A pp lication Study on R econ struction of Chao tic T im e Series and P rediction of Shanghai Stock Index

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1 : (2003) ,, (, ) :,,, ; , : ; ; ; ; : C : A A pp lication Study on R econ struction of Chao tic T im e Series and P rediction of Shanghai Stock Index M A Jun2hai, Q I E r2sh i, M O X in (M anagem ent Co llege, T ianjin U niversity, T ianjin , Ch ina) Abstract: H igher accurate param eters identification can be m et by m ethod of associating neural netw o rk w ith w avelet theo ry, w here non linear au to2co rrelated chao tic m odel is app lied. Effect of reconstruction and p rediction can be imp roved by p re2treatm ent of the chao tic tim e series and use of Fourier w ave filteṙ In th is paper, w e establish non linear auto2co rrelated chao tic models w ith data of opening, m axim um, m inim um and clo sing p rices fo r stock coded of Shanghai Security m arket as w ell as identify the param eterṡ A s it can been seen, results of p rediction are comparably accurate. Key words: nonlinear auto2co rrelated chao tic model; w avelet neural netw o rk; stock data of Shanghai Secu rity M arket; param eter iden tification; tim e series p rdiction 1 :, ; ( [ 1-13 ) ],,, [ ],, [ 18-23, ], (P ing Chen [ 24 ] ),,, : : ( ). 863 (2003AA 4Z2040) : (1965- ),,,, E2m ail: hai2m ajun@ 263. net

ianjin 300072, Ch ina) Abstract: H igher accurate param eters identification can be m et by m ethod of associating neural netw o rk w ith w avelet theo ry, w here non linear au to2co rrelated chao

2 12 87,,,, [ 25 ],,, 2. [ y i (i= 1, 2, 3,, N ), ] x n = G [x n- 1, x n- 2,, x n- M ] + kεn, G, M. : x n = 6 M am P m (n) + Εn (1) P 0 (n) = 1, m Ε 0 P m (n) = x n- i m 1, x n- i m 2,, x n- i m j, {im j }, (1) m axm j ({im j }), (1), : x n = 6 M g mw m (n) + kεn (2) W m (n) P m (n), W 0 (n) = 1, g 0= x θ, gm (3) : : : : (9) : (3) Αm r = Αm = 6 M J = x n - 6 M gmw m (n) W m (n) = P m (n) - 6 m - 1 P m (n), gm = (W r (n) ) 2 i= m V i = - 6 i- 1 2 (3) Αm rw r (n) (4) r= 0 y nw m (n) (5) (W m (n) ) 2 g iv i, V (6) r= m ΑirV r, i = m + 1, m + 2,,M (7) Q (m ) = g 2 m Q (M ) M - 1 y 2 n - 6 g md 2 (m, m ) W 2 m (n) (8) J = x n - 6 M 2 gmw n (n) = f (Α1, 0, Α2, 0, Α2, 1,, Αm, m - 1,, ΑM, M - 1) (10) Αm, m - 1, 2,,M,. 1g2 > 2 N (9)

W 0 (n) = 1, g 0= x θ, gm (3) : : : : (9) : (3) Αm r = Αm = 6 M J = x n - 6 M gmw m (n) W m (n) = P m (n) - 6 m - 1 P m (n), gm = (W r (n) ) 2 i= m V i = - 6 i- 1 2 (3) Αm rw r (n) (4)

3 : g7δ (w ) g R 7 ( ) gw gdw <. 5 = { ak 7 (akx - bk) : ak R +, bk R, k Z }, A > 0 B <, f L 2 (R ), : A f 2 Φ 6 g Υ, f g 2 Φ B f 2 (11) Υ 5 5 L 2 (R ), 5 g (x ) = 6 L i= 1 w i7 (a T i x - bi), Υi 5 (12) L 2 (R ), a i, bi Ζ, 1Ζ Sigm o id, Ζ N, (12) f (x ) L 2 (R ) Ζ 7 (x ),, (4) θ g, : 1 g (x ) = 6 L w i7 (a T i x - bi) + g θ (13) Q inghua Zhang [ 2 ], A lbert Benven iste (13), : g (x ) = 6 L i= 1 i= 1 w i7 [D i (t - ti) ] + g θ (14) ti, D t, D i= d iag (d i), g θ g (x ) Ζ L, 7 : w (s) = (1 - s 2 ) e - s2 g2 x i ( i= 1, 2, 3,, N ) m, (12) L = m, : a = m in{x (i), 1 Φ i Φ N }, b = m ax{x (i), 1 Φ i Φ N }, g θ = i= 1 (15) x (i) N 7 : 7 (x ) = w (x 1)w (x 2) w (x m ) (16) M { (x m, y m ) : y m = f (x m ),, 2,,M }, Η w i a i bi (i= 1, 2,, N ), g Η(x ) (15), : E (Η) = M (g Η(x m ) - y m ) ) 2 (17) Η 3, : Η(k + 1) = Η(k) - Κ> 0, E (Η) E (Η) Η, Κ E (Η(k) ) (18) E (Η) = 6 M (g Η(x m ) - y m ) g 5 5 Η g Η (x m ) (19) g Η(x m ) (15), E (Η)

o id, Ζ N, (12) f (x ) L 2 (R ) Ζ 7 (x ),, (4) θ g, : 1 g (x ) = 6 L w i7 (a T i x - bi) + g θ (13) Q inghua Zhang [ 2 ], A lbert Benven iste (13), : g (x ) = 6 L i= 1 i= 1 w i7 [D i (t - ti) ] + g θ

4 E (Η) 5w j 5 E (Η) 5 a j 5 E (Η) 5 bj = 6 M = 6 M = - 6 M (g Η(x m ) - y m ) g 7 (a jx m - bj) (g Η(x m ) - y m ) gw j7 (a jx m - bj) g x m (g Η(x m ) - y m ) g w j7 (a jx m - bj), j = 1, 2,, N (20) (20) (18) Η 3 Ζ, Ζ Ζ 4,, : x x i- xθ i= Ρ ( i= 0, 1, 2,, N )., Ζ ( 687 ) 683, a, b, c, d, eζ a 2, b 3, c 4, d 5, e (9) a b c d e 5, (2) 5 a b c d 687, 683 (2),,, 5,, :

5 x n = a1x n- 1 + a2x 2 n- 1 + a3x 3 n- 1 + a4x n- 2 + a5x 2 n- 2 + a6x 3 n- 2 + a 7x n- 3 + a8x 2 n- 3 + a9x 3 n- 3 + a10x n- 4 + a11x 2 n- 4 + a12x n- 5 + a13x 2 n- 5 + a14x n- 2x n- 1 + a15x n- 3x n- 1 + a16x n- 3x n- 2 + a17x n- 4x n- 1 + a18x n- 4x n- 2 + a19x n- 4x n- 3 (21) + a20x n- 5x n- 1 + a21x n- 5x n- 2 + a22x n- 5x n- 3 + a23x n- 5x n- 4 a b c d a a a a a a a a a a a a a a a a a a a a a a a a , 683, (21),, ,

n- 5x n- 1 + a21x n- 5x n- 2 + a22x n- 5x n- 3 + a23x n- 5x n- 4 a b c d 1 3 5 7 1 a a1 0. 097550 a7 0. 153103 a13-0. 038153 a19 0. 065333 a2 0. 077617 a8-0. 010280 a14-0. 056888 a20 0. 127818 a3-0.

6 %, %, % %,,, 2 a,, x t Y t (y i- x i) gx i % % % % b a a a a a a a a a a a a a a a a a a a a a a a a , 683, (21),, , %, %, % %,,, 4 b,, x t Y t (y i - x i) gx i % % % %

007090 a9 0. 000812 a15 0. 069272 a21-0. 078307 a4 0. 132646 a10 0. 121645 a16 0. 051582 a22-0. 052082 a5-0. 003979 a11-0. 044396 a17 0. 046946 a23 0. 047470 a6 0. 003202 a12 0. 067969 a18 0.

7 c a a a a a a a a a a a a a a a a a a a a a a a a , 683, (21),, , %, %, % %,,, 6 c,, x t Y t (y i - x i) gx i % % % % d a a a a a a a a a a a a a a a a a a a a a a a a , 683, (21),, , %, %, % %,,,

34 %, - 1. 64%, - 3. 70% - 8. 13%,,, 6 c,, x t Y t (y i - x i) gx i 679 11. 91 680 11. 34 681 11. 40 682 11. 75 683 11. 97 684 12. 01 12. 051071 5 0. 34 % 685 12. 26 12. 058727-1. 64% 686 12. 53 12.

8 d,, x t Y t (y i - x i) gx i % % % % ,,,, a, b, c, d 2. 5%, 0. 14% ; 4. 34%, 0. 62%,, (6 ), ,, 676, 1% ; 1. 45% A R (4) ,,, ( ),,,,, : [1 ] L iang Yue Cao, Y iguang Hong, H aip ing Fang, Guow ei H e. P redicting chao tic tim eseries w ith w avelet netw o rk s[j ]. Phyṡ D, (8): [2 ] Zhang Q inghua. W avelet N etw o rk s[j ]. IEE T ransections on N eural N etw o rk ṡ 1992 (11), 6: [3 ] D iam bra L, P lastino A. M odelling tim e series using info rm ation theo ry[j ]. Phyṡ L ett A (216) 3: [4 ] Castillo E, Gutierrez J M. N onlinear tim e series modeling and p rediction using functional netw o rk ṡ extracting info rm ation m asked by chao s[j ]. Phyṡ L ett A, 1998, 244(5): [5 ] Kevin Judd, A listair M eeṡ Em bedding as a modeling p roblem [J ]. Phys D, 1998, 120 (4): [6 ] Kugium tzis D, L ingjxrde O C, Ch ristophersen N. R egularized local linear p rediction of chao tic tim e setries[j ]. Phys D, 1998, 112 (6): [7 ] N ijm erijer H. A dynam ical contro l view on synch ronization[j ]. Phys D, 2001, 154 (6): [8 ] Kevin Judd, A listair M eeṡ M odeling chao tic mo tions of a string from experim ental data[j ]. Phys D, 1996, 92 (8):

45% A R (4) 600062 600063,,, ( ),,,,, 4 6 5 : [1 ] L iang Yue Cao, Y iguang Hong, H aip ing Fang, Guow ei H e. P redicting chao tic tim eseries w ith w avelet netw o rk s[j ].

9 [9 ] M cguire, N abeel b. A zar, M ark Shelham er, R ecurrence m atrices and the p reservation of dynam ical p roperties[j ]. Phys L ett A, 1997, 237 (10): [10 ] Ch ristian G. Sch roer, T im Sauer, Edw ard O tt, Jam es A. Yo rke. P redicting chao tic mo st of the tim e from em beddings w ith self2in tersection s[j ]. Phys R ev L ett, 1998, 80 (7): [11 ] Q retavik S, Carretero2Gouzalez R, Stark J. E stim ation of intensive quantities in spatio2tempo ral system s from tim e series[j ]. Phys D, 2000, 147 (11): [12 ] Sato sh i k itoh,m ah ito k im ura, T akao M o ri, Kenji T akezaw a. A fundam ental bias in calculating dim ension from finite data sets[j ]. Phys D, 2000, 141 (10): [ 13 ],,. [J ] , 20 (11): [14 ],,. [J ]., 1998, 19 (6): [15 ] M a Junhai. T h resho ld value fo r diagno sis of chao tic nature of the data obtained in nonlinear dynam ic analysis[j ]. A pp lied M athem atics and M echanics, 1998, 19 (6): [16 ] M a Junhai. T he influence of the different distributed phase2random ized on the experim ental data obtained in dynam ic analysis[j ]. A pp lied M athem atics and M echan ics, 1998, 19 (11): [17 ] M a Junhai. T he m atric algo rithm of lyapunov exponent fo r the experim ental data obtained in dynam ic analysis[j ]. A pp lied M athem atics and M echanics, 1999, 20 (9): [18 ] M a Junhai, Chen Yushu. A n analytic and app lication to state space reconstruction about chao tic tim e series [J ]. A pp lied M athem atics and M echanicṡ 2000, 21 (11): [19 ] M a Junhai, Chen Yushu. Study on the p rediction m ethod of low 2dim ension tim e series that arise from the intrinsic non2linear dynam ics[j ]. A pp lied M athem atics and M echan ics, 2001, 22 (5): [20 ] M a Junhai, Chen Yushu. Study fo r the bifurcation topo logical structure and the global comp licated character of a k ind of nonlinear finance system ( I) [J ]. A pp lied M athem atics and M echanics, 2001, 22 (11): [21 ] M a Junhai, Chen Yushu. Study fo r the bifurcation topo logical structure and the global comp licated character of a k ind of nonlinear finance system ( II) [J ]. A pp lied M athem atics and M echanics, 2001, 22 (12): [22 ] Yushu Chen and A ndrew Y. T. L eung. B ifurcation and Chao s in Engineering[M ]. Sp ringer2verlag Berlin H eidelberg N ew Yo rk [23 ]. [M ]. : [24 ] P ing Chen. A R andom w alk o r co lo r chao s on the stock m arket? tim e2freqency analysis of S&P indexes[j ]. N onlinear D ynam ics and Econom etrics, 1996, 1 (2): [25 ] K i H Chon. D etection of chao tic determ inism in tim e series from random ly fo rced m ap s[j ]. Phys D, 1997, 99 (4):

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