SYMMETRIC UNIMODAL MODELS FOR DIRECTIONAL DATA MOTIVATED BY INVERSE STEREOGRAPHIC PROJECTION

Transcription

1 J. Japan Statist. Soc. Vol. 40 No SYMMETRIC UNIMODAL MODELS FOR DIRECTIONAL DATA MOTIVATED BY INVERSE STEREOGRAPHIC PROJECTION Toshihiro Abe*, Kunio Shimizu** and Arthur Pewsey*** In this paper, a modified inverse stereographic projection, from the real line to the circle, is used as the motivation for a means of resolving a discontinuity in the Minh Farnum family of circular distributions. A four-parameter family of symmetric unimodal distributions which extends both the Minh Farnum and Jones Pewsey families is proposed. The normalizing constant of the density can be expressed in terms of Appell s function or, equivalently, the Gauss hypergeometric function. Important special cases of the family are identified, expressions for its trigonometric moments are obtained, and methods for simulating random variates from it are described. Parameter estimation based on method of moments and maximum likelihood techniques is discussed, and the latter approach is used to fit the family of distributions to an illustrative data set. A further extension to a family of rotationally symmetric distributions on the sphere is briefly made. Key words and phrases: Appell s function, Gauss hypergeometric function, Jones Pewsey family, maximum likelihood estimation, Möbius transformation, von Mises distribution. 1. Introduction As the papers of Shimizu and Iida 00, Minh and Farnum 003 and Jones and Pewsey 005 attest, recent years have seen renewed interest in the development of flexible models for directional data distributed on the unit circle or sphere. Historically, four general approaches have been used to generate circular distributions. These are projection or offsetting, conditioning, wrapping and inverse stereographic projection. The reader is directed to the books by Mardia and Jupp 1999, Section 3 and Jammalamadaka and SenGupta 001, Section.1.1 for the details of these various constructions. In this paper, we consider families of distributions that are either directly generated, or are motivated, by the latter of these general techniques, inverse stereographic projection. Minh and Farnum 003 used inverse stereographic projection, or equivalently bilinear Möbius transformation, of distributions defined on the real line to induce distributions on the circle. Their construction based on this approach proceeds as follows. Let fx denote the probability density function pdf of the t-distribution on the real line with m a positive integer degrees of freedom, Received April 7, 009. Revised February 6, 010. Accepted April 7, 010. *School of Fundamental Science and Technology, Keio University, Yokohama, Japan. **Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan. ***Mathematics Department, Escuela Politécnica, University of Extremadura, Cáceres, Spain.

2 46 TOSHIHIRO ABE ET AL. i.e. 1.1 fx Γm +1/ πmγm/1 + x /m m+1/, <x<. Then, applying inverse stereographic projection, defined by the generally oneto-one mapping, 1. sin θ x u + v u + v tanθ/, π θ<π, 1 + cos θ with u 0 and v m, and writing m n + 1, for n 0, 1,..., leads to the pdf on the circle of radius v, Γn fθ n+1 πγn +1/ 1 + cos θn. Given its construction, the distribution with this density might be referred to as a type of t-distribution on the circle, or the circular t-distribution induced by inverse stereographic projection if one were being more precise. However, the distribution with density 1.3 already has a name in the literature, namely Cartwright s power-of-cosine distribution. When n 0 i.e. m 1, the circular uniform distribution is obtained, clearly induced by inverse stereographic projection of the Cauchy distribution on the real line. The three-parameter family of symmetric circular distributions proposed by Jones and Pewsey 005 has density 1.4 fθ coshκψ1/ψ 1 + tanhκψ cosθ µ 1/ψ, πp 1/ψ coshκψ where π µ<πis a location parameter equal, in general, to the mean/modal/median direction, κ 0 is a concentration parameter, ψ R is a shape index, and P 1/ψ is the associated Legendre function of the first kind of degree 1/ψ and order 0. The family with density 1.4 is an important one because it includes most of the standard symmetric circular distributions. Specifically, it contains the circular uniform κ 0,orψ ± and κ finite, von Mises ψ 0, wrapped Cauchy ψ 1 and cardioid ψ 1 distributions, as well as Minh and Farnum s 003 or, equivalently, Cartwright s power-of-cosine distribution, with the range of n now extended to include the whole of the positive real line, ψ >0,κ. Due to the relation that exists between the associated Legendre function and the Gauss hypergeometric function a combination of equations and of Gradshteyn and Ryzhik 007, pp. 970 and 1009, namely 1.5 P 1/ψ z z 1/ψ F 1 1/ψ, 1 1/ψ/; 1; z 1/z, 1.4 can be re-expressed as fθ 1 + tanhκψ cosθ µ 1/ψ π F 1 1/ψ, 1 1/ψ/; 1, tanh κψ.

3 INVERSE STEREOGRAPHIC PROJECTION 47 Here the Gauss hypergeometric function is defined as or, alternatively, as where F 1 a, b; c; z F 1 a, b; c; z 1 1 t b 1 1 t c b 1 Bb, c b 0 1 tz a dt r0 a r b r z r, z < 1, c r r! { aa +1 a + r 1, r 1, a r 1, r 0, is referred to as Pochhammer s symbol. Note that in 1.5 the value of F 1 is positive if ψ<0 from the definition of the Gauss hypergeometric function, and is also positive if ψ>0 due to the transformation formula Abramowitz and Stegun 197, , p. 559 F 1 a, b; c; z 1 z c a b F 1 c a, c b; c; z. Making use of the substitutions 1/ψ 1+n/ and tanhκψ κ n ρ r/n, where n is a positive integer, ρ>0, r>0 and κ n ρ r ρr/{1+ρ + r /n}, this last representation of the density of the Jones Pewsey distribution can be seen to be closely related to the pdf of the t-distribution on the circle proposed by Shimizu and Iida 00, which is given by fθ {1 /nκ n ρ r cosθ µ} n/ 1 π F 1 n/4+1/,n/4 + 1; 1; κ n ρ r/n. Thus, the Jones Pewsey family also includes the t-distribution on the circle of Shimizu and Iida 00, covered as it is by ψ-values in the range 1, 0. The distribution with density 1.3 lacks continuity, in the sense that 1.1 tends to the standard normal pdf as m but a degenerate distribution concentrated at the point θ 0 is obtained when n in 1.3. In order to resolve this problem we employ the transformation 1. with u 0 and v>0 independent of n. A geometric representation of this transformation is portrayed in Figure 1. Now inverse stereographic projection of the t-distribution on the real line yields a different circular t-distribution from that given in 1.3, with density 1.6 fθ v mbm/, 1/ 1 + tan θ/ 1 + v tan θ//m m+1/, where m n + 1 and Bp, q denotes the beta function, defined by Bp, q 1 0 tp 1 1 t q 1 dt. In this paper, we shall refer to the distribution with density 1.6 as the modified Minh Farnum distribution. Asm, the pdf in 1.6 tends to fθ v θ 1 + tan exp v θ 1.7 π tan.

4 48 TOSHIHIRO ABE ET AL. a b Figure 1. Graphical representation of the stereographic projection x v tanθ/ for a 0 <θ<π/; b π/ <θ<π. This is the density of the distribution obtained when the modified inverse stereographic projection described above is applied to the standard normal distribution on the real line. Thus, the transformation 1. with v>0independent of n has continuity from the t to the normal distributions on the circle. The density 1.7 is unimodal if v, and is bimodal if v<. An alternative representation of 1.6 is 1.8 fθ v 0 Bn +1/, 1/1 + v0 1 + cos θn n+1 1 z cos θ n+1, where v 0 v/ n + 1 and z v0 1/v Finally, we note that the Jones Pewsey density tends to that of a Minh Farnum distribution when κ. The remainder of the paper is structured as follows. In Section we propose a new family of circular distributions which extends both the Jones Pewsey and unimodal modified Minh Farnum distributions. The basic properties of this new family, such as its unimodality, special cases and trigonometric moments, are considered in Section 3. We also describe how random variates can be simulated from it using the inversion and acceptance-rejection methods. Section 4 addresses the estimation of the distribution s parameters. There we discuss the use of the method of moments and maximum likelihood approaches. Maximum likelihood techniques are then employed to fit the family to an illustrative data set in Section 5. A form of stereographic projection which can be used to generate distributions on the sphere of arbitrary dimension is introduced in Section 6. The paper concludes, in Section 7, with a p-dimensional extension of the new family of circular distributions proposed in Section.. A new extended family of unimodal symmetric circular distributions In this section we introduce the new family of unimodal symmetric distributions on the circle which extends both the Jones Pewsey family of distributions as well as the unimodal distributions contained within the modified Minh Farnum family. Its density results on combining the defining structure of the pdf in 1.8,

5 INVERSE STEREOGRAPHIC PROJECTION 49 a b c d Figure. Density plots of.1 for µ 0 and: a ψ 0.5, κ 0; b ψ 0.5, κ 1; c ψ 1,κ 0; d ψ 1,κ 1. The six densities plotted in each panel are those for κ 1,, 1.5, 1, 0.5, 0 in order of decreasing height at 0. The densities for κ 1 0, in each panel are the densities for the bounding Jones Pewsey and modified Minh Farnum distributions, respectively. The horizontal lines in panels a and c delimit the density of the circular uniform distribution. Note that the vertical scales vary between panels. derived by inverse stereographic projection, with the transformation technique based on the hyperbolic tangent function employed in the construction of Jones and Pewsey 005. The resulting density is given by.1 fθ; µ, κ 1,κ,ψC ψ κ 1,κ 1 + tanhκ 1ψ cosθ µ 1/ψ 1 tanhκ ψ cosθ µ 1/ψ+1, where π µ < πis a location parameter which, in general, corresponds to the distribution s mean/modal/median direction, κ 1,κ 0 are concentration parameters and ψ>0is an index which also effects the peakedness and tails of the distribution. Note that fθ; µ, κ 1, κ,ψfθ; µ+π, κ 1,κ,ψif π µ<0, fθ; µ π, κ 1,κ,ψif0 µ<π. The overall shape of the distribution is determined by the last three of these parameters. The normalizing constant C ψ κ 1,κ is the reciprocal of the integral. π π 1 + tanhκ 1 ψ cos θ 1/ψ 1 tanhκ ψ cos θ 1 + tanhκ 1ψ 1/ψ 1 tanhκ ψ 1/ψ+1 1/ψ+1 dθ z 1 t 1/ψ 1 z t 1/ψ+1 t 1/ 1 t 1/ dt, where z 1 tanhκ 1 ψ/1+tanhκ 1 ψ and z tanhκ ψ/1 tanhκ ψ. Here, the transformation cos θ 1 t has been used to obtain the right-hand

6 50 TOSHIHIRO ABE ET AL. side of.. The last integral is expressible in terms of Appell s function, F 1, and the Gauss hypergeometric function, F 1,as 1 1 z 1 t 1/ψ 1 1 z t 1/ψ+1 t 1/ 1 t 1/ dt πf 1 ; 1 ψ, 1+ 1 ψ ;1;z 1,z 0 π1 z 1/ F 1 1, 1 ψ ;1;z 1 z 1 z due to the relation Appell and Kampé de Fériet 196, Eq. 8, p. 4 F 1 a; b 1,b ; b 1 + b ; z 1,z 1 z a F 1 a, b 1 ; b 1 + b ;z 1 z /1 z, where 1 1 t a 1 1 t c a 1 F 1 a; b 1,b ; c; z 1,z Ba, c a 0 1 z 1 t b 1 1 z t b dt. The flexibility of density.1 is illustrated in Figure for µ 0 and various combinations of the other three parameters. Note how, as ψ and κ increase, κ 1 has increasingly less effect on the shape of the density. 3. Basic properties 3.1. Unimodality Given that the two concentration parameters are constrained so that κ 1,κ 0, it follows that distributions with density.1 are always unimodal with fµ > fµ ± π, apart from the special case of the circular uniform distribution which, formally, has no mode. The circular uniform distribution is obtained when κ 1 κ 0orψ and κ 1 and κ are finite. To show that, for all other cases, distributions from the proposed family are unimodal with mode at µ, we set, without loss of generality, µ equal to 0 and consider the derivative of.1 with respect to θ, i.e. d dθ 1 + tanhκ1 ψ cos θ 1/ψ 1 tanhκ ψ cos θ 1/ψ+1 sin θ1 + tanhκ 1ψ cos θ 1/ψ 1 1 tanhκ ψ cos θ 1/ψ+ tanhκ 1 ψ tanhκ ψ cos θ ψ tanhκ ψ + tanhκ 1 ψ ψ tanhκ 1 ψ tanhκ ψ It follows from the constraints on the parameters that the density is unimodal if [1 + ψ tanhκ ψ + tanhκ 1 ψ]/[ψ tanhκ 1 ψ tanhκ ψ] > 1. Now, 1 + tanhκ 1 ψ 1/ψ > 1 tanhκ 1 ψ 1/ψ and 1 tanhκ ψ 1/ψ+1 < 1 + tanhκ ψ 1/ψ+1. Thus, 1 + tanhκ 1 ψ 1/ψ /1 tanhκ ψ 1/ψ+1 > 1 tanhκ 1 ψ 1/ψ /1 + tanhκ ψ 1/ψ+1. This implies that f0 >f±π. In summary, then, distributions with density.1 are either circular uniform or, more generally, unimodal with mode at µ. The fact that they are also symmetric about µ follows directly from.1. We note that when κ 1,1/ψ m 1/, κ 1/ψ log m/v, m 1 and v>0, the unimodality condition reduces to that of the modified,.

7 INVERSE STEREOGRAPHIC PROJECTION 51 Minh Farnum distribution, namely [m1 v ]/[m v ] 1. In addition, as m, this becomes the unimodality condition of the circular standard normal distribution induced by modified inverse stereographic projection. 3.. Special cases In this subsection we consider four special cases of the new family. The first three cover the Jones Pewsey family and the unimodal distributions contained within the modified Minh Farnum family, whilst the last one corresponds to a distribution with a pole at 0. We will assume throughout, without loss of generality, that µ 0 in.1. Case 1. κ 0, the Jones Pewsey distribution with positive power 1/ψ. For this value of κ, the density becomes fθ;0,κ 1, 0,ψ Cκ 1, 01 + tanhκ 1 ψ cos θ 1/ψ, where Cκ 1, 0 coshκ 1 ψ 1/ψ /[πp 1/ψ coshκ 1 ψ]. Case. κ 1 0, the Jones Pewsey distribution with negative power 1 + 1/ψ. For this choice of κ 1, the density can be written as fθ;0, 0,κ,ψ 1 tanhκ ψ 1/ψ+1 1 tanhκ ψ cos θ 1/ψ 1 π F 1 1/ψ +1, 1/; 1; z 1 tanhκ ψ cos θ 1/ψ 1 π F 1 1/ψ+1/, 1/ψ + 1; 1; tanh κ ψ, where the quadratic transformation formula Abramowitz and Stegun 197, , p. 560 F 1 a, b;b; z 1 z/ a F 1 a/, a +1/; b +1/; z z has been employed to obtain the second equality. Case 3. κ 1, the unimodal distributions contained in the modified Minh Farnum family with density 1.6. When κ 1, fθ;0,,κ,ψ C ψ,κ 1 + cos θ 1/ψ /1 tanhκ ψ cos θ 1/ψ+1, where 1 tanhκ ψ 1/ψ+1 C ψ,κ 1/ψ+1 B1/, 1/ψ +1/ F 1 1/ψ +1, 1/; 1/ψ +1;z 1 tanhκ ψ 1/ψ tanhκ ψ/ 1/ψ+1 B1/, 1/ψ +1/ 1 tanhκ ψ/. For this case note that f±π 0. Substituting, 1/ψ m 1/ and κ 1/ψ log m/v for m 1 and v>0, gives C ψ, log m/v m 1/ v mbm/, 1/1 + v /m m+1/. The density.1 can then be expressed as f θ;0,, m 1 m log, v m 1 v1 + tan θ/ mbm/, 1/1 + v tan θ//m m+1/,

8 5 TOSHIHIRO ABE ET AL. which is identical to that given previously in 1.6. Case 4. κ +, a new distribution with a pole at 0. When κ +, fθ;0,κ 1, +,ψc ψ κ 1, tanhκ 1 ψ cos θ 1/ψ /1 cos θ 1/ψ+1, where C ψ κ 1, tanhκ 1 ψ/ 1/ψ F 1 1/ψ, 1/ψ 1/; 1/ψ; z 1 B 1/ψ 1/, 1/ e κ1ψ [/1 tanhκ 1 ψ] 1/ψ Γ 1/ψ/[ πγ 1/ψ 1/]. Note that the ratio of gamma functions in this last expression is non-negative since Γ 1/ψ 1 n+1 Γn +1 1/ψ/[1/ψ1/ψ 1 1/ψ n] and Γ 1/ψ 1/ 1 n+1 Γn +1/ 1/ψ/[1/ψ +1/1/ψ 1/ 1/ψ + 1/ n], for n<1/ψ < n +1n, 3, 4,..., and hence that the normalizing constant is indeed non-negative Trigonometric moments As in the previous subsection, we also assume here that µ 0 in.1. The trigonometric moments of the distribution are given by {φ m : m 1,, 3,...}, where φ m α m + iβ m, with α m Ecos mθ and β m Esin mθ being the mth order cosine and sine moments of the random angle Θ, respectively. Because the distribution is symmetric about µ 0, it follows that the sine moments are all zero. Thus, φ m α m. To obtain the cosine moments, we will make use of the relationships that exist between them and the standard moments of cos Θ. The latter can be expressed as integral expressions involving Appell s function, F 1, or the Gauss hypergeometric function, F 1. Specifically, in terms of Appell s function, π Ecos m cos θ m 1 + tanhκ 1 ψ cos θ 1/ψ ΘC ψ κ 1,κ 0 1 tanhκ ψ cos θ 1/ψ+1 dθ 1 πf 1 1/, 1/ψ, 1+1/ψ;1,z 1,z m mc k k k0 F 1 k + 1, 1 ψ, 1+ 1 ψ ; k +1,z 1,z B k + 1, 1, for all m 0, 1,,..., and, in terms of the Gauss hypergeometric function, Ecos m Θ C ψκ 1,κ 1 + tanhκ 1 ψ 1/ψ 1 tanhκ ψ 1/ψ+1 m n0 k0 F ψ,n+ k + 1 ; n + k +1;z 1 ψ n k z n 1 n! B n + k + 1, 1 Using these expressions, the first three cosine moments of Θ are given by α 1 1 z 1/ π F 1 1/, 1/ψ;1;z 1 z /1 z 1 1C k k k0 mc k.

9 INVERSE STEREOGRAPHIC PROJECTION 53 F 1 k + 1, 1 ψ, 1+ 1 ψ ; k +1,z 1,z B k + 1, z 1/ F 1 3/, 1/ψ, 1+1/ψ;,z 1,z, F 1 1/, 1/ψ;1;z 1 z /1 z α Ecos Θ 1 1 z 1/ C k k π F 1 1/, 1/ψ;1;z 1 z /1 z k0 F 1 k + 1, 1 ψ, 1+ 1 ψ ; k +1,z 1,z B k + 1, z 1/ F 1 3/, 1/ψ, 1+1/ψ;,z 1,z F 1 1/, 1/ψ;1;z 1 z /1 z + 31 z 1/ F 1 5/, 1/ψ, 1+1/ψ;3,z 1,z F 1 1/, 1/ψ;1;z 1 z /1 z and α 3 4Ecos 3 Θ 3Ecos Θ 41 z 1/ 3 3C k k π F 1 1/, 1/ψ;1;z 1 z /1 z k0 F 1 k + 1, 1 ψ, 1+ 1 ψ ; k +1,z 1,z B k + 1, z 1/ F 1 3/, 1/ψ, 1+1/ψ;,z 1,z F 1 1/, 1/ψ;1;z 1 z /1 z 1 91 z 1/ F 1 3/, 1/ψ, 1+1/ψ;,z 1,z F 1 1/, 1/ψ;1;z 1 z /1 z C k k 1 z 1/ F 1 k +1/, 1/ψ, 1+1/ψ; k +1,z 1,z k F 1 1/, 1/ψ;1;z 1 z /1 z B k + 1, 1. Higher-order moments can be obtained similarly Simulation Lacking any obvious direct construction which leads to the distribution with density.1, here we present two approaches to simulating pseudo-random variates from it based on the more generally applicable inversion and acceptancerejection methods. Applying the inversion method, if U is a uniform random number on 0,1 then, in order to generate a variate on π, π from the distribution with parameters µ 0,κ 1, κ and ψ, one is required to solve numerically for θ in the equation F θ U, where F θ θ π fφdφ is the value of the distribution function at θ. Trivially, the equation required to be solved can be re-expressed as F θ U 0 and thus numerical methods of root finding can be used to solve

10 54 TOSHIHIRO ABE ET AL. for θ. For instance, the approximation to the root at the n + 1st iteration of the Newton Raphson method is given by θ n+1 θ n F θ n U/fθ n. As the distribution is symmetric about µ 0, an initial value for the root, θ 0, can be chosen on the basis of whether U<1/ θ 0 π, 0 or U>1/ θ 0 0,π. Of course, if U 1/ then θ 0. From our experience, the Newton Raphson method works well, except for when the density is very peaked. Other standard numerical methods for obtaining the roots of non-linear equations which do not make use of derivatives are more reliable in such circumstances. For cases of the distribution which are not overly peaked, an approach based on the acceptance-rejection method will generally be more efficient than the one described above. In its crudest form, such an approach requires the generation of two uniform random numbers: U 1 on π, π, and U on 0,f m, where f m C ψ κ 1,κ 1+tanhκ 1 ψ 1/ψ /1 tanhκ ψ 1/ψ+1 is the modal value of the density. If U >fu 1 then U 1 is rejected. Otherwise it is accepted as a random variate from the distribution. For fixed values of κ 1, κ and ψ, this approach only requires the initial computation of f m and, for each acceptance-rejection step, the two uniform numbers U 1 and U and the value of fu 1. Whilst the computational effort required at each acceptance-rejection step is obviously less than that for the inversion method, the overall efficiency of this approach will depend on the acceptance rate, 1/πf m. At the expense of increasing the computational effort, the acceptance rate can be increased by using a non-rectangular envelope which more closely bounds the target density. For the simulation of variates from cases of the distribution with non-zero values of µ, one simply follows either of the recipes described above to obtain θ and then takes θ + µ mod π. 4. Parameter estimation Our extended family of distributions with density.1 has four parameters which, in general, will all require estimation. Let θ 1,...,θ n be an independent and identically distributed sample drawn from the distribution with density.1. In this section we will consider the estimation of the four parameters based on the method of moments and maximum likelihood approaches Method of moments In principle, method of moments MM estimation would proceed by equating the sample mean direction θ with µ, the first three sample cosine moments about θ, i.e. ā 1,ā and ā 3, where ā m n 1 n i1 cosmθ i θ, with the expressions for α 1, α and α 3 given in Subsection 3.3, and solving for µ, κ 1, κ and ψ. Whilst, from the first of these equations, the MM estimate of µ is clearly µ θ, the MM estimates of the other three parameters would involve the solution of a system of three highly complicated non-linear simultaneous equations in κ 1, κ and ψ. For this reason, we do not feel encouraged to promote this particular approach to point estimation.

11 INVERSE STEREOGRAPHIC PROJECTION Maximum likelihood Using the results presented in Section, the log-likelihood function can be expressed as 4.1 lµ, κ 1,κ,ψ n logπ n ψ log1 + tanhκ 1ψ + n log1 + tanhκ ψ n log + n 1 F 1, 1 ψ ;1; 1 ψ + 1 log1 tanhκ ψ tanhκ 1 ψ + tanhκ ψ 1 + tanhκ 1 ψ1 + tanhκ ψ + 1 n log1 + tanhκ 1 ψ cosθ i µ ψ i n log1 tanhκ ψ cosθ i µ. ψ i1 Maximum likelihood ML point estimation then reduces to the constrained maximization of 4.1 over the points in the parameter space. The elements of the score vector are just the first-order partial derivatives of 4.1 with respect to each of the parameters. The expressions for some of these partial derivatives are rather involved, and so, with space restrictions in mind, we do not present them here. For those interested in deriving them, we would recommend the use of a symbolic mathematical computing package, such as Mathematica. In addition to its symbolic capabilities, we used Mathematica s NMaximize command to carry out the actual numerical optimization of 4.1. This command has four options which determine the method used to perform the maximization. These are: i the Nelder Mead simplex, ii simulated annealing, iii differential evolution and iv random search. The default option is the Nelder Mead simplex, which we found to be fairly robust. From our experience, NMaximize generally has no difficulty in homing in on the ML solution. However, for some of the samples that we analyzed, the log-likelihood surface was found to have multiple maxima. For others, the algorithm sometimes converged to a point on the boundary of the parameter space. Thus it is advisable to explore the parameter space carefully so as to try and ensure that the maximum likelihood solution really is identified. This can be achieved by using a wide range of different initial values from which to start the iterative search off from. Standard asymptotic theory applies for interior points of the parameter space, but not on its boundaries. The calculations involved in obtaining the second-order partial derivatives of the log-likelihood function required for the observed and expected information matrices are yet more involved than those for the first-order partial derivatives and so, again, we would recommend the use of a symbolic mathematical computing package to obtain them. Denoting the expected values of minus the second derivatives of the log-likelihood by i µµ, i µκ1,

12 56 TOSHIHIRO ABE ET AL. etc., it transpires that i µκ1 i µκ i µψ 0. Thus, the maximum likelihood estimate of the location parameter µ is asymptotically independent of those for the other three parameters. However, the other elements of the expected information matrix are generally non-zero. Estimates of the asymptotic covariance matrix for the ML estimates can be obtained by inverting the information matrices evaluated at the maximum likelihood estimates. These matrices can then be used together with standard asymptotic normal theory for ML estimators to calculate confidence intervals and regions for the parameters. Alternatively, profile likelihood methods together with standard asymptotic chi-squared theory for the likelihood ratio test can be employed. The latter requires only minor modifications to the numerical routines used to obtain the ML estimates and so will generally be less computationally demanding. However, the decision as to which approach to use will generally depend on the software available to the user. 5. An example As our illustrative example, we present an analysis of the same grouped data set of n 714 observations as was considered in Jones and Pewsey 005. The data consist of the vanishing angles of non-migratory British mallard ducks taken from Table 1 of Mardia and Jupp As reported in Jones and Pewsey 005, the test of Pewsey 00 provides no evidence against circular reflective symmetry for these data p-value The results obtained from fitting the new family of distributions and the Jones Pewsey and modified Minh Farnum submodels are presented in Table 1. A histogram of the data together with the densities for the three fits are presented in Figure 3. From the latter it can be seen that the density of the fit for the full family is bounded above and below by the densities of the fits for the two limiting submodels. Moreover, there is evidence, particularly around the mode, that the fit for the full family is more similar to the fit for the modified Minh Farnum distribution than that for the Jones Pewsey. This visual impression is confirmed by the results for the maximum values of the log-likelihood presented in Table 1. The maximum likelihood solution for the full family is an interior point of the parameter space, and so applying standard asymptotic distribution theory for the likelihood ratio test, there is some evidence that it offers a significant improvement in fit over the Jones Pewsey family p-value 0.070, but not over the modified Minh Farnum family p-value Focusing on the results for the two model selection criteria included in Table 1, the AIC identifies the full family as providing a slightly better fit than the modified Minh Farnum submodel. The BIC penalizes parameter-heavy models more and identifies both the modified Minh-Farnum and Jones-Pewsey threeparameter submodels as providing better fits than the full four-parameter family. Chi-squared goodness-of-fit tests at the 5% significance level marginally rejected the fit of the Jones Pewsey submodel p-value 0.047, but did not reject the fit of the modified Minh Farnum submodel p-value 0.10 nor that of the full family p-value Again, these p-values provide more evidence of the

13 INVERSE STEREOGRAPHIC PROJECTION 57 Figure 3. Histogram of the vanishing angles of 714 British mallard ducks together with the maximum likelihood fits for the proposed model solid and the modified Minh Farnum dashed and Jones Pewsey dot-dashed distributions. The data and densities are plotted on approximately ˆµ π, ˆµ + π. Table 1. Maximum likelihood estimates for three fits to the duck data, and the corresponding maximized log-likelihood MLL, AIC and BIC values. Distribution ˆκ 1 ˆκ ˆµ ˆψ MLL AIC BIC Jones Pewsey Proposed Model Modified Minh Farnum superior fit of the modified Minh Farnum submodel for these data. Finally, approximate 95% confidence intervals for µ, κ 1, κ and ψ, obtained from their respective profile log-likelihood functions, are 0.87, 0.77, 0,, 1.35, 0.8 and.35,.40. Of course, the interpretation of these confidence intervals is hampered by the fact that they only provide information about each individual parameter, not their relations with the others. Nevertheless, the intervals for µ, κ and ψ give fairly tight information regarding the probable values for these three parameters. The interval for κ 1 simply indicates that models across the whole range from the Jones Pewsey to the modified Minh Farnum are potential generators of the data. Obviously, confidence regions for different combinations of the parameters would provide insight as to the probable joint values of the parameters. 6. A p-dimensional version of stereographic projection A three-dimensional version of stereographic projection which preserves angles is mentioned by Watson However, when applying it, the Jacobian does not take a simple form. Thus, here we propose a projection which maps the exterior point x R p 1 into the southern hemisphere of the p-dimensional sphere of radius v, Sp, v and the interior point x R p 1 into the northern hemisphere of

14 58 TOSHIHIRO ABE ET AL. a b Figure 4. The p-dimensional generalization of stereographic projection from the point x in R p 1 to the point ξ on the surface of the sphere in R p for a 0 <θ p <π/; b π/ <θ p <π. S v p. As such, this projection is somewhat different from that of Watson. Figure 4 provides a graphical representation of the proposed projection. Let p 3, v > 0, and S v p : {ξ R p ; ξ v}. For π φ π and 0 θ i π i 1,...,p, we consider the polar coordinates 6.1 ξ 1 v cos φ sin θ 1 sin θ p ξ v sin φ sin θ 1 sin θ p... ξ p 1 v cos θ p 3 sin θ p ξ p v cos θ p Then, for any x x 1,...,x p 1 R p 1, there exists a unique point ξ ξ 1,...,ξ p S v p such that vξ1 x 1,...,x p 1,..., vξ p 1 v + ξ p v + ξ p v cos φ sin θ1...sin θ p,..., v cos θ p 3 sin θ p, 1 + cos θ p 1 + cos θ p where ξ p v. The Jacobian is x 1,x,...,x p 1 φ, θ 1,...,θ p v sinp θ p x 1,x,...,x p cos θ p p 1 v, φ, θ 1,...,θ p 3 p 1 v sin θ 1 sin p θ p. 1 + cos θ p This leads to the following proposition. Proposition 6.1. For i 1,...,p, suppose φ and θ i are randomly chosen angles in the intervals π, π and 0,π, respectively. Let gφ, θ 1,...,θ p

15 INVERSE STEREOGRAPHIC PROJECTION 59 denote the pdf of φ, θ 1,...,θ p. Then 6.1 is a random point on the sphere of radius v. Letx x 1,...,x p 1 R p 1 and fx denote the pdf of x. Then, g can be written in terms of f as x 1,x,...,x p 1 gφ, θ 1,...,θ p fx φ, θ 1,...,θ p v cos φ sin θ1 sin θ p f,..., v cos θ p 3 sin θ p J, 1 + cos θ p 1 + cos θ p where p 1 v J sin θ 1 sin p θ p. 1 + cos θ p We note that, when p, the projection proposed above is equivalent to that in Theorem 1 of Minh and Farnum The p-dimensional generalization of density.1 In this section, we derive the p-dimensional generalization of the pdf.1, namely 7.1 gφ, θ 1,...,θ p C ψ,p κ 1,κ 1 + tanhκ 1ψ cosθ p µ 1/ψ+1 p/ 1 tanhκ ψ cosθ p µ 1/ψ+p/ p i1 sin i θ i, where ψ>0, κ 1, κ 0, p 3, and the normalizing constant C ψ,p κ 1,κ is the reciprocal of π π π 0 π 1 + tanhκ 1 ψ cos θ p 1+1/ψ p/ 0 1 tanhκ ψ cos θ p 1/ψ+p/ p 1 + tanhκ 1 ψ 1+1/ψ p/ ω p 1 1 tanhκ ψ 1/ψ+p/ B F 1 p 1 ; 1+ 1 ψ p p i1 p 1, p 1, 1 ψ + p,p 1; z 1; z p 1 + tanhκ 1 ψ 1+1/ψ p/ ω p 1 1 tanhκ ψ 1/ψ+p/ 1 z p 1/ B p 1 F 1 1+, 1 ψ p ; p 1; z 1 z, 1 z sin i θ i dθ 1 dθ p dφ p 1, p 1 where ω p 1 is the surface area of the unit sphere in R p 1, given by ω p 1 π p 1/ /Γp 1/. This reduces to a simpler form when p 3, namely π π π tanhκ 1 ψ cos θ 1/+1/ψ sin θdθdφ 1 tanhκ ψ cos θ 3/+1/ψ

16 60 TOSHIHIRO ABE ET AL. 8π 1 + tanhκ 1ψ 1/+1/ψ 1 tanhκ ψ 3/+1/ψ F 1 1; 1 1 ψ, ψ ;;z 1,z 8π 1 + tanhκ 1ψ 1/+1/ψ 1 1 tanhκ ψ 3/+1/ψ 1/+1/ψz 1 z 1 1/+1/ψ z z In p 1-dimensional Euclidean space, R p 1, a multivariate Pearson type VII distribution has pdf Γν +p 1/ f p 1 x πa p 1/ 1+ x x p 1/ ν Γν A, where x R p 1, A 0, ν > 0. By an application of Proposition 6.1, the corresponding pdf gφ, θ 1,...,θ p onsp v is gφ, θ 1,...,θ p Γν +p 1/ Γν p 1/ ν πa 1+ v sin p 1/ θp J. A1 + cos θ p A special case of the Pearson type VII distribution, with ν n/ and A n, is the multivariate t-distribution with n degrees of freedom. When transformed onto the sphere using inverse stereographic projection, the resulting distribution has density 7. gφ, θ 1,...,θ p Γn + p 1/ Γn/ 1+ 1 v sin θp n 1 + cos θ p πn p 1/ n+p 1/ J. Note that as n, this pdf converges to a density on Sp v induced by the p 1-dimensional standard normal distribution, N0,I, namely { 1 gφ, θ 1,...,θ p exp v sin } θ p πp 1/ 1 + cos θ p J. Formally, p 3. However, if we set p, we obtain the normal distribution on the circle with density { v gθ 0 π 1/ 1 + cos θ 0 exp v sin } θ cos θ 0. Replacing θ 0 by θ, this density is equivalent to that in 1.7, which is the density of the circular normal induced by the modified inverse stereographic projection.

17 INVERSE STEREOGRAPHIC PROJECTION 61 An alternative form for 7. is gφ, θ 1,...,θ p Γn + p 1/v p cos θ p n p 1/ p Γn/πn p 1/ 1 + u 0 n+p 1/ 1 w cos θ p n+p 1/ sin i θ i, where u 0 v/ n and w u 0 1/u Transforming the two cosine components appearing in this last density using tanhκ 1 ψ and tanhκ ψ, respectively, and adjusting the normalizing constant appropriately, we obtain the density in 7.1. Thus, 7.1 can indeed be considered to be the p-dimensional generalization of the density in.1. Acknowledgements The work of Dr. Shimizu was supported in part by The Ministry of Education, Culture, Sports, Science and Technology of Japan under a Grant-in-Aid of the 1st Century Center of Excellence COE for Integrative Mathematical Sciences: Progress in Mathematics Motivated by Social and Natural Sciences. Dr. Pewsey would like to express his most sincere gratitude to the Department of Mathematics at Keio University for its warm hospitality and COE funding during the research visit which led to the production of this paper. References Abramowitz, M. and Stegun, I. A. Eds Handbook of Mathematical Functions, Dover, New York. Appell, P. and Kampé de Fériet, J Fonctions Hypergéométriques et Hypersphériques: Polynomes d Hermite, Gauthier Villars, Paris. Gradshteyn, I. S. and Ryzhik, I. M Table of Integrals, Series, and Products, seventh ed, Academic Press, London. Jammalamadaka, S. R. and SenGupta, A Topics in Circular Statistics, World Scientific, Singapore. Jones, M. C. and Pewsey, A A family of symmetric distributions on the circle, J. Amer. Stat. Assoc., 100, Mardia, K. V. and Jupp, P. E Directional Statistics, Wiley, Chichester. Minh, D. L. P. and Farnum, N. R Using bilinear transformations to induce probability distributions, Commun. Stat. Theory Meth., 3, 1 9. Pewsey, A. 00. Testing circular symmetry, Canadian J. Stat., 30, Shimizu, K. and Iida, K. 00. Pearson type VII distributions on spheres, Commun. Stat. Theory Meth., 31, Watson, G. S Statistics on Spheres, Wiley, New York. i1