Symmetric Unimodal Models For Directional Data Mot
Symmetric Unimodal Models For Directional Data Mot
Symmetric Unimodal Models For Directional Data Mot
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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 fam-
ilies 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 tech-
niques 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 (2002), Minh and Farnum (2003) and
Jones and Pewsey (2005) 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 circu-
lar 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 (2001, Section
2.1.1) 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 (2003) used inverse stereographic projection, or equiva-
lently 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 f (x) 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, 2009. Revised February 26, 2010. Accepted April 27, 2010.
*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, 10003 Cáceres, Spain.
46 TOSHIHIRO ABE ET AL.
i.e.
Γ((m + 1)/2)
(1.1) f (x) = √ , −∞ < x < ∞.
πmΓ(m/2)(1 + x2 /m)(m+1)/2
Then, applying inverse stereographic projection, defined by the (generally) one-
to-one mapping,
sin θ
(1.2) x=u+v = u + v tan(θ/2), −π ≤ θ < π,
1 + cos θ
√
with u = 0 and v = m, and writing m = 2n + 1, for n = 0, 1, . . ., leads to the
pdf on the circle of radius v,
Γ(n + 1)
(1.3) f (θ) = √ (1 + cos θ)n .
2n+1 πΓ(n + 1/2)
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 cir-
cular 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 (2005) has density
where
a(a + 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 2 F1 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 (1972), 15.3.3, p. 559) 2 F1 (a, b; c; z) = (1 − z)c−a−b 2 F1 (c − a, c − b; c; z).
Making use of the substitutions 1/ψ = −(1+n/2) and tanh(κψ) = −2κn (ρ | r)/n,
where n is a positive integer, ρ > 0, r > 0 and κn (ρ | r) = ρr/{1 + (ρ2 + r2 )/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 (2002), which is given by
v (1 + tan2 (θ/2))
(1.6) f (θ) = √ × ,
2 mB(m/2, 1/2) (1 + v 2 tan2 (θ/2)/m)(m+1)/2
(a) (b)
This is the density of the distribution obtained when the modified inverse stereo-
graphic projection described above is applied to the standard normal distribution
on the real line. Thus, the transformation (1.2) with v > 0 independent of n has
continuity from the t to the normal distributions
√ on the circle. √
The density (1.7) is unimodal if v ≥ 2, and is bimodal if v < 2. An
alternative representation of (1.6) is
v0 (1 + cos θ)n
(1.8) f (θ) = × ,
B(n + 1/2, 1/2)(1 + v02 )n+1 (1 − z cos θ)n+1
√
where v0 = v/ 2n + 1 and z = (v02 −1)/(v02 +1). 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 2 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 2.
(a) (b)
(c) (d)
Figure 2. Density plots of (2.1) for µ = 0 and: (a) ψ = 0.5, κ2 = 0; (b) ψ = 0.5, κ2 = 1;
(c) ψ = 1, κ2 = 0; (d) ψ = 1, κ2 = 1. The six densities plotted in each panel are those
for κ1 = ∞, 2, 1.5, 1, 0.5, 0 in order of decreasing height at 0. The densities for κ1 = 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.
3. Basic properties
3.1. Unimodality
Given that the two concentration parameters are constrained so that κ1 , κ2 ≥
0, it follows that distributions with density (2.1) are always unimodal with f (µ) >
f (µ ± π), apart from the special case of the circular uniform distribution which,
formally, has no mode. The circular uniform distribution is obtained when κ1 =
κ2 = 0 or ψ → ∞ and κ1 and κ2 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 (2.1) with respect to θ, i.e.
d (1 + tanh(κ1 ψ) cos θ)1/ψ
dθ (1 − tanh(κ2 ψ) cos θ)1/ψ+1
sin θ(1 + tanh(κ1 ψ) cos θ)1/ψ−1
=−
(1 − tanh(κ2 ψ) cos θ)1/ψ+2
(1 + ψ) tanh(κ2 ψ) + tanh(κ1 ψ)
× tanh(κ1 ψ) tanh(κ2 ψ) cos θ + .
ψ tanh(κ1 ψ) tanh(κ2 ψ)
It follows from the constraints on the parameters that the density is unimodal
if |[(1 + ψ) tanh(κ2 ψ) + tanh(κ1 ψ)]/[ψ tanh(κ1 ψ) tanh(κ2 ψ)]| > 1. Now, (1 +
tanh(κ1 ψ))1/ψ > (1 − tanh(κ1 ψ))1/ψ and (1 − tanh(κ2 ψ))1/ψ+1 <
(1 + tanh(κ2 ψ))1/ψ+1 . Thus, (1 + tanh(κ1 ψ))1/ψ /(1 − tanh(κ2 ψ))1/ψ+1 > (1 −
tanh(κ1 ψ))1/ψ /(1 + tanh(κ2 ψ))1/ψ+1 . This implies that f (0) > f (±π). In sum-
mary, then, distributions with density (2.1) are either circular uniform or, more
generally, unimodal with mode at µ. The fact that they are also symmetric about
µ follows directly from (2.1).
√
We note that when κ1 → ∞, 1/ψ = (m − 1)/2, κ2 = −1/ψ log( m/v),
m ≥ 1 and v > 0, the unimodality condition reduces to that of the modified
INVERSE STEREOGRAPHIC PROJECTION 51
Note that the ratio of gamma functions in this last expression is non-negative
since Γ(−1/ψ) = (−1)n+1 Γ(n + 1 − 1/ψ)/[(1/ψ)(1/ψ − 1) · · · (1/ψ − n)] and
Γ(−1/ψ − 1/2) = (−1)n+1 Γ(n + 1/2 − 1/ψ)/[(1/ψ + 1/2)(1/ψ − 1/2) · · · (1/ψ +
1/2 − n)], for n < 1/ψ < n + 1 (n = 2, 3, 4, . . .), and hence that the normalizing
constant is indeed non-negative.
(1 − z2 )1/2 1
k
α1 = 1 Ck (−2)
π 2 F1 (1/2, −1/ψ; 1; (z1 − z2 )/(1 − z2 )) k=0
INVERSE STEREOGRAPHIC PROJECTION 53
1 1 1 1 1
× F1 k + , − , 1 + ; k + 1, z1 , z2 B k + ,
2 ψ ψ 2 2
(1 − z2 ) F1 (3/2, −1/ψ, 1 + 1/ψ; 2, z1 , z2 )
1/2
= 1− ,
2 F1 (1/2, −1/ψ; 1; (z1 − z2 )/(1 − z2 ))
α2 = 2E(cos2 Θ) − 1
2(1 − z2 )1/2 2
k
= 2 Ck (−2)
π 2 F1 (1/2, −1/ψ; 1; (z1 − z2 )/(1 − z2 )) k=0
1 1 1 1 1
× F1 k + , − , 1 + ; k + 1, z1 , z2 B k + , −1
2 ψ ψ 2 2
4(1 − z2 ) F1 (3/2, −1/ψ, 1 + 1/ψ; 2, z1 , z2 )
1/2
= 1−
2 F1 (1/2, −1/ψ; 1; (z1 − z2 )/(1 − z2 ))
3(1 − z2 )1/2 F1 (5/2, −1/ψ, 1 + 1/ψ; 3, z1 , z2 )
+
2 F1 (1/2, −1/ψ; 1; (z1 − z2 )/(1 − z2 ))
and
α3 = 4E(cos3 Θ) − 3E(cos Θ)
4(1 − z2 )1/2 3
k
= 3 Ck (−2)
π 2 F1 (1/2, −1/ψ; 1; (z1 − z2 )/(1 − z2 )) k=0
1 1 1 1 1
× F1 k + , − , 1 + ; k + 1, z1 , z2 B k + ,
2 ψ ψ 2 2
(1 − z2 )1/2 F1 (3/2, −1/ψ, 1 + 1/ψ; 2, z1 , z2 )
−3 1−
2 F1 (1/2, −1/ψ; 1; (z1 − z2 )/(1 − z2 ))
3.4. Simulation
Lacking any obvious direct construction which leads to the distribution with
density (2.1), here we present two approaches to simulating pseudo-random vari-
ates from it based on the more generally applicable inversion and acceptance-
rejection 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 pa-
rameters µ = 0, κ1 , κ2 and ψ, one is required to solve numerically for θ in the
θ
equation F (θ) = U , where F (θ) = −π f (φ)dφ 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
54 TOSHIHIRO ABE ET AL.
for θ. For instance, the approximation to the root at the (n + 1)st iteration of
the Newton–Raphson method is given by θn+1 = θn − (F (θn ) − U )/f (θn ). As the
distribution is symmetric about µ = 0, an initial value for the root, θ0 , can be
chosen on the basis of whether U < 1/2 (θ0 ∈ (−π, 0)) or U > 1/2 (θ0 ∈ (0, π)).
Of course, if U = 1/2 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 genera-
tion of two uniform random numbers: U1 on (−π, π), and U2 on (0, fm ), where
fm = Cψ (κ1 , κ2 )(1+tanh(κ1 ψ))1/ψ /(1−tanh(κ2 ψ))1/ψ+1 is the modal value of the
density. If U2 > f (U1 ) then U1 is rejected. Otherwise it is accepted as a random
variate from the distribution. For fixed values of κ1 , κ2 and ψ, this approach only
requires the initial computation of fm and, for each acceptance-rejection step,
the two uniform numbers U1 and U2 and the value of f (U1 ). Whilst the computa-
tional 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/(2πfm ). 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 2π).
4. Parameter estimation
Our extended family of distributions with density (2.1) has four parameters
which, in general, will all require estimation. Let θ1 , . . . , θn be an independent
and identically distributed sample drawn from the distribution with density (2.1).
In this section we will consider the estimation of the four parameters based on
the method of moments and maximum likelihood approaches.
(4.1) l(µ, κ1 , κ2 , ψ)
n n
= −n log(2π) − log(1 + tanh(κ1 ψ)) + log(1 + tanh(κ2 ψ))
ψ 2
1 1 2(tanh(κ1 ψ) + tanh(κ2 ψ))
− n log 2 F1 , − ; 1;
2 ψ (1 + tanh(κ1 ψ))(1 + tanh(κ2 ψ))
1 1
+n + log(1 − tanh(κ2 ψ))
ψ 2
1 n
+ log(1 + tanh(κ1 ψ) cos(θi − µ))
ψ i=1
n
1
− 1+ log(1 − tanh(κ2 ψ) cos(θi − µ)).
ψ i=1
Maximum likelihood (ML) point estimation then reduces to the constrained max-
imization 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 iµµ , iµκ1 ,
56 TOSHIHIRO ABE ET AL.
etc., it transpires that iµκ1 = iµκ2 = iµψ = 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 infor-
mation matrix are generally non-zero. Estimates of the asymptotic covariance
matrix for the ML estimates can be obtained by inverting the information ma-
trices 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, pro-
file likelihood methods together with standard asymptotic chi-squared theory for
the likelihood ratio test can be employed. The latter requires only minor modi-
fications 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 (2005).
The data consist of the ‘vanishing angles’ of non-migratory British mallard ducks
taken from Table 1 of Mardia and Jupp (1999). As reported in Jones and Pewsey
(2005), the test of Pewsey (2002) provides no evidence against circular reflective
symmetry for these data (p-value = 0.124).
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 = 0.123).
Focusing on the results for the two model selection criteria included in Ta-
ble 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 three-
parameter submodels as providing better fits than the full four-parameter family.
Chi-squared goodness-of-fit tests at the 5% significance level marginally re-
jected 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 = 0.120) nor that of the
full family (p-value = 0.082). Again, these p-values provide more evidence of the
INVERSE STEREOGRAPHIC PROJECTION 57
Figure 3. Histogram of the vanishing angles of 714 British mallard ducks together with the max-
imum 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 approxi-
mately (µ̂ − π, µ̂ + π).
Table 1. Maximum likelihood estimates for three fits to the duck data, and the corresponding
maximized log-likelihood (MLL), AIC and BIC values.
(a) (b)
Figure 4. The p-dimensional generalization of stereographic projection from the point x in Rp−1
to the point ξ on the surface of the sphere in Rp for a) 0 < θp−2 < π/2; b) π/2 < θp−2 < π.
Spv . 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 Spv := {ξ ∈ Rp ; |ξ| = v}. For −π ≤ φ ≤ π and
0 ≤ θi ≤ π (i = 1, . . . , p − 2), we consider the polar coordinates
⎛ ⎞ ⎛ ⎞
ξ1 v cos φ sin θ1 · · · sin θp−2
⎜ ⎟ ⎜ ⎟
⎜ ξ2 ⎟ ⎜ v sin φ sin θ1 · · · sin θp−2 ⎟
⎜ ⎟ ⎜ ⎟
⎜ .. ⎟ ⎜ .. ⎟
(6.1) ⎜ ⎟=⎜ ⎟.
⎜ . ⎟ ⎜ . ⎟
⎜ ⎟ ⎜ ⎟
⎜ξ ⎟ ⎜ v cos θp−3 sin θp−2 ⎟
⎝ p−1 ⎠ ⎝ ⎠
ξp v cos θp−2
Then, for any x = (x1 , . . . , xp−1 ) ∈ Rp−1 , there exists a unique point ξ =
(ξ1 , . . . , ξp ) ∈ Spv such that
vξ1 vξp−1
(x1 , . . . , xp−1 ) = ,...,
v + ξp v + ξp
v cos φ sin θ1 . . . sin θp−2 v cos θp−3 sin θp−2
= ,..., ,
1 + cos θp−2 1 + cos θp−2
denote the pdf of (φ, θ1 , . . . , θp−2 ). Then (6.1) is a random point on the sphere
of radius v. Let x = (x1 , . . . , xp−1 ) ∈ Rp−1 and f (x) denote the pdf of x. Then,
g can be written in terms of f as
∂(x , x , . . . , x
1 2 p−1 )
g(φ, θ1 , . . . , θp−2 ) = f (x)
∂(φ, θ1 , . . . , θp−2 )
v cos φ sin θ1 · · · sin θp−2 v cos θp−3 sin θp−2
=f ,..., J,
1 + cos θp−2 1 + cos θp−2
where p−1
v
J= sin θ1 · · · sinp−2 θp−2 .
1 + cos θp−2
where ωp−1 is the surface area of the unit sphere in Rp−1 , given by ωp−1 =
2π (p−1)/2 /Γ((p − 1)/2). This reduces to a simpler form when p = 3, namely
π π
(1 + tanh(κ1 ψ) cos θ)−1/2+1/ψ
sin θdθdφ
−π 0 (1 − tanh(κ2 ψ) cos θ)3/2+1/ψ
60 TOSHIHIRO ABE ET AL.
(1 + tanh(κ1 ψ))−1/2+1/ψ 1 1 3 1
= 8π F1 1; − , + ; 2; z1 , z2
(1 − tanh(κ2 ψ))3/2+1/ψ 2 ψ 2 ψ
(1 + tanh(κ1 ψ))−1/2+1/ψ 1
= 8π
(1 − tanh(κ2 ψ))3/2+1/ψ (1/2 + 1/ψ)(z1 − z2 )
1/2+1/ψ
1 − z1
× 1− .
1 − z2
g(φ, θ1 , . . . , θp−2 )
⎛ 2 ⎞−(p−1)/2−ν
Γ(ν + (p − 1)/2) v sin θp−2
= (πA2 )−(p−1)/2 ⎝1 + ⎠ J.
Γ(ν) A(1 + cos θp−2 )
A special case of the Pearson type VII distribution, with ν = n/2 and A2 = 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
g(φ, θ1 , . . . , θp−2 )
(1 + cos θp−2 )(n−(p−1))/2
p−2
Γ((n + p − 1)/2)v p−1
= sini θi ,
Γ(n/2)(πn)(p−1)/2 (1 + u20 )(n+p−1)/2 (1 − w cos θp−2 )(n+p−1)/2 i=1
√
where u0 = v/ n and w = (u20 − 1)/(u20 + 1). Transforming the two cosine
components appearing in this last density using tanh(κ1 ψ) and tanh(κ2 ψ), re-
spectively, 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 (2.1).
Acknowledgements
The work of Dr. Shimizu was supported in part by The Ministry of Edu-
cation, Culture, Sports, Science and Technology of Japan under a Grant-in-Aid
of the 21st 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