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where
I
p
(
·
) represents the modified Bessel function of the first kind and order
p
, and is defined as (1)
I
p
(
κ
)=
k
κ
2
2
k
+
p
1
Γ(
p
+
k
+1)
k
!
,
≥
0
where Γ(
) is the well-known Gamma function.
The density
f
(
x
·
μ, κ
) is parameterized by the mean direction
μ
,andthe
concentration
parameter
κ
, so-called because it characterizes how strongly the
unit vectors drawn according to
f
(
x
|
μ, κ
) are concentrated about the mean
direction
μ
. Larger values of
κ
imply stronger concentration about the mean
direction. In particular when
κ
=0,
f
(
x
|
|
μ, κ
) reduces to the uniform density
d−
1
,andas
κ
on
μ, κ
) tends to a point density. The interested
reader is referred to (30), (24), or (21) for details on vMF distributions.
The vMF distribution is one of the simplest parametric distributions for
directional data, and has properties analogous to those of the multivariate
Gaussian distribution for data in
S
→∞
,
f
(
x
|
d
. For example, the maximum entropy
R
d−
1
subject to the constraint that
E
[
x
]isfixedisavMFdensity
density on
S
6.3.2 Maximum Likelihood Estimates
In this section we look briefly at maximum likelihood estimates for the
parameters of a single vMF distribution.
The detailed derivations can be
found in (5). Let
X
be a finite set of sample unit vectors drawn independently
following
f
(
x
|
μ, κ
) (6.1), i.e.,
d−
1
X
=
{
x
i
∈
S
|
x
i
drawn following
f
(
x
|
μ, κ
)for1
≤
i
≤
n
}
.
Given
we want to find maximum likelihood estimates for the parameters
μ
and
κ
of the distribution
f
(
x
X
μ, κ
). Assuming the
x
i
to be independent and
identically distributed, we can write the log-likelihood of
|
X
as
μ, κ
)=
n
ln
c
d
(
κ
)+
κμ
T
r
,
(6.3)
where
r
=
i
x
i
. To obtain the maximum likelihood estimates of
μ
and
κ
,
we have to maximize (6.3) subject to the constraints
μ
T
μ
=1and
κ
ln
P
(
X|
≥
0. A
simple calculation (5) shows that the MLE solutions
μ
and
κ
may be obtained
from the following equations:
i
=1
x
i
r
μ
=
=
i
=1
x
i
,
(6.4)
r
κ
)
I
d/
2
−
1
(
I
d/
2
(
κ
)
=
r
n
and
=
r.
(6.5)
Since computing
κ
involves an implicit equation (6.5) that is a ratio of Bessel
functions, it is not possible to obtain an analytic solution, and we have to
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