论文研究-灰色多变量GM(1,Nγ\,r)模型及其粒子群优化算法.pdf-资料库

29 10 2009 10 Systems Engineering | Theory & Practice Vol.29, No.10 Oct., 2009 : 1000-6788(2009)10-0145-07 GM(1; N j; r ) !"#$ ( , 210016) , '()+*,)-.+/102345+/6 '( 9;:;<;=?>A@?BAC;D;E;F;G;E?HAI;J;K;L;M?NAO;P?QAR H\T^]\_\`\a\b\cedAf;g\h?i;j^k;l?d T\o\N\p^q\res m\n T\{^\\\ N^;?^ t \\ F\G\¥eH\N^}\~\¦e§ \\\\ e¡^¢\£\¤ ¨\©\ªe«^¬\Y U\V GM(1; N j ,r ) ; _\`\a\b ¢\£ \\\e A , , , , , ; ; ze{^|\}\~\\\e Z\l ®\¯\° ±\²\³\´\µ C931 ¶\·\¸\¹\º . H;TAU;V;@?BAC;W;X GM T^u\vew^x\M\y\`eN . \L\\ . \\\\\\e^ Z\I\J\e\N^\ ¥eH^}\~ Grey GM(1; N j ,r ) model and its particle swarm optimization algorithm (College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China) HUANG Ji Abstract The multi-variables grey model is based on the classical grey theory, which mainly solves the practical forecast problems of multi-input of variable, single variable or multi-variables output. For the multivariable control input-output systems with time-varying delay, this paper proposes grey model GM with time-varying delay, which separately produces the parameter estimate formula, the approximate time corresponding formula and the derivative models. In order to get the highest forecast precision by the determined value, the article establishes optimized problems which take the mean error as the goal, and uses the particle swarm optimization to seek optimum. The practice indicates that this model has a higher precision. Keywords grey GM(1; N j ,r ) model; parameter estimate; particle swarm optimization; science and technology prediction À;Ë\Ì , , GM(1,N ) \Ä\Å\Æ\Ç\È\É\Ê \ Ç\ß ã\ä\å\Ø\ÙeÚ^æ\ç\Û\Ü\Ý û\ü\Ç\þ\É\ÿ\Ä û\ü\ý â\ú ¾ GM(1,N ) ö û\ü\þ\É\Ç ()+*,-.\Ü0/21\ê\Å\Æ û\ü ?@ABCDE \ú Í\ÎeÏ^Ð\Å\Æ\Ñ\Ò\Ó\Ô\É\Ê\Õ\Ö\Ç Ç\ß\à\á; \è\é;ê\ë;Ç\Û\â;Ý Â\Ã ¾ !" û\ü\ÇFGH7IJ ú ß;Ç KJL\Ô\Ý GM(1,N ) GM(1,N ) GM(1,1) [2], û\ü\Ç4\ó5678 [5]; Ø\ÙeÚ^Û\Ü\Ý ì\í\å\î\Ô\ï \ú #$\ú\Ô%&' 9:;< Ç\à\áMN [1] , [4]; , , , , , , À\× Ç\ß 1 . »½¼ ¾\¿\À\Á\Â\Ã\¿ Ç\ß\à\á\Û\â\Ý ê\ñ\Ç\ò\ó ô\õ\ö\÷\ø\ù É\Ý ú\û\ü\Ç\à\á=> OQPSRUT VQWQXSY eQfQgQh GM(1,N ) : 2008-08-05 [3]; : : (2007GXQ4D172) (1977{), ZQ[Q\Q]Q^Q_Q`QaQbQcSd iQj , , mQnQoQpQqQr , sQtQ_Q`Qu : , _Q`QvSw ]QxQyQz , E-mail: whst1612@163.com. % & 7 8 S Y Z [ Y Z t Y Y Z Þ Þ Þ Þ ð ¾ 3 Þ k l

146 , n [7{8] à\á %\û\üB\Å\Æ\û\ü ¥¦ ê\ ô\õ\ö²³´ #$\ú \ì\Å\\Ç\Ô\Ý \å3.\Ü Û\Ü\Ý ¡\Ç\å\Ô\ç% :§¨©©\ ª«\ÝH5\ê ¬5® [9{10] x(0)(k) + az(1)(k ) = bkr; GM(1,N ), MGM(1,N ) {}|}~}z}}}} '\û\ü #$\û\ü\Ç ) ¾ K\Ý ¯5\ÇB °\È5 ±§¨® ¤¶· û\ü Û\â\Ý Ç¢<+£ GM(1,1j; r) GM(1,N ) GM(1,N ) k + 2; 2 f0; 1; g; r 2 R [6]. ( ( ) , , . , . : û\ü\Ç û\ü\å\ç ¤ . (1) â\ú , r û\ü . 29 a 2 (2; 2) 5Ë¢;ÇÃ ËÔ , Ç4\ó ßà J\þ\É ï\Ã \û\ü\Ç , , , . . , ||GM(1,N j; r) ¸¹\\\Ç®ª« K¬5ª«\ÝH'\Û\Ü\Çº\É Í\Î\ú'Ä\ÉÅ\ Æ ÇÏÈ;; 'Ö\û\ü Bª«\ÝH ýÃ\û\ü\Ç ÇÀÁÂ« ÃÌ ÈÍÎ;Ô;Ý ý5;Ó;é;Ç â\ì¨\Ç'Ö\û;ü À\Ë\Ì ÈÍÎ Ø\Å ±\ú\Ô\Ý Ô\Ý \\à\á\û\üåJLæçèÂéêë áâãä ¤ GM(1,N ) Ï^Ð\û\ü û\ü Àñ , ¾ 'Ö\\\\ \Ç¾¿¬5\\ óôB\óô õ÷ö÷ø÷ù÷ú÷ûü}ý 1 = (x(0) x(0) 1 (1); x(0) K\\\Ç1\ê%& 1 (2); ; x(0) \\ \;Ç® #\û , (i = 1; 2; ; N ); z(1) 1 (n)) x(0) j 7ð (j = 2; ; N ) x(1) 1 1 1 , , (j = 2; 3; ; N ) 1-AGO 2 GM(1,N j ,r ) ã Ç\ß NXi=2 bix(1) i GM(1,N j ,r ) . û\ü , 0; N = 1 = 0; r = 0 , GM(1,N j ,r ) GM(1,1) þÿ û\ü\Ç 2 ìè . · ì , GM(1,N j ,r ) û\ü\ÝK\Ç dx(1) 1 (t) dt GM(1,1) û;üKê û\ü NXi=2 , GM(1,1) GM(1,1j; r) \û\ü Å\Æ¸» ÇÇÈ Û;Ü; ;Û;â;Ç®¯ÐËKÑÒ \û\ü×5 ÃÜ É¸»\û\üÊ Ù\â\û\ü\Ç\þ\ÉÚÛ\ý û\ü¼½\ú¾¿\\É û\ü¬ À;Ë û\ü Ý4HÞ îª éìí;ß\Ç\à;áMN û\ü#\ûá\é GM(1,N j ,r ) GM(1,1j; r) GM(1,N ) :Ó , , , , , . , x(0) , j = (x(0) \ j (1); x(0) j (2), , x(0) Û\â . x(1) x(0) 1 i j (n)) x(0) i Û\Üª GM(1,N ) , û;ü GM(1,N j ,r ) . % r > 0; = 0; r = GM(1,N ) û\üK 1 (k) + az(1) x(0) 1 (k ) = (k )(k )r; r > 0 (2) û\üB (3) , ^ = + az(1) 1 (t ) = bix(1) i (t )(t )r , . K¾¿;É , bi K;É , bix(1) i (k ) K z(1) z(1) 1 (2 ) 1 (3 ) x(1) 2 (2 )(2 )r x(1) 2 (3 )(3 )r N (2 )(2 )r N (3 )(3 )r x(1) x(1) ... x(1) 2 (n )(n )r N (n )(n )r 377775 GM(1,N j ,r ) [a; b2; ; bN ]T , a û\ü\ÇH ´ GM(1,N j ,r ) K\þ\É "# û;üÌµ þ! 1 . 266664 266664 , B = Y = n = N + 1 n > N + 1 , 1) 2) z(1) 1 (n ) x(1) x(0) 1 (2) x(0) 1 (3) ... x(0) 1 (n) P = B1Y; jBj 6= 0; P = (BTB)1BTY ; 377775 P = [a; b2; ; bN ]T Þ K ù é Ë J Þ Þ Þ Ç µ ù ¶ µ Þ Ì Õ ù Þ Ã Þ ¤ Æ ¾ Ê ò ù â þ ÿ K É Ê K Ç K K ª Ì ª K ª ª

10 3) n < N + 1 , <=?>@ABCDE MNOP : &(' )(*(+(,(- P = BT(BBT)1Y . GM(1; N j; r ) .(/(0(1(2(3(4(5(6(7(8 147 . k = 2; 3; ; n x(0) 1 (k) + az(1) 1 (k ) = bix(1) i (k )(k )r GHIJKL NXi=2 NXi=2 NXi=2 NXi=2 x(0) 1 (2) + az(1) 1 (2 ) = bix(1) i x(0) 1 (3) + az(1) 1 (3 ) = bix(1) i (2 )(2 )r (3 )(3 )r ... x(0) 1 (n) + az(1) 1 (n ) = bix(1) i (n )(n )r BP = Y n < N + 1 , B STUV: , B XUVYZS B = DC B X\]^_` B+ "ab D = In1( cd_` ), ;W 2 ef GM(1,N j ,r ,x(0)): GM(1,N j ,r ) B+ = C T(CC T)1(DTD)1DT P = C T(CC T)1(DTD)1DTY B = DC = In1C B = C P = BT(BBT)1Y GM(1,N j ,r ) x(1) hijk x(0) , Ym#S Ll GM(1,N j ,r ,x(1)) x(0) 1 (k) = x(0) 1 (k) = NXi=2 NXi=2 <=onp z(1) 1 (k) bix(1) i bix(0) i (k )(k )r 0:5ax(1) 1 (k ) 0:5ax(1) 1 (k 1) (k )(k )r 0:5ax(0) 1 (k ) 0:5ax(0) 1 (k 1) + x(0) 1 (k 1) (4) (5) Xq]Wz(1) 1 (k ) = 0:5[x(1) 1 (k ) + x(1) 1 (k 1)] GH rs GM(1,N j ,r ) M x(0) 1 (k) + 0:5a[x(1) 1 (k ) + x(1) 1 (k 1)] = bix(1) i (k )(k )r NXi=2 x(0) 1 (k) = (k )(k )r 0:5ax(1) 1 (k ) 0:5ax(1) 1 (k 1) i bix(1) NXi=2 k = j 1; j M bix(1) i tuvwxy , Ymz NXi=2 x(0) 1 (j 1) = (j 1)(j 1)r 0:5ax(1) 1 (j 1) 0:5ax(1) 1 (j 2) $ % 9 : ; F Q 9 : R W [ : S g X L l

= = NXi=2 NXi=2 NXi=2 NXi=2 NXi=2 lq bx(0) = 2 148 x(0) 1 (j) = {}|}~}}}}} bix(1) i (j )(j )r 0:5ax(1) 1 (j ) 0:5ax(1) 1 (j 1) bi[x(1) i (j 1) + x(0) 1 (j )](j )r 0:5a[x(1) 1 (j 1) + x(0) 1 (j ) + x(1) 1 (j 2) + x(0) 1 (j 1)] bix(1) i bix(0) i bix(0) i (j 1)(j 1)r 0:5ax(1) 1 (j 1) 0:5ax(1) 1 (j 2)+ (j )(j )r 0:5ax(0) 1 (j ) 0:5ax(0) 1 (j 1) (j )(j )r 0:5ax(0) 1 (j ) 0:5ax(0) 1 (j 1) + x(0) 1 (j 1) , GM(1,N j ,r ) KLX: (k )(k )r az(1) 1 (k ) 1 (k) = 1 q k 2. bix(1) i bix(1) i NXi=2 NXi=2 = 1, GM(1,N j; r ; x(1)) 1 (k) x(0) 1 (k) KLXS = (k )(k )r 0:5ax(1) 1 (k ) 0:5ax(1) 1 (k 1) b i = bix(1) bx(0) NXi=2 s KLXZh IX¥ nXk=3 [bx(0) NXi=2 nXk=3 min Q(r) = KL = f : (k 1)(k 1)r 0:5ax(1) 1 (k 2) 0:5ax(1) 1 (k 1) x(0) 1 (k) sN S¡ X¢£¤¥ KLZ , GM(1,N j; r ) KL§¨ 1 (k) x(0) 1 (k)]2 bix(1) i (k 1)(k 1)r 0:5ax(1) 1 (k 2) 0:5ax(1) 1 (k 1) x(0) 1 (k)g2 29 (6) (7) (8) (9) s:t [a; b2; ; bN ]T = (BTB)1BTY GM(1,N j; r ; x(0)) , , ¦« min Q(r) = KL 1 (k) x(0) 1 (k)]2 ;X¥ KLS r 0 nXk=3 [bx(0) nXk=3( NXi=2 = s:t [a; b2; ; bN ]T = (BTB)1BTY r 0 (8) lKL (9), ¬®¯° x± ²³ . Z GM(1,N j ,r ) ¦«KL 3 ´¶µ PSO ÃSÄ Mà ÈÉÞß ÆÇ²³ CÅ \áX bix(0) i (k 1)(k 1)r 0:5ax(0) 1 (k 2) + (1 0:5a)x(0) 1 (k 1) x(0) 1 (k))2 ·¶¸¶¹¶º¶»}¼}½¾}¿}ÀÁ}Â ²Í [11]. WÈÉ¥Ê ©Ì , , cÏÎÑÐÒÓÔÕ ÖR×ØÙÚXÛ , [Ü $ w S y x ; S ¦ © ª ¥ Ë S Ý t

10 : GM(1; N j; r ) &(' ¦¡ äåæ tð , Gñ (Pi). Mà XZ . PSO ²³âã Äêîï Ãêëóï Mà , , Sóï )(*(+(,(- Pçè , .(/(0(1(2(3(4(5(6(7(8 . np éê° Äêóïú x Vi (t + 1) = wVi (t) + c1r1 (Pi Xi (t)) + c2r2 (Pg Xi (t)) Äêóïúû° ¥Xóï éêêëZìíÄê° êóïôõö÷ùø W° npò ±y IJ XdlÓÔ Xêëóï õö÷ éê° (Pg). , , : ¡ ä êóï , éê° ±y ¡üýþ Xÿ Äê Xi (t + 1) = Xi (t) + Vi (t + 1) t xt Vi (t) i c2 x± GM(1,N j ,r ) ê° , c1Î KL° åæ , np , Pg Sóï 1) t hTß ¦«éê° q a$óïlêëóï 2) 3) ( , Xi (t) i : :XÓÔ ÓÖR ²³ Xë , ZKL ê° c1 = c2 = 2; r1Î ²O , C «ýþúÄX !"#HKL ), Å 1 ; , t xt r2 ú ± (8), np :Xd t [0,1] K ¡ ä 6) , 2) ; /; 4) ©% 5) 6) U&Ø (10) 1kóï (11) s2 Äq(Ô)*U& +,- 0 õö° ¦XZ XÓÔld 3k+, !" , . , , Pi S çè ; i ê° . w Xêëó ú¤ , Ì Xîï éê° 5) ; 149 (10) (11) . ; , 4 4656768696:<;>=>?<@>A>:C>D¹>E>FH IJKL , XMHNOkú , XïRqXNZ[\]X1kXYX^Q_ Màmn dijX , Ì IJ cd Jk OlU Ol Ìxy [ö Jk [ö 1 2001{2006 QPRqX , , , úÄêSTX¢£¤U IJ `RõØú Mmqr Jk MHNOkWaQ"¤ Xís Ý{| OlzØ &Xop opN} bcd[ö ©ª«¬ ³´µ¶ ¥¦§¨ ©ª±² £¤ ¢ op ¡¢ 1.2166 1.1022 1.1250 1.2227 1.2794 ®¯° , . úÄêVLX bcd yuvtw W ê1HXY tef Igh . np ^_ 1. ÓÔ~Q R&D ³´µ 1.3911 ©ª«¬ ³·¸¶ 1.2547 0.2166 0.1022 0.1250 0.2227 0.2794 0.3911 0.2547 ¿ 1.1940 1.2098 0.1940 0.2098 IJKL ÄqXÐ¤ ìkg ÝQÂ IJKL , QQÑQÒQÓ , ÈÉ ¹º»¼½¾ ¹º ½¾ 1 y g Jk OlX ÊËbcd ÄqXQÐ vt Ã OlMHOksÙÚWÛÜX TâãYä IJðñ Nîï [ö ZKL , rs SýþúÄX x± . Ý ²³ G÷ £¤úûl KL S1H òæ GM(1,N j ,r ) GM(1, N ) oô° ¤ó r = 0:236 ú9 , . uÀ DE op ÎÄOQÅÆQÇÐH !QhQÕQÖ . MHØQPQs , , . Ol ÝÂ 2006 x ÓÔúQ[« , ª bcd[ö Oï uÎ Jk ÎÄ} ËMHØPX} uQÔ .¤ QÃ ïX IQJKQL §¨ 2001 x QÃ XMHQÌÔÍQ XQaQ"¤ ÓÔ Ø"} ¦ [åæ OlOïS1k 20, ð G-S ÃN ÏÎ G-N 0.9898 104, Ì hb° ÃN abbcçè X Jk 50 2. S , ð ÐÒ S . : , ÓÔ ÝQÂQéê ~ KLX P = [a; b2; b3; b4]T = (2:4051; 6:5431; 7:4590; 229:6480)T . úÁ ÿX 2QÈQÉQÃ ÖQ zØQOk Ëbcd[ö , . NOkXQ} Q× CDE tbcd[ ÓÔX , vt uQu Jk îá Îíç ² ¤¬ \ Dõ 3. è ©ö 3 ª [0.4, 0.9] 1. :øk ¬Éù ÛÝÞÉXYR1HN1kWa"¤XßàQU ëQìk x± GM(1,N j ,r ) IQJKQL KL , ÝÂ ° $ % Ä Z x x h s O y x x à r x y ° x w ø y S A S A A x ï X Ì © I ° x C ± x ² x ' à T A . Å T A . w l x A . É N Å O y b t b ² n h s à t ë ö X } n À ï } n t : n n ª . Ï % C ª t | Ï X } n } n [ C Å S Î | | Ö È ¥ ³ p © x - t £ ñ Ì ² Ë q ð % : ² ÷ % m Ì ² ÷ %

150 2 {}|}~}}}}} ýþÿ 2002 2001 (2001{2006 ) 2003 2004 2005 29 ( ) : 2006 ©ª«¬³·¸¶ ©ª«¬³ ´µ ¥¦§¨©ª±² ª 42.60 38.09 2.07 48.09 41.55 2.16 50.33 51.95 3.37 63.30 62.04 3.57 71.31 77.02 3.72 82.63 73.47 5.24 428.61 497.35 572.23 641.00 837.02 1110.69 1 )* 1 2 3 4 5 6 3 2 !"# $%&'( GM(1,N ) GM(1,N j ,r ) +,)* 428.61 497.35 572.23 641.00 837.02 -./ 0 34 409.915 489.7727 558.7492 5678 4.36% 1.52% 2.36% 34 428.61 12 5678 0 425.2065 14.50% 723.0480 26.35% 731.6074 14.14% 1034.7555 56.54% 805.6954 976.0206 16.60% 34 428.61 484.11 577.02 667.13 837.27 1110.69 1088.6698 1328.2828 19.59% 1097.86 3.47% 1.98% 12 5678 0 2.66% 0.83% 4.07% 0.03% 1.15% ¹º 5678 9: 4.86% 0.8765 26.72% 0.6154 1.74% 0.9556 3 y X÷ g ìk ï=A=B=Cò=D=A=B=@=E=F=G=H=I=J a=b , GM(1,N ) , GM(1,N j ,r ) , . K=L KLXâã÷ L=M=NPORQ=S=T=U=V=W , , àm GM(1,N ) Ô[ 98 >=?=c=d=e=f=g=h=i=j=kml=nmo=pmqmr=@msmt=\ GM(1,N j ,r ) 56.54%, K=L=u=v=q=r=w=x . , ª=< >=?=X=Y=Z=[=\P]R^=>=? , . û=>=?=@âã _=` |=@ =Y=Z=[=\P]R^=>=?= y=z={ 2004 >=?==dPORQ=@=S= , . = 5 ==e=f==s=t=\=@ x=~===I=J J=====q=r=w== =======@===>=? === \=X=°=±=¤=² Ä=Å=Æ Ô=Õ=Ö=× à=á =@ \=ê=Ý ³=´=µ=¶=·=¬=P¸RY=¹=Kmº ==?=Ç==È=É=@=Ê=º=Ë=S=²=Ì=N @=u=v=ØP¸ ==?=â=ã=ä ë=ì=Y=ZP]R^=_=`=o=í=Y===@=o=í Ù=Ú=ÛPO =å GM(1,1 j ,r ) GM(1,N ) =d , , . . , , GM(1,N j ,r ) ||GM(1,N j ,r ) , GM(1,N ) GM(1,1) ==>=? , ==?= =?=@ =¢=£=_=e=f==Y=¤=¥=¦=§=¨=©=ª=«=¬==@=®=¯ =?=X , . m» I=t=Ã =?=@=S=¡ m¤m¥=@m¼=½mJ=¾mXm¿=ÀmE=¬ Ä=Í GM(1,N j ,r ) j=k=Ü=Ý=@=l=n =? u=v , × . === =Á====?=@=I=S=Â =?=Î=U=g=h=Ï=Ð=j=k=V=Ñ=Ò=Ó==k {=ß , Å £=_=e=f=¬=PæRç=è=K=p=Ø=±=@ u=v==d=´mÓ=@=ÞmG , d=©=ª=«=\=C=s=tm\=K=pm@=¬m ü ¾ % h s % ' ; s u K } C Ã Õ » » é Á

10 : ðñ òóôõö GM(1; N j; r ) ÷øùúûüýþÿ 151 [1] [2] [3] [4] [5] [6] The Basis of Grey Technology and Its Application[M] Beijing: Science Press, 2005 [M] : , 2002. , Xiao X P, Song Z M, Li F Deng J L. The Basis of Grey Theory[M] , ò%ùú&' ò #$ !" ./012 3ô45òó÷ø ,- . , . Wuhan: HUST Press, 2002. [M]. : , 2005. () GM(1,N ) % [J]. * + Zhang L T, Luo Y X. Multi-factored grey GM(1, N ) model of experimentation data processing and its applica- tion[J]. Journal of Machine Design, 2003(3): 23{25. 6789 ù&' , . <=" :; Ye Z, Chen K M. Integrated forecasting of GM(1, 1) and GM(1, N ) based on self-correlation theory[J]. J of University of Shanghai for Science and Technology, 2002(1): 17{20. >@?AB 3 HI JK DEFG GM(1,1) GM(1,N ) [J]. , 2002(1): 17{20. , 2003(3): 23{25. . , OP A new method of set up GM(1, N ) forecasting model[J] System Sciences and Comprehensive Studies RSTUVWEXY , 1997(4): 17{22. GM(1, N ) LMN He M X in Agriculture, 1997(4): 17{22. FG÷ø3Q [J]. ]^$ Z[\ Qiu W J, Liu S F 28(11): 1679{1681. . GM(1,N ) [J]. , 2002, 28(11): 1679{1681. Dispersed structure solve of model GM(1,N )[J]. Systems Engineering and Electronics, 2002, ÷ø3_`ÿabc TUJdV@e ü% [7] Wu W Y, Chen S P. A prediction method using the grey model GMC(1, N ) combined with the grey relational analysis: A case study on internet access population forecast[J]. Applied Mathematics and Computation, 2005, 169: 198{217. [8] Tien T L. The indirect measurement of tensile strength of material by the grey prediction model GMC(1, N )[J]. Measurement Science and Technology, 2005, 16: 1322{1328. [9] Deng J L. A novel grey model GM(1,1j , ): Generalizing GM(1,1)[J]. Journal of Grey System, 2001, 13(1): 1{8. [10] Deng J L. Solution of grey dierential equation for GM(1,1j , ) in matrix train[J]. Journal of Grey System, 2002, 14(1): 105{110. [11] . [M]. 2006. f%g Wuhan Science & Technology Bureau. Wuhan Statistics Yearbook on Science & Technology[M]. 2006. fU9hi î ï C

资料库

论文研究-灰色多变量GM(1,Nγ\,r)模型及其粒子群优化算法.pdf

相关推荐

开发技术

热门标签

最新资料