 ⌋  
 =2
(−1)
 −1 (  + 2)!
(  + 2 − 2  + 1)!
1
(2  )2 
=
⌊
C+2
2 ⌋  
 =1
(−1)
 −1 (  + 2)!
(  + 2 − 2  + 1)!
1
(2  )2 
which demonstrates (12).
Equally, we have the recurrence relation
  ( ) = −
1
2  
−
 (  − 1)
(2  )2   (  − 2) (15)
which implies that   (2) = − 1
2   and   (3) = − 1
2   + 3!
(2  )3 which satisfy (13).
Assuming the formula is valid for   = 2,...,  then
  (  + 2) = −
1
2  
−
(  + 2)(  + 1)
(2  )2   ( ) from (15)
= −
1
2  
−
(  + 2)(  + 1)
(2  )2
⌊
C+1
2 ⌋  
 =1
(−1)   !
(  − 2  + 2)!
1
(2  )2 −1
from the induction assumption
= −
1
2  
−
⌊
C+1
2 ⌋  
 =1
(  + 2)!
(  − 2  + 2)!
1
(2  )2 +1
=
⌊
C+2+1
2 ⌋  
 =1
(−1)  (  + 2)!
(  + 2 − 2  + 2)!
1
(2  )2 −1
which demonstrates (13) and proves the theorem. □
4. Estimation for the Circular Bernstein Polynomial
Given a sample of   data generated from a linear variable,  , with support
[0,1], then from (2), the natural Bernstein polynomial estimator of order   for
7the density of   is given by,
ˆ   ( ) =
   
 =1
 
ˆ   
 
 
 
 
− ˆ   
 
  − 1
 
  
 (  ∣  ,  −   + 1),
where ˆ   (⋅) 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 function
ℎ( ) =
   
 =1
 
ˆ   
 
 
 
 
− ˆ   
 
  − 1
 
  
 [
j−1
k ,
j
k] ( ),
where    ( ) is the indicator function in the set  .
Suppose that we have a sample of   circular data. Then the Bernstein
polynomial density approximation can be ﬁtted in various steps. Firstly, for a
given order,  , we need to select an origin. Analogous to (5), it may be that
there exist multiple origins,  , such that the empirical distribution function,
ˆ   (⋅), satisﬁes the condition
ˆ   
 
2 
 
 
= 1 − ˆ   
 
2 (  − 1)
 
 
(16)
when the ﬁtted Bernstein polynomial density is from (4)
ˆ  
ˆ  
 ( ) =
1
2 
   
 =1
 
ˆ  
ˆ  
 
2  
 
 
− ˆ  
ˆ  
 
2 (  − 1)
 
  
 
 
 
2 
 
 
 
  ,  −   + 1
 
and ˆ   is chosen to maximize the log-likelihood estimate
   
 =1
log
   
 =1
 
ˆ   
 
2  
 
 
− ˆ   
 
2 (  − 1)
 
  
 
 
  
2 
 
 
 
  ,  −   + 1
 
.
However, as the observed data are discrete, the existence of an origin which
satisﬁes (16) is not guaranteed and if no such origin exists, an origin ˆ   is chosen
such that the distance,
 
 
   
ˆ   
 
2 
 
 
+ ˆ   
 
2 (  − 1)
 
 
− 1
 
 
   
is minimized. Then the Bernstein polynomial density estimate is given by
ˆ   ˆ  
 ( ) =
1
2 
 
1
2
 
ˆ   ˆ  
 
2 
 
 
+ 1 − ˆ   ˆ  
 
2 (  − 1)
 
  
 
 
 
2 
 
 
 
 1, 
 
+
8 −1  
 =2
 
ˆ   ˆ  
 
2  
 
 
− ˆ   ˆ  
 
2 (  − 1)
 
  
 
 
 
2 
 
 
 
  ,  −   + 1
 
+
1
2
 
ˆ  
ˆ  
 
2 
 
 
+ 1 − ˆ  
ˆ  
 
2 (  − 1)
 
  
 
 
 
2 
 
 
 
  ,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 minimizing
enclosure in operational research problems, see e.g. Lutterkort et al. (2001).
The problem of choosing the order,  , of the Bernstein polynomial can be
viewed as similar to the problem of choosing the number of bars for the his-
togram. Small values of   will lead to a very poor approximation and high
values will lead to overﬁtting. One possibility is to choose   = ⌈
√
 ⌉. The
goodness of ﬁt of the ﬁtted Bernstein polynomial density can then be tested
using standard tests such as those of Watson (1961) or Kuiper (1960) and if
the ﬁtted model is rejected at a given signiﬁcance level (e.g. 10%),   can be
increased until the ﬁtted model is accepted.
5. Example
Here we consider data which correspond to the twenty four hour clock
times of 1297 crimes perpetrated in Chicago on May 11th, 2007, obtained from
www.chicagocrime.org. A Bernstein polynomial density approximation of or-
der   = 37 = ⌈
√
1297⌉ was ﬁtted to these data. Figure 5 shows a histogram
of the data and the ﬁtted density. The data are plotted so that the origin is
set to the ﬁtted origin time, ˆ   = 06 : 20 hours. The ﬁtted and empirical mean
direction and 17:39 hours and 17:42 hours and the ﬁtted and empirical mean
circular resultant lengths are 0.1998 and 0.1880 respectively. Thus, there is good
agreement between the ﬁtted and empirical moments.
Both the Kuiper and Watson tests were used to test the goodness of ﬁt of
the Bernstein polynomial density and in both cases, the model was not rejected
at a 10% level.
906:20 12:20 18:20 00:20 06:20
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
θ
f
(
θ
)
Figure 1: Histogram of the Chicago crime data with ﬁtted Bernstein polynomial density
approximation
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trigonometric sums. Biometrics, 60, 499–503.
[3] Kakizawa, Y. (2004). Bernstein polynomial probability density estimation.
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