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# Families Of Distributions On The Circle - A Review

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### Families Of Distributions On The Circle - A Review

1. 1. 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, 2010Families of distributions on the circle — A review 1/ 23
2. 2. Theory and Methods of Statistical Inference F.Rotolo 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 BibliographyFamilies of distributions on the circle — A review 2/ 23
3. 3. Theory and Methods of Statistical Inference F.Rotolo 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.)Families of distributions on the circle — A review 3/ 23
4. 4. Theory and Methods of Statistical Inference F.Rotolo 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.) 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)]Families of distributions on the circle — A review 3/ 23
5. 5. Theory and Methods of Statistical Inference F.Rotolo 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.) 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)Families of distributions on the circle — A review 3/ 23
6. 6. Theory and Methods of Statistical Inference F.Rotolo 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).Families of distributions on the circle — A review 4/ 23
7. 7. Theory and Methods of Statistical Inference F.Rotolo Introduction These simple circular distributions are symmetric and unimodal, so their ﬂexibility is quite limited.Families of distributions on the circle — A review 5/ 23
8. 8. Theory and Methods of Statistical Inference F.Rotolo Introduction These simple circular distributions are symmetric and unimodal, so their ﬂexibility is quite limited. ⇓ Recently some more general families of circular distributions have been proposed:Families of distributions on the circle — A review 5/ 23
9. 9. Theory and Methods of Statistical Inference F.Rotolo Introduction These simple circular distributions are symmetric and unimodal, so their ﬂexibility is quite limited. ⇓ 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)]Families of distributions on the circle — A review 5/ 23
10. 10. Theory and Methods of Statistical Inference F.Rotolo The Jones & Pewsey distribution The ﬁrst proposed family of circular distributions [Jones & Pewsey(2005)] is the three-parameter Jones & Pewsey distribution JP(µ, κ, ψ)Families of distributions on the circle — A review 6/ 23
11. 11. Theory and Methods of Statistical Inference F.Rotolo The Jones & Pewsey distribution 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.Families of distributions on the circle — A review 6/ 23
12. 12. Theory and Methods of Statistical Inference F.Rotolo The Jones & Pewsey distribution 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. All the vM, wC, ca, Cpc and cU distributions can be obtained as special cases of it.Families of distributions on the circle — A review 6/ 23
13. 13. Theory and Methods of Statistical Inference F.Rotolo The Jones & Pewsey distribution 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. 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.Families of distributions on the circle — A review 6/ 23
14. 14. Theory and Methods of Statistical Inference F.Rotolo The Jones & Pewsey distribution Properties The JP family is symmetric unimodal.Families of distributions on the circle — A review 7/ 23
15. 15. Theory and Methods of Statistical Inference F.Rotolo The Jones & Pewsey distribution Properties The JP family is symmetric unimodal. MLE: ˆ ˆ µ is asymptotically independent of (ψ, κ), ˆ ˆ ˆ no reparametrization is available to reduce corr(ψ, κ).Families of distributions on the circle — A review 7/ 23
16. 16. Theory and Methods of Statistical Inference F.Rotolo The Jones & Pewsey distribution Properties The JP family is symmetric unimodal. 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=−0.6Families of distributions on the circle — A review 7/ 23
17. 17. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution 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)]Families of distributions on the circle — A review 8/ 23
18. 18. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution 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.Families of distributions on the circle — A review 8/ 23
19. 19. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution Properties The GvM family can be asymmetric and it can have one or two maxima.Families of distributions on the circle — A review 9/ 23
20. 20. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution Properties The GvM family can be asymmetric and it can have one or two maxima. The skewness and the maxima location are mainly controlled by µ1 and µ2 ,Families of distributions on the circle — A review 9/ 23
21. 21. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution Properties The GvM family can be asymmetric and it can have one or two maxima. The skewness and the maxima location are mainly controlled by µ1 and µ2 , the kurtosis mostly by κ1 and κ2 .Families of distributions on the circle — A review 9/ 23
22. 22. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution Properties The GvM family can be asymmetric and it can have one or two maxima. The skewness and the maxima location are mainly controlled by 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=4.1Families of distributions on the circle — A review 9/ 23
23. 23. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises 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 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.Families of distributions on the circle — A review 10/ 23
24. 24. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises 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 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.Families of distributions on the circle — A review 10/ 23
25. 25. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises 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 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.Families of distributions on the circle — A review 10/ 23
26. 26. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises 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 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.Families of distributions on the circle — A review 10/ 23
27. 27. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution 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 )TFamilies of distributions on the circle — A review 11/ 23
28. 28. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution 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 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.Families of distributions on the circle — A review 11/ 23
29. 29. Theory and Methods of Statistical Inference F.Rotolo The Generalized von Mises distribution 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 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.Families of distributions on the circle — A review 11/ 23
30. 30. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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)] .Families of distributions on the circle — A review 12/ 23
31. 31. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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)] . 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.Families of distributions on the circle — A review 12/ 23
32. 32. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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)] . 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 =Families of distributions on the circle — A review 12/ 23
33. 33. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Properties The KJ distribution can be symmetric or asymmetric.Families of distributions on the circle — A review 13/ 23
34. 34. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Properties The KJ distribution can be symmetric or asymmetric. It includes the vM, the wC and the cU models as special cases.Families of distributions on the circle — A review 13/ 23
35. 35. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Properties 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 for unimodality are not straigthforward.Families of distributions on the circle — A review 13/ 23
36. 36. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Properties 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 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=2.3Families of distributions on the circle — A review 13/ 23
37. 37. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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. For each variable the value chosen for the graph where it is absent is shown by the grey lines.Families of distributions on the circle — A review 14/ 23
38. 38. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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. 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 bimodal distribution is appreciably smaller than in the GvM caseFamilies of distributions on the circle — A review 14/ 23
39. 39. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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. 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 bimodal distribution is appreciably smaller than in the GvM case → better for general applications.Families of distributions on the circle — A review 14/ 23
40. 40. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Circle-circle regression The most interesting property of the KJ distribution is its role in circular regression.Families of distributions on the circle — A review 15/ 23
41. 41. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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.Families of distributions on the circle — A review 15/ 23
42. 42. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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. 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)].Families of distributions on the circle — A review 15/ 23
43. 43. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution 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. 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.Families of distributions on the circle — A review 15/ 23
44. 44. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Two particular submodels ν = 0 Symmetric and unimodal πν=± Asymmetric and uni/bi-modal 2Families of distributions on the circle — A review 16/ 23
45. 45. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Two particular submodels ν = 0 Symmetric and unimodal The kurtosis varies, the skewness being ﬁxed πν=± Asymmetric and uni/bi-modal 2Families of distributions on the circle — A review 16/ 23
46. 46. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Two particular submodels ν = 0 Symmetric and unimodal The kurtosis varies, the skewness being ﬁxed It includes the vM, wC and cU distributions πν=± Asymmetric and uni/bi-modal 2Families of distributions on the circle — A review 16/ 23
47. 47. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Two particular submodels ν = 0 Symmetric and unimodal The kurtosis varies, the skewness being ﬁxed It includes the vM, wC and cU distributions As for the JP distribution, µ is asymptotically independent of ˆ ˆ and κ r ˆ πν=± Asymmetric and uni/bi-modal 2Families of distributions on the circle — A review 16/ 23
48. 48. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Two particular submodels ν = 0 Symmetric and unimodal The kurtosis varies, the skewness being ﬁxed It includes the vM, wC and cU distributions As for the JP distribution, µ is asymptotically independent of 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 πν=± Asymmetric and uni/bi-modal 2Families of distributions on the circle — A review 16/ 23
49. 49. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Two particular submodels ν = 0 Symmetric and unimodal The kurtosis varies, the skewness being ﬁxed It includes the vM, wC and cU distributions As for the JP distribution, µ is asymptotically independent of 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 πν=± Asymmetric and uni/bi-modal 2 The skewness varies, the kurtosis being ﬁxedFamilies of distributions on the circle — A review 16/ 23
50. 50. Theory and Methods of Statistical Inference F.Rotolo The Kato & Jones distribution Two particular submodels ν = 0 Symmetric and unimodal The kurtosis varies, the skewness being ﬁxed It includes the vM, wC and cU distributions As for the JP distribution, µ is asymptotically independent of 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 πν=± Asymmetric and uni/bi-modal 2 The skewness varies, the kurtosis being ﬁxed Good performances in modelling real dataFamilies of distributions on the circle — A review 16/ 23
51. 51. Theory and Methods of Statistical Inference F.Rotolo Comparison Generality Generality: in terms of known densities comprised as special cases.Families of distributions on the circle — A review 17/ 23
52. 52. Theory and Methods of Statistical Inference F.Rotolo Comparison Generality Generality: in terms of known densities comprised as special cases. Special cases JP GvM KJ von Mises • • • circular Uniform • • • wrapped Cauchy • ◦ • cardioid • ◦ ◦ Cartwright’s power-of-cosine • ◦ ◦ •=Yes; ◦=NoFamilies of distributions on the circle — A review 17/ 23
53. 53. Theory and Methods of Statistical Inference F.Rotolo Comparison Generality Generality: in terms of known densities comprised as special cases. Special cases JP GvM KJ von Mises • • • circular Uniform • • • wrapped Cauchy • ◦ • cardioid • ◦ ◦ Cartwright’s power-of-cosine • ◦ ◦ •=Yes; ◦=No • The JP model is the most generalFamilies of distributions on the circle — A review 17/ 23
54. 54. Theory and Methods of Statistical Inference F.Rotolo Comparison Generality Generality: in terms of known densities comprised as special cases. Special cases JP GvM KJ von 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 widely used one, belongs to all of the three modelsFamilies of distributions on the circle — A review 17/ 23
55. 55. Theory and Methods of Statistical Inference F.Rotolo Comparison Generality Generality: in terms of known densities comprised as special cases. Special cases JP GvM KJ von 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 widely used one, belongs to all of the three models • The poorest family, in this sense, is the GvM modelFamilies of distributions on the circle — A review 17/ 23
56. 56. Theory and Methods of Statistical Inference F.Rotolo Comparison Data modelling Very few empirical examples are available. [Further work would be useful in future in this sense...] JP [Jones & Pewsey(2005)]Families of distributions on the circle — A review 18/ 23
57. 57. Theory and Methods of Statistical Inference F.Rotolo Comparison Data modelling Very few empirical examples are available. [Further work would be useful in future in this sense...] JP [Jones & Pewsey(2005)] with symmetric data, JP distribution performs signiﬁcantly better than the vM, ca and wC modelsFamilies of distributions on the circle — A review 18/ 23
58. 58. Theory and Methods of Statistical Inference F.Rotolo Comparison Data modelling Very few empirical examples are available. [Further work would be useful in future in this sense...] JP [Jones & Pewsey(2005)] with symmetric data, JP distribution performs signiﬁcantly better than the vM, ca and wC models its advantage is no more signiﬁcant in presence of heavy tails, requiring a mixture model with a cU distributionFamilies of distributions on the circle — A review 18/ 23
59. 59. Theory and Methods of Statistical Inference F.Rotolo Comparison Data modelling Very few empirical examples are available. [Further work would be useful in future in this sense...] JP [Jones & Pewsey(2005)] with symmetric data, JP distribution performs signiﬁcantly better than the vM, ca and wC models its advantage is no more signiﬁcant in presence of heavy tails, requiring a mixture model with a cU distribution KJ [Kato & Jones(2010)]Families of distributions on the circle — A review 18/ 23
60. 60. Theory and Methods of Statistical Inference F.Rotolo Comparison Data modelling Very few empirical examples are available. [Further work would be useful in future in this sense...] JP [Jones & Pewsey(2005)] with symmetric data, JP distribution performs signiﬁcantly better than the vM, ca and wC models its advantage is no more signiﬁcant in presence of heavy tails, requiring a mixture model with a cU distribution KJ [Kato & Jones(2010)] with asymmetric data, the GvM model ﬁts better than simpler distributions and the KJ model and its asymmetric submodel are even better.Families of distributions on the circle — A review 18/ 23
61. 61. Theory and Methods of Statistical Inference F.Rotolo Comparison Data modelling Very few empirical examples are available. [Further work would be useful in future in this sense...] JP [Jones & Pewsey(2005)] with symmetric data, JP distribution performs signiﬁcantly better than the vM, ca and wC models its advantage is no more signiﬁcant in presence of heavy tails, requiring a mixture model with a cU distribution KJ [Kato & Jones(2010)] 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 distributionsFamilies of distributions on the circle — A review 18/ 23
62. 62. Theory and Methods of Statistical Inference F.Rotolo Comparison Inferential aspects • The GvM distribution belongs to the exponential family ⇒ if the MLE exists, then it is unique ⇒ even numerical solutions are very reliableFamilies of distributions on the circle — A review 19/ 23
63. 63. Theory and Methods of Statistical Inference F.Rotolo 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 Explicit estimates exists for some parameters in some casesFamilies of distributions on the circle — A review 19/ 23
64. 64. Theory and Methods of Statistical Inference F.Rotolo 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 Explicit estimates exists for some parameters in some cases • The two other models have no particularly good properties in generalFamilies of distributions on the circle — A review 19/ 23
65. 65. Theory and Methods of Statistical Inference F.Rotolo 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 Explicit estimates exists for some parameters in some cases • 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 ˆFamilies of distributions on the circle — A review 19/ 23
66. 66. Theory and Methods of Statistical Inference F.Rotolo 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.Families of distributions on the circle — A review 20/ 23
67. 67. Theory and Methods of Statistical Inference F.Rotolo 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.Families of distributions on the circle — A review 21/ 23
68. 68. Theory and Methods of Statistical Inference F.Rotolo 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.Families of distributions on the circle — A review 22/ 23