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Samples from the cosine variant of the bivariate von Mises distribution. The green points are sampled from a distribution with high concentration and high correlation (
κ
1
=
κ
2
=
200
{\displaystyle \kappa _{1}=\kappa _{2}=200}
,
κ
3
=
0
{\displaystyle \kappa _{3}=0}
), the blue points are sampled from a distribution with high concentration and negative correlation (
κ
1
=
κ
2
=
200
{\displaystyle \kappa _{1}=\kappa _{2}=200}
,
κ
3
=
100
{\displaystyle \kappa _{3}=100}
), and the red points are sampled from a distribution with low concentration and no correlation (
κ
1
=
κ
2
=
20
,
κ
3
=
0
{\displaystyle \kappa _{1}=\kappa _{2}=20,\kappa _{3}=0}
).
In probability theory and statistics , the bivariate von Mises distribution is a probability distribution describing values on a torus . It may be thought of as an analogue on the torus of the bivariate normal distribution . The distribution belongs to the field of directional statistics . The general bivariate von Mises distribution was first proposed by Kanti Mardia in 1975. One of its variants is today used in the field of bioinformatics to formulate a probabilistic model of protein structure in atomic detail.[1]
Definition
The bivariate von Mises distribution is a probability distribution defined on the torus ,
S
1
×
S
1
{\displaystyle S^{1}\times S^{1}}
in
R
3
{\displaystyle \mathbb {R} ^{3}}
.
The probability density function of the general bivariate von Mises distribution for the angles
ϕ
,
ψ
∈
[
0
,
2
π
]
{\displaystyle \phi ,\psi \in [0,2\pi ]}
is given by[2]
f
(
ϕ
,
ψ
)
∝
exp
[
κ
1
cos
(
ϕ
−
μ
)
+
κ
2
cos
(
ψ
−
ν
)
+
(
cos
(
ϕ
−
μ
)
,
sin
(
ϕ
−
μ
)
)
A
(
cos
(
ϕ
−
ν
)
,
sin
(
ϕ
−
ν
)
)
T
]
,
{\displaystyle f(\phi ,\psi )\propto \exp[\kappa _{1}\cos(\phi -\mu )+\kappa _{2}\cos(\psi -\nu )+(\cos(\phi -\mu ),\sin(\phi -\mu ))\mathbf {A} (\cos(\phi -\nu ),\sin(\phi -\nu ))^{T}],}
where
μ
{\displaystyle \mu }
and
ν
{\displaystyle \nu }
are the means for
ϕ
{\displaystyle \phi }
and
ψ
{\displaystyle \psi }
,
κ
1
{\displaystyle \kappa _{1}}
and
κ
2
{\displaystyle \kappa _{2}}
their concentration and the matrix
A
∈
M
(
2
,
2
)
{\displaystyle \mathbf {A} \in \mathbb {M} (2,2)}
is related to their correlation.
Two commonly used variants of the bivariate von Mises distribution are the sine and cosine variant.
The cosine variant of the bivariate von Mises distribution[1] has the probability density function
f
(
ϕ
,
ψ
)
=
Z
c
(
κ
1
,
κ
2
,
κ
3
)
exp
[
κ
1
cos
(
ϕ
−
μ
)
+
κ
2
cos
(
ψ
−
ν
)
−
κ
3
cos
(
ϕ
−
μ
−
ψ
+
ν
)
]
,
{\displaystyle f(\phi ,\psi )=Z_{c}(\kappa _{1},\kappa _{2},\kappa _{3})\ \exp[\kappa _{1}\cos(\phi -\mu )+\kappa _{2}\cos(\psi -\nu )-\kappa _{3}\cos(\phi -\mu -\psi +\nu )],}
where
μ
{\displaystyle \mu }
and
ν
{\displaystyle \nu }
are the means for
ϕ
{\displaystyle \phi }
and
ψ
{\displaystyle \psi }
,
κ
1
{\displaystyle \kappa _{1}}
and
κ
2
{\displaystyle \kappa _{2}}
their concentration and
κ
3
{\displaystyle \kappa _{3}}
is related to their correlation.
Z
c
{\displaystyle Z_{c}}
is the normalization constant.
The sine variant has the probability density function[3]
f
(
ϕ
,
ψ
)
=
Z
s
(
κ
1
,
κ
2
,
κ
3
)
exp
[
κ
1
cos
(
ϕ
−
μ
)
+
κ
2
cos
(
ψ
−
ν
)
+
κ
3
sin
(
ϕ
−
μ
)
sin
(
ψ
−
ν
)
]
,
{\displaystyle f(\phi ,\psi )=Z_{s}(\kappa _{1},\kappa _{2},\kappa _{3})\ \exp[\kappa _{1}\cos(\phi -\mu )+\kappa _{2}\cos(\psi -\nu )+\kappa _{3}\sin(\phi -\mu )\sin(\psi -\nu )],}
where the parameters have the same interpretation.
See also
References
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families