Acta Phys. Si. Vol. 61, No. 4 (212) 456 * 1)2) 1) 1) (, 213 ) 2) (, 3325 ) ( 211 4 18 ; 211 7 6 ),., ; Rossler,,, ;. :,,, PACS: 5.45.Pq, 5.45.Tp 1,,. Takes [1] [2 5],,,,.,, [6]., [7] Lyapuov [8],,,. (extreme learig machie, ELM) Huag [9] - (MP) [1],.,.,,.., [11],. Rossler, [7],. 2 L X 1, X 2,, X L, X l = (x l1, x l2,...,x ln ), l = 1, 2,...,L, ( L = 1), [6], V (m 1,...,m i,...,m L ) =(x 1,, x 1, τ1,...,x 1, (m1 1)τ 1 ; * (: 687491), (: SJ296), (: 213223113), (: 8KJD5122), (: BK21526), (:NY2921) (: CXZZ11 4). E-mail: jiaggp@jupt.edu.c c 212 Chiese Physical Society http://wulixb.iphy.ac.c 456-1
Acta Phys. Si. Vol. 61, No. 4 (212) 456 x 2,, x 2, τ2,...,x 2, (m2 1)τ 2 ;. x L,, x L, τl,...,x L, (ml 1)τ L ), (1), = H, H + 1,, N, H = max 1 l L ((m l 1)τ l + 1), τ l m l l. Takes, m = m 1 + m 2 + + m L > 2d(d ), F : R m R m, V +1 = F(V ), (2) x i,+1 = F(V ), i = 1, 2,, L. (3) V V +1, [12].,,,. [7], [6], [13]. 3,,.,. 3.1 (composite chaos). x +1 = F(x ) = f (I ) (x ), I = B( wf 1 (y ) mod r) B( ef 1 (y ) mod t), f, f 1, I f, B( ),, w, r, e t, x +1 = F(x ) = f (I ) (x ) f, f 1. logistic x +1 = f 1 (x ) = 4x (1 x ) (4), y +1 = f (y ) = 4 si 2 (y 2.5) (5). x +1 = F(x ) = f (I ) (x ), (6) I = B( 157y mod 6) B( 213y mod 8). (7) (x, y ) (.45,.15). Lyapuov [14] 1 1 λ = lim l df(x) dx. (8) x=xi i= Lyapuov λ = 1.599, Lyapuov, (6). Lyapuov. Lyapuov,. (approximate etropy) Picus [15],.,. 1 Tet, logistic [14] Lyapuov.,,. 1 Lyapuov Tet.8434.6127 Logistic.6914.643 [14].8589.8564 1.599 1.699 3.2, : 1), ; 2) ; 3),.., Huag [9] - (MP) [1] (ELM),, 456-2
Acta Phys. Si. Vol. 61, No. 4 (212) 456,.. m, M,, g(x), b i. N (x i, t i ), 1 i N, x i = [x i1, x i2,..., x im ] T R m, t i = [t i1, t i2,...,t i ] T R, 1. x i1 x i2 x im w 11 w mm g x b 1 g x b 2 g x b M β 11 βm 1 : M β i g(w i x i + b i ) = o j, j = 1, 2,, N, (9) i=1, w i = [w 1i, w 2i,, w mi ] i ; β i = [β i1, β i2,, β i ] T i ; o j = [o j1, o j2,, o j ] T. E E(S, β) = t i1 t i2 t i N o j t j, (1) j=1, S = (w i, b i, i = 1, 2,, M),. Huag [9] S, β,, mi(e(s, β)). mi(e(s, β)) mi E(S, β) = mi w i,b i,β H(w1,, w M, b 1,, b M, x 1,, x N )β T, (11), H, β, T. H, β, T : H(w 1,, w M, b 1,, b M, x 1,, x N ) g(w 1 x 1 + b 1 ) g(w M x 1 + b M ) =.. g(w 1 x N + b 1 ) g(w M x N + b M ) β =. β T 1 β T M M t T 1, T =. t T N N N M, (12). (13), (11).,, H, Hβ = T ˆβ, ˆβ = H T (14) H H MP. ˆβ,. 3.3,,,., (CC- ELM), S, β MP. CC-ELM, (11) N mie(s) = o j t j. (15) j=1 S = (w i, b i, i = 1, 2,, M) S = (z i, i = 1, 2,, L), z i CC-ELM, a i z i b i. z i, mi E.,,. [16]. Bartlett [17],,., β., β E. 456-3
Acta Phys. Si. Vol. 61, No. 4 (212) 456 CC-ELM : 1. k =, r =, x k i = x i (), x i = x i(), a r i = a i, b r i = b i, i = 1, 2,, L, x i () (6), E β. 2 x k i z i, z k i = ar i + (br i ar i )xk i. (16) 3, (12),(14) β zi, (9),(15) E(z k i ), E β, ε ( ): E = E(z k i ), β = β zi, x i = xk i, E E(z k i ) > ε; E = E(z k i ), β = β zi, x i = xk i, E E(z k i ) < ε, β zi < β ; E = E, β = β,. (17) 4 k := k + 1, x k i := F(xk 1 i ) = f (Ik 1 i ) (x k 1 i ). (18) 5 2 4, E T, R (R > T ), 6. 6 z i, a r+1 i = zi Γ(b r i a r i), b r+1 i = zi + Γ(b r i a r i), (19) Γ (,.5), zi = ar i + x i (br i ar i ), z i. 7 x i xk i, x i = (z i a r+1 i )/(b r+1 i a r+1 i ), x k i := (1 α)x i + αxk i, (2) α,.1. 8 r := r + 1. r > P E Q,, mi E., α := α.1, 2., P r, Q E. 3.4 CC-ELM : 1), (m l /τ l ). 2) (1),. 3) CC-ELM,. 4) CC-ELM, CC-ELM. 5),. 3.5 CC-ELM,., K, L, m 1, m 2,, m L, L, m l l, m = m 1 + m 2 + + m L, O(K(m 1 + m 2 + + m L )) = O(Km). CC-ELM, O(1);, P ;, R, O(RMN),, M, N ; O(1), O(1) + O(P(RMN + 1)) = O(PRMN). 4 Lorez, Heo, Rossler MG [2 5]. Rossler Rossler, Lorez ;,, [7] Rossler. e AAE e NMSE, e AAE = 1 N e NMSE = 1 N N ˆx l, x l,, (21) =1 N (ˆx l, x l, ) 2 =1 σ 2, (22) ˆx l, x l,, σ 2 x l,. 456-4
Acta Phys. Si. Vol. 61, No. 4 (212) 456 4.1 1 Rossler. ẋ 1 = ω 1 y 1 z 1 + ε(x 2 x 1 ), ẏ 1 = ω 1 x 1 +.15y 1, ż 1 =.2 + z 1 (x 1 1), ẋ 2 = ω 2 y 2 z 2 + ε(x 1 x 2 ), ẏ 2 = ω 2 x 2 +.15y 2, ż 2 =.2 + z 2 (x 2 1), (23) ω 1 =.99, ω 2 =.95, x 1, =.1, y 1, =.2, z 1, =.3, x 2, =, y 2, =, z 2, = 15, ε=.5. 15 1 4.5 4.4 5 4.3 y1 x2 5 1 15 1 5 5 1 (b1) 15 2 4 6 8 1 15 1 5 5 1 (a1) 15 2 4 6 8 1 15 (c1) 2 2 4 6 8 1 x2 y1 4.2 4.1 4 3.9 3.8 657 658 659 66 12.5 12.48 12.46 12.44 12.42 12.4 12.38 12.36 12.34 12.32 (b2) 12.3 657 658 659 66 11.6 11.58 11.56 11.54 11.52 11.5 11.48 11.46 (a2) (c2) 657 658 659 66 2 1 (a1) x 1 ; (a2) (a1) ; (b1) x 2 ; (b2) (b1) ; (c1) y 1 ; (c2) (c1) 2 ẋ 1 = ω 1 y 1 z 1 + ε(x 2 x 1 ), ẏ 1 = ω 1 x 1 +.15y 1, ż 1 =.2 + z 1 (x 1 1), ẋ 2 = ω 2 +.25x 2 + z 2 + ε(x 1 x 2 ), ẏ 2 = 3 + y 2 ω 2, 456-5
Acta Phys. Si. Vol. 61, No. 4 (212) 456 ż 2 =.5y 2 +.5z 2, ω 2 = x 2 y 2, (24) ω 1 =.925, x 1, =.1, y 1, =.2, z 1, =.3, x 2, =, y 2, =, z 2, = 15, ω 2, = 2, ε =.8., 17, 1. -, h =.1, 5, 1. 4.2 1 x 1, x 1, x 2, x 1, x 2, y 1, x 1, x 2, y 1, y 2., 2. 2, e AAE e NMSE, [7]. 2 x 1, x 2, y 1. 2 1 1 2 5 (a1) 1 1 x 2 5 2 y 1 2 8.42 8.4 8.38 8.36 8.34 8.32 8.3 21.5 (a2) x 2 21 5 4.95 2.5 5.2 5.15 5.1 5.5 y1 2 1 1 2 5 (b1) 1 1 x 2 y 5 2 1 2 8.42 8.4 8.38 8.36 8.34 8.32 8.3 21.5 (b2) x 2 21 5 4.95 2.5 5.2 5.15 5.1 5.5 y1 2 1 1 2 5 (c1) 1 1 x 2 y 1 5 2 2 8.42 8.4 8.38 8.36 8.34 8.32 8.3 21.5 (c2) 5 4.95 21 x 2 2.5 5.2 5.15 5.1 5.5 y 1 3 2 (a1) 1 ; (a2) (a1) ; (b1) 2 ; (b2) (b1) ; (c1) 3 ; (c2) (c1) 456-6
Acta Phys. Si. Vol. 61, No. 4 (212) 456 2 1 ( J = 2) (m l /τ l ) e AAE e NMSE [7] [7] x 1 6/17.9.198.1899 1 7.5354 1 5 x 1 4/17.55.488.631 1 6.1241 1 3 x 2 3/14.64.76.8767 1 6.2675 1 3 x 1 1/17.384.497.322 1 4.716 1 3 x 2 4/14.394.762.4315 1 4.1951 1 3 y 1 2/11.293.384.1717 1 4.564 1 3 x 1 1/17.433.662.4596 1 4.1575 1 3 x 2 4/14.357.79.2497 1 4.1712 1 3 y 1 2/11.338.562.2958 1 4.1215 1 3 y 2 1/16.241.6.156 1 4.18 1 3 4.3 2 x 1, x 2, ω 2, y 1, s = 1, 2, 3. 3 [7]., 3,, [7]. 3 s = 1, 2, 3 x 1, x 2, y 1. 3 2 ( J = 5) (m l /τ l ) s e AAE e NMSE [7] [7] x 1 3/15 1.196.775.19 1 4.4749 1 3 x 2 5/12.465.1522.17 1 4.8856 1 3 ω 2 1/15.398.432.4892 1 4.56 1 3 y 1 1/15.214.61.1122 1 4.2344 1 3 x 1 3/15 2.298.1546.256 1 3.19 x 2 5/12.658.329.24 1 3.35 ω 2 1/15.957.849.2145 1 3.2 y 1 1/15.327.1211.294 1 3.9 x 1 3/15 3.46.2312.466 1 3.42 x 2 5/12.822.452.45 1 3.78 ω 2 1/15.1161.1257.3833 1 3.4 y 1 1/15.445.186.576 1 3.2 4 2 ( J = 3) (m l /τ l ) e AAE e NMSE [7] [7] x 1 5/15.66.786.855 1 5.1851 1 3 x 2 3/12.373.841.4133 1 5.2185 1 3 x 1 5/15 N(,1 1 3 ).48.11.3536 1 4.2797 1 3 x 2 3/12 N(,1 1 3 ).96.1111.2992 1 4.3484 1 3 x 1 5/15 N(,1 1 2 ).198.28.2553 1 3.12 x 2 3/12 N(,1 1 2 ).1138.1154.461 1 3.13 x 1 5/15 N(,1 1 1 ).3334.5998.24.97 x 2 3/12 N(,1 1 1 ).3736.6573.4.115 456-7
Acta Phys. Si. Vol. 61, No. 4 (212) 456 enmse /1-5 6 4 2 (a) enmse /1-5 6 4 2 (b) enmse /1-5 8 6 4 2 (c) 1 2 3 4 5 J 1 2 3 4 5 J 1 2 3 4 5 J 4 1 x 1, x 2, y 1 e NMSE J (a) x 1 ; (b) x 2 ; (c) y 1 4.4, 2 x 1 (), x 2 (), 4 [7]., e AAE e NMSE,, [7],. 4.5 J J. J, J, J,, J. 4 J 1 x 1, x 2, y 1 e NMSE., J = 4, x 1, x 2, y 1 e NMSE, J, e NMSE. 5, : 1),, ; 2), (CC-ELM) ; 3) CC-ELM,,. [1] Takes F 1981 I Lecture otes i mathematics, Vol.898 Dyamical systems ad turbulece(berli:spriger)p366 [2] Ya H, Wei P, Xiao X C 29 Chi. Phys. B 18 3287 [3] Samata B 211 Expert Syst. with Appl. 38 1146 [4] Zhag C T, Ma Q L, Peg H 21 Acta Phys. Si. 59 7623 (i Chiese) [,, 21 59 7623] [5] Fag F, Wag H Y, Yag Z M 211 Appl. Mech. Mater. 47 318 [6] Cao L Y, Mees A, Judd K 1998 Phys. D 121 75 [7] Zhag Y, Guag W 29 Acta Phys. Si. 58 756 (i Chiese) [, 29 58 756] [8] Lu S, Wag H Y 26 Acta Phys.Si. 55 572 (i Chiese) [, 26 55 572] [9] Huag G B, Zhu Q Y, Siew C K 26 Neuro Computig 7 489 [1] Serre D 22 Matrices:Theory ad Appkicatios (New York: Spriger) p145 [11] Zhag T, Wag H W, Wag Z C 1999 Cotrol ad Decisio 14 285 (i Chiese) [,, 1999 14 285] [12] Sauer T, Yorke J A, Casdagli M 1991 J. Stat. Phys. 65 579 [13] Wag H Y, Sheg Z H, Zhag J 23 J. South. Uiv. (Natural Sciece Editio) 33 115 (i Chiese) [,, 23 ( ) 33 115] [14] Tog X J, Cui M G 29 Sciece i Chia F 39 588 (i Chiese) [, 29 (F ) 39 588] [15] Picus S 1995 Chaos 5 11 [16] Zhu Q Y, Qi A K, Sugatha P N, Huag G B 25 Patter Recogitio 38 1759 [17] Bartlett P L 1998 IEEE Tras. Iform. Theory 44 525 456-8
Acta Phys. Si. Vol. 61, No. 4 (212) 456 Predictio of multivariable chaotic time series usig optimized extreme learig machie Gao Guag-Yog 1)2) Jiag Guo-Pig 1) 1) ( Ceter for Cotrol & Itelligece Techology, Najig Uiversity of Posts ad Telecommuicatios, Najig 213, Chia ) 2) ( School of Iiformatio Sciece & Techology Jiujiag Uiversity, Jiujiag 3325, Chia ) ( Received 18 April 211; revised mauscript received 6 July 211 ) Abstract A predictio algorithm of multivariable chaotic time series is proposed based o optimized extreme learig machie (ELM). I this algorithm, a preseted composite chaos system ad mutative scale chaos method are utilized first to search ad optimize the parameters of ELM for improvig the geeralizatio performace. The the optimized ELM is used to predict the multivariable chaotic time series of Rossler coupled system for sigle step ad muti-step, ad the scheme is compared with the cogeeric method, which shows the validity ad stroger ability agaist oise of the developed algorithm. Fially, the relatio betwee predictio result ad umber of hidde euros is discussed. Keywords: extreme learig machie, multivariable chaotic time series, predictio of chaotic time series, composite chaos optimizatio PACS: 5.45.Pq, 5.45.Tp * Project supported by the Natioal Natural Sciece Foudatio of Chia (Grat No. 687491), the Six Projects Sposorig Talet Summits of Jiagsu Provice, Chia (Grat No. SJ296), the Research Fud for the Doctoral Program of Higher Educatio of Chia(Grat No. 213223113), the Natural Sciece Basic Research Project for Uiversities of Jiagsu Provice, Chia (Grat No. 8KJD5122), the Natural Sciece Foudatio of Jiagsu Provice, Chia (Grat No. BK21526), the Project for Itroduced Talet i Najig Uiversity of Posts ad Telecommuicatios, Chia (Grat No. NY2921), ad the Scietific Research Iovatio Program for the Graduat Studets i Jiagsu Provice, Chia (Grat No. CXZZ11 4). E-mail: jiaggp@jupt.edu.c 456-9