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至酆
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3 (nonstationary) (nonlinear) Huang et al. (1998) (empirial mode deomposition EMD) (intrinsi mode funtion IMF) (Nasdaq index) (Dow Jones industrial average index)

5 1 IMF..19 IMF..20 IMF IMF..22 IMF..23 IMF..24

6 HHT IMF IMF Hilber IMF IMF4 48 IMF IMF (a) 7 (b) IMF,,,,, IMF IMF,,,,,,

7 IMF IMF,,,,,,,, IMF (a) 1 (b) IMF, IMF IMF, IMF (a) 1 (b)

8 31. IMF,,,,

9 1.1 Fama(1965) (random walk) (no memory) (harts) (tehnial analysis) (Random walk) (sample mean) 0 (autoorrelation oeffiient) (lag) (lag) 1

10 (volatility) Engle(1982) (autoregressive onditionally heterosedasti ARCH) Taylor(1986) ARCH (stohasti volatility) (stationary) Kokoszka & Leipus (2000) (hange point)arch Dahlhaus (1996) (loally stationary fourier LSF) Kim (1998) LSF (time sale) (wavelets) (multisale) Piotr (2002) Nason et al. (2000) (loally stationary wavelet LSW) (power spetrum) Huang et 2

11 al.(1998) Hilbert (empirial mode deomposition EMD) (intrinsi mode funtion IMF) HHT 1.2 Haung et al.(1998) HHT HHT Huang et al.(1998) EMD ( ) ( ) 3

12 ( ) Huang et al.(1998)emd (HHT) Salisbury & Wimbush (2002) HHT (SOI) HHT 4

13 2.1 (EMD) Huang et al.(1998)emd (time sale) (intrinsi mode funtion IMF) (Hilbert transform) (loal energy) (instantaneous frequeny) - - (energy-frequeny-time distribution) IMF( C t) i 1 Ci (') t Yi () t = dt' π (2.1) t t' (analyti signal) Z t C t iy t a t e θ i i () t i() = i() + () = i() (2.2) a () t θ () t i i Y 2 i () t ai() t = [ C i() t + Yi ()] t θ i ( t) = artan( ) (2.3) C () t i 5

14 dθi () t ω i = (2.4) dt (amplitude) - (frequeny-time distribution) (Hilbert amplitude spetrum)( H ω, t) (Hilbert spetrum) (Hilbert amplitude spetrum) (Hilbert energy spetrum) Huang et al.(1998) IMF (Instataneous frequeny) IMF (1) 0 (Zero rossing) (2) (envelope) 0 IMF Huang et al. EMD IMF EMD (1) - (2) (time sale) (time lapse) (3) (infletion point) 6

15 x() t IMF x() t (ubi line) (upper envelope) (lower envelope) m () t 1 x() t m 1 () t h() t 1 h() t = x() t m () t (2.5) 1 1 4a- 4b h() t 1 h() t 1 m t h () t = h() t m () t 11 () h 11 () t (shifting) h 1 () t () t x() t x() t () t 1 1 p 7

16 (the first residue) r() t = x() t () t (2.6) 1 1 (residue) r 1 () t (shifting proess) rs j r( t) ( t) = r ( t),..., r ( t) ( t) = r ( t) (2.7) n 1 n n (shifting proess) (1) n rn (2) (monotoni funtion) (2.6)(2.7) n x() t = i() t + rn() t (2.8) i= 1 n empirial modes r (trend) IMF (expansion) IMF (orthogonal) (omplete basis) IMF n 8

17 (expansion) (adaptive) (non-linear) (non-stationary) (loality) (adaptivity) 2.2 Salisbury & Wimbush (2002)EMD (SOI) (5) Hilbert ENSO 3.6 (6) 4 IMF 4 IMF (7) 4 IMF (polynomial prediting funtion) Salisbury & Wimbush (2002) 48 (in sample errors) (out of sample IMF ( 8) 9

18 3.1 (EMD) (IMF) (9) ± %( /10000 ) (10) ± 500 ( 10%) ( 10) IMF ab b a (data) (signal) (noise) (runs up and down) 10

19 Kolmogorov-smirnov(KS) IMF ( 1a) (1b) 21 ( ) m (m+1) 21 (SRPMSE) m k SRPMSE (1) 6 IMF,,,,, KS p IMF (11) (12) (EMD) (IMF) (13) 11 9

20 ± 500 ( 5 10%) (14) IMF (2a) ( 2b) 21 ( ) SRPMSE (2) 7 IMF 6, 7, 8, 2, 4, 5, 9 + KS p IMF (15) (16) (EMD) (IMF) (17) 12 9

21 ±1000 ( 40%) (18) IMF (3a) (3b) 21 ( ) SRPMSE (3) 9 IMF,,,,,,, , KS p 0 1 IMF (19) (20)

22 4.1 (EMD) (IMF) (21) ± % 5 (22) ± 0.04 ( 40%) (22) IMF (4a) (4b) 21 ( ) SRPMSE (4) 2 IMF,

23 (23) 10 (24) (EMD) (IMF) (25) 13 ± % 2.5 (26) 2.5 IMF (5a) (5b) 21 ( ) SRPMSE (5) 2 IMF,

24 (27) 10 (28) (EMD) (IMF) (29) ± % (30) ± 0.05 ( 33%) (30) IMF (6a) (5b) 12 2

25 21 ( ) SRPMSE (5) 5 IMF,,,, ,,,, (31) 11 (32)

26 SRPMSE IMF IMF IMF IMF IMF IMF IMF IMF

27 1 IMF (a) (b) IMF 1 1, 4 1, 4, 5 1, 4, 5, () k IMF m

28 2 IMF (a) (b) IMF () k IMF m

29 3 IMF (a) (b) IMF () k IMF m

30 4 IMF (a) IMF (b) () k IMF m

31 5 IMF (a) (b) IMF () k IMF m

32 6 IMF (a) (b) IMF IMF () k IMF m

33 (a) (b)

34 (a) (b)

35 (a) (b)

36 (a) (b) () m() t 1 4 HHT (a) (b) m () 1 t (C) m() t 1 28

37 5 8 IMF (Salisbury & Wimbush, 2002) 29

38 (a) (b) 6 (SOI) (a)4 IMF (b) Hilbert (Salisbury & Wimbush, 2002) IMF 30

39 EMD IMF4 ()48 ( )IMF4 () 31

40 9 7 IMF 32

41 (a) (b)

42 11 6 IMF,,,,, ( ) 34

43 13 8 IMF 35

44 IMF,,,,,,

45 16 ( ) 37

46 17 9 IMF 38

47 IMF,,,,,,,,

48 20 ( ) 40

49 11 IMF 41

50 (a) (b) 22 (a) 1 (b) 2 42

51 23 IMF, ( ) 43

52 12 IMF 44

53 IMF,

54 28 10 ( ) 46

55 11 IMF 47

56 (a) (b)

57 31 IMF,,,, ( ) 49

58 Bollerslev, T. (1986). Generalized autoregressive onditional heteroskedastiity. J. Eonometris, 31, Dahlhaus, R. (1996). On the Kullbak-Leibler information divergene of loally stationary proesses. Stohasti Proess. Appl., 62, Engle, R. F. (1982). Autoregressive onditional heterosedastiity with estimates of the variane of United Kingdom inflation. Eonometria, 50, Fama, E. F. (1965), The behavior of Stok Market Pries. Journal of Business 38, Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H.-H., Zheng, Q., Yen, N.-C., Tung, C.-C., and Liu, H.-H. (1998). The empirial mode deomposition and the Hilbert spetrum for nonlinear and nonstationary time series analysis. Pro. Roy. So. London Ser. A, 454, Kim, W. (1998). Eonometri analysis of loally stationary time series models. Manusript, Yale University. 50

59 Kokoszka, P., and Leipus, R. (2002). Change-point estimation in ARCH models. Bernoulli, 6, Nason, G. P., von Sahs, R., and Kroisandt, G. (2000). Wavelet proesses and adaptive estimation of the evolutionary wavelet spetrum. Journal of the Royal Statistial Soiety. Series B, 62, Piotr, F., Modeling and foreasting finanial log-returns as loally stationary wavelet proesses. Submitted for publiation. Salisbury, J. I., and Wimbush, M. (2002). Using modern time series analysis tehniques to predit ENSO events from the SOI time series. Nonlinear Proesses in Geophysis, 9, Taylor, S. J. (1986). Modeling Finanial Time Series. Chihester: Wiley. 51

22 Journal of Taiwan Land Research Vol. 10, No.2 pp. 1~22 * ** 摘要 ARCH GARCH Markov-Switch-ARCH SWARCH ARCH-M *

22 Journal of Taiwan Land Research Vol. 10, No.2 pp. 1~22 * ** 0 0 0 20 摘要 ARCH GARCH Markov-Switch-ARCH SWARCH ARCH-M * E-mail: trista2@mail.stut.edu.tw ** E-mail: mcchen@finance.nsysu.edu.tw Housing