1 3 1.1............................... 3 1.2.............................. 4 1.3.............................. 6 2 Self-Organizing Map:SOM 7 2.1 SOM..

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1 3 1.1............................... 3 1.2.............................. 4 1.3.............................. 6 2 Self-Organizing Map:SOM 7 2.1 SOM................................ 7 2.2 SOM.............................. 8 2.2.1 SOM................... 12 2.3 SOM.................. 13 2.3.1................. 13 2.4 (S-SOM)................... 15 2.4.1 S-SOM............ 16 2.4.2 S-SOM.................. 18 2.4.3 S-SOM U-matrix................... 21 3 24 3.1 (HMM).................. 24 3.2 HMM..................... 24 3.2.1 Forward algorithm....................... 25 3.2.2 Backward algorithm...................... 30 3.2.3 Viterbi algorithm........................ 32 3.2.4 Baum-Welch algorithm..................... 33 3.3 HMM..................... 43 4 (HMM-S-SOM) 51 1

4.1 HMM-S-SOM................. 52 4.2........................ 54 4.2.1 HMM-S-SOM....... 58 5 (F-HMM-S-SOM) 61 5.1 F-HMM-S-SOM........ 67 5.1.1 F-HMM-S-SOM...... 67 5.2 F-HMM-S-SOM.......... 68 5.2.1 DNA............. 68 5.2.2.............. 71 6 83 6.1.................. 87 6.1.1................... 89 6.1.2 2................... 91 6.1.3 3................... 93 7 96 98 99 101 2

1 1.1,,,,,,,,,,,,,k-means,,,, 3

1.2,,,,,,,,,,,, Siri,,,,,,, [1][2],,, [3],,,,,,,,,,, 4

,,,,,,, [4], [5], 5

1.3,,,,,, [6][7],,,,,,,, 3,, 3 6

2 Self-Organizing Map:SOM 2.1 SOM, (TKohonen) SOM,,,, 1, data 2, data3, (data1, data2),,,, 1: SOM 7

2.2 SOM, ( 2) SOM,,,data2 data3,data2,data3,data1 2 2: SOM SOM,data1, 2,SOM data1data3, 6 6 36,, SOM,,1 1,,,, ( 3), 8

,, 3: SOM,data1,data1 3, (winner node) data1, 4,,,, data1 data1 data1,data2, 9

4: data1 data2, data1 5, data2 data2, data2 data3 data2 6,,,data3 data2,data2 data1,,,,,, 10

5: data2 6: data3 11

, SOM,,, SOM 2.2.1 SOM STEP 1: ()v j SOM STEP 2: v j w N j k N j k N j k = min f j (k) (1) k f j (k) = n (a jt w kt ) 2 (2) STEP 3: N j k w k v j t=1 h(p jx, P jx ) = exp( P jx P jx 2 σ 2 ) (3) w k = β h(p jx, P jx )f j (k) (4) w new k = w k + w k (5) 12

P jx N j k P jx P jx P jx σ β 7 7: 24 w 2.3 SOM 2.3.1 8 Data1Data5 0255 4 R,G,B SOM Data2 Data5,, 13

, SOM SOMPlane-SOM,,,, SOM ( 9:), SOM,, SOM ( 9:), 14

8: SOM 2.4 (S-SOM) SOM SOM [8][9][10][11] [9] OpenGL Windows Windows API DirectX [12][13] 15

9: SOM 2.4.1 S-SOM DirectX D 10, 16

10: 11 11: S-SOM SOM 17

SOM SOM S-SOM SOM SOM 12 12: S-SOM 2.4.2 S-SOM n m v j = {a j1, a j2, a j3,..., a jn }(j = 1, 2, 3,..., m) S- SOM l N k (k = 1, 2, 3,..., l) w k = {w k1, w k2, w k3,..., w kn } 18

S-SOM STEP 1: w w STEP 2: v v j SOM STEP 3: v j w N j k N j k N j k = min k f j (k) (6) f j (k) = n (a jt w kt ) 2 (7) t=1 STEP 4: N j k w k v j h(p jx, P jx ) = exp( P jx P jx θ 2 ) (8) w k = β h(p jx, P jx )f j (k) (9) w new k = w k + w k (10) P jx N j k P jx P jx P jx θ β 19

θ 13β 13: S-SOM 20

2.4.3 S-SOM U-matrix SOM S-SOM 14 1 0 14: S-SOM U-matrix 21

S-SOM U-matrix S-SOM n n-1 S-SOM n 16 15 U-matrix T g U g ab U g ab N a N b U g ab = U g ba U g ab, U g ba U g N k (k = 1, 2, 3,..., l) w k = {w k1, w k2, w k3,..., w kn } SOM U g ab n U g ab = (w aj w bj ) 2 (11) j=1 T g N a V a, V g L V g = L (αu g + max h(u h ) M min h (U h ) ) (12) max h (U h ) min h (U h ) V a = L 1 6 j (αu g aj + max h(u h ) M min h (U h ) max h (U h ) min h (U h ) ) (13) α = M 1 max h (U h ) min h (U h ) (14) V g min(u h ) 0, max (U h ) M U g L h h 22

M L V a N a 6 U a L 15: S-SOM U-matrix 16: S-SOM U-matrix 23

3 3.1 (HMM) Baum-Welch algorithm Viterbi algorithm 3.2 HMM HMM Q = {q 1, q 2, q 3,..., q k } : q a i,j : q i q j j a i,j = 1 s i (x) : i x x s i (x) = 1 w[t] : t 24

Θ : HMM 3.2.1 Forward algorithm Forward algorithm HMM w w N R N R R M R R m (W ) = {r m [0], r m [1], r m [2],..., r m [N]} (m = 1, 2, 3,..., M) (15) r m [t] R m (W ) t S L w M N L w = s g(m,t 1) (w[t]) a g(m,t 1),j g(m, t) = r m [t] (16) m=1 t=1 17 A, B, C HMM w AABC w 1 25

17: 26

1: w R 1 R 3 w R 1 : a 0,0 s 0 (w[1]) a 0,1 s 0 (w[2]) a 1,2 s 1 (w[3]) a 2,3 s 2 (w[4]) = a 0,0 s 0 (A) a 0,1 s 0 (A) a 1,2 s 1 (B) a 2,3 s 2 (C) = 0.2 0.3 0.8 0.3 0.5 0.1 0.3 0.2 = 0.0000432 (17) R 2 : a 0,1 s 0 (w[1]) a 1,1 s 1 (w[2]) a 1,2 s 1 (w[3]) a 2,3 s 2 (w[4]) = a 0,1 s 0 (A) a 1,1 s 1 (A) a 1,2 s 1 (B) a 2,3 s 2 (C) = 0.8 0.3 0.5 0.7 0.5 0.1 0.3 0.2 = 0.0002520 (18) R 3 : 27

a 0,1 s 0 (w[1]) a 1,2 s 1 (w[2]) a 2,2 s 2 (w[3]) a 2,3 s 2 (w[4]) = a 0,1 s 0 (A) a 1,2 s 1 (A) a 2,2 s 2 (B) a 2,3 s 2 (C) = 0.8 0.3 0.5 0.7 0.7 0.3 0.3 0.2 = 0.0010584 (19) w L w 17 19 L w = 0.0000432 + 0.0002520 + 0.0010584 = 0.0013536 (20) 17 19 Forward algorithm f j (t) = s i (w[t]) q i Q f i (t 1)a i,j t 1 (21) f 0 (0) = 1 f j (t) t q j w = {w[1], w[2],..., w[t]} 28

Forward algorithm, L w f 0 (0) = 1.0 f 0 (1) = f 0 (0) a 0,0 s 0 (w[1]) = f 0 (0) a 0,0 s 0 (A) = 1.0 0.2 0.3 = 0.06 f 1 (1) = f 0 (0) a 0,1 s 0 (w[1]) = f 0 (0) a 0,1 s 0 (A) = 1.0 0.8 0.3 = 0.24 f 1 (2) = f 0 (1) a 0,1 s 0 (w[2]) + f 1 (1) a 1,1 s 1 (w[2]) = f 0 (1) a 0,1 s 0 (A) + f 1 (1) a 1,1 s 1 (A) = 0.06 0.8 0.3 + 0.24 0.5 0.7 = 0.0984 f 2 (2) = f 1 (1) a 1,2 s 1 (w[2]) = f 1 (1) a 1,2 s 1 (A) = 0.24 0.5 0.7 = 0.084 f 2 (3) = f 1 (2) a 1,2 s 1 (w[3]) + f 2 (2) a 2,2 s 2 (w[3]) = f 1 (2) a 1,2 s 1 (B) + f 2 (2) a 2,2 s 2 (B) = 0.0984 0.5 0.1 + 0.084 0.7 0.3 = 0.02256 f 3 (4) = f 2 (3) a 2,3 s 2 (w[4]) = f 2 (3) a 2,3 s 2 (C) = 0.02256 0.3 0.2 = 0.0013536 L w = f 3 (4) = 0.0013536 20 29

3.2.2 Backward algorithm Forward algorithm w q j Backward algorithm q j w = {w[n], w[n 1],..., w[t + 1]} b j (t) b i (t) = q j Q b j (t + 1)a i,j s i (w[t + 1]) (22) 17 AABC L w Backward algorithm 30

b 3 (4) = 1.0 b 2 (3) = b 3 (4) a 2,3 s 2 (w[4]) = b 3 (4) a 2,3 s 2 (C) = 1.0 0.3 0.2 = 0.06 b 2 (2) = b 2 (3) a 2,2 s 2 (w[3]) = b 2 (3) a 2,2 s 2 (B) = 0.06 0.7 0.3 = 0.0126 b 1 (2) = b 2 (3) a 1,2 s 1 (w[3]) = b 2 (3) a 1,2 s 1 (B) = 0.06 0.5 0.1 = 0.003 b 1 (1) = b 2 (2) a 1,2 s 1 (w[2]) + b 1 (2) a 1,1 s 1 (w[2]) = b 2 (2) a 1,2 s 1 (A) + b 1 (2) a 1,1 s 1 (A) = 0.0126 0.5 0.7 + 0.003 0.5 0.7 = 0.00546 b 0 (1) = b 1 (2) a 0,1 s 0 (w[2]) = b 1 (2) a 0,1 s 0 (A) = 0.003 0.8 0.3 = 0.00072 b 0 (0) = b 1 (1) a 0,1 s 0 (w[1]) + b 0 (1) a 0,0 s 0 (w[1]) = b 1 (1) a 0,1 s 0 (A) + b 0 (1) a 0,0 s 0 (A) = 0.00546 0.8 0.3 + 0.00072 0.2 0.3 = 0.0013536 L w = b 0 (0) = 0.0013536 Backward algorithm b 0 (0) Forward algorithm f 3 (4) b 0 (0) = f 3 (4) 31

3.2.3 Viterbi algorithm Forward algorithm f j (t) = s i (w[t]) f i (t 1)a i,j q i Q w Viterbi algorithm W Rm(w) N L w = arg max s g(m,t) (w[t]) a g(m,t),j g(m, t) = r m [t] (23) t=0 f j (t) = s i (w[t]) max q Q f i(t 1)a i,j (24) 32

3.2.4 Baum-Welch algorithm HMM Viterbi algorithm HMM Θ Θ w Viterbi algorithm w Θ Baum-Welch algorithm Θ HMM w Θ Θ Forward algorithm L w 18 left-to-light 33

18: Baum-Welch algorithm Baum-Welch algorithm O i,j : W = {w 1, w 2,..., w l } q i q j E i (x) : W = {w 1, w 2,..., w l } x q i P (w k Θ) : W = {w 1, w 2,..., w l } Θ w k L w P (w k Θ) = L w W P (W Θ) P (W Θ) = l k=1 L k l 34

, O i,j E i (x) t 1 O i,j = l k=1 1 fi k (t 1) a i,j s i (w k [t]) b k j (t) (25) P (w k Θ) t E i (x) = l k=1 1 P (w k Θ) t:w k [t]=x f k i (t 1) b k i (t 1) (26) O i,j E i (x) a i,j s i (x) a i,j = s i (x) = O i,j j O i,j E i(x) x E i (x ) (27) (28) 35

17 AABC AABC L w O i,j = = = t 1 1 f i (t 1) a i,j s i (w 1 [t]) b j (t) P (w k Θ) k=1 t 1 f i (t 1) a i,j s i (w 1 [t]) b j (t) P (w 1 Θ) t f i (t 1) a i,j s i (w 1 [t]) b j (t) L w t q i q j ξ t (i, j) ξ t (i, j) = f i(t 1) a i,j s i (w k [t]) b j (t) L w O i,j = t ξ t (i, j) PROCESS1PROCESS4 36

PROCESS 1 ξ t (i, j) ξ t (i, j) ξ 1 (0, 0) = f 0(0) a 0,0 s 0 (w 1 [1]) b 0 (1) = f 0(0) a 0,0 s 0 (A) b 0 (1) L w L w 1.0 0.2 0.3 0.00072 = = 0.03191489 0.00135360 ξ 1 (0, 1) = f 0(0) a 0,1 s 0 (w 1 [1]) b 1 (1) = f 0(0) a 0,1 s 0 (A) b 1 (1) L w L w 1.0 0.8 0.3 0.00546 = = 0.96808510 0.00135360 ξ 2 (0, 1) = f 0(1) a 0,1 s 0 (w 1 [2]) b 1 (2) = f 0(1) a 0,1 s 0 (A) b 1 (2) L w L w 0.06 0.8 0.3 0.003 = = 0.03191489 0.00135360 ξ 2 (1, 1) = f 1(1) a 1,1 s 1 (w 1 [2]) b 1 (2) = f 1(1) a 1,1 s 1 (A) b 1 (2) L w L w 0.24 0.5 0.7 0.003 = = 0.18617021 0.00135360 ξ 2 (1, 2) = f 1(1) a 1,2 s 1 (w 1 [2]) b 2 (2) = f 1(1) a 1,2 s 1 (A) b 2 (2) L w L w 0.24 0.5 0.7 0.0126 = = 0.78191489 0.00135360 ξ 3 (1, 2) = f 1(2) a 1,2 s 1 (w 1 [3]) b 2 (3) = f 1(2) a 1,2 s 1 (B) b 2 (3) L w L w 0.0984 0.5 0.1 0.06 = = 0.21808510 0.00135360 ξ 3 (2, 2) = f 2(2) a 2,2 s 2 (w 1 [3]) b 2 (3) = f 2(2) a 2,2 s 2 (B) b 2 (3) L w L w 0.084 0.7 0.3 0.06 = = 0.78191489 0.00135360 ξ 4 (2, 3) = f 2(3) a 2,3 s 2 (w 1 [4]) b 3 (4) = f 2(3) a 2,3 s 2 (C) b 3 (4) L w L w 0.02256 0.3 0.2 1.0 = = 1.0 0.00135360 37

ξ 1 (0, 0) + ξ 1 (0, 1) = 1.0 ξ 2 (0, 1) + ξ 2 (1, 1) + ξ 2 (1, 2) = 1.0 ξ 3 (1, 2) + ξ 3 (2, 2) = 1.0 ξ 4 (2, 3) = 1.0 PROCESS 2 a i,j PROCESS 1 ξ t (i, j) a i,j = O i,j j O i,j O i,j O i,j = ξ t (i, j) t j 29 O i,j = t ξ t (i, j) a i,j j a i,j = t ξ t(i, j) j ξ t(i, j) t (29) a i,j 38

a 0,0 = = a 0,1 = = a 1,1 = = a 1,2 = = a 2,2 = = a 2,3 = = ξ 1 (0, 0) ξ 1 (0, 0) + ξ 1 (0, 1) + ξ 2 (0, 1) 0.03191489 0.03191489 + 0.96808510 + 0.03191489 0.03092783 ξ 1 (0, 1) + ξ 2 (0, 1) ξ 1 (0, 0) + ξ 1 (0, 1) + ξ 2 (0, 1) 0.03191489 0.03191489 + 0.96808510 + 0.03191489 0.96907217 ξ 2 (1, 1) ξ 2 (1, 1) + ξ 2 (1, 2) + ξ 3 (1, 2) 0.18617021 0.18617021 + 0.78191489 + 0.21808510 0.15695067 ξ 2 (1, 2) + ξ 3 (1, 2) ξ 2 (1, 1) + ξ 2 (1, 2) + ξ 3 (1, 2) 0.78191489 + 0.21808510 0.18617021 + 0.78191489 + 0.21808510 0.84304933 ξ 3 (2, 2) ξ 3 (2, 2) + ξ 4 (2, 3) 0.78191489 0.78191489 + 1.0 0.43880597 ξ 4 (2, 3) ξ 3 (2, 2) + ξ 4 (2, 3) 1.0 0.78191489 + 1.0 0.56119403 a i,j a i,j 39

PROCESS 3 s i (x) s i (x) l 1 L w P (w k Θ) k=1 l 1 1 P (w k Θ) = 1 P (w 1 Θ) = 1 L w k=1 k=1 26 E i (x) = 1 L w t:w k [t]=x f i (t 1) b i (t 1) (30) 3022 E i (x) = 1 L w = 1 = = t:w k [t]=x L w t:w k [t]=x f i (t 1) b i (t 1) f i (t 1) b j (t)a i,j s i (w[t]) q j Q t:w k [t]=x f i(t 1) q j Q b j(t)a i,j s i (w[t]) t:w k [t]=x q j Q L w f i (t 1) b j (t)a i,j s i (w[t]) L w (31) ξ t (i, j) = f i(t 1) a i,j s i (w k [t]) b j (t) L w 31 E i (x) = t:w k [t]=x q j Q f i (t 1) b j (t)a i,j s i (w[t]) L w = t:w k [t]=x q j Q ξ t (i, j) (32) 40

28 s i (x) = = = E i (x) x E i (x ) t:w k [t]=x x t:w k [t]=x t t:w k [t]=x q j Q ξ t(i, j) q j Q ξ t(i, j) q j Q ξ t(i, j) q j Q ξ t(i, j) (33) 41

33 s i (x) s 0 (A) = ξ 1(0, 0) + ξ 1 (0, 1) + ξ 2 (0, 1) ξ 1 (0, 0) + ξ 1 (0, 1) + ξ 2 (0, 1) = 1.0 s 0 (B) = 0 ξ 1 (0, 0) + ξ 1 (0, 1) + ξ 2 (0, 1) = 0.0 s 0 (C) = 0 ξ 1 (0, 0) + ξ 1 (0, 1) + ξ 2 (0, 1) = 0.0 s 1 (A) = = s 1 (B) = = s 1 (C) = s 2 (A) = s 2 (B) = = s 2 (C) = = ξ 2 (1, 1) + ξ 2 (1, 2) ξ 2 (1, 1) + ξ 2 (1, 2) + ξ 3 (1, 2) 0.18617021 + 0.78191489 0.18617021 + 0.78191489 + 0.21808510 0.81614350 ξ 3 (1, 2) ξ 2 (1, 1) + ξ 2 (1, 2) + ξ 3 (1, 2) 0.21808510 0.18617021 + 0.78191489 + 0.21808510 0.18385650 0 ξ 2 (1, 1) + ξ 2 (1, 2) + ξ 3 (1, 2) = 0.0 0 ξ 3 (2, 2) + ξ 4 (2, 3) = 0.0 ξ 3 (2, 2) ξ 3 (2, 2) + ξ 4 (2, 3) 0.78191489 0.78191489 + 1.0 0.43880597 ξ 4 (2, 3) ξ 3 (2, 2) + ξ 4 (2, 3) 1.0 0.78191489 + 1.0 0.56119403 42

3.3 HMM Forward algorithmbackward algorithmbaum-welch algorithm 17 AABC 1 t G = {G 1, G 2,...G t,...g l } k G t t k k G t = Γ t (0, 0) Γ t (0, 1) Γ t (0, k 1) Γ t (1, 0) Γ t (1, 1) Γ t (1, k 1)...... Γ t (k 1, 0) Γ t (k 1, 1) Γ t (k 1, k 1) 4 k = 4 G t Γ t (i, j) t i s i (w[t]) i j 0 i s i (w[t]) i j 1 0 1 t = 1 A i A Γ 1 (i, j) = 1, i 43

A i j Γ 1 (i, j) = 1 i j t = 1 t = 4 44

t = 1 t = 1 0 i = 0 G 1 i 0 1 t = 4 t = 4 4 i 3(... k 1 = 4 1 = 3) a i,3 0 i 1, 0 i = 0 1 Γ 1 (0, 0) = 0 (... s 0 (A) a 0,0 0) Γ 1 (0, 1) = 0 (... s 0 (A) a 0,1 0) Γ 1 (0, 2) = 1 (... s 0 (A) a 0,1 = 0) Γ 1 (0, 3) = 1 (... s 0 (A) a 0,1 = 0) 45

Γ 1 (1, 0) = 1 Γ 1 (2, 0) = 1 Γ 1 (3, 0) = 1 Γ 1 (1, 1) = 1 Γ 1 (2, 1) = 1 Γ 1 (3, 1) = 1 Γ 1 (1, 2) = 1 Γ 1 (2, 2) = 1 Γ 1 (3, 2) = 1 Γ 1 (1, 3) = 1 Γ 1 (2, 3) = 1 Γ 1 (3, 3) = 1 t G t 19 46

19: G t G t STEP 1 G 1 j 1 2 G 2 j 1 20 G 1 G 2 G 3, G 4 t = 1 G t j 1 G t+1 j 1 t = l 1 G 47

20: G STEP t = l G t i 1 G t 1 i 1 t = 2 G 21 48

21: G STEP STEP t i j G t Γ t (i, j) 0 1 0 1 AABC 22 1 49

22: G 50

4 隠れマルコフ球面自己組織化マップ (HMM-S-SOM) 隠れマルコフ球面自己組織化マップとは,S-SOM のノードに隠れマルコフモデル を用いたものであり 図 23, 入力データには, ベクトルではなく文字列集合を用い る. ノード上に内包された隠れマルコフモデルは, すべて同じ構造を持つ この自 己組織化マップは, データを直接分類するのではなくデータの背景にある確率モデ ルにもとづいて, 確率モデルを球面上にクラスタリングし, その分類結果を視覚的に も容易に提示することのできる学習モデルである S-SOM と HMM-S-SOM の違い については表 2 に示す ノードや入力データの形式の違い以外に, 勝者ノードの決 定方法や, ノードの更新の方法が異なる その詳細については, 次の HMM-S-SOM の学習アルゴリズムの説明に記す 図 23: 隠れマルコフ球面自己組織化マップ 51

2: S-SOM HMM-S-SOM 4.1 HMM-S-SOM STEP 0: HMM Θ k STEP 1: STEP 2: Baum-Welch algorithm STEP 3,, STEP 3: HMM Baum-Welch algorithm, HMM SOM 52

SOM, STEP1STEP3, STEP2,,,, 24,,,Baum-Welch algorithm HMM,,,,Baum-Welch algorithm HMM,STEP3 24: 53

4.2, left to right, Ergodic, U-matrix, U-matrix U-matrix,,,, A,B,C 3 HMM HMM M 1, M 3,..., M 10,,,, 25 312, 1322 54

3: M 1 4: M 2 5: M 3 6: M 4 7: M 5 8: M 6 55

9: M 7 10: M 8 11: M 9 12: M 10 13: M 1 14: M 2 56

15: M 3 16: M 4 17: M 5 18: M 6 19: M 7 20: M 8 57

21: M 9 22: M 10 25: 4.2.1 HMM-S-SOM HMM-S-SOM 26 26 M 1 Data1 M 1 M 10 M 1 M 10 M 3, M 8, M 9, M 7, M 2, M 5, M 6, M 4 M 1,U-matrix HMM, U-matrix HMM 58

26:, HMM,, 27,,,,model 1 model 4,HMM -matrix,, 59

27: 60

5 (F-HMM-S-SOM), Baum-Welch algorithm,,som,hmm,, SOM HMM, (F-HMM-S-SOM) ( 28) 28: F-HMM-S-SOM,, HMM,, 61

, F-S-HMM-SOM, A,B,C, n, n + 1,A,B,C,,, A,B,C, l, l 1 l, 12 (l 1) m, l max, (m 2 + m)(l max 1) ACCB, BCC,,, 2,ACCB, AC, 29 AC, 0, 000001000000, CC 000000000001, CB 000000000010 ACCB 62

000001000000000000000001000000000010,BCC 000000001000000000000001 29:,,, 30,, 0( 30: ) Θ k : HMMS-SOMHMMΘ k = {A k, S k } 63

30: A k = a k 01 a k 02 a k 0j a k 11 a k 12 a k 1j...... S k = s k 01 s k 02 s k 0h s k 11 a k 12 s k 1h...... a k i1 a k i2 a k ij s k i1 s k i2 s k ih a i,j i j s i,h i x h X : X = {x 1, x 2, x 3,..., x h } V = {v 1, v 2, v 3,..., v h } STEP 1: HMM Θ k STEP 2: W n Baum-Welch algorithm N i HMM L i, N i E i STEP 3:, 64

(1 E i E max ) L i,e max E i STEP 4: Baum-Welch algorithm, STEP 5:,SOM N n k Θ k STEP 4 Nk n HMM Θ n k = {An k, Sn k } A = β h(p nx, P nx )(A n k A k) A new k = A k + A S = β h(p nx, P nx )(Sk n S k) S new k = S k + S Θ new k Θ k Θ new k = {A new k, Sk new } h(p nx, P nx ) = exp( P nx P nx θ 2 ) P jx N j k P jx 65

P jx P jx β,baum-welch algorithm STEP 6: SOM SOM STEP2STEP5,,, U-matrix, 66

5.1 F-HMM-S-SOM,, HMM-S-SOM, 5.1.1 F-HMM-S-SOM HMM U-matrix 31,, 31: HMM U-matrix, ( 32),, 67

32: U-matrix 5.2 F-HMM-S-SOM,,,DNA (HMM-S-SOM) 5.2.1 DNA, DNA ATGC,ATGC [14], 1024, 6 1024, (Hsall), Mmuall, (Cfaall), Ecoall, (Dmaall), (Osaall),HMM- S-SOM, U-matrix 33 68

33: HMM-S-SOM,,.. F-HMM-S-SOM,HMM U-matrix, ( 34), ( 35),,,,,,,,,, 69

34: F-HMM-S-SOM : U-matrix 35: F-HMM-S-SOM : U-matrix 70

5.2.2, 2010 36,,,,,,, 36,,,, U, D, S, U D,, F-HMM-S-SOM,F- HMM-S-SOM HMM, 37 4, U,S,D 3,Umatrix, 71

36: 72

37:,,. t t+1, HMM-S-SOM F-HMM-S-SOM 38 39 0 38, 22 46 39, S-HMM-SOM, F-S-HMM-SOM,, S-HMM-SOM,U-matrix,,F-S-HMM-SOM, 73

38: 74

39: 75

, U-matrix,,,, F-S-HMM-SOM 1.377,S-HMM-SOM 2.623,,., 40,, S-HMM-SOM 40,,,, F-S-HMM-SOM 40,,,,,,,,,,,,,,,,, 76

40: 77

,, 41,,,, 0 23 42, 24 46 43 5,,,,, 4,,,,,, 78

41: 79

図 42: 移動平均線との比較 80

図 43: 移動平均線との比較 81

230 36 23 23: 82

6,,,,,,,,,,,,, 44, 1,2, SOM v j x i 1, (mask-vector)w i w i,, 0 1 1 0,1,0 83

44:, 1, 0 1,1 2, 30 5,,,, 45: SOM,, 45, h a, 84

STEP 1: 1 x i = {x i0, x i1,..., x in }, h a = {h a0, h a1,..., h an } 2 w i = {w i0, w i1,..., w in } 1 STEP 2: i,l 1 [argmin] i {w ij (h aj v aj w ij x ij )} L 2 (34) j L = j w2 ij w i w i STEP 3: wi,, µ a µ a µ ai ω 46 46: 85

(35) m [argmin] m { 1 m m k=2 ( ω k ω m ) 2 } ϕ (35),ϕ 0.5 µ t ω k f(t) = k, f(t) m w new it = w old it + γ(1 w old it ) (36) m < f(t) w new it = w old it + γ(0 w old it ) (37),γ 0 < γ < 1, wi w i 1 0 h a STEP 4:, x new j = x old j + α(v a x old j ) (38) w new j = w old j + β(w i w old j ) (39) α,,0 < α < 1, β 0 < β < 1 86

STEP2STEP4,,, h a, 6.1, h a, 1 3,,, noise Mersenne Twister[15] 01 1x,y,z y = 3 x + noise z = noise 2x 0.5, x,y,z x,y,z 87

47: y = 3 x + noise z = noise (x 0.5) y = 2 x + noise z = noise (x > 0.5) 3x 0.5,x,z, x 0.5,y z y = 3 x + 2 noise z = noise (x 0.5) 88

48: 2 y = 2 z + noise + 2 x = noise (x > 0.5) 6.1.1 50,, x y model 1 100 z, x y 0.819, SOM 51 SOM, 8, 89

49: 3 50: 90

51: SOM, x y, 6.1.2 2 52, x y, x y z, 95. 0.335, SOM 53,, x y z, x y 2,model 91

52: 53: SOM 92

6.1.3 3 3, 54 55 54 xy x y, y,x,y,y z x 0.5 0.5,, 78, 54: yz,xy, 56 93

55: 56: 94

56, 54 yz,yz, yz,xy,x, yz,x,, xy,,, SOM, 57 57: SOM,, x z SOM, 95

7,,,,,,,, DNA,,,,,,,,,,,,,,,,,,,,.,,,, 96

, k-means,., 2,,,.,, SOM F-HMM-S-SOM,, 97

,,,,,,,,,.,,, 98

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[10],,,, : SOM,, Vol.2010-MPS-77 No.29,2010. [11] Ying Xin WU SOM Vol.19, No.6, pp.611-617,2007. [12]N2Factory DirectX. [13]Jeffrey Richter, Christophe Nasarre:Advanced Windows 5, BP,2008. [14]Hiroshi Dozono. et.al:a Design Method of DNA Chips for sequence Analysis Using Self Organizing Maps, Proceeding of WSOM 2003,pp.116-121,2003. [15]M. Matsumoto and T. Nishimura:A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. on Modeling and Computer Simulation Vol. 8, No. 1, January pp.3-30,1998. 100

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