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R.GuoAn univariate DEMR modelling on repair effects - RTA # 3-4, 2007, December - Special Issue⎧ ( δ + β ) α⎪ A0=22⎪ω + ( β + δ )⎨⎪αω⎪B0= −2⎩ ω + ( β + δ )2(21)In theory, the expressions of A0and B0willdetermine the particular solutionxpδt= A0esin( ωt+ ϖ )x pδ+ B et cos( ωt+ )(22)0ϖwhich will result in the general solution to Eq. (14) as−βtδtx = c1e+ A0esin( ωt+ ϖ )δ+ B et cos( ωt+ )(23)0ϖNote that for the unequal-gapped data sequence,( ) ( ) (( ) ) (X 0 = ( x 0 t 0 ( ) 0)1 , x t2, L , x ( tn)), the coupling (ortranslation) rule is slightly different from the equalgappeddata sequence.Table 2. Coupling Principle in unequal gappedGM(1,1) Model.Term Motivated DE Coupling REGModel FormationIntrinsic ContinuousDiscretefeatureIndependenttVariablet kResponse ( 0x) ( t)1 st -orderDerivative2 nd -orderDerivativePrimitivefunction( 0x ) ( t k )dx (1) ( t)/dt ( 0 ) ( )x t k0 02 (1) 2d x ( t)/dt x( )( ) − x( ) ( t )x(1) () tParameter ( )DynamiclawDynamics(Solution)Filtering(Prediction)t ktk− tk−1(1) ( ) z t kData Assimilation in Modelα, β( a,b)dxδt+β x =αe sin( ω t+ϖ)dt( 1 ) ( )δt0δt0−βt1x t = ce( t )( t )+ A e sin ω +ϖ+ B e cos ω +ϖ( 0 ) −βtx ( t)= −βce1δt+ ( A0ω+ B0δ) e cos( ω t+ϖ)δt+ ( A δ−B ω) e sin( ω t+ϖ)0 0The coupling regression isx(0)( tk)=kα δ tke sin( ωt+ ϖ )k −1( )( 0x )( tk) =αe ( 1sin( ω t ) )k+ϖ +β −z ( tk)δt k( 1 ) ( )kδtk0δtk0−βtk1x t = ce( tk)( t )+ A e sin ω +ϖ+ B e cos ω +ϖ( 0 ) −βt x ( t )kk = −βce1δt+ ( A0ω+ B0δ) e cos( ω tk+ϖ)δt+ ( A δ−B ω) e sin( ω t +ϖ)0 0kkwherez =z(1)( −z( )) ε ,+ β + k = 2,3,4,K,n ,(24)(1)(0)( t1)z ( t1)t1t kk( t − t )(1)(1)(0)( tk) = z ( tk−1) + z ( tk)k k −1k = 2,3,4,K,n . (25)The parameter pair ( αβ , ) is obtained by least-squareestimation ( , ) ( ) −1⎡e⎢⎢eX =⎢⎢⎢⎣e⎡z⎢⎢zY =⎢⎢⎢⎣zδt2δt3δtn(0)(0)(0)T T Ta b = X X X Y , wheresin( ωtsin( ωtMsin( ωt( t( tM(12t n) ⎤⎥) ⎥⎥⎥) ⎥⎦12n+ ϖ )+ ϖ )+ ϖ )− z− z− z(1)(1)M(1)( t( t( t12n) ⎤⎥) ⎥,⎥⎥) ⎥⎦(26)since δ and ω are given (in a manner by trials anderrors).Formally, we have a DEMR model as⎧dx⎪ + βx= αedt⎨⎪(0)⎪⎩x ( tk) = αeδtδtsin( ωt+ ϖ )sin( ωt+ ϖ ) + β4. Fuzzy repair effect structure(1)( − z ( t ))k+ ε .k(27)In standard regression modelling exercises, it is oftento assume that the error terms εi, i= 1,2, L , n arerandom with zero mean and constant variance, i.e.,2E ε = 0 and VAR [ ε ] =σ , i= 1,2, L , n. It is[ ]iitypically assuming a normal distribution with zero2mean and constant variance, i.e., ( 0, )N σ .Furthermore, as we pointed out that a grey differentialequation model is a motivated differential equationmotivated regression, which takes the form translatedfrom the motivated differential equation, as shown inTable 1 for GM(1,1) case. However, we should befully aware that translation back and forward betweenthe motivated differential equation and the coupling- 92 -