Appl. Math. Inf. Sc. 6 No. 3 745-750 (0) 745 Appled Matheatcs & Inforaton Scences An Internatonal Journal The Non-equdstant New Inforaton Optzng MGM(n) Based on a Step by Step Optu Constructng Background Value Rubao Zou College of Scences Hunan Agrculture Unversty Changsha 408 Chna Receved: Mar. 8 0; Revsed May 4 0; Accepted Jun. 7 0 Publshed onlne: Sep. 0 Abstract: Grey syste s a theory whch studes poor nforaton specally and t possesses wde sutablty. Applyng a step by step optu new nforaton odelng ethod to construct new nforaton background value of ult-varable non-equdstance new nforaton Grey odel MGM(n) takng the th coponent of each varable as ntal value of response functon takng the ean relatve error as objectve functon and takng the odfed values of response functon ntal value as desgn varables the ult-varable non-equdstance new nforaton Grey odel MGM(n) was bult. The proposed odel can be used to buld odel n non-equal nterval and equal nterval te seres. It enlarges the scope of applcaton and has hgh precson and easy to use. Exaple valdates the practcablty and relablty of the proposed odel. Keywords: Multvarable background value non-equdstance sequence a step by step optu odelng new nforaton optzng non-equdstance MGM(n) least square ethod.. Introducton The theory of Grey syste s the study of the grey syste analyss odelng predcton decson akng and control theory. Gray odel s an portant content of the grey syste theory In the search for laws between data t akes up for the lack of avalable data nng ethod and provdes a new scentfc ethod for data nng. Snce professor Deng Ju Long brng about the grey syste theory n 98 gray odel s wdely appled n any areas []. Grey odel has ore types anly ncludng GM () GM (N) MGM (N) etc. where GM () has been used wdely and researched deeply GM(N) Can only be used for the qualtatve analyss and cannot be used to predct. Beng an extenson of GM () odel n case of n varable MGM(N) odel s nether a sple cobnaton of GM () odel also dffers fro GM(n) odel.there are n dfferental equatons contaned n eleents In MGM(N) odel but just a sngle frst order dfferental equaton contaned n eleent n GM(N) odel then we can fnd ther sultaneous soluton and paraeters n odel MGM (N) can reflect the nterrelatng and nteractng relatonshp aong ultple varables. Because Study on MGM (N) odel s uch less than that on GM () odel so far studyng deeply on t has portant theory sgnfcance and applcaton value. Lterature [] have corrected and establshed optzaton MGM(N) odel that regard the frst coponent of the sequence as the ntal condtons of grey dfferental equaton. Accordng to the new nforaton prorty prncple of grey syste theory Lterature [4] have corrected and establshed ultvarate varables new nforaton optzaton MGM (N) odel that regard the nth coponent of the sequence X () as the ntal condtons of grey dfferental equaton. Lterature [4] have establshed ultvarate new nforaton MGM ( N) odel that regard the nth coponent of X () as ntal condtons of the grey dfferental equaton and ade optal correcton for the ntal value and background value coeffcent(background value s ntroduced n the for of z () = q (k + ) + ( q) (k) (q [0 ]) but these MGM (N) odel are equdstant odel. Lterature [6] bult non equdstant ultvarable MGM (N) by eans of hoogeneous ex- Correspondng author: e-al: rbzou@63.co
746 Rubao Zou: The Non-equdstant New Inforaton Optzng MGM(n)... ponental functon fttng background value. However nonhoogeneous exponental functon s ore unversal then there are nherent defects n echans of the odelng. Lterature [7] bult non equdstant ultvarable MGM (N). But ts background value was generated by usng of the ean value so the precson of the odel needs to be further proved. Lterature [8] bult non equdstant ultvarable MGM(N) through non hoogeneous exponental functon fttng background value provng the accuracy of the odel. However the paraeters only use least square ethod wthout akng a response functon optzaton. Lterature [9] analyzed the structure ethod of background value of ultvarable grey odel MGM () puttng forward to utlzng ratonal nterpolaton and nuercal ntegraton of trapezod forula and extrapolaton to reconstruct background value by usng of the vector valued contnued fracton theory and provng the precson of sulatng and forecastng odel effectvely for ultvarate nterval MGM () odel. Lterature [0] proposed a gradual optzaton new nforaton equdstant GM () odel by usng of grey syste odelng ethod and new nforaton prncple. Based on the GM () odelng the odel used the nth coponent of the orgnal data as ntal condtons of the grey dfferental equaton. Paraeters of the odel were estated through the optzaton of background value and dfference adjustent coeffcent. In ths paper takng the th coponent of the data as the ntal value of soluton of grey dfferental equaton absorbng new nforaton odelng and a step by step optzaton ethod fro lterature [0] constructng new nforaton background value of ultvarable non equdstant grey new-nforaton odel MGM(n) takng nu relatve error as objectve functon and the correcton of ntal value as desgn varables ultvarable non-equdstance grey new nforaton optzaton odel MGM(n) s establshed. The odel s not only sutable for the equdstant odel as well as ncludng non equdstance odelng. It extends the scope of applcaton of the grey odel. In addton to hgh precson the odel has great theoretcal value and appled value.. Non-equdstant ultvarate new nforaton optzaton odel based on a step by step optzaton odelng to construct new nforaton background values Defnton : Suppose the sequence = [ (t ) (t ) (t )] where = n. If t j = t t j cost ( j ) then s called non-equdstant sequence. Defnton : Let X () denoted by X () = [ (t ) (t ) (t )] the sequence X () s called the frst-order accuulated generatng operaton of non equdstant sequence f (t ) = (t ) = (t j ) + t j where = n j = 3 t j s sae to Defnton. Set the orgnal data atrx of ultvarable = [ X(0) X(0) n ] T (t ) (t ) (t ) = (t ) (t ) (t ) () n (t ) n (t ) n (t ) where ( = n j = ) s the observed values of varable n the oents of t j [ (t ) (t ) (t )] s non equdstant sequence naely the dstance of t j t j (j = 3 ) s not constant. In order to establsh odel frst takng the accuulated raw data a new atrx can be obtaned naely X () = [X () X() X() n ] T (t ) (t ) (t ) = (t ) (t ) (t ) () n (t ) n (t ) n (t ) where ( = n) eet the defnton naely = j (t ) + (t k ) t k (j = ) (3) k= (t ) (j = ). Multvarable non equdstant MGM ( n) odel s frst order dfferental equatons contanng n eleent d = p x () + p + + p n n + q d = p + p + + p n n + q (4) d n = p n + p n + + p nn n + q n. Note p p p n p A = p p n B = p n p n p nn q q.. q n
Appl. Math. Inf. Sc. 6 No. 3 745-750 (0) / www.naturalspublshng.co/journals.asp 747 then (4) ay be wrtten as dx () (t) = AX () (t) + B. (5) Accordng to new nforaton prorty prncple of grey syste theory we regard the frst coponent of the sequence (j = ) as ntal condtons of the grey dfferental equaton causng that new nforaton cannot be appled fully If we regard th coponent of the sequence (j = ) as ntal condtons of the grey dfferental equaton causng that new nforaton can be appled fully n whch contnuous te response of (5) s X () (t) = e At X () (t ) + A (e At I)B (6) where e At A k = I + k! tk I s unt atrx. k= In order to obtan A and B we ntegral the both sde on the nterval [t j t j ] n t j = p k + q t j (7) = Let n k= k= t j t j p k z () = (t j ) t j + q. (8) t j (t j ) t j The tradtonal background value calculaton forula actually use trapezodal areaz () t j thus there s bg error. But we obtan Paraeter atrx  and ˆB by eans of regardng z () = t j t j as background value calculaton t j n the nterval [t j t j ] that s ore sutable for whtenng equaton (4). Accordng to exponental rule of gray forecast odel and odelng theory and ethod [0] of a step by step optzaton new nforaton non-equdstant GM ( ) odel we take (t) = a e bt + c where a b c are the undeterned coeffcent. The lterature [0] analyzed thought and ethod of a step by step optzaton new nforaton non-equdstance GM ( ) odel and the key of odelng s whtenng grey dervatve dx(). If we choose reasonable whtenng grey dervatve accuracy of odelng wll be proved. The ethod to obtan whtenng grey dervatve wth the ost ntutonst and easy to understand s a dfferental quotent nstead of dervatve. (t j) d (t j+ ) x() (t j+ ). (9) t j+ t j d x() (t j+ ). (0) t j+ t j In fact on the prese of X(t) beng a dervatve the Lagrange ean value theore shows x() t j+ t j s a dervatve value n a pont of the nterval (t k t k+ ) that s dervatve value can be thought as beng known value the correspondng varable values are nterval gray nubers (t j t j+ ). For the ntroducton of grey dervatve correcton coeffcent correcton of grey dervatve we adopt grey dervatve correcton coeffcent ρ and ξ to correct x grey dervatve and construct odelng ρ () (t j+) (t j) t j+ t j. Actually we don t knowξ and ρ. We take step by step optzaton ethod and the steps are as follows: ) Obtan orgnal data. )Take ntal value of the teraton step nuber repeatedly s = 0 a (s) = a (0) = 0 so ξ (s) = ea (s) e a = (s) a (s) ρ (s) = a (s)( + e a (s) ) ( e a (s) ) =. (t j+) (t j) Takng lnear regresson to whtenng values = ([ ξ ] + ξ ) (t j+ ) We obtan the whtenng values of paraeter: â (s+) = ρ (s) S (s)xy S (s)xx () where (s) (t j) = [ ξ (s) (s) (t j)] + ξ (s) (s) (t j+) (s) = (s) (t j) y (s) (t j ) = x() (s) (t j+) (s) (t j) t j+ t j ȳ (s) = y (s) (t j ) S (s)xx = [ (s) (t j) (s) ] S (s)xy = [ (s) x() (s) ][y (s)(t j ) ȳ (s) ]. Then we construct the lnear regresson to get ndex odel M s+ : ˆ (s) (t j+) = ĉ (s+) e â (s+)(t j t ) + ˆb (s+) ()
748 Rubao Zou: The Non-equdstant New Inforaton Optzng MGM(n)... where ĉ (s+) = [e â (s+)(t j t ) e â (s+)(t j t ) ] { ( [e â (s+)(t j t ) e â (s+)(t j t (3) ) ] [ (s) (t j) ˆb(s+) = ĉ (s+) (s) (t j)] ) } (s) (t j) e â (s+)(t j t ). (4) After fndng out â ˆb ĉ z () can be solved and then get z () (t j ) = t j (t j ) =. t j Set p = (p p p n q ) T ( = n) p can be to acheved the value ˆp by the least square ethod. ˆp = (ˆp ˆp ˆp n ˆq ) T = (L T L) L T Y ( = n) (5) where z () (t ) z () (t ) z () n (t ) L = z () (t 3) z () (t 3) z n () (t 3 ) (6) z () (t ) z () (t ) z n () (t ) Y = [ (t ) (t 3 ) (t )] T. (7) We can get dscrnaton value of A and B: Â = ˆp ˆp ˆp n ˆp ˆp ˆp n ˆp n ˆp n ˆp nn ˆB = ˆq ˆq.. ˆq n Coputaton value of new nforaton MGM ( n) ode s ˆX () = eâ(tj t) X () (t )+ Â (eâ(t j t ) I) ˆB. (8) We use the th coponent of the data as ntal value of the grey dfferental equaton and odfy ntal value naely (t ) + β nstead of (t ) whch β s vector β = [β β β n ] T. The fttng values of the orgnal data are obtaned. ˆ (t j ) l = ˆX () t 0 X () () (t j) ˆX (t j ) t j t j (t ) X ()(t t) (j = ) t (j = 3 ). (9) The absolute error of th varable s ˆ. Relatve error of th varables s e = ˆx(0) 00. (0) (t j ) The ean value of relatve error of th varable s e. The average error of all the data s ē = n n = e. Takng the average error f as objectve functon and as desgn varables usng the optzaton functon of Matlab 7.5 optzaton ethod or other ethod for solvng all paraeters are obtaned. 3. Precson nspectng for the odel The nspectng eans contan resdual analyss correlaton degree analyss and post-error analyss [7 3]. The dsplaceent relatve degree the speed related degree the acceleraton degree and the total related degree are calculated sultanety. These knds of related degrees are called related degrees of C- type [4] whch can be used to both of the whole analyss and the dynac analyss. The followng related degree nspecton of C-type s eployed n ths paper. ) To calculate the three-layer related degrees. Dsplaceent related degree d (0) (t j ): d (0) (t j ) = x(0) (t j ) ˆ (t j ) (0) wherej =. Speed related degree d () (t j ): d () (t j ) = x(0) (t j+ ) (t j ) ˆ (t j+ ) ˆ (t j ) where j =. ()
Appl. Math. Inf. Sc. 6 No. 3 745-750 (0) / www.naturalspublshng.co/journals.asp 749 Acceleraton related degree d () (t j ): d () (t j ) = x(0) (t j+ ) (t j ) + (t j ) ˆ (t j+ ) ˆ (t j ) + ˆ (t j ) where j =. () ) To calculate the three-layer related coprehensve degree at t j : D(t ) = d() (t ) + d (0) (t ) D(t ) = d (0) (t ) (3) D(t j ) = d(0) (t j)+d () (t j)+d () (t j) 3 where j = 3. (4) 3) To calculate the total related degree of the odel ˆ (t j ) : D = D(t j ) j =. (5) When 0.6 < D 5 3 the precson of the odel s Good. When 0.30 D 0.60 the precson of the odel s better. When D < 0.30 or D > 5 3 the precson of the odel s bad [4]. 4. Model applcatons Exaple : On the contactng strength calculaton prncpal curvature functon F (ρ) and the coeffcent a b wth the pont contactng ellpse length and short radus a b. Paraeters are generally processed by consultng table.the data extracted fro the table []. Table The values of F (ρ) a and b No. 3 4 5 F (ρ) 0.9995 0.9990 0.9980 0.9970 0.9960 a 3.95 8.53 4.5.6.0 b 0.63 0.85 0. 0.8 0.4 No. 6 7 8 9 0 F (ρ) 0.9950 0.9940 0.9930 0.990 0.990 a 0.5 9.46 8.9 8.47 8.0 b 0.5 0.60 0.68 0.75 0.8 No. 3 4 5 F (ρ) 0.9900 0.9890 0.9880 0.9870 0.9860 a 7.76 7.49 7.5 7.0 6.84 b 0.87 0.9 0.97 0.30 0.305 No. 6 7 8 9 0 F (ρ) 0.9850 0.9840 0.9830 0.980 0.980 a 6.64 6.47 6.33 6.9 6.06 b 0.30 0.34 0.37 0.3 0.35 No. 3 4 5 F (ρ) 0.9800 0.9790 0.9780 0.9770 a 5.95 5.83 5.7 5.63 b 0.38 0.33 0.335 0.338 The coeffcent b of the oval short radus b s noted as t j prncpal curvature functon F (ρ) s noted as X the coeffcent a of the oval short radus a s noted as X. Establshng non-equdstance new nforaton optu GM() odel wth the proposed ethod the odel paraeters are as follows:. A = [ 0.3787 0.043 6.4555 8.58 β = ] [ ] 0.7945 B = 96.88 [ ] 0.00505. 0.070706 Prncpal curvature functon fttng value s ˆF (ρ) = [.0000.00040 0.99990 0.99837 0.9968 0.99534 0.99396 0.996 0.9936 0.990 0.989 0.98808 0.9870 0.986 0.98539 0.98446 0.9835 0.9876 0.9800 0.98 0.98034 0.97955 0.97875 0.97806]. The absolute error of prncpal curvature functon s q = [ 0.677.483.8995.374 0.805 0.3375 0.0406 0.3793 0.6374 0.784 0.88 0.936 0.8975 0.7944 0.6079 0.5405 0.4899 0.399 0.0003 0.95 0.3396 0.5505 0.753.063]. The relatve error (percent) of prncpal curvature functon s: e = [ 0.067753 0.497 0.9039 0.3786 0.08379 0.03390 0.004080 0.03897 0.06459 0.078950 0.08906 0.093386 0.090839 0.080488 0.06650 0.05487 0.049785 0.04408 0.00006 0.078 0.034655 0.0563 0.07708 0.0880]. The average value of relatve error s 0.06973%. The precson of the odel s Good. The average value of relatve error of wthout optzaton for new nforaton odel s 0.4657%. So optzaton odel has very hgh precson. Exaple : Refer to reference [7] of data for water absorpton rate affectng the pure PA66 echancs perforance accordng to echancs perforance test on PA66 saples wth dfferent water absorpton rate get the bendng strength and flexural odulus of PA66 and the experental data of the tensle strength changng wth the water absorpton rate. t j s water absorpton rate s the bendng strength (Mpa) s the flexural odulus (Gpa) 3 s the tensle strength (Mpa). The data s shown n table. Table Absorpton rate s nfluence for echancs perforance of pure PA66
750 Rubao Zou: The Non-equdstant New Inforaton Optzng MGM(n)... No. 3 4 5 t j 0 0.0607 0.07 0.66 0.069 83.4 84.9 84.5 84. 84.4.63.64.6.65.66 3 84. 84.4 86.3 84.3 8.3 No. 5 6 7 8 9 t j 0.4344 0.543 0.854 0.9756 78.4 75.4 59.5 54..5.3.90.7 3 74.9 75.7 73. 66.9 Accordng to the ethod entoned n ths paper set up the non-equdstant odel MGM (). The odel paraeters are shown as follows: 0.55 4.74 0.07 A = 0 4 0.0046 0.66 0.0007 0.09.063 0.0043 B = 85.780 6.876 97.965 The ftted value of 3 s ρ =.6073 0.03880..4395 ˆ 3 = [ 87.3769 85.685 84.890 8.7884 8.503 78.83 74.765 70.7947 67.0939]. The absolute error of 3 s q = [ 3.769.85.0.56 0.03 3.383 0.9385.4053 0.939]. The relatve error (percent) of 3 s e = [ 3.7730.58.446.793 0.499 4.544.398 3.859 0.899]. The average value of relatve error s.39%. The precson of the odel s Good. The average value of relatve error of wthout optzaton for new nforaton odel s 3.653%. So optzaton odel has very hgh precson. 5. Concluson In vew of ultvarable non-equdstance sequence that ultple varables affect restrct utually we construct the ultvarable non-equdstance new nforaton optzaton grey odel MGM(n). In odelng applyng a step by step optu new nforaton odelng ethod to construct new nforaton background value of ult-varable non-equdstance new nforaton Grey odel MGM(n) regardng th coponent of the data as ntal condtons of the grey dfferental equaton takng the nu relatve error as objectve functon and takng revsng correcton values of the ntal value as desgn varables. The new odel s not only sutable for equdstance odelng also sutable for non-equdstant odel. It enlarge the scope of applcaton of the grey odel and has hgh precson easy to use and so on. Actual exaples show that the odel s practcal and relable wth portant practcal and theoretcal sgnfcance. It s worth usng wdely. References [] Y. X. LUO and J. Y. LI Proc. Internatonal Conference on Mechatroncs and Autoaton 403(009). [] Y. X. LUO Wu M. L and A. H. Ca Kybernetes 38 435 (009). [3] Z. Jun and J. M. Sheng Systes Engneerng Theory and Practve 7 09 (997). [4] Z. M. HE and Y. X. LUO Proc. Internatonal Conference on Intellgent Coputaton Technology and Autoaton 0(009). [5] Y. X. LUO and W. Y. Xao Proc. Internatonal Conference on Intellgent Coputaton Technology and Autoaton 86(009). [6] F. X. WANG Systes Engneerng and Electroncs 9 388 (007). [7] P. P. Xong Y. G. Dang and H. Zhu Hu Control and Decson 6 49 (0). [8] P. P. Xong Y. G. Dang and Y. YANG Proc. 9th Chnese Conference on Grey Systes 77(00). [9] L. Z. Cu S. F. Lu and Z. P. Wu Systes Engneerng 6 47 (008). [0] Y. X. LUO Systes Engneerng Theory and Practce 30 54 (00). [] Z. H. Han and X. H. DONG Journal of Machne Desgn 5 8 (008). [] Y. X. Luo D. G. Lao and Q. X. Tang Machne tool and Hydraulcs 6 9(00). [3] S. F. Lu T. B. Guo and Y. G. Dang Grey systes and Applcatons (Thrd Edton n Chnese) (Chna Scence Press Bejng 004). [4] Q. Y. Wang Uncertanty atheatcal odel of forecast and decson-akng (n Chnese)( Metallurgcal ndustry Press Bejng 00). Rubao Zou s wth college of scence Hunan Agrculture Unversty Changsha n Chna. He receved the BS degree n 987 and MS degree n 005 fro Hunan noral Unversty. Hs current research nterests nvolve grey syste theory and ts applcaton fractal theory syste engneerng theory and practse and the desgn and pleentaton of teachng anageent syste. He publshed over 0 papers.