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2 1 2 3 ARMA 4 BN 5 BN 6

3 (Ω, F, µ) Ω: F Ω σ 1 Ω, ϕ F 2 A, B F = A B, A B, A\B F F µ F 1 µ(ϕ) = 0 2 A F = µ(a) 0 3 A, B F, A B = ϕ = µ(a B) = µ(a) + µ(b) µ(ω) = 1

4 X : µ X : X x 1,, x n X (Ω) x 1,, x n µ X ( ) µ X x 1,, x n ( )

5 : ARMA BN ARMA BN BN ( BN )

6 X, Y : µ Y X : Y X X Y x x µ Y X (y x) = µ Y X (y x ), y Y (Ω) n X (Ω) Y (Ω) <

7 : Y X X (Ω) Y (Ω) = {, } X (Ω) = G 1 G 2 G 3 G 4 G 5 75 > 75 G 3 Yes No G 1 G 2 G 4 G 5

8 : BN X i X j, j π(i) π(i) {1,, i 1} 1 X 2 X 1 2 X 3 X 1, X 2 3 X 4 X 2, X 3 4 X 5 X 1, X 4 X 1 X 3 X 4 X 5 X 2

9 : n z n := (z 1,, z n ) Z n (Ω) X (Ω) z i := (x i, y i ) Z(Ω) := X (Ω) Y (Ω) 2 ( ) ( )

10 : Quinlan Q45 (b) (a) 75 > 75 3 Yes No 75 > 75 Yes No Yes No (c) (d) 4 75 > Yes No

11 z n Z n (Ω) I (G, z n ) := H(G, z n ) + k(g) 2 d n G H(G, z n ): ( z n G ) k(g): (G ) d n 0: d n n 0 d n = log n BIC/MDL d n = 2 AIC

12 (n ) (O(1) < d n < o(n)) (MDL/BIC etc) AIC (d n = 2) {d n } {d n }? (Suzuki, 2006) d n = 2 log log n ( )

13 G : G G µ{ω Ω I (G, Z n (ω)) < I (G, Z n (ω))} (K(G) K(G ))d n f K(G) K(G )(x)dx f l : l χ 2 G G n 0

14 ARMA ARMA k 0 {λ j } k j=1 : λ i R σ 2 R >0 ARMA (Autoregressive Moving Average, ) {X n } n= : X n + k j=1 λ jx n j = ϵ i N (0, σ 2 ) ARMA n k {λ j } k j=1 k k {λ j } k j=1

15 ARMA Yule-Walker k {ˆλ j,k } k j=1 ˆσ2 k x := 1 n n i=1 x i n j c j := 1 (x i x)(x i+j x), j = 0,, k n i=1 1 c 1 c 2 c k 0 c 0 c 1 c k 1 0 c 1 c 0 c k 2 0 c k 1 c k 2 c 0 ˆσ 2 k ˆλ 1,k ˆλ 2,k ˆλ k,k = c 0 c 1 c 2 c k

16 ARMA k x n X n (Ω) I (k, x n ) k ˆσ k 2 : Yule-Walker d n 0: d n n 0 d n = log n BIC/MDL d n = 2 AIC I (k, x n ) := 1 2 log ˆσ2 k + k 2 d n d n = 2 log log n Hannan-Quinn (1979) = {d n } Suzuki (2006) Hannan-Quinn (1979)

17 ARMA (, ARMA) k : µ{ω Ω I (k, X n (ω)) < I (k, X n (ω))} k > k (, ARMA) n 0 (Hannan-Quinn, 1979)

18 ARMA ARMA ( G) ( k) 2 log log n 2 log log n d n (Suzuki, 2006) (Hannan-Quinn, 1979) 0 0 ( ) (Suzuki, 2006) (Hannan-Quinn, 1979) f K(G) K(G )(x)dx (K(G) K(G ))d n f k k (x)dx (k k )d n ( ) (Suzuki, 2006)? k < k (, ARMA) (k k )d n f k k (x)dx

19 ARMA k = k + 1, k + 2, 1 2{I (k, x n ) I (k 1, x n )} = nˆλ 2 k,k + d n (Hannan-Quinn, 1979) µ k := nˆλ k,k N (0, 1) k k 2{I (k, x n ) I (k 1, x n )} = µ 2 j χ 2 k k j=k +1 j=k +1

20 ARMA {X i } i= : S n := n j=1 X j Hyde, E[X 0 ] = 0, E[X 2 0 ] < X 0 G (G F) 2 j=1 E[X je[x N G]] N 1 3 j=j E[X je[x N G]] J N 0 = S n /(σ n) N (0, 1)

21 BN BN 1 X 2 = ϵ (2) N (0, σ 2 2 ) 2 λ (3) 1 X 1 + λ (3) 2 X 2 + X 3 = ϵ (3) N (0, σ 2 3 ) 3 λ (4) 2 X 2 + λ (4) 3 X 3 + X 4 = ϵ (4) N (0, σ 2 4 ) 4 λ (5) 1 X 1 + λ (5) 4 X 4 + X 5 = ϵ (5) N (0, σ 2 5 ) X 1 X 3 X 4 X 5 X 2

22 BN BN i = 1,, N j π(i) λ (i) j X j + X i = ϵ (i) n x n = (x 1,, x n ) x m = (x m,1,, x m,n ) X 1 (Ω) X N (Ω), m = 1,, n Yule-Walker c j,h := 1 n n m=1 x m,jx m,h, j, h π(i) {i} j π(i) λ (i) j c j,h + c i,h = σi 2 δ i,h, h π(i) {i} ( π(i) + 1 π(i) + 1 )

23 BN BN π (i) = π(i) π (i) π(i) π (i) π(i) d n d n = 2 log log n ( ) ( ) n 0 f π(i) π (i) (x)dx ( π(i) π (i) )d n

24 BN ARMA BN : ARMA ( ) : BN d n ( ) ( )