ASReml-S reference manual - VSN International

4.4 Variance model functions 48CholeskykDetailsmodelnumber of parameters(f) form: f() fv() fh()cor 1 2 1 + ncorb k − 1 k 2k − 1corg n(n − 1)/2 1 + n(n − 1)/2 n + n(n − 1)/2diagnus n(n + 1)/2chol 1 n(n + 1)/2cholc 1 n(n + 1)/2ante 1 n(n + 1)/2fakn + n1 chol, cholc and ante models have (k + 1)(n − k/2) parameters but n(n + 1)/2 initialvalues row-wise from the lower triangle of an unstructured matrix are given andconverted to the appropriate parameterization.the number of subdiagonal bands for corbthe order of the Cholesky decoposition for chol and cholcthe order of antedependence (ante) and factor analytic models (fa).The k-factor Cholesky structure models Σ ω×ω asΣ = LDL ′cholwhere L ω×ω is a unit lower triangular matrix and D = diag(d 1 , . . . , d ω ).In the chol(,k) factorization L has k non-zero (unequal) bands below the diagonal, thatis, the elements {l ij } of L arel ii = 1l ij = v ij . 1 ≤ i − j ≤ kl ij = 0, otherwisecholcIn the cholc(,k) factorization L has columns l i = (l 1i , . . . , l ωi ) ′ wherel ii = 1l ij = 0 for i < j, k < j < iFor example, if a factor Site has 4 levels thenasreml(. . . ,cholc(Site,1). . . )generates Σ = LDL ′ where⎡1⎢ lL = 21 1⎣l 31 0 1l 41 0 0 1⎤⎥⎦ D =⎡⎢⎣⎤d 1 0 0 00 d 2 0 0 ⎥⎦0 0 d 3 00 0 0 d 4This form is similar to a Factor Analytic model. In asreml the initial parameters forboth Cholesky factorizations are given as the lower triangle row-wise of an unstructuredmatrix and converted internally to the appropriate factorization. So, if⎡⎤σ 11⎢ σΣ = ⎣21σ 22⎥⎦σ 31σ 32σ 33σ 41σ 42σ 43σ 44