Голяндина Н.Э. Метод Гусеница-SSA: анализ временных рядов
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L ×K
L ×K X = [X
1
: . . . : X
K
] X
X =
d
X
i=1
p
λ
i
U
i
V
T
i
,
λ
i
i = 1, . . . , L S = XX
T
d = max{i, λ
i
> 0} = rank X,
{U
1
, . . . , U
d
}
S V
i
= X
T
U
i
/
√
λ
i
√
λ
i
U
i
V
i
X (
√
λ
i
, U
i
, V
i
) i
X
X
i
=
√
λ
i
U
i
V
T
i
X
λ
i
X
i
X
X
1
, . . . , X
K
V
1
, . . . , V
d
X
T
X
λ
i
U
i
X
V
i
X
X L K
U
i
V
i
i U
i
i
Z
i
=
√
λ
i
V
i
X
i
i
i
m = 2 I
1
= I = {i
1
. . . , i
r
}
I
2
= {1, . . . , d} \ I 1 ≤ i
1
< . . . < i
r
≤ d
F
F
(1)
F
(2)
f
i
= f
(1)
i
+ f
(2)
i
i = 0, . . . , N −1
L X X
(1)
X
(2)
L F F
(1)
F
(2)
X F
(1)
F
(2)
I ⊂ {1, . . . , d} X
(1)
=
P
i∈I
X
i
X
(2)
=
P
i/∈I
X
i
X
(1)
X = X
(1)
+ X
(2)
P
i∈I
λ
i
.
P
L
i=1
λ
i
I = I
1
X
I
1
X
I
2
= X − X
I
1
X
I
1
X
I
2
F
(1)
F
(2)
F = F
(1)
+ F
(2)
X
I
1
X
I
2
I
1
, . . . , I
m
X
I
1
, . . . , X
I
m
X = [X
1
: . . . : X
K
]
F F = F
(1)
+ F
(2)
F
(1)
F
(2)
{1, . . . , d} I {1, . . . , d}\I m = 2
U
1
, . . . , U
d
L
(L)
L
(L)
L
(L,1)
= L(U
i
, i ∈ I)
L
(L,2)
= L(U
i
, i 6∈ I) U
i
, i ∈ I U
i
, i 6∈ I
F
(1)
F
(2)
X
I
X
1
, . . . , X
K
L
(L,1)
F
(1)
U
i
V
i
λ
i
X
i
m = 2
I
k n
e
f
(k)
n
x
(k)
ij
{(i, j), i + j = n + 2}
X
I
k
e
F
(k)
X
I
k
e
Y Y
Y
e
Y Y
H
X =
e
X
I
1
+ . . . +
e
X
I
m
,
e
X
I
l
= HX
I
l
X
I
k
e
F
(k)
e
F
(k)
I
k
X
I
k
e
F
(k)
I
k
F m
m
L
m
F
F
(1)
N
F
(2)
N
N F
N
= F
(1)
N
+ F
(2)
N
L F
(1)
N
F
(2)
N
F
N
L X
(1)
X
(2)
X
L
(L,1)
L
(L,2)
X
(1)
X
(2)
L
(K,1)
L
(K,2)
(X
(1)
)
T
(X
(2)
)
T
K = N − L + 1
F
(1)
N
F
(2)
N
L
L
(L,1)
⊥ L
(L,2)
L
(K,1)
⊥ L
(K,2)
F
(1)
N
= (f
(1)
0
, . . . , f
(1)
N−1
) F
(2)
N
= (f
(2)
0
, . . . , f
(2)
N−1
)
K = N − L + 1 F
(1)
N
F
(2)
N
f
(1)
k
f
(2)
m
+ . . . + f
(1)
k+L−1
f
(2)
m+L−1
= 0, 0 ≤ k, m ≤ K − 1,
f
(1)
k
f
(2)
m
+ . . . + f
(1)
k+K−1
f
(2)
m+K−1
= 0, 0 ≤ k, m ≤ L −1.
L
F
(1)
L F
(2)
K
F
(1)
K F
(2)
F
(1)
N
F
(2)
N
L
X
(1)
X
(2)
X
(1)
=
X
k
p
λ
1k
U
1k
V
T
1k
, X
(2)
=
X
k
p
λ
2k
U
2k
V
T
2k
.
X = X
(1)
+ X
(2)
=
X
k
p
λ
1k
U
1k
V
T
1k
+
X
m
p
λ
2m
U
2m
V
T
2m
F
N
= F
(1)
N
+ F
(2)
N
X
F
(1)
N
F
(2)
N
L
X F
N
F
(1)
N
F
(2)
N
F
(1)
N
F
(2)
N
L
λ
1k
6= λ
2m
k m
F
(1)
N
F
(2)
N
L
F
(1)
N
F
(2)
N
L
X F
N
F
(1)
N
F
(2)
N
+
L K = N − L + 1
cos exp exp cos ak + b
− + − − −
cos + + − − −
exp − − − + −
exp cos − − + + −
ak + b − − − − −
F
(1)
N
f
(1)
n
≡ c 6= 0
0 ≤ n ≤N −1
F
(2)
N
F
(2)
N
T L/T K/T
f
(2)
0
+ . . . + f
(2)
T −1
= 0
f
(2)
n
= cos(2πn/T + φ) N
(N + 1)/T F
(2)
N
L < N T
f
(1)
n
= e
αn
F
(2)
N
e
αn
F
(2)
N
T L/T K/T
T −1
P
m=0
e
αm
f
(2)
m
= 0
f
(2)
n
= e
−αn
cos(2πn/T + φ) N
(N + 1)/T F
(2)
N
L < N T
f
(1)
n
= cos(2πωn + φ)
0 < ω < 1/2 L, K > 2 T = 1/ω
f
(2)
n
= cos(2πeωn +
e
φ)
e
T = 1/eω F
(1)
ω 6= eω
L K N + 1 T
e
T
f
(1)
n
= e
αn
cos (2πωn + φ)
f
(2)
n
= e
−αn
cos(2πeωn +
e
φ)
e
T = 1/eω
F
(1)
ω 6= eω L K N + 1
T
e
T
f
(1)
n
= an+b a 6= 0
F
N
= (f
0
, . . . , f
N−1
)
F
i,j
= (f
i−1
, . . . , f
j−1
), 1 ≤ i ≤ j < N.
F
(1)
N
= (f
(1)
0
, . . . , f
(1)
N−1
) F
(2)
N
= (f
(2)
0
, . . . , f
(2)
N−1
)
i, j ≥ 1 M ≤ N − 1 −max(i, j)
ρ
(M)
i,j
=
³
F
(1)
i,i+M−1
, F
(2)
j,j+M−1
´
||F
(1)
i,i+M−1
||||F
(2)
j,j+M−1
||
( ·, ·)
|| · ||
ρ
(M)
i,j
= 0
ρ
(M)
i,j
F
(1)
i,i+M−1
F
(2)
j,j+M−1
ρ
(M)
i,j
F
(1)
N
, F
(2)
N
ε
L
ρ
(L,K)
def
= max
µ
max
1≤i,j≤K
|ρ
(L)
i,j
)|, max
1≤i,j≤L
|ρ
(K)
i,j
)|
¶
< ε,
K = N − L + 1 ε
F
(1)
N
F
(2)
N
F
(1)
N
F
(2)
N