本文主要是记录与这两个概念相关的概念。看中文或者英文时，尝尝容易弄混。
内容
- 1 covariance和correleation
- 2 covariance matrix和correlation matrix
- 3 cross-covariance和cross-correlation
- 4 autocovariance和autocorrelation
对于两个随机信号X, Y
1 covariance和correleation
covariance:
c
o
v
X
Y
=
σ
X
Y
=
E
[
(
X
−
μ
X
)
(
Y
−
μ
Y
)
]
cov_{XY}=\sigma _{XY}=E\left[ \left( X-\mu _X \right) \left( Y-\mu _Y \right) \right]
covXY=σXY=E[(X−μX)(Y−μY)]
correlation:
c
o
r
r
X
Y
=
ρ
X
Y
=
E
[
(
X
−
μ
X
)
(
Y
−
μ
Y
)
]
/
(
σ
X
σ
Y
)
corr_{XY}=\rho _{XY}=E\left[ \left( X-\mu _X \right) \left( Y-\mu _Y \right) \right] /\left( \sigma _X\sigma _Y \right)
corrXY=ρXY=E[(X−μX)(Y−μY)]/(σXσY)
因此
ρ
X
Y
=
σ
X
Y
/
(
σ
X
σ
Y
)
\rho _{XY}=\sigma _{XY}/\left( \sigma _X\sigma _Y \right)
ρXY=σXY/(σXσY)
如果Y=X，那么 σ X Y = σ X X = ( σ X ) 2 \sigma _{XY}=\sigma _{XX}=(\sigma_{X})^2 σXY=σXX=(σX)2, ρ X Y = 1 \rho_{XY}=1 ρXY=1
2 covariance matrix和correlation matrix
这个是对于多维随机变量来说的，也就是
X
=
[
X
1
,
X
2
,
.
.
.
,
X
m
]
X=[X_1,X_2,...,X_m]
X=[X1,X2,...,Xm],
Y
=
[
Y
1
,
Y
2
,
.
.
.
,
Y
n
]
Y=[Y_1,Y_2,...,Y_n]
Y=[Y1,Y2,...,Yn]，那么
c
o
v
X
Y
=
[
c
o
v
X
i
Y
j
]
(
m
,
n
)
c
o
r
r
X
Y
=
[
c
o
r
r
X
i
Y
j
]
(
m
,
n
)
cov_{XY}=\left[ cov_{X_iY_j} \right] _{\left( m,n \right)} \\ corr_{XY}=\left[ corr_{X_iY_j} \right] _{\left( m,n \right)}
covXY=[covXiYj](m,n)corrXY=[corrXiYj](m,n)
3 cross-covariance和cross-correlation
这是对于平稳随机信号来说的，也就是信号的平均值和方差随时间是不变的
E
(
X
n
)
=
E
(
x
n
+
m
)
E(X_n)=E(x_{n+m})
E(Xn)=E(xn+m),
V
a
r
(
X
n
)
=
V
a
r
(
X
n
+
m
)
Var(X_n)=Var(X_{n+m})
Var(Xn)=Var(Xn+m)
cross-covariance
σ
X
Y
(
m
)
=
E
[
(
X
n
−
μ
X
)
(
Y
n
+
m
−
μ
Y
)
]
\sigma _{XY}(m)=E\left[ \left( X_n-\mu _X \right) \left( Y_{n+m}-\mu _Y \right) \right]
σXY(m)=E[(Xn−μX)(Yn+m−μY)]
cross-correlation
ρ
X
Y
(
m
)
=
E
[
(
X
n
−
μ
X
)
(
Y
n
+
m
−
μ
Y
)
]
/
(
σ
X
σ
Y
)
\rho _{XY}(m)=E\left[ \left( X_n-\mu _X \right) \left( Y_{n+m}-\mu _Y \right) \right] /\left( \sigma _X\sigma _Y \right)
ρXY(m)=E[(Xn−μX)(Yn+m−μY)]/(σXσY)
cross-covariance, cross-correlation与n的位置无关，只与时刻之间的间隔m有关
4 autocovariance和autocorrelation
autocovariance
σ
X
X
(
m
)
=
E
[
(
X
n
−
μ
X
)
(
X
n
+
m
−
μ
X
)
]
\sigma _{XX}\left( m \right) =E\left[ \left( X_n-\mu _X \right) \left( X_{n+m}-\mu _X \right) \right]
σXX(m)=E[(Xn−μX)(Xn+m−μX)]
autocorrelation
ρ
X
X
(
m
)
=
E
[
(
X
n
−
μ
X
)
(
X
n
+
m
−
μ
X
)
]
/
(
σ
X
2
)
\rho _{XX}\left( m \right) =E\left[ \left( X_n-\mu _X \right) \left( X_{n+m}-\mu _X \right) \right] /\left( \sigma _{X}^{2} \right)
ρXX(m)=E[(Xn−μX)(Xn+m−μX)]/(σX2)
