1 IntroductionNCmachinetoolsarekeyequipmento fmoderntimesandmakesprecisionresultsingearproduction.ThecomplicatederrorsystemofarunningNCmachin etoolincludesgeometryerror,heaterrorandloaddistortionerroretc .Thissystemisagraysystembecauseonlypartinformationisknown .Erroranalysisanddataprocessingofagraysystem istousetheknowninformationtounderstandtheer rorcharacteristic,stateanddevelopmenttogettodecisionmakingonitsabilitytowork .Thegraysystemtheorywasdevelopedinthe 1 980’sandiswidelyappliedintheeconomicsociety .Asaresultoftheoreticalcharacteristicsandnoveltyofthegraysystem ,thegrayforecast,theassociateanalysis ,thegrayclusteringandgraydecision makingarealllikelytobecometoolsoffailurediag nosisanderroranalysis.Theprocessoferroranal ysisanddisposalistoforecast,toestimateandtomakedecisionthroughknowninformationsystemthatcontainunknowninformation .Thispaperstudiestheapplicationofgrayforecas tmeansandgrayassociateanalysisforgeometryerrordatainamachiningcenter.2 GM(1,1 )modelandwaystodisposeoferrordataTheGM( 1 ,1 )modeloftraditionalgraysystemtheoryrequiresthedisposeddatabepositivenumbers ,butNCmachinetoolerrordatausuallyhavepositiv eandnegativenumbers,andcurrentlycannotusedirectmodeling;therefore ,itfirstrequirestotransformtoapositivenumberforthedata ,namelyeverylaster rordataaddingoneconstantW(W >0 ) ;inthisway ,originalerrordatatranslatesintoapluslist{xi}.Forbestowedvariablexi,yi,i =1 ,2 ,… ,N ,fetchyi+ 1-yi=a =constant (i=1 ,2 ,3,… ,N - 1 ) ,thenvariableyicorrespondstonaturalnumberlist {t},t=1 ,2 ,3,… ,N ,andvariablexibywayoftransformcanbetranslatedintoapositivenumberlist:x(0 )(t) , t=1 ,2 ,3,… ,Nthejustlist x(0 )(t) currentlycannotbeusedindirectmodeling ,sincethelistisusuallystochastic ,dis crete ;thereforeaddupandobtain :x(1)(t) =∑tk=1x(0 )(t) ( 1 )therebygaininganewlist:{x(1)(t) }, t=1 ,2 ,3,… ,N ( 2 )Where ,( 0 ) -originaldata( 1 ) -newlistoforiginaldatacreatedBythisprocesswe canweakentherandomnessoftheoriginallist.Inthisway,xibecomespositiveornegativeindirectmodeling ,whilexiispositiveandnegative ,itneedsaddingaplusconstantWtogetwholexitobecomepositive .TherearetwowaysofchoosingW ,oneistooptimize(interpolationerrorleastofsinglevariable) ,anotherisexperience .Variablex(0 ) ={x(0 )(1) ,x(0 )(2 ) ,… ,x(0 )(N) },byonceaddingupandobtain :x(1) ={x(1)(1) ,x(1)(2 ) ,… ,x(1)(N) }( 3)Relevantdifferentialequation :dx(1) dt+ax(1) =u ( 4)Theformofresult:x(1)(t) =
e-at +u a ( 5)Thedispersalbecome :x(1)(k) =e-a(k- 1) +u a ( 6 )intheformula :u ,a -parameterofwaitfordiscernandendogenesisvariable .Marktheparameterlistofwaitfordiscern : ^a =au ( 7)Usingtheleast squaremethodtoseektheresultfora :^a= T =(BT B) - 1BT YN ( 8)intheformula :YN =T ( 9)B =- 2 1- 2 1 - 2 1( 1 0 )Withafillinequation( 6 ) ,andletx(1)(0 ) =x(0 )(1) ,obtaintheforecastfunctionoftheGM( 1 ,1 )model:x(1)(k+ 1) =(x(0 )(1) -u a)e-ak +u a ( 1 1 )Thisresultregressesbysubtractioninsuccession ,x(0 )(k+ 1) =x(1)(k+ 1) -x(1)(k) ( 1 2 )krevertstoyi,x(0 )(k) revertstoxi,obtainthefittingfunctionofxi,yi.Workouttheresultandcheckuponmodelprecision.Testmeansasfollows:x(k)isthefullsizeofthetimeofk ,x(0 )(k) iscalculatednumericalvalue ,thentheoffsetofthetimeofkisq(k) :q(k) =x(k) -x(0 )(k)Mark xforx(k) (k =1 ,2 ,… ,N) : x =1N∑Nk =1x(k)Mark qforq(k) : q =1N∑Nk =1q(k)S21isthemeansquarevalueofthepracticedata : S21=1N∑Nk =1(x(k) - x) 2S22 isthemeansquarevalueoftheoffset: S22 =1N∑Nk=1(q(k) - q) 2CandPareacoupleofguidelinestotestfinalerror:C =S2 S1P =P{｜q(k) - q｜<0 .6 745S1}P -smallerrorprobabilityLesserC ,theresultisbetter.ClessermeansalargerS1andsmallerS2 ,andthemeansquarevalueoftheoriginaldataislarger,theoriginaldataisdispersed ,themeansquarevalueoftheoffsetissmaller,andtheoffsetismass .BiggerP ,theresultisbetter.Pbiggermeanstheoddsofoffs etandthemeansquarevalueoftheoffsetislessthanthebestowedvalue0 .6 745S1.Wecansyntheticallyjudgetheprecisionofthemodelacco