笔记3之Wishart矩阵最大特征值分布
作者:互联网
利用如下定理推导最大特征值的分布。
An expression for the distribution function of the largest root of
S
S
S follows from the following theorem due to Constantine ( 1963).
THEOREM 9.7.1. If
A
A
A is
W
m
(
n
,
Σ
)
,
(
n
>
m
−
I
)
W_m(n,\Sigma), ( n > m - I )
Wm(n,Σ),(n>m−I) and
Ω
\Omega
Ω is an
m
×
m
m \times m
m×m positive definite matrix (
Ω
>
0
\Omega >0
Ω>0) then the probability that
Ω
−
A
\Omega -A
Ω−A is positive definite (
A
<
Ω
A<\Omega
A<Ω is
P
(
A
<
Ω
)
=
Γ
m
[
1
2
(
m
+
1
)
]
Γ
m
[
1
2
(
n
+
m
+
1
)
]
⋅
det
(
1
2
Σ
−
1
Ω
)
n
/
2
1
F
1
(
1
2
n
;
1
2
(
n
+
m
+
1
)
;
−
1
2
Σ
−
1
Ω
)
(1)
\begin{aligned} P(A<\Omega)=& {\Gamma_m \left[{1\over2}(m+1)\right]\over \Gamma_m \left[{1\over2}(n+m+1)\right]} \\&\\ & \cdot \det \left(\small{1\over2}\Sigma^{-1}\Omega\right) ^{n/2}{}_1F_1(\small{1\over2}n;\small{1\over2}(n+m+1);-\small{1\over2}\Sigma^{-1}\Omega) \end{aligned} \tag 1
P(A<Ω)=Γm[21(n+m+1)]Γm[21(m+1)]⋅det(21Σ−1Ω)n/21F1(21n;21(n+m+1);−21Σ−1Ω)(1)
where
1
F
1
(
a
;
c
;
X
)
=
∑
k
=
0
∞
∑
κ
(
a
)
κ
(
c
)
κ
C
κ
(
X
)
k
!
\begin{aligned} _1F_1(a;c;\mathrm X)=\sum_{k=0}^\infty \sum_\kappa \frac{(a)_\kappa}{(c)_\kappa} \frac{C_\kappa(\mathrm X)}{k!} \end{aligned}
1F1(a;c;X)=k=0∑∞κ∑(c)κ(a)κk!Cκ(X)
证明:根据
W
m
(
n
,
Σ
)
W_m(n,\Sigma)
Wm(n,Σ) 的概率密度函数,
A
<
Ω
A<\Omega
A<Ω 的概率为
P
(
A
<
Ω
)
=
1
2
m
n
/
2
Γ
m
(
1
2
n
)
(
det
Σ
)
n
/
2
∫
0
<
A
<
Ω
e
t
r
(
−
1
2
Σ
−
1
A
)
⋅
det
(
A
)
(
n
−
m
−
1
)
/
2
(
d
A
)
\begin{aligned} P(A<\Omega)=& {1 \over2^{mn/2} \Gamma_m \left({1\over2}n\right) (\det \Sigma)^{n/2}} \int_{0<A<\Omega}\mathrm{etr} (-\small{1\over2}\Sigma^{-1}A) \\&\\ & \cdot \det (A) ^{(n-m-1)/2}(dA)\end{aligned}
P(A<Ω)=2mn/2Γm(21n)(detΣ)n/21∫0<A<Ωetr(−21Σ−1A)⋅det(A)(n−m−1)/2(dA)
令
A
=
Ω
1
/
2
X
Ω
1
/
2
A=\Omega^{1/2}X\Omega^{1/2}
A=Ω1/2XΩ1/2,则
(
d
A
)
=
(
det
Ω
)
m
+
1
/
2
(
d
X
)
(dA)=(\det\Omega)^{m+1/2}(dX)
(dA)=(detΩ)m+1/2(dX),因此
P
(
A
<
Ω
)
=
(
det
1
2
Σ
−
1
Ω
)
n
/
2
Γ
m
(
1
2
n
)
∫
0
<
X
<
I
e
t
r
(
−
1
2
Ω
1
/
2
Σ
−
1
Ω
1
/
2
X
)
⋅
det
(
X
)
(
n
−
m
−
1
)
/
2
(
d
X
)
=
(
det
1
2
Σ
−
1
Ω
)
n
/
2
Γ
m
(
1
2
n
)
∑
k
=
0
∞
∑
κ
1
k
!
∫
0
<
X
<
I
det
(
X
)
(
n
−
m
−
1
)
/
2
⋅
C
κ
(
−
1
2
Ω
1
/
2
Σ
−
1
Ω
1
/
2
X
)
(
d
X
)
\begin{aligned} P(A<\Omega)=&{(\det \frac{1}{2}\Sigma^{-1}\Omega)^{n/2} \over \Gamma_m \left({1\over2}n\right)} \int_{0<X<I}\mathrm{etr} (-\small{1\over2}\Omega^{1/2}\Sigma^{-1}\Omega^{1/2}X) \\&\\ & \cdot \det (X) ^{(n-m-1)/2}(dX)\\&\\ =&{(\det \frac{1}{2}\Sigma^{-1}\Omega)^{n/2} \over \Gamma_m \left({1\over2}n\right)} \sum_{k=0}^\infty\sum_{\kappa}{1\over k!}\int_{0<X<I}\det (X) ^{(n-m-1)/2}\\&\\& \cdot C_{\kappa}(-\small{1\over2}\Omega^{1/2}\Sigma^{-1}\Omega^{1/2}X)(dX) \end{aligned}
P(A<Ω)==Γm(21n)(det21Σ−1Ω)n/2∫0<X<Ietr(−21Ω1/2Σ−1Ω1/2X)⋅det(X)(n−m−1)/2(dX)Γm(21n)(det21Σ−1Ω)n/2k=0∑∞κ∑k!1∫0<X<Idet(X)(n−m−1)/2⋅Cκ(−21Ω1/2Σ−1Ω1/2X)(dX)
这里采用了关系式
e
t
r
(
R
)
=
0
F
0
(
R
)
=
∑
k
=
0
∞
∑
κ
C
κ
(
R
)
k
!
\mathrm{etr} (R)={}_0F_0(R)=\sum_{k=0}^\infty\sum_{\kappa}{C_{\kappa}(R)\over k!}
etr(R)=0F0(R)=k=0∑∞κ∑k!Cκ(R)
根据积分公式(定理7.2.10)
∫
0
<
Y
<
I
m
(
det
Y
)
a
−
(
m
+
1
)
/
2
det
(
I
−
Y
)
b
−
(
m
+
1
)
/
2
C
κ
(
X
Y
)
(
d
Y
)
=
(
a
)
κ
(
a
+
b
)
κ
Γ
m
(
a
)
Γ
m
(
b
)
Γ
m
(
a
+
b
)
C
κ
(
X
)
\begin{aligned} \int_{0<Y<I_m} & (\det Y)^{a-(m+1)/2} \det (I-Y)^{b-(m+1)/2} C_\kappa(XY)(dY) \\ &\\&={(a)_\kappa\over(a+b)_\kappa}{\Gamma_m(a) \Gamma_m(b)\over\Gamma_m(a+b)}C_\kappa(X)\end{aligned}
∫0<Y<Im(detY)a−(m+1)/2det(I−Y)b−(m+1)/2Cκ(XY)(dY)=(a+b)κ(a)κΓm(a+b)Γm(a)Γm(b)Cκ(X)
这里令
a
=
n
/
2
,
b
=
(
m
+
1
)
/
2
a=n/2, \quad b=(m+1)/2
a=n/2,b=(m+1)/2
P
(
A
<
Ω
)
=
Γ
m
[
1
2
(
m
+
1
)
]
Γ
m
[
1
2
(
n
+
m
+
1
)
]
⋅
det
(
1
2
Σ
−
1
Ω
)
n
/
2
⋅
∑
k
=
0
∞
∑
κ
(
1
2
n
)
κ
(
1
2
(
n
+
m
+
1
)
)
κ
C
κ
(
−
1
2
Σ
−
1
Ω
)
k
!
\begin{aligned} P(A<\Omega)=& {\Gamma_m \left[{1\over2}(m+1)\right]\over \Gamma_m \left[{1\over2}(n+m+1)\right]} \cdot \det \left(\small{1\over2}\Sigma^{-1}\Omega\right) ^{n/2}\\&\\ &\cdot \sum_{k=0}^\infty\sum_{\kappa} {(\frac{1}{2}n)_\kappa\over (\small{1\over2}(n+m+1))_\kappa}{C_\kappa (-\frac{1}{2}\Sigma^{-1}\Omega)\over k!} \end{aligned}
P(A<Ω)=Γm[21(n+m+1)]Γm[21(m+1)]⋅det(21Σ−1Ω)n/2⋅k=0∑∞κ∑(21(n+m+1))κ(21n)κk!Cκ(−21Σ−1Ω)
注意到
∑
k
=
0
∞
∑
κ
(
1
2
n
)
κ
(
1
2
(
n
+
m
+
1
)
)
κ
C
κ
(
−
1
2
Σ
−
1
Ω
)
k
!
=
1
F
1
(
1
2
n
;
1
2
(
n
+
m
+
1
)
;
−
1
2
Σ
−
1
Ω
)
\sum_{k=0}^\infty\sum_{\kappa} {(\frac{1}{2}n)_\kappa\over (\small{1\over2}(n+m+1))_\kappa}{C_\kappa (-\frac{1}{2}\Sigma^{-1}\Omega)\over k!}={}_1F_1(\small{1\over2}n;\small{1\over2}(n+m+1);-\small{1\over2}\Sigma^{-1}\Omega)
k=0∑∞κ∑(21(n+m+1))κ(21n)κk!Cκ(−21Σ−1Ω)=1F1(21n;21(n+m+1);−21Σ−1Ω)
证毕。
利用这个定理可推导出最大特征值的分布
如果
l
1
l_1
l1 是 矩阵
S
S
S 的最大特征根,
A
=
n
S
A = nS
A=nS 是
W
m
(
n
,
Σ
)
,
(
n
>
m
−
I
)
W_m(n,\Sigma), ( n > m - I )
Wm(n,Σ),(n>m−I) 分布,那么
l
1
l_1
l1 的概率分布函数为
P
Σ
(
l
1
<
x
)
=
Γ
m
[
1
2
(
m
+
1
)
]
Γ
m
[
1
2
(
n
+
m
+
1
)
]
det
(
1
2
n
x
Σ
−
1
)
n
/
2
⋅
1
F
1
(
1
2
n
;
1
2
(
n
+
m
+
1
)
;
−
1
2
n
x
Σ
−
1
)
(2)
\begin{aligned} P_\Sigma(l_1<x)=& {\Gamma_m \left[{1\over2}(m+1)\right]\over \Gamma_m \left[{1\over2}(n+m+1)\right]} \det \left(\small{1\over2}nx\Sigma^{-1}\right) ^{n/2}\\&\\&\cdot{}_1F_1(\small{1\over2}n;\small{1\over2}(n+m+1);-\small{1\over2}nx\Sigma^{-1})\end{aligned}\tag 2
PΣ(l1<x)=Γm[21(n+m+1)]Γm[21(m+1)]det(21nxΣ−1)n/2⋅1F1(21n;21(n+m+1);−21nxΣ−1)(2)
在 (1) 中令
Ω
=
n
x
I
\Omega=nxI
Ω=nxI 可直接得到 (2)式。
l
1
<
x
l_1<x
l1<x 意味着
A
<
n
x
I
A<nxI
A<nxI。
标签:特征值,kappa,21,Sigma,sum,矩阵,Wishart,Omega,over2 来源: https://blog.csdn.net/wubyatseu/article/details/123306990