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1 27 F13J011D

2 ( ) Rotational Random Shuffling ,2 1

3 (CHPCA) (RMT) Rotational Random Shuffling(RRS) A (PCA) 38 A B 42 B B B C 49 D 50 D D D E PCA CHPCA 54 2

4 1 1.1 (PCA) (RMT) [1 6] PCA RMT RMT Rotational Random Shuffling(RRS) [7] PCA (CHPCA) [8, 9] CHPCA ( ) 3

5 2000 [10] 350 [11] CHPCA [12] CHPCA RMT RRS [12] [19] ( ) 4

6 2 2.1 (CHPCA) (CHPCA) [8,9] x µ (t) (µ = 1,, N) y µ (t) ξ µ (t) = x µ (t) + iy µ (t), (2.1.1) y µ (t) x µ (t) y µ (t) = 1 π x µ (u) du. (2.1.2) t u u. y µ (t) x µ (t) π/2 {ξ µ (t)} C = 1 T ΞΞ, (2.1.3) Ξ = ξ 1 (t) ξ 2 (t). ξ n (t). (2.1.4) Ξ ξ µ (t) N T Ξ Ξ C Cα l = λ l α l, (2.1.5) α l α m = δ lm. (2.1.6) (CHPCA) λ l α l C 5

7 x µ µ = 1 µ = t 2.1: sin cos C C = N λ l = N, (2.1.7) l=1 N λ l αα, (2.1.8) l=1 λ 1 > λ 2 > > λ N CHPCA sin cos x 1 (t) = sin ( ) ( ) πt πt, x 2 (t) = cos (2.1.9) t = 0, 1, 2,, 99, 2.1 PCA ( ) C =, (2.1.10) C 12 = CHPCA ( ) e C π =. (2.1.11) 0.985e 0.515π 1 6

8 ( λ 1 = 1.985, α 1 = 1 2 ( λ = 0.015, α 2 = e 0.515π 1 e π ) ), (2.1.12), (2.1.13) 0.515π 0.5π C 12 = sin cos CHPCA 2.2 (RMT) ρ(λ) RMT ρ(λ) RMT λ λ 2.2: RMT N % 3. 7

9 C m = λ m N i=1 λ. (2.2.1) i m 70% 70% 70% RMT λ Q = T/(2N) > 1 N, T 1 ρ(λ) = Q (λ+ λ)(λ λ ), (2.2.2) 2π λ λ ± = ( 1 ± 1 Q ) 2. (2.2.3) [λ, λ + ] 2.2 λ Rotational Random Shuffling(RRS) RMT RMT 2.2 RMT Rotational Random Shuffling(RRS) RMT RRS 1 PCA Q = T/N CHPCA RMT Q = T/(2N) [13] 8

10 2.3: Rotational Random Shuffling RRS RMT 9

11 3 8 N = T = 360 (CPI) 2 45 (PPI) 3 23 (IPI) 3 10 (USD/JPY) 4 (Leading Index) 5 (Coincident Index) 5 (Lagging Index) 5 (Money Stock) 6 (CPI) (PPI) (PPI) ( ) ( / NEEDS 5 Composite Index M2 M2+CD, M2+CD 10

12 ) (IPI) (IPI) ( CIF ) (CI) (DI) CI ( ) DI ( ) CI DI 3 ( ) 1 r µ (t) = logx µ (t) logx µ (t 1). (3.0.1) r µ (t) = r µ(t) r µ σ µ (3.0.2) 11

13 CPI CGPI IPI USD/JPY Leading Index Coincident Index Lagging Index year 3.1: ( ) 0 1 r µ r µ σ µ r µ Phillips-Perron 4CPI 5CPI 6CPI 24CPI 25CPI 12

14 1: 1 CPI 46 PPI 2 CPI 47 PPI 3 CPI 48 PPI 4 CPI 49 PPI 5 CPI 50 PPI 6 CPI 51 PPI 7 CPI 52 PPI 8 CPI 53 PPI 9 CPI 54 PPI 10 CPI 55 PPI 11 CPI 56 PPI 12 CPI 57 PPI 13 CPI 58 PPI 14 CPI 59 PPI 15 CPI 60 PPI 16 CPI 61 PPI 17 CPI 62 PPI 18 CPI 63 PPI 19 CPI 64 PPI 20 CPI 65 PPI 21 CPI 66 PPI 22 CPI 67 PPI 23 CPI 68 PPI 24 CPI 69 IPI 25 CPI 70 IPI 26 CPI 71 IPI 27 CPI 72 IPI 28 CPI 73 IPI 29 CPI 74 IPI 30 CPI 75 IPI 31 CPI 76 IPI 32 CPI 77 IPI 33 CPI 78 IPI 34 CPI 79 USD/JPY 35 CPI 80 Leading Index 36 CPI 81 Coincident Index 37 CPI 82 Lagging Index 38 CPI 83 Money Stock 39 CPI 40 CPI 41 CPI 42 CPI 43 CPI 44 CPI 45 CPI 13

15 4 4.1 ρ(λ) λ RMT 4.1: 4.1 RMT λ + (= 2.82) 6 (λ 1 = 10.43, λ 2 = 5.97, λ 3 = 5.68, λ 4 = 3.98, λ 5 = 3.04, λ 6 = 2.86) 6 (31.97/83 = 0.385) 38.5% 6 RMT RRS 4.2 RRS RMT RRS 1000 λ l + 3σ l λ l RRS l σ l l l RRS λ l + 3σ l 6 RMT 14

16 ρ(λ) RMT λ l actual RRS RRS+3σ λ l 4.2: RRS (1000 ) α l 2 = N α li 2 = N. (4.2.1) i=1 4.3 Mode (84 ) CHPCA ±2σ 15

17 Eigenvector No.1 Eigenvector No.2 Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Re Eigenvector No.3 Eigenvector No.4 Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Re Eigenvector No.5 Eigenvector No.6 Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Re 4.3: 16

18 +2σ 4.4 ( )e µ α l N N ξ(t) = ξ µ (t)e µ = a l (t)α l, (4.4.1) µ=1 l=1 a l (t) = α l ξ(t). (4.4.2) a l (t) l a la m = δ lm λ l. (4.4.3) a l (t) 3 6 ( ) 5 cosine similarity cos θ lα = α l α m α l α m. (4.4.4) θ lα 2 α l α m 4.4 α l α m cosine similarity N(=83) p

19 Eigenvector No.1 Eigenvector No Eigenvector No.3 Eigenvector No Eigenvector No.5 Eigenvector No : x y 84 ( ) (1000 ±2σ) 18

20 Mode signal No. 1 Mode signal Re Im time Mode signal No. 2 Mode signal time Re Im Mode signal No. 3 Mode signal Re Im time Mode signal No. 4 Mode signal time Re Im Mode signal No. 5 Mode signal time Re Im Mode signal No. 6 Mode signal time Re Im 4.5: 6 19

21 Mode signal No.1 Mode signal Year Mode signal No.2 Mode signal Year Mode signal No.3 Mode signal Year Mode signal No.4 Mode signal Year Mode signal No.5 Mode signal Year Mode signal No.6 Mode signal Year 4.6: 20

22 Mode signal No.1 Mode signal No Period Period Mode signal No.3 Mode signal No Period Period Mode signal No.5 Mode signal No Period Period : 21

23 2: α l (l = 1,, 6) α m (m = 1,, 5) cosine similarity N(=83) p l\m p cos θ % 2 22

24 π π x y ( ) ( ) ( ) (CPI,PPI) 68 CHPCA ( ) 23

25 3: 1 2 Abs 1 2 Mode 1 Mode 2 Abs θ[rad] Category Abs θ[rad] Category Leading Index Leading Index USD/JPY PPI PPI IPI IPI IPI IPI Coincident Index IPI IPI IPI PPI IPI IPI IPI IPI PPI USD/JPY IPI CPI IPI IPI IPI IPI IPI IPI Coincident Index IPI PPI CPI CPI Lagging Index PPI PPI PPI CPI CPI PPI Lagging Index PPI PPI CPI PPI PPI CPI PPI CPI PPI PPI PPI PPI CPI PPI PPI CPI CPI PPI PPI PPI PPI PPI CPI CPI CPI CPI CPI PPI PPI CPI CPI CPI CPI PPI PPI PPI CPI CPI PPI PPI CPI CPI PPI CPI CPI CPI PPI CPI CPI CPI CPI CPI PPI CPI CPI CPI CPI 24

26 Eigenvector No π 2 π 3π 2 2π θ Eigenvector No π 2 π 3π 2 2π θ 4.8: 1 2 x y π 25

27 θ θ 1 4.9: x 1 y 2 4: α l α m cosine similarity l\m

28 4.7 I l (t) S l (t) I l (t) = a l (t) 2 N l=1 a l(t) 2. (4.7.1) S l (t) = t t =1 I l(t ) T t =1 I l(t ). (4.7.2) 4.11 I l (t) S l (t) l ξ lµ (t) = a l (t)α lµ. (4.7.3) ξ µ (t) (a) (1985.3) 1 2 (b) (1991.3) (c) (1995.9) (d) ( ) (e)

29 Mode signal No.1 Relative Intensity Year Mode signal No.2 Relative Intensity Year Mode signal No.3 Relative Intensity Year Mode signal No.4 Relative Intensity Year Mode signal No.5 Relative Intensity Year Mode signal No.6 Relative Intensity Year 4.10: 28

30 Mode signal No.1 Mode signal No.2 Cumulative Intensity (a) (c) (d) (e) Cumulative Intensity (b) (d) Year Year 4.11: 1 (a) (b) (c) (d) (e) , ( 1 2 )

31 Mode 1 Mode 2 (a) Mode 1 USD/JPY Mode 2 USD/JPY Month Month (b) Mode 1 USD/JPY Mode 2 USD/JPY Month Month (c) Mode 1 USD/JPY Mode 2 USD/JPY Month Month (d) Mode 1 USD/JPY Mode 2 USD/JPY Month Month (e) Mode 1 USD/JPY Month Mode 2 USD/JPY Month 4.12: ( ) 1 2 (a): (1985.9) (b): (1991.3) (c): (d): ( ) (e): 30

32 Mode 1 Mode 2 (a) Mode 1 Coincident Index Mode 2 Coincident Index Month Month (b) Mode 1 Coincident Index Mode 2 Coincident Index Month Month (c) Mode 1 Coincident Index Mode 2 Coincident Index Month Month (d) Mode 1 Coincident Index Mode 2 Coincident Index Month Month (e) Mode 1 Coincident Index Month Mode 2 Coincident Index Month 4.13: ( ) 1 2 (a): (1985.9) (b): (1991.3) (c): (d): ( ) (e): 31

33 Mode 1 + Mode 2 Mode 1 Mode 2 USD/JPY Month USD/JPY 4.14: , ,2 19.8% IT ,2 7 32

34 Mode Year Mode Year Mode 2 Mode Year 4.15:

35 Year Mode 1 + Mode 2 Mode 1 Mode 2 Coincident Index Coincident Index 4.16: 5: IT

36 RMT RRS ,2 CHPCA

37 2 1 2 IT

38 4 OB 37

39 A (PCA) ( ) P {x p }(p = 1, 2,, P ) {x p } M(M P ) {z m } z m = P w pm x p (m = 1, 2,, M) (A.0.1) p=1 z m m {w pm } 1 z 1 {x p } 1 m {z m } {z m }(m = 1, 2,, m 1) 1 P wpm 2 = 1 (A.0.2) p=1 A.1 P N {x np}(n = 1, 2,, N; p = 1, 2,, P ) { x p } {x np } x np = x np x p X x 11 x 12 x 1P x 21 x 22 x 2P X = (A.1.1) (A.1.2) x N1 x N2 x NP 1 z 1 (A.0.1) w 11 w 21 w 1 =. (A.1.3) w P 1 38

40 n 1 z 1 t n1 x n = ( x n1 x n2 x np ) (A.1.4) t n1 = P w p1 x np p=1 = x n w 1 (A.1.5) 1 z 1 t n1 1 N 1 1 t 11 t 21 t 1 = (A.1.6). t N1 t 1 = Xw 1 (A.1.7) 1 t 1 t 1 = 1 N = 1 N = 1 N = 1 N N n=1 t n1 N x n w 1 n=1 ( N P ) w p1 x np n=1 p=1 ( P N ) w p1 x np p=1 1 z 1 σ 2 z 1 n=1 = 0 (A.1.8) σz 2 1 = 1 N 1 t 1 t 1 = 1 N 1 (Xw 1) (Xw 1 ) = w1 Vw 1 0 (A.1.9) 39

41 V V = 1 N 1 X X (A.1.10) (i, j) v ij v ij = = 1 N 1 1 N 1 N x ni x nj n=1 N (x ni x i )(x nj x j ) (A.1.11) n=1 v ij = v ji V = V 1 z 1 (A.0.2) σ 2 z 1 Lagrange Lagrange λ J 1 = w 1 Vw λ(w 1 w 1 1) (A.1.12) J 1 w 1 J 1 w 1 0 J 1 w 1 = = J 1 w 11 J 1 w 21. J 1 w P 1 2 P p=1 v 1pw p1 2 P p=1 v 2pw p1. 2 2λ P p=1 v P pw p1 = 2Vw 1 2λw 1 w 11 w 21. w P 1 = 0 (A.1.13) (V λi)w 1 = 0 (A.1.14) Lagrange λ det V λi = 0 (A.1.15) 40

42 Lagrange λ 1 z 1 w 1 V V P λ P 1 z 1 σ 2 z 1 ( )w 1 1 z 1 σ 2 z 1 (A.1.9) w 1 : (A.1.15) (A.1.9) (A.1.15) w 1 w 1 = 1 σ 2 z 1 = w 1 Vw 1 = w 1 λw 1 = λ (A.1.16) 1 z 1 σ 2 z 1 V λ 1 z 1 σ 2 z 1 V w 1 2 {w m }(m = 2, 3,, M) 1 m z m (m = 2, 3,, M) σ 2 z m {z m }(m = 1, 2,, m 1) 1 m (A.0.2) m w m m 1 {w i }(i = 1, 2,, m 1) (V λ i I)w i = 0 (A.1.17) { wi T 1 (i = j) w j = (A.1.18) 0 (i j) m w m Lagrange J m = w mvw m λ m (w mw m 1) m 1 i=1 µ i w mw i (A.1.19) J m w m m z m (m = 2,, M) z m (m = 1, 2,, m 1) 41

43 J m w m 0 m 1 J m = 2Vw m 2λ m w m µw i w m i=1 = 0 (A.1.20) w j (j = 1, 2,, m 1) (A.1.18) w j Vw m µ j = 0 (j = 1, 2,, m 1) (A.1.21) 1 V = V (A.1.17) (A.1.21) w j Vw m = w mvw j = w mλ j w j = 0 (j = 1, 2,, m 1) (A.1.22) µ j = 0 (j = 1, 2,, m 1) (A.1.23) (A.1.20) (V λ m I)w m = 0 (A.1.24) (A.1.17) m z m σ 2 z m V m 1 m 1 m z m σ 2 z m V m w m B B.1 T N 1 N T H H C = HH C N N 42

44 ρ(λ) 0.4 P(u) B.1: λ Gauss u Porter-Thomas N(0, 1) H = [h ij ] h ij N(0, 1) (B.1.1) C = HH Gauss Wishart λ B.1(a) λ B.1(b) = 10 6 N j=1 u2 ij = N, N = 1000 B.1(b) 1956 C.E.Porter R.G.Thomas [14] Porter-Thomas Gauss P (u i,j ) = 1 ( ) exp u2 i,j (B.1.2) 2π 2 Gauss U( 0.5, 0.5) N = H = [h ij ], h ij U( 0.5, 0.5) (B.1.3) C = HH λ B.2(a) B.1(a) λ Gauss 43

45 ρ(λ) 6 4 P(u) B.2: λ u Porter-Thomas B.2(b) = 10 6 N j=1 u2 ij = N, N = 1000 B.1(b) u ij Porter-Thomas B.1(a) B.2(a) 1988 A.Edelman [15] A.M.Sengupta P.P.Mitra [16] ρ(λ) Q := T/N N T ρ(λ) = Q (λ+ λ)(λ λ ) 2πσ 2 λ (B.1.4) σ 2 H N(0, 1) σ 2 = 1 U( 0.5, 0.5) σ 2 = 1/12 ( λ ± = σ ) 1 Q ± 2 Q (B.1.5) λ [λ, λ + ] [17] (B.1.4) B.1.1 (B.1.4) Dirac Dirac 44

46 Dirac 1 ϵ δ(x) = lim (B.1.6) ϵ 0 π x 2 + ϵ 2 δ(x) = lim I 1 1 (B.1.7) ϵ 0 π x iϵ I z i z = Rz + iiz Rz z I z δ(x) = 1 e ikx dk (B.1.8) 2π N A = [a ij ] (i, j) ã ij A det(a) N det(a) = a ij ã ij (B.1.9) i=1 j Gauss Gauss a π dxe ax2 = (B.1.10) a A N ( ) N N dx i exp x i a ij x j = πn/2 = π N det(a 1 ) (B.1.11) det (A) i=1 i,j=1 A 1 A A 1 = det(a) ij = ( 1) i+j minor(a) ij (B.1.12) (B.1.13) minor(a) ij A i j A 1 ij = ( 1)i+j minor(a) ij det(a) (B.1.14) 45

47 2 Banach Banach Hilbert N T T N N N H Hu = λu (B.1.15) N u 0 λ H u λ λ H N v (λ1 H)u = v (B.1.16) u 1 u = (λ1 H) 1 v (B.1.17) H Banach X D(H) z Hu = zu u D(H) u = 0 (z1 H) 1 X z z R(z) := (z1 H) 1 (B.1.18) H Hu = zu u D(H) u 0 Hu = zu u D(H) u = 0 (z1 H) 1 X z (Spectrum) z H B.1.2 H = [h ij ] N N λ i (i = 1, 2,, N) ρ(λ) ρ(λ) = 1 N N δ(λ λ i ) i=1 46 (B.1.19)

48 H G(λ) G ij (λ) = (λ1 H) 1 ij (B.1.20) G(λ) (Trace ) TrG(λ) = N G ii (λ) = i=1 N 1 λ λ i i=1 (B.1.21) (B.1.7) (B.1.19) ρ(λ) = 1 N N i=1 1 lim ϵ 0 π I 1 λ λ i iϵ 1 = lim ITrG(λ iϵ) ϵ 0 Nπ (B.1.22) G(λ) (B.1.22) ρ(λ) (B.1.21) TrG(λ) = N 1 = λ λ i=1 i λ log (λ λ i ) i = log det(λ1 C) := λ λ Z(λ) (B.1.23) Z(λ) := log det(λ1 C) (B.1.22) (B.1.23) ρ(λ) Gauss (B.1.11) Z(λ) = log det(λ1 C) = 2 log I(λ), (B.1.24) ( ) I(λ) := exp λ N x 2 i 1 N T N ( ) dxi x i x j h ik h jk 2 2 2π i=1 i,j=1 k=1 i=1 (B.1.25) (B.1.23) TrG(λ) TrG(λ) N H (B.1.24) H (B.1.24) N (B.1.24) Z(λ) = 2 log I(λ) = 2 log I(λ) (B.1.26) 47

49 (B.1.26) N T h ik 0 σ 2 /T Gauss (B.1.25) (B.1.26) exp ( 1 2 N i,j=1 k=1 ) ( T x i x j h ik h jk = 1 σ2 T ) T/2 N x 2 i (B.1.27) i=1 q := σ2 T N i=1 x 2 i (B.1.28) (B.1.8) Dirac ( ) N [ ( δ q σ2 1 x 2 i := T 2π exp iζ q σ2 T i=1 N i=1 x 2 i )] dζ (B.1.29) (B.1.25) x i [ T i { Z(λ) = 2 log exp N 4π i 2 ( log(λ σ 2 z) + Q log(1 q) + Qqz )} ] dqdz z := 2iζ/T Q := T/N Q T N (B.1.30) (B.1.30) Qq = z = σ 2 λ σ 2 z, 1 1 q (B.1.31) (B.1.32) (B.1.31) (B.1.32) q(λ) q(λ) = σ2 (1 Q) + Qλ ± [σ 2 (1 Q) + Qλ] 2 4σ 2 Qλ 2Qλ (B.1.33) (B.1.30) (B.1.23) G(λ) = N λ σ 2 z(λ) = NQq(λ) σ 2 (B.1.34) 48

50 (B.1.22) 4σ 2 Qλ [σ 2 (1 Q) + Qλ] 2 ρ(λ) = 2πλσ 2 Q (λ+ λ)(λ λ ) = 2πσ 2 λ ( λ ± = σ ) 1 Q ± 2 Q (B.1.35) (B.1.36) (B.1.37) (B.1.4) λ [λ, λ + ] λ + C x(t) Hilbert y(t) y(t) = 1 x(u) π t u du, (C.0.38) du Hilbert x(t) : x(ω) = x(t) = 1 2π x(t)e iωt dt, x(ω)e iωt dω. y(t) y(ω) y(ω) = = = 1 π = 1 π y(t)e iωt dt ) dt e iωt ( 1 π du x(u) t u dωx(u)e iωu dt e iω(t u) 1 t u du x(u)e iωu dt 1 t e iωt = 1 π x(ω) dt 1 t e iωt 49 (C.0.39) (C.0.40) (C.0.41) (C.0.42) (C.0.43) (C.0.44) (C.0.45)

51 dt 1 t e iωt = { iπ (ω > 0) iπ (ω < 0) 1 (ω > 0) y(ω) = ix(ω) sgn(ω), sgn = 0 (ω = 0) 1 (ω < 0). y(t) = 1 2π (C.0.46) (C.0.47) x(ω) e i π 2 e iωt sgn(ω) dω, (C.0.48) y(t) x(t) Hilbert Hilbert x t T 1 x k = x t = 1 T t=0 T 1 k=0 2πkt i x t e T x k e i 2πkt T Hilbert T 1 2πkt i x(t) = x k e T x k e i π T 2 sgn(k 2 t=0 sgn(k T 2 ) = (C.0.49) (C.0.50) ), (C.0.51) 1 (k > T ) 2 0 (k = T ) 2 (C.0.52) 1 (k < T ) 2 x k π/2 π/2 y t x t π/2 D 50

52 D.1 x µ (t) = x µ(t) x µ (t 12). (D.1.1) N(0, 1) 12 D.2 ρ(λ) RMT λ D.1: D.1 RMT λ + (= 2.85) 5 (λ 1 = 10.5, λ 2 = 6.26, λ 3 = 3.56, λ 4 = 3.07, λ 5 = 2.85) D.2 RRS 1000 λ l + 3σ l λ l RRS l σ l 5 51

53 ρ(λ) RMT λ l actual RRS RRS+3σ λ l D.2: RRS (1000 ) D.3 D

54 Eigenvector No.1 Eigenvector No CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Eigenvector No.3 Re Eigenvector No CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Re Eigenvector No CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re D.3: 5 53

55 E PCA CHPCA Eigenvector No.1 PCA Eigenvector No.1 Im Im φ = 0.02π Re PCA Eigenvector No Re Eigenvector No Im Re Im Re φ = 0.4π PCA Eigenvector No E.1: CHPCA PCA CHPCA PCA 2 PCA RMT RRS 8 E.1 CHPCA PCA CHPCA PCA CHPCA ϕ Re Im Re[ α l ] Im[ α l ] PCA α k cosine similarity 54

56 Re[ α 1 ] α 1 Re[ α 1 ] α 1 Im[ α 1 ] α 2 Im[ α 1 ] α 2 Re[ α 2 ] α 2 Re[ α 2 ] α 2 Im[ α 2 ] α 4 Im[ α 2 ] α 4 = = = = CHPCA PCA CHPCA 1 Re PCA 1 Im PCA 2 PCA CHPCA π/2 1 PCA 1 CHPCA ϕ Re Im PCA cosine similarity Re[ α i ] α j Re[ α i ] α j + Im[ α i] α k Im[ α i ] α k (E.0.1) j, k 55

57 [1] L. Laloux, P. Cizeau, J. P. Bouchaud, and M. Potters, Phys. Rev. Lett. 83, 1467(1999). [2] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, T. Guhr, and H. E. Stanley, Phys. Rev. E 65, (2002) [3] A. Utsugi, K. Ino, and M. Oshikawa, Phys. Rev. E 70, (2004) [4] D. H. Kim and H. Jeong, Phys. Rev. E 72, (2005) [5] V. Kulkarni and N. Deo, Eur. Phys. J. B. 60, 101(2007) [6] R. K. Pan and S. Sinba, Phys. Rev. E 76, (2007) [7] H. Iyetomi et al., Phys. Rev. E83, (2011) [8] T. Barnett, Mon. Wea. Rev. 111, 756(1983). [9] A. Hannachi, I. T. Jolliffe, and D. B. Stephenson, Int. J. Climatol. 27, 1119(2007) [10] Klenow, Peter J and Benjamin A Malin. Handbook of Monetary Economics 3A, North-Holland, 2011, chapter 6. [11] Journal of Political Economy, Octoberr 2004, 112, [12] Yoshikawa, Hiroshi and Aoyama, Hideaki and Fujiwara, Yoshi and Iyetomi, Hiroshi, Deflation/Inflation Dynamics: Analysis Based on Micro Prices (February 15, 2015). Available at SSRN: or [13] Y. Arai, T. Yoshikawa and H. Iyetomi, Frontiers in Arti cial Intelligence and Applications, vol. 255, pp , [14] C.E.Porter and R.G.Thomas. Fluctuations of nucrear reaction widths. Physical Review, Vol. 104, No. 2, pp , Oct [15] A.Edelman. Eigenvalues and Condition Numbers of Random Matrices. SIAM Journal on Matrix Analysis and Applictions, Vol. 9, No. 4, pp , [16] A.M.Sengupta and P.P.Mitra. Distributions of singular values for some random matrices. Physical Review E, Vol. 60, No. 3, pp ,

58 [17] J.P.Bouchud and M.Potters. Theory of Financial Risk and Derivate Pricing: From Statistical Physics to Rsik Management. Cambridge University Press, 2nd edition, ( ) [18] 2008 [19] :

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meiji_resume_1.PDF β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E

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