Голяндина Н.Э. Метод Гусеница-SSA: анализ временных рядов
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L K = N −L + 1
A L × K
X F X
∗
= X − A
S
∗
= X
∗
(X
∗
)
T
λ
i
U
i
i = 1, . . . , d
S
∗
V
i
= (X
∗
)
T
U
i
/
√
λ
i
X = A +
d
X
i=1
X
∗
i
X
∗
i
=
√
λ
i
U
i
V
T
i
A
A X
∗
i
||X||
2
M
= ||A||
2
M
+
d
X
i=1
||X
∗
i
||
2
M
A
A = A(X) = [E
1
(X) : . . . : E
1
(X)],
i E
1
(X) (i = 1, . . . , L)
i X
1
, . . . , X
K
span(X
1
, . . . , X
K
) span(X
(c)
1
, . . . , X
(c)
K
)
X
(c)
i
= X
i
−E
1
(X)
A = A(X) + B(X)
B(X) = [E
12
(X) : . . . : E
12
(X)]
T
j E
12
(X)
(j = 1, . . . , K) X
(c)
j
A
U
0(1)
= E
1
(X)/||E
1
(X)|| U
i
A = U
0(1)
Z
T
0(1)
Z
0(1)
= ||E
1
(X)||1
K
1
K
∈ IR
K
U
0(1)
U
0(2)
= 1
L
/
√
L
A = U
0(1)
Z
T
0(1)
+ U
0(2)
Z
T
0(2)
Z
0(2)
=
√
LE
12
(X)
λ
0(1)
= ||Z
0(1)
|| = ||E
1
(X)||
√
K λ
0(2)
= ||Z
0(2)
|| = ||E
12
(X)||
√
L.
V
0(1)
= Z
0(1)
/
p
λ
0(1)
V
0(2)
= Z
0(2)
/
p
λ
0(2)
³
U
0(i)
), V
0(i)
, λ
0(i)
´
(i = 1, 2)
A(X) B(X)
||X||
2
M
= λ
0(1)
+ λ
0(2)
+
d
X
i=1
λ
i
λ
0(2)
λ
0(1)
/||X||
2
M
, λ
0(2)
/||X||
2
M
λ
i
/||X||
2
M
.
F F = F
(1)
+ F
(2)
F
(1)
F
(2)
F
N
f
n
= an + b + 5 sin(2πn/T ) a = 1 b = 0 T = 10
N = 19 L
f
(1)
n
= n
a b
ea = 0.77
e
b = 2.22
N F
F
S = XX
T
L C = S/K S
X
S C
S K
S C
c
ij
= c
ij
(N) L
C K → ∞
lim c
ij
= R
f
(|i −j|) N → ∞ R
f
(k)
k
L
ec
ij
|i − j| = k
ec
ij
→ R
f
(|i −j|)
F = (f
0
, . . . , f
N−1
)
L
e
C
ec
ij
=
1
N − |i −j|
N−|i−j|−1
X
m=0
f
m
f
m+|i−j|
, 1 ≤ i, j ≤ L,
L
C = S/K
c
ij
=
1
K
K−1
X
m=0
f
m+i−1
f
m+j−1
, 1 ≤ i, j ≤ L.
L
e
C
H
1
, . . . , H
L
X =
L
X
i=1
H
i
Z
T
i
,
Z
i
= X
T
H
i
λ
i
= ||Z
i
||
2
Q
i
= Z
i
/
√
λ
i
e
C
X
λ
i
e
C
L
L
1
L
2
L
1
L
L
L
1
L
2
X L > 1 K > 1
S
def
= XX
T
L ×L
S L
U
1
, . . . , U
L
SU
i
= λ
i
U
i
,
λ
i
S U
i
i = 1, . . . , L
U
i
(U
i
, U
j
) = 0 i 6= j
||U
i
|| = 1 (X, Y )
||X|| =
p
(X, X) X
S
λ
i
≥ 0 i = 1, . . . , L
λ
i
λ
1
≥ λ
2
≥ . . . ≥ λ
L
≥ 0
d
S d < L λ
d
> 0 λ
d+1
= 0
d λ
L
> 0
d = L d X
d ≤ min(L, K)
1 ≤ i ≤ d
V
i
=
1
√
λ
i
X
T
U
i
.
U
i
V
i
1 1 ≤ i, j ≤ d (V
i
, V
j
) = 0 i 6= j ||V
i
|| = 1
i > d X
T
U
i
= 0
K
∈ IR
K
, 0
K
2 V
i
X
T
X
λ
i
.
3 1 ≤ i ≤ d
U
i
=
1
√
λ
i
XV
i
.
4 K > d K −d
X
T
X
5
X =
d
X
i=1
p
λ
i
U
i
V
T
i
.
X
√
λ
i
X U
i
V
i
X (
√
λ
i
, U
i
, V
i
)
i X
(−U
i
)
S λ
i
(U
i
, V
i
) (−U
i
, −V
i
)
λ
def
= λ
1
= λ
2
> λ
3
λ
U
1
U
2
U
1
U
2
V
1
V
2
λ
1
= λ
2
> λ
3
√
λ
1
U
1
V
T
1
√
λ
2
U
2
V
T
2
U
1
U
2
P
1
, . . . , P
L
Q
1
, . . . , Q
L
IR
L
IR
K
c
1
≥ . . . ≥ c
L
≥ 0
X =
L
X
i=1
c
i
P
i
Q
T
i
.
X
1 c
d
> 0 c
d+1
= . . . = c
L
= 0.
2 c
2
i
= λ
i
1 ≤ i ≤ d.
3 i = 1, . . . , d P
i
XX
T
λ
i
.
4 Q
i
= X
T
P
i
/
√
λ
i
(i = 1, . . . , d).
5 c
i
(24) (23)
U
i
V
i
I ⊂ {1, . . . , d} J = {1, . . . , d} \ I
X
I
=
X
i∈I
p
λ
i
U
i
V
T
i
X
J
def
= X − X
I
X
J
=
X
i∈J
p
λ
i
U
i
V
T
i
X
J
U
d
= [U
1
: . . . : U
d
]
V
d
= [V
1
: . . . : V
d
] Λ
d
d × d
λ
i
i
X = U
d
Λ
1/2
d
V
T
d
,