Воеводин Вл. В. Численные методы, параллельные вычисления и информационные технологии
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//
//
//
//
U
T
U + U
T
R + R
T
U
//
y
i
(t), i = 1, . . . m,
x
j
(t), j = 1, . . . , n.
y(t) = Ax(t) + ξ(t), (∗)
y(t) x(t)
A
t
A x
i
(t)
BSS
ξ
i
(t)
x
i
(t)
E[x(t)x
∗
(t)],
E[ · ]
Φ
y
= E[y(t)y
∗
(t)] = E[Ax(t)x
∗
(t)A
∗
] = AE[x(t)x
∗
(t)]A
∗
= ADA
∗
,
A A
Φ
y
= UΛU
∗
A
D Λ.
A
A = UΛ
1/2
WΛ
†/2
,
W U
A
PCA
ICA
MILCA SNICA
w
W
x
i
(t) = (w
i
, y(t))
ICA
BSS
(∗).
ICA
(∗)
UΛ
1/2
WΛ
†/2
x(t) = y(t),
Λ U
Φ
y
, Λ
†
W
x(t) := Λ
†/2
x(t)
Wx(t) = z(t), z(t) = Λ
†/2
U
∗
y(t).
ICA
BSS
PCA
33
W.
x(t) = W
∗
z(t), x
i
(t) =
m
X
j=1
w
ji
z
j
(t) = (w
i
, z(t)), i = 1, . . . , n,
w
i
i W.
x
i
(t)
x
i
(t)
y
i
(t) z
i
(t)
w
ξ = w
∗
z = w
∗
Wx = q
∗
x, q = W
∗
w.
M
(ξ)
ξ
M
(ξ) =
M
(q
∗
x) =
M
n
X
i=1
q
i
x
i
!
Lsig
1
(t)
16001400120010008006004002000
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Bsig
1
(t)
16001400120010008006004002000
0.055
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
Msig
1
(t)
16001400120010008006004002000
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
Lsig
2
(t)
16001400120010008006004002000
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
Bsig
2
(t)
16001400120010008006004002000
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
Msig
2
(t)
16001400120010008006004002000
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
ica
P ca0
1
(t) [bss 0.98]
16001400120010008006004002000
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
ica
P ca0
2
(t) [bss 0.98]
16001400120010008006004002000
3
2
1
0
-1
-2
-3
-4
ica
P ca0
3
(t) [bss 0.98]
16001400120010008006004002000
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
from ica
true
Sig1
1
(t) [bss 0.98]
16001400120010008006004002000
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
from ica
true
Sig1
12
(t) [bss 0.98]
16001400120010008006004002000
0.055
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
from ica
true
Sig1
23
(t) [bss 0.98]
16001400120010008006004002000
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
from ica
true
Sig1
2
(t) [bss 0.98]
16001400120010008006004002000
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
from ica
true
Sig1
13
(t) [bss 0.98]
16001400120010008006004002000
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
from ica
true
Sig1
24
(t) [bss 0.98]
16001400120010008006004002000
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
x
i
.
kwk = 1, W
q kqk = kW
∗
wk = kwk = 1.
ξ q,
w
M
(w
∗
z) = W
q
M
(q
∗
x) = We
i
= w
i
,
w,
W.
z(t),
W.
M
(ξ) =
M
(w
∗
z) = E[G(|w
∗
z|
2
)],
G( · )
G(ξ) = ξ
2
,
(y) = E[|y|
4
] − E[yy
∗
]E[yy
∗
] − E[yy]E[y
∗
y
∗
] − E[yy
∗
]E[y
∗
y],
y ∈ C
(y) = E[|y|
4
] − 2
E[|y|
2
2
−
E[y
2
]
2
= E[|y|
4
] − 2.
G
w.