Families Of Distributions On The Circle - A Review

Theory and Methods of Statistical Inference F.Rotolo

Families of distributions on the circle
A review

Federico Rotolo
federico.rotolo@stat.unipd.it

Department of Statistical Sciences
University of Padua

September 7, 2010

Families of distributions on the circle — A review 1/ 23


Introduction

Families of Circular Distributions
The Jones & Pewsey distribution
The Generalized von Mises distribution
The Kato & Jones distribution
Two particular submodels

Comparison
Generality
Data modelling
Inferential aspects

Bibliography



Introduction
Data on the circle are present in many applications, whenever
directional data are observed.
(wind direction, earthquake propagation, waves action on moving ships, etc.)



Introduction

Distributions on the real line are not suitable for direction, so new
models are needed.
[Jammalamadaka & SenGupta(2001), Mardia & Jupp(1999), Fisher(1993), Mardia(1972)]



Introduction

Distributions on the real line are not suitable for direction, so new
models are needed.
[Jammalamadaka & SenGupta(2001), Mardia & Jupp(1999), Fisher(1993), Mardia(1972)]

The most popular circular distributions are:
• von Mises vM(µ, κ)
• wrapped Cauchy wC(µ, ρ)
• Carthwright’s power-of-cosine Cpc(µ, ψ)
• cardioid ca(µ, ρ)
• circular Uniform cU(0; 2π)


Introduction
0.6
0.5 An example

π/2
0.4
Density

π 0
0.3
0.2

3/2π
0.1
0.0

−3 −2 −1 0 1 2 3

Angle

vM(0.48π,1.8) (dash), wC(-0.45π,0.6) (dot),
Cpc(-0.16π,0.6) (long dash), ca(0.89π,0.2) (dot-dash).


Introduction

These simple circular distributions are symmetric and unimodal,
so their ﬂexibility is quite limited.



Introduction


⇓
Recently some more general families of circular distributions
have been proposed:



Introduction


⇓
Recently some more general families of circular distributions
have been proposed:
• Jones & Pewsey [Jones & Pewsey(2005)]

• Generalized von Mises [Gatto & Jammalamadaka(2007)]

• Kato & Jones [Kato & Jones(2010)]



The ﬁrst proposed family of circular distributions [Jones & Pewsey(2005)] is
the three-parameter Jones & Pewsey distribution JP(µ, κ, ψ)



with density

(cosh(κψ) + sinh(κψ) cos(θ − µ))1/ψ
fJP (θ) =
2πP1/ψ (cosh(κψ))
0 ≤ θ < 2π, 0 ≤ µ < 2π, κ ≥ 0, ψ ∈ R, and P1/ψ (·) is the associated Legendre function of the ﬁrst kind

and order 0.



with density

fJP (θ) =

and order 0.

All the vM, wC, ca, Cpc and cU distributions can be obtained
as special cases of it.



with density

fJP (θ) =

and order 0.

All the vM, wC, ca, Cpc and cU distributions can be obtained
as special cases of it.

Two other distributions, the wrapped Normal [Stephens(1963)] and the
wrapped symmetric stable [Mardia(1972)], can be well approximated by
the JP model.


Properties

The JP family is symmetric unimodal.



Properties

MLE:
ˆ ˆ
µ is asymptotically independent of (ψ, κ),
ˆ
ˆ ˆ
no reparametrization is available to reduce corr(ψ, κ).



Properties

MLE:
ˆ ˆ
µ is asymptotically independent of (ψ, κ),
ˆ
ˆ ˆ
no reparametrization is available to reduce corr(ψ, κ).
0.6

π/2
Density

0.4

π 0
0.2

3/2π
0.0

−3 −2 −1 0 1 2 3

Angle µ=4.1, κ=1.8, ψ=−0.6




A ﬁve-parameter class of distributions comprising the vM was
proposed by Maksimov in 1967.

An interesting subclass of it is the four-parameter Generalized
von Mises distribution GvM(µ1 , µ2 , κ1 , κ2 ) [Gatto & Jammalamadaka(2007)]




A ﬁve-parameter class of distributions comprising the vM was
proposed by Maksimov in 1967.

An interesting subclass of it is the four-parameter Generalized
von Mises distribution GvM(µ1 , µ2 , κ1 , κ2 ) [Gatto & Jammalamadaka(2007)]
with density
1
fGvM (θ) = exp{κ1 cos(θ − µ1 ) + κ2 cos 2(θ − µ2 )}
2πG0 (δ, κ1 , κ2 )
0 ≤ θ < 2π, 0 ≤ µ1 < 2π, 0 ≤ µ2 < π, κ1 , κ2 ≥ 0, δ = (µ1 − µ2 )modπ, G0 (δ, κ1 , κ2 ) is the normalizing

constant.



Properties

The GvM family can be asymmetric and it can have one or two
maxima.



Properties

maxima.
The skewness and the maxima location are mainly controlled by
µ1 and µ2 ,



Properties

maxima.
µ1 and µ2 , the kurtosis mostly by κ1 and κ2 .



Properties

maxima.
µ1 and µ2 , the kurtosis mostly by κ1 and κ2 .
1.0
0.8

π/2
0.6
Density

π 0
0.4

3/2π
0.2
0.0

−3 −2 −1 0 1 2 3

Angle µ1=0.8π, µ2=π, κ1=4.8, κ2=4.1



3.0

3.0

3.0
2.0

2.0

2.0
κ1

κ2

κ2
1.0

1.0

1.0
0.0

0.0

0.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0.0 1.0 2.0 3.0

µ2 µ2 κ1

Unimodality(white)/bimodality(green) of the GvM density with µ1 = 0.
For each variable the value chosen for the graph where it is absent is shown by the grey lines.



3.0

3.0

3.0
2.0

2.0

2.0
κ1

κ2

κ2
1.0

1.0

1.0
0.0

0.0

0.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0.0 1.0 2.0 3.0

µ2 µ2 κ1


! A big portion of the parameter space gives a bimodal distribution.



3.0

3.0

3.0
2.0

2.0

2.0
κ1

κ2

κ2
1.0

1.0

1.0
0.0

0.0

0.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0.0 1.0 2.0 3.0

µ2 µ2 κ1



In general there is no reason to expect bimodality
→ maybe misleading results.



3.0

3.0

3.0
2.0

2.0

2.0
κ1

κ2

κ2
1.0

1.0

1.0
0.0

0.0

0.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0.0 1.0 2.0 3.0

µ2 µ2 κ1



In general there is no reason to expect bimodality
→ maybe misleading results.

When bimodality is expected (e.g. with two groups of data)
→ good model: simpler inference w.r.t. mixture models.



Membership of the Exponential Family

The most interesting property of the GvM model is that the
reparametrization

λ = (κ1 cos µ1 , κ1 sin µ1 , κ2 cos 2µ2 , κ2 sin 2µ2 )T




reparametrization


makes it possible to express the density as

fGvM (θ | λ) = exp{λT t(θ) − k(θ)},

a four-parameter exponential family, indexed by λ ∈ [−1, 1]4 .
t(θ) = (cos θ, sin θ, cos 2θ, sin 2θ)T , k(λ) = log(2π) + log G0 (δ, λ(1) , λ(2) ), λ(1) = (λ1 , λ2 )T ,
(2) T (1) (2)
λ = (λ3 , λ4 ) and δ = (arg λ − arg λ /2)modπ.




reparametrization


makes it possible to express the density as

fGvM (θ | λ) = exp{λT t(θ) − k(θ)},

a four-parameter exponential family, indexed by λ ∈ [−1, 1]4 .
t(θ) = (cos θ, sin θ, cos 2θ, sin 2θ)T , k(λ) = log(2π) + log G0 (δ, λ(1) , λ(2) ), λ(1) = (λ1 , λ2 )T ,
(2) T (1) (2)
λ = (λ3 , λ4 ) and δ = (arg λ − arg λ /2)modπ.

Thus it has many good inferential properties, like the uniqueness
of the MLEs, when they exist, and the asymptotic normality of
the estimator.


The four-parameter Kato & jones distribution KJ is obtained by
applying a M¨bius transformation to a vM-distributed random
o
variable [Kato & Jones(2010)] .



o
The M¨bius transformation is a (closed under composition)
o
circle-to-circle function M¨µ,ν,r : Ξ → Θ given by
o

e iΞ + re iν
e iΘ = e iµ ,
re i(Ξ−ν) + 1
with 0 ≤ µ, ν < 2π and 0 ≤ r < 1.



o
The M¨bius transformation is a (closed under composition)
o
circle-to-circle function M¨µ,ν,r : Ξ → Θ given by
o

e iΞ + re iν
e iΘ = e iµ ,
re i(Ξ−ν) + 1
with 0 ≤ µ, ν < 2π and 0 ≤ r < 1.

If Ξ ∼ vM(0, κ), then Θ = M¨µ,ν,r (Ξ) ∼ KJ(µ, ν, r , κ) has density
o
κ{ξ cos(θ−η)−2r cos ν}
1 − r 2 exp 1+r 2 −2r cos(θ−γ)
fKJ (θ) = 2 − 2r cos(θ − γ)
,
2πI0 (κ) 1 + r
r 4 + 2r 2 cos(2ν) + 1, η = µ + arg[r 2 {cos(2ν) + i sin(2ν)} + 1], γ = µ + ν.
p
0 ≤ θ < 2π, ξ =



Properties

The KJ distribution can be symmetric or asymmetric.



Properties

It includes the vM, the wC and the cU models as special cases.



Properties

It can also be either unimodal or bimodal, but conditions for
unimodality are not straigthforward.



Properties

It can also be either unimodal or bimodal, but conditions for
unimodality are not straigthforward.

π/2
0.3
Density

0.2

π 0
0.1

3/2π
0.0

−3 −2 −1 0 1 2 3

Angle µ=0.3π, ν=0.95π, r=0.7, κ=2.3




3.0

3.0
0.8

2.0

2.0
κ

κ
r

0.4

1.0

1.0
0.0

0.0

0.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

ν ν r

Unimodality(white)/bimodality(yellow) of the KJ density.




3.0

3.0
0.8

2.0

2.0
κ

κ
r

0.4

1.0

1.0
0.0

0.0

0.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

ν ν r


! The portion of the parameter space originating a bimodal
distribution is appreciably smaller than in the GvM case




3.0

3.0
0.8

2.0

2.0
κ

κ
r

0.4

1.0

1.0
0.0

0.0

0.0
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

ν ν r


! The portion of the parameter space originating a bimodal
distribution is appreciably smaller than in the GvM case
→ better for general applications.



Circle-circle regression

The most interesting property of the KJ distribution is its role in
circular regression.




The considered regression model [Downs & Mardia(2002)] is
xj + β1
Yj = β0 ¯ εj , xj ∈ Ω,
β1 xj + 1
with β0 ∈ Ω, β1 ∈ C and {arg(εj )} independent angular errors.




xj + β1
Yj = β0 ¯ εj , xj ∈ Ω,
β1 xj + 1
The use of the KJ distribution for circular errors is a general
extention of the model with vM and the wC distributions, in use
untill now [Downs & Mardia(2002), Kato et al.(2008)].




xj + β1
Yj = β0 ¯ εj , xj ∈ Ω,
β1 xj + 1
The use of the KJ distribution for circular errors is a general
extention of the model with vM and the wC distributions, in use
untill now [Downs & Mardia(2002), Kato et al.(2008)].
Since both the the regression curve and the KJ distribution are
expressed in terms of M¨bius transformations this framework
o
seems very promising.



ν = 0 Symmetric and unimodal

π
ν=± Asymmetric and uni/bi-modal
2




The kurtosis varies, the skewness being ﬁxed

π
2




It includes the vM, wC and cU distributions

π
2




As for the JP distribution, µ is asymptotically independent of
ˆ
ˆ and κ
r ˆ

π
2




ˆ
ˆ and κ
r ˆ
A reparametrization (r , κ) → (s(r , κ), κ) is proposed which
reduces both the asymptotic correlation between ˆ and κ and
s ˆ
the asymptotic variance of κ.
ˆ
π
2




ˆ
ˆ and κ
r ˆ
s ˆ
ˆ
π
2
The skewness varies, the kurtosis being ﬁxed




ˆ
ˆ and κ
r ˆ
s ˆ
ˆ
π
2
The skewness varies, the kurtosis being ﬁxed
Good performances in modelling real data



Comparison
Generality

Generality: in terms of known densities comprised as special cases.



Comparison
Generality


Special cases JP GvM KJ
von Mises • • •
circular Uniform • • •
wrapped Cauchy • ◦ •
cardioid • ◦ ◦
Cartwright’s power-of-cosine • ◦ ◦
•=Yes; ◦=No



Comparison
Generality


•=Yes; ◦=No

• The JP model is the most general



Comparison
Generality


•=Yes; ◦=No

• The vM distribution, which is the most important and widely
used one, belongs to all of the three models



Comparison
Generality


•=Yes; ◦=No

• The vM distribution, which is the most important and widely
used one, belongs to all of the three models
• The poorest family, in this sense, is the GvM model


Comparison
Data modelling

Very few empirical examples are available.
[Further work would be useful in future in this sense...]

JP [Jones & Pewsey(2005)]



Comparison
Data modelling


with symmetric data, JP distribution performs signiﬁcantly
better than the vM, ca and wC models



Comparison
Data modelling


its advantage is no more signiﬁcant in presence of heavy
tails, requiring a mixture model with a cU distribution



Comparison
Data modelling


KJ [Kato & Jones(2010)]



Comparison
Data modelling


with asymmetric data, the GvM model ﬁts better than
simpler distributions and the KJ model and its asymmetric
submodel are even better.



Comparison
Data modelling


with asymmetric data, the GvM model ﬁts better than
simpler distributions and the KJ model and its asymmetric
submodel are even better.
circular-circular regression: improvement in performances for
the KJ model w.r.t. its submodels, but no comparison with
other distributions



Comparison
Inferential aspects

• The GvM distribution belongs to the exponential family
⇒ if the MLE exists, then it is unique
⇒ even numerical solutions are very reliable



Comparison
Inferential aspects

Explicit estimates exists for some parameters in some cases



Comparison
Inferential aspects


• The two other models have no particularly good properties
in general



Comparison
Inferential aspects


• The two other models have no particularly good properties
in general

• The KJ distribution has a slight advantage in the
reparametrization (r , κ) → (s(r , κ), κ) useful in general to
reduce both the asymptotic correlation with and the
asymptotic variance of κ ˆ



Bibliography I

Downs, T. D. & Mardia, K. V. (2002).
Circular regression.
Biometrika 89, 683–697.
Fisher, N. I. (1993).
Statistical Analysis of Circular Data.
Cambridge: Cambridge University Press.
Gatto, R. & Jammalamadaka, S. R. (2007).
The generalized von Mises distribution.
Statistical Methodology 4, 341–353.
Jammalamadaka, S. R. & SenGupta, A. (2001).
Topics in circular statistics.
Singapore: World Scientiﬁc.



Bibliography II

Jones, M. C. & Pewsey, A. (2005).
A family of simmetric distributions on the circle.
J. Am. Statist. Assoc. 100, 1422–1428.
Kato, S. & Jones, M. C. (2010).
A family of distributions on the circle with links to, and
applications arising from, M¨bius transformation.
o
J. Am. Statist. Assoc. 105, 249–262.
Kato, S., Shimizu, K. & Shieh, G. S. (2008).
A circular-circular regression model.
Statistica Sinica 18, 633–645.



Bibliography III
Maksimov, V. M. (1967).
Necessary and suﬃcient conditions for the family of shifts of
probability distributions on the continuous bicompact groups.
Theoria Verojatna 12, 307–321.
Mardia, K. V. (1972).
Statistics of directional data.
London: Academic Press.
Mardia, K. V. & Jupp, P. E. (1999).
Directional statistics.
Chichester: Wiley.
Stephens, M. A. (1963).
Random walk on a circle.
Biometrika 50, 385–390.


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