ASReml-S reference manual - VSN International

B Available variance models 126Details of the available variance modelsfunctionnamedescriptionalgebraicformnumber of parameterscorr hom hetvariance varianceisp() spherical C ij= 1 − 3 2 θ ij + 1 2 θ3 ij0 < φ 11 2 1 + ωcir()circular (Websterand Oliver [2001])C ij= 1− 2 (θ π ij√1 − θ2ij+ sin −1 θ ij )1 2 1 + ω0 < φ 1aexp() anisotropic exponentialC ii= 1C ij= φ |x i−x j |1 φ |y i−y j |2x and y vectors of coordinates|φ 1| < 1, |φ 2| < 12 3 2+ωagau()anisotropicgaussianC ii= 1C ij= φ (x i−x j ) 21 φ (y i−y j ) 22x and y vectors of coordinates|φ 1| < 1, |φ 2| < 12 3 2 + ωmtrn() Matérn withfirst 1 ≤ k ≤ 5parametersspecified by theuserC ij=Matérn: see Section 4.4.3φ > 0 range, ν shape(0.5)δ > 0 anisotropy ratio(1),α anisotropy angle(0),λ(1|2) metric(2)k k+1 k + ωHeterogeneous variance modelsdiag() diagonal = idh() Σ ii= φ iΣ ij= 0, i ≠ j - - ωus()unstructuredgeneral covariancematrixΣ ij = φ ij- -ω(ω+1)2ante(,k)chol(,k)antedependenceorder k1 ≤ order ≤ω − 1choleskyorder k1 ≤ order ≤ω − 1fa(,k) factor analyticorder kΣ −1 = UDU ′D ii= d i, D ij= 0, i ≠ jU ii= 1, U ij= u ij, 1 ≤ j − i ≤orderU ij = 0, i > jΣ = LDL ′D ii= d i, D ij= 0, i ≠ jL ii= 1, L ij= l ij, 1 ≤ i−j ≤ orderΣ = ΓΓ ′ + Ψ,Γ contains covariance factorsΨ contains specific variance- - (k + 1)(ω − k/2)- - (k + 1)(ω − k/2)- - kω + ω