- Source: Projected normal distribution
In directional statistics, the projected normal distribution (also known as offset normal distribution, angular normal distribution or angular Gaussian distribution) is a probability distribution over directions that describes the radial projection of a random variable with n-variate normal distribution over the unit (n-1)-sphere.
Definition and properties
Given a random variable
X
∈
R
n
{\displaystyle {\boldsymbol {X}}\in \mathbb {R} ^{n}}
that follows a multivariate normal distribution
N
n
(
μ
,
Σ
)
{\displaystyle {\mathcal {N}}_{n}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}
, the projected normal distribution
P
N
n
(
μ
,
Σ
)
{\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}
represents the distribution of the random variable
Y
=
X
‖
X
‖
{\displaystyle {\boldsymbol {Y}}={\frac {\boldsymbol {X}}{\lVert {\boldsymbol {X}}\rVert }}}
obtained projecting
X
{\displaystyle {\boldsymbol {X}}}
over the unit sphere. In the general case, the projected normal distribution can be asymmetric and multimodal. In case
μ
{\displaystyle {\boldsymbol {\mu }}}
is orthogonal to an eigenvector of
Σ
{\displaystyle {\boldsymbol {\Sigma }}}
, the distribution is symmetric. The first version of such distribution was introduced in Pukkila and Rao (1988).
Density function
The density of the projected normal distribution
P
N
n
(
μ
,
Σ
)
{\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}
can be constructed from the density of its generator n-variate normal distribution
N
n
(
μ
,
Σ
)
{\displaystyle {\mathcal {N}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}
by re-parametrising to n-dimensional spherical coordinates and then integrating over the radial coordinate.
In spherical coordinates with radial component
r
∈
[
0
,
∞
)
{\displaystyle r\in [0,\infty )}
and angles
θ
=
(
θ
1
,
…
,
θ
n
−
1
)
∈
[
0
,
π
]
n
−
2
×
[
0
,
2
π
)
{\displaystyle {\boldsymbol {\theta }}=(\theta _{1},\dots ,\theta _{n-1})\in [0,\pi ]^{n-2}\times [0,2\pi )}
, a point
x
=
(
x
1
,
…
,
x
n
)
∈
R
n
{\displaystyle {\boldsymbol {x}}=(x_{1},\dots ,x_{n})\in \mathbb {R} ^{n}}
can be written as
x
=
r
v
{\displaystyle {\boldsymbol {x}}=r{\boldsymbol {v}}}
, with
‖
v
‖
=
1
{\displaystyle \lVert {\boldsymbol {v}}\rVert =1}
. The joint density becomes
p
(
r
,
θ
|
μ
,
Σ
)
=
r
n
−
1
|
Σ
|
(
2
π
)
n
2
e
−
1
2
(
r
v
−
μ
)
⊤
Σ
−
1
(
r
v
−
μ
)
{\displaystyle p(r,{\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {r^{n-1}}{{\sqrt {|{\boldsymbol {\Sigma }}|}}(2\pi )^{\frac {n}{2}}}}e^{-{\frac {1}{2}}(r{\boldsymbol {v}}-{\boldsymbol {\mu }})^{\top }\Sigma ^{-1}(r{\boldsymbol {v}}-{\boldsymbol {\mu }})}}
and the density of
P
N
n
(
μ
,
Σ
)
{\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}
can then be obtained as
p
(
θ
|
μ
,
Σ
)
=
∫
0
∞
p
(
r
,
θ
|
μ
,
Σ
)
d
r
.
{\displaystyle p({\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})=\int _{0}^{\infty }p(r,{\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})dr.}
The same density had been previously obtained in Pukkila and Rao (1988, Eq. (2.4)) using a different notation.
= Circular distribution
=Parametrising the position on the unit circle in polar coordinates as
v
=
(
cos
θ
,
sin
θ
)
{\displaystyle {\boldsymbol {v}}=(\cos \theta ,\sin \theta )}
, the density function can be written with respect to the parameters
μ
{\displaystyle {\boldsymbol {\mu }}}
and
Σ
{\displaystyle {\boldsymbol {\Sigma }}}
of the initial normal distribution as
p
(
θ
|
μ
,
Σ
)
=
e
−
1
2
μ
⊤
Σ
−
1
μ
2
π
|
Σ
|
v
⊤
Σ
−
1
v
(
1
+
T
(
θ
)
Φ
(
T
(
θ
)
)
ϕ
(
T
(
θ
)
)
)
I
[
0
,
2
π
)
(
θ
)
{\displaystyle p(\theta |{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {e^{-{\frac {1}{2}}{\boldsymbol {\mu }}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{2\pi {\sqrt {|{\boldsymbol {\Sigma }}|}}{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}}}\left(1+T(\theta ){\frac {\Phi (T(\theta ))}{\phi (T(\theta ))}}\right)I_{[0,2\pi )}(\theta )}
where
ϕ
{\displaystyle \phi }
and
Φ
{\displaystyle \Phi }
are the density and cumulative distribution of a standard normal distribution,
T
(
θ
)
=
v
⊤
Σ
−
1
μ
v
⊤
Σ
−
1
v
{\displaystyle T(\theta )={\frac {{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}{\sqrt {{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}}}}}
, and
I
{\displaystyle I}
is the indicator function.
In the circular case, if the mean vector
μ
{\displaystyle {\boldsymbol {\mu }}}
is parallel to the eigenvector associated to the largest eigenvalue of the covariance, the distribution is symmetric and has a mode at
θ
=
α
{\displaystyle \theta =\alpha }
and either a mode or an antimode at
θ
=
α
+
π
{\displaystyle \theta =\alpha +\pi }
, where
α
{\displaystyle \alpha }
is the polar angle of
μ
=
(
r
cos
α
,
r
sin
α
)
{\displaystyle {\boldsymbol {\mu }}=(r\cos \alpha ,r\sin \alpha )}
. If the mean is parallel to the eigenvector associated to the smallest eigenvalue instead, the distribution is also symmetric but has either a mode or an antimode at
θ
=
α
{\displaystyle \theta =\alpha }
and an antimode at
θ
=
α
+
π
{\displaystyle \theta =\alpha +\pi }
.
= Spherical distribution
=Parametrising the position on the unit sphere in spherical coordinates as
v
=
(
cos
θ
1
sin
θ
2
,
sin
θ
1
sin
θ
2
,
cos
θ
2
)
{\displaystyle {\boldsymbol {v}}=(\cos \theta _{1}\sin \theta _{2},\sin \theta _{1}\sin \theta _{2},\cos \theta _{2})}
where
θ
=
(
θ
1
,
θ
2
)
{\displaystyle {\boldsymbol {\theta }}=(\theta _{1},\theta _{2})}
are the azimuth
θ
1
∈
[
0
,
2
π
)
{\displaystyle \theta _{1}\in [0,2\pi )}
and inclination
θ
2
∈
[
0
,
π
]
{\displaystyle \theta _{2}\in [0,\pi ]}
angles respectively, the density function becomes
p
(
θ
|
μ
,
Σ
)
=
e
−
1
2
μ
⊤
Σ
−
1
μ
|
Σ
|
(
2
π
v
⊤
Σ
−
1
v
)
3
2
(
Φ
(
T
(
θ
)
)
ϕ
(
T
(
θ
)
)
+
T
(
θ
)
(
1
+
T
(
θ
)
Φ
(
T
(
θ
)
)
ϕ
(
T
(
θ
)
)
)
)
I
[
0
,
2
π
)
(
θ
1
)
I
[
0
,
π
]
(
θ
2
)
{\displaystyle p({\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {e^{-{\frac {1}{2}}{\boldsymbol {\mu }}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{{\sqrt {|{\boldsymbol {\Sigma }}|}}\left(2\pi {\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}\right)^{\frac {3}{2}}}}\left({\frac {\Phi (T({\boldsymbol {\theta }}))}{\phi (T({\boldsymbol {\theta }}))}}+T({\boldsymbol {\theta }})\left(1+T({\boldsymbol {\theta }}){\frac {\Phi (T({\boldsymbol {\theta }}))}{\phi (T({\boldsymbol {\theta }}))}}\right)\right)I_{[0,2\pi )}(\theta _{1})I_{[0,\pi ]}(\theta _{2})}
where
ϕ
{\displaystyle \phi }
,
Φ
{\displaystyle \Phi }
,
T
{\displaystyle T}
, and
I
{\displaystyle I}
have the same meaning as the circular case.
See also
Directional statistics
Multivariate normal distribution
References
Sources
Pukkila, Tarmo M.; Rao, C. Radhakrishna (1988). "Pattern recognition based on scale invariant discriminant functions". Information Sciences. 45 (3): 379–389.
Hernandez-Stumpfhauser, Daniel; Breidt, F. Jay; van der Woerd, Mark J. (2017). "The General Projected Normal Distribution of Arbitrary Dimension: Modeling and Bayesian Inference". Bayesian Analysis. 12 (1): 113–133.
Wang, Fangpo; Gelfand, Alan E (2013). "Directional data analysis under the general projected normal distribution". Statistical methodology. 10 (1). Elsevier: 113–127.
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