Families Of Distributions On The Circle - A Review

Theory and Methods of Statistical Inference F.Rotolo

Families of distributions on the circleA review

Federico [email protected]

Department of Statistical SciencesUniversity of Padua

September 7, 2010

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


Introduction

Families of Circular DistributionsThe Jones & Pewsey distributionThe Generalized von Mises distributionThe Kato & Jones distribution

Two particular submodels

ComparisonGeneralityData modellingInferential aspects

Bibliography



Introduction

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

Distributions on the real line are not suitable for direction, so newmodels 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











Introduction











IntroductionAn example

−3 −2 −1 0 1 2 3

0.0

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Den

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0

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π

3/2π

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.

⇓Recently some more general families of circular distributionshave been proposed:

• Jones & Pewsey [Jones & Pewsey(2005)]

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

• Kato & Jones [Kato & Jones(2010)]



Introduction








Introduction








The Jones & Pewsey distribution

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

with density

fJP(θ) =(cosh(κψ) + sinh(κψ) cos(θ − µ))1/ψ

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.

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

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




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




and order 0.










and order 0.










and order 0.





The Jones & Pewsey distributionProperties

The JP family is symmetric unimodal.

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

−3 −2 −1 0 1 2 3

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sity

µ=4.1, κ=1.8, ψ=−0.6

0

π/2

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3/2π






−3 −2 −1 0 1 2 3

0.0

0.2

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Den

sity

µ=4.1, κ=1.8, ψ=−0.6

0

π/2

π

3/2π






−3 −2 −1 0 1 2 3

0.0

0.2

0.4

0.6

Angle

Den

sity

µ=4.1, κ=1.8, ψ=−0.6

0

π/2

π

3/2π



The Generalized von Mises distribution

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

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

with density

fGvM(θ) =1

2πG0(δ, κ1, κ2)expκ1 cos(θ − µ1) + κ2 cos 2(θ − µ2)

0 ≤ θ < 2π, 0 ≤ µ1 < 2π, 0 ≤ µ2 < π, κ1, κ2 ≥ 0, δ = (µ1 − µ2)modπ, G0(δ, κ1, κ2) is the normalizing

constant.




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

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

with density

fGvM(θ) =1

2πG0(δ, κ1, κ2)expκ1 cos(θ − µ1) + κ2 cos 2(θ − µ2)

0 ≤ θ < 2π, 0 ≤ µ1 < 2π, 0 ≤ µ2 < π, κ1, κ2 ≥ 0, δ = (µ1 − µ2)modπ, G0(δ, κ1, κ2) is the normalizing

constant.



The Generalized von Mises distributionProperties

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

The skewness and the maxima location are mainly controlled byµ1 and µ2, the kurtosis mostly by κ1 and κ2.

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µ1=0.8π, µ2=π, κ1=4.8, κ2=4.1

0

π/2

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3/2π





The skewness and the maxima location are mainly controlled byµ1 and µ2,

the kurtosis mostly by κ1 and κ2.

−3 −2 −1 0 1 2 3

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µ1=0.8π, µ2=π, κ1=4.8, κ2=4.1

0

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3/2π






−3 −2 −1 0 1 2 3

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Den

sity

µ1=0.8π, µ2=π, κ1=4.8, κ2=4.1

0

π/2

π

3/2π






−3 −2 −1 0 1 2 3

0.0

0.2

0.4

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0.8

1.0

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Den

sity

µ1=0.8π, µ2=π, κ1=4.8, κ2=4.1

0

π/2

π

3/2π




0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

µ2

κ 1

0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

µ2

κ 2

0.0 1.0 2.0 3.0

0.0

1.0

2.0

3.0

κ1

κ 2

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.

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

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.




0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

µ2

κ 1

0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

µ2

κ 2

0.0 1.0 2.0 3.0

0.0

1.0

2.0

3.0

κ1

κ 2








0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

µ2

κ 1

0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

µ2

κ 2

0.0 1.0 2.0 3.0

0.0

1.0

2.0

3.0

κ1

κ 2








0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

µ2

κ 1

0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

µ2

κ 2

0.0 1.0 2.0 3.0

0.0

1.0

2.0

3.0

κ1

κ 2







The Generalized von Mises distributionMembership of the Exponential Family

The most interesting property of the GvM model is that thereparametrization

λ = (κ1 cosµ1, κ1 sinµ1, κ2 cos 2µ2, κ2 sin 2µ2)T

makes it possible to express the density as

fGvM(θ | λ) = expλTt(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,

‖λ(2)‖ = (λ3, λ4)T and δ = (arg λ(1) − arg λ(2)/2)modπ.

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























The Kato & Jones distributionThe four-parameter Kato & jones distribution KJ is obtained byapplying a Mobius transformation to a vM-distributed randomvariable [Kato & Jones(2010)] .

The Mobius transformation is a (closed under composition)circle-to-circle function Moµ,ν,r : Ξ 7→ Θ given by

e iΘ = e iµ e iΞ + re iν

re i(Ξ−ν) + 1,

with 0 ≤ µ, ν < 2π and 0 ≤ r < 1.

If Ξ ∼ vM(0, κ), then Θ = Moµ,ν,r (Ξ) ∼ KJ(µ, ν, r , κ) has density

fKJ(θ) =1− r2

2πI0(κ)

exp[κξ cos(θ−η)−2r cos ν

1+r2−2r cos(θ−γ)

]1 + r2 − 2r cos(θ − γ)

,

0 ≤ θ < 2π, ξ =p

r4 + 2r2 cos(2ν) + 1, η = µ + arg[r2cos(2ν) + i sin(2ν) + 1], γ = µ + ν.






re i(Ξ−ν) + 1,

with 0 ≤ µ, ν < 2π and 0 ≤ r < 1.


fKJ(θ) =1− r2

2πI0(κ)



]1 + r2 − 2r cos(θ − γ)

,

0 ≤ θ < 2π, ξ =p







re i(Ξ−ν) + 1,

with 0 ≤ µ, ν < 2π and 0 ≤ r < 1.


fKJ(θ) =1− r2

2πI0(κ)



]1 + r2 − 2r cos(θ − γ)

,

0 ≤ θ < 2π, ξ =p




The Kato & Jones distributionProperties

The KJ distribution can be symmetric or asymmetric.

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

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

−3 −2 −1 0 1 2 3

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µ=0.3π, ν=0.95π, r=0.7, κ=2.3

0

π/2

π

3/2π




The KJ distribution can be symmetric or asymmetric.It includes the vM, the wC and the cU models as special cases.


−3 −2 −1 0 1 2 3

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Den

sity

µ=0.3π, ν=0.95π, r=0.7, κ=2.3

0

π/2

π

3/2π






−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

Angle

Den

sity

µ=0.3π, ν=0.95π, r=0.7, κ=2.3

0

π/2

π

3/2π






−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

Angle

Den

sity

µ=0.3π, ν=0.95π, r=0.7, κ=2.3

0

π/2

π

3/2π



The Kato & Jones distribution

0 1 2 3 4 5 6

0.0

0.4

0.8

ν

r

0 1 2 3 4 5 60.

01.

02.

03.

0

ν

κ

0 1 2 3 4 5 6

0.0

1.0

2.0

3.0

r

κ

Unimodality(white)/bimodality(yellow) of the KJ density.For each variable the value chosen for the graph where it is absent is shown by the grey lines.

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

→ better for general applications.




0 1 2 3 4 5 6

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r

0 1 2 3 4 5 60.

01.

02.

03.

0

ν

κ

0 1 2 3 4 5 6

0.0

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r

κ







0 1 2 3 4 5 6

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0.8

ν

r

0 1 2 3 4 5 60.

01.

02.

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0

ν

κ

0 1 2 3 4 5 6

0.0

1.0

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r

κ






The Kato & Jones distributionCircle-circle regression

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

The considered regression model [Downs & Mardia(2002)] is

Yj = β0xj + β1

β1xj + 1εj , xj ∈ Ω,

with β0 ∈ Ω, β1 ∈ C and arg(εj) independent angular errors.

The use of the KJ distribution for circular errors is a generalextention of the model with vM and the wC distributions, in useuntill now [Downs & Mardia(2002), Kato et al.(2008)].

Since both the the regression curve and the KJ distribution areexpressed in terms of Mobius transformations this frameworkseems very promising.






Yj = β0xj + β1










Yj = β0xj + β1










Yj = β0xj + β1







The Kato & Jones distributionTwo particular submodels

ν = 0 Symmetric and unimodal

I The kurtosis varies, the skewness being fixedI It includes the vM, wC and cU distributionsI As for the JP distribution, µ is asymptotically independent of

r and κI A reparametrization (r , κ) 7→ (s(r , κ), κ) is proposed which

reduces both the asymptotic correlation between s and κ andthe asymptotic variance of κ.

ν = ±π2

Asymmetric and uni/bi-modal

I The skewness varies, the kurtosis being fixedI Good performances in modelling real data





I The kurtosis varies, the skewness being fixed

I It includes the vM, wC and cU distributionsI As for the JP distribution, µ is asymptotically independent of



ν = ±π2







I The kurtosis varies, the skewness being fixedI It includes the vM, wC and cU distributions

I As for the JP distribution, µ is asymptotically independent ofr and κ

I A reparametrization (r , κ) 7→ (s(r , κ), κ) is proposed whichreduces both the asymptotic correlation between s and κ andthe asymptotic variance of κ.

ν = ±π2








r and κ

I A reparametrization (r , κ) 7→ (s(r , κ), κ) is proposed whichreduces both the asymptotic correlation between s and κ andthe asymptotic variance of κ.

ν = ±π2










ν = ±π2










ν = ±π2


I The skewness varies, the kurtosis being fixed

I Good performances in modelling real data








ν = ±π2





ComparisonGenerality

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

Special cases JP GvM KJvon Mises • • •circular Uniform • • •wrapped Cauchy • •cardioid • Cartwright’s power-of-cosine •

•=Yes; =No

• The JP model is the most general

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

• The poorest family, in this sense, is the GvM model






•=Yes; =No









•=Yes; =No









•=Yes; =No









•=Yes; =No






ComparisonData modelling

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

JP [Jones & Pewsey(2005)]

I with symmetric data, JP distribution performs significantlybetter than the vM, ca and wC models

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

KJ [Kato & Jones(2010)]

I with asymmetric data, the GvM model fits better thansimpler distributions and the KJ model and its asymmetricsubmodel are even better.

I circular-circular regression: improvement in performances forthe KJ model w.r.t. its submodels, but no comparison withother distributions





JP [Jones & Pewsey(2005)]I with symmetric data, JP distribution performs significantly

better than the vM, ca and wC models

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









better than the vM, ca and wC modelsI its advantage is no more significant in presence of heavy

tails, requiring a mixture model with a cU distribution





















KJ [Kato & Jones(2010)]I with asymmetric data, the GvM model fits better than

simpler distributions and the KJ model and its asymmetricsubmodel are even better.









KJ [Kato & Jones(2010)]I with asymmetric data, the GvM model fits better than

simpler distributions and the KJ model and its asymmetricsubmodel are even better.




ComparisonInferential aspects

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

Explicit estimates exists for some parameters in some cases

• The two other models have no particularly good propertiesin general

• The KJ distribution has a slight advantage in thereparametrization (r , κ) 7→ (s(r , κ), κ) useful in general toreduce both the asymptotic correlation with and theasymptotic variance of κ
























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, andapplications arising from, Mobius transformation.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 ofprobability 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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Families Of Distributions On The Circle - A Review