An introduction to statistical methods for circular data

89

2. Circular densities

This section will be devoted to present some classical parametric

models for circular data, progressing from simple parametric

formulations to more flexible approaches. Along this section, denote

by Θ a circular random variable, measured in radians in [0

,

2

π

).

Jammalamadaka and SenGupta (2001, Section 2.1) specify the

conditions for a function

f

to be a circular density (

f

(

θ

)

≥

0,

R

2

π

0

f

(

θ

)

dθ

= 1 and

f

(

θ

) =

f

(

θ

+ 2

πk

), for

θ

∈

[0

,

2

π

) and

k

∈

Z

).

2.1. Parametric models

The most widely used parametric circular density is the von

Mises, a symmetric unimodal function, satisfying that the maximum

likelihood estimator of the location parameter is the sample mean

(it is also known as the circular normal). The von Mises density is

given by:

f

µ,κ

(

θ

) =

1

2

πI

0

(

κ

)

exp (

κ

cos(

θ

−

µ

))

,

0

≤

θ <

2

π,

(2.1)

where

I

0

denotes the modified Bessel function of the first kind

and order zero (which is used as a normalizing constant in the

density expression). As it can be seen in Figure 2 (left plot, solid

line), the von Mises is a symmetric model with mean/mode at

µ

and concentration controlled by

κ

. Taking

κ

= 0 gives the

uniform density on the circle,

f

(

θ

) = (2

π

)

−

1

. There are other

popular circular models, such as the cardiod (which also contains

the uniform as a particular case), but a general way of constructing

circular densities is by using a wrapping procedure. That is, if

X

is a random variable on

R

with density

g

, then

θ

=

X

(mod(2

π

))

has density

f

(

θ

) =

P

∞

k

=

−∞

g

(

θ

+ 2

πk

). Some examples are the

wrapped–normal and wrapped–Cauchy, two of them depicted in

Figure 2, left plot. All of these models are symmetric densities