Голяндина Н.Э. Метод Гусеница-SSA: анализ временных рядов
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F
(1)
= (f
(1)
0
, . . . , f
(1)
n
, . . . )
F
(2)
= (f
(2)
0
, . . . , f
(2)
n
, . . . ) N > 2 F
(1)
N
F
(2)
N
N F
(1)
F
(2)
1 < L = L(N) < N
ρ
N
= ρ
(L(N),K(N))
ρ
(L(N),K(N))
→ 0
L = L(N) N → ∞ F
(1)
F
(2)
L(N) F
(1)
F
(2)
L(N)
L(N) L(N) → ∞ K(N) → ∞
cos exp exp cos ak + b
− + + + −
cos + + + + +
exp + + + + +
exp cos + + + + +
ak + b − + + + −
k
α 6= 0 β 6= 0 ω
1
∈ (0, 0.5]
ω
2
∈ (0, 0.5]
cos(2πω
2
k) exp(βk)
− 1/ min(L, K) 1/
p
min(L, K)
cos(2πω
1
k) 1/ min(L, K) 1/ min(L, K) 1/
p
min(L, K)
(ω
1
6= ω
2
)
exp(αk) 1/
p
min(L, K) 1/
p
min(L, K) e
−C min(L,K)
(αβ < 0)
L K
L K
f
(1)
n
= const f
(2)
n
= e
αn
|α|
(|α|min(L, K))
−1/2
α L K
f
(1)
n
= cos(2πω
1
n) f
(2)
n
= cos(2πω
2
n) ω
1
ω
2
C|ω
1
−ω
2
|/ min(L, K)
(Ω, F, P) F
(1)
F
(2)
L = L(N) ρ
(L,K)
N → ∞
F
(1)
F
(2)
F
(1)
F
(2)
F
(2)
= (f
(2)
1
, f
(2)
2
, . . .)
f
(2)
i
F
(1)
F
(2)
L, K → ∞ L/K → a > 0
F
N
= (f
0
, . . . , f
N−1
) N
f
n
= c
0
+
[N/2]
X
k=1
³
c
k
cos(2πn k/N) + s
k
sin(2πn k/N)
´
,
0 ≤ n < N s
N/2
= 0 N
F
N
c
0
N
f
n
= c
0
+
[(N−1)/2]
X
k=1
q
c
2
k
+ s
2
k
cos(2πn k/N + ϕ
k
) + c
N/2
(−1)
k
.
c
0
0 c
N/2
0.5
F
N
= (f
0
, . . . , f
N−1
)
Π
N
f
{k/N, k = 0, ...[N/2]}
Π
N
f
(k/N) =
N
2
2c
2
0
k = 0,
c
2
k
+ s
2
k
0 < k < N/2,
2c
2
N/2
k = N/2.
N
F
(1)
F
(2)
N c
(j)
k
s
(j)
k
j = 1, 2
d
k
=
(
c
(1)
k
c
(2)
k
+ s
(1)
k
s
(2)
k
k 6= 0 N/2,
2c
(1)
k
c
(2)
k
k = 0 N/2,
¡
F
(1)
, F
(2)
¢
def
=
N−1
X
k=0
f
(1)
n
f
(2)
n
=
N
2
[N/2]
X
k=0
d
k
.
||F || =
p
(F, F )
||F ||
2
=
[N/2]
X
k=0
Π
N
f
(k/N).
k/N
ω = k/N
N
−1
Π
N
f
(k/N)
k/N ω
k
= k/N
[a, b]
F
N
= F
(1)
N
+F
(2)
N
N L K = N − L + 1
F
(1)
N
L
F
(2)
N
L
K F
(1)
N
F
(2)
N
T
1
T
2
L = k
1
T
1
= k
2
T
2
, K = m
1
T
1
= m
2
T
2
k
1
, k
2
, m
1
, m
2
i i = 1, 2
L ω
(L)
i
=
1/k
i
K
ω
(K)
i
= 1/m
i
N = 399 a = 1.005 T = 200
F
(1)
F
(2)
f
(1)
n
= a
−n
, f
(2)
n
= a
n
cos(2πn/T ).
L = 200 K = 200
F
(1)
N
F
(2)
N
ρ
(Π)
12
def
=
q
Π
N
f
1
(k/N) Π
N
f
2
(k/N)
s
[N/2]
P
k=0
Π
N
f
1
(k/N)
s
[N/2]
P
k=0
Π
N
f
2
(k/N)
F
(1)
N
F
(2)
N
F
(1)
L = L(N) F
(2)
L N → ∞
K = K(N)
F
(1)
F
(2)
F
F
N
N → ∞ F
L → ∞ K → ∞
L
∗
= min(L, K) K
∗
= max(L, K)
w
i
=
i + 1 0 ≤ i ≤ L
∗
− 1,
L
∗
L
∗
≤ i < K
∗
,
N − i K
∗
≤ i ≤ N − 1.
F
(1)
N
F
(2)
N
N
¡
F
(1)
N
, F
(2)
N
¢
w
def
=
N−1
X
i=0
w
i
f
(1)
i
f
(2)
i
.
F
(1)
N
F
(2)
N
w
L
¡
F
(1)
N
, F
(2)
N
¢
w
= 0.
L w
i
w
i
w
i
L
N L ≈ N/2
N = 3 L = K = 2
F
(1)
= (1, −a, a
2
), F
(2)
= (1, a
−1
, a
−2
), a 6= 0.
X
(1)
X
(2)
X
(1)
=
µ
1 −a
−a a
2
¶
, X
(2)
=
µ
1 a
−1
a
−1
a
−2
¶
.
X
(1)
(X
(1)
)
T
X
(2)
(X
(2)
)
T
X
(1)
X
(2)
p
λ
(1)
= 1 + a
2
p
λ
(2)
= 1 + a
−2
a 6= 1 F
(1)
F
(2)
a = 1
a 6= 1
X F
(1)
+ F
(2)
X =
µ
2 −a + a
−1
−a + a
−1
a
2
+ a
−2
¶
=
= (1 + a
2
) U
1
V
T
1
+ (1 + a
−2
) U
2
V
T
2
= X
(1)
+ X
(2)
,
U
1
= V
1
= (1, −a)
T
/
p
(1 + a
2
) U
2
=V
2
=(1, a
−1
)
T
/
p
(1 + a
−2
)
a = 1
X
X =
µ
2 0
0 2
¶
= 2 U
1
V
T
1
+ 2 U
2
V
T
2
,
{U
1
, U
2
} IR
2
V
1
V
2
V
1
= X
T
U
1
/2 V
2
= X
T
U
2
/2
U
1
= (1, 1)
T
/
√
2 U
2
= (1, −1)
T
/
√
2
X = X
(1)
+ X
(2)
XX
T
L K
F
N
= F
(1)
N
+F
(2)
N
F
(1)
N
F
(2)
N
F
N
F
(1)
N
F
(2)
N
F
N
= (f
0
, . . . , f
N−1
) N ≥ 3
L (1 < L < N)
X
(L)
i
= X
i
= (f
i−1
, . . . , f
i+L−2
)
T
, i = 1, . . . , K,
L
(L)
= L
(L)
(F
N
)
def
= span(X
1
, . . . , X
K
)
F
N
0 ≤ d ≤ L dim L
(L)
= d
F
N
L d
rank
L
(F
N
) = d dim L
(L)
= 0
rank
L
(F
N
) = d
d ≤ min(L, K).
d L
L N −L + 1
rank
L
(F
N
) = d < N/2
L F
N
d rank(F
N
) = d d F
N