Download pdf guide - VSN International

2 Some theory 14where y i is the ‘working’ variate for κ i and is given byy i = H i P y= H i R −1 ẽ= R i R −1 ẽ, κ i ∈ φ= ZG i G −1 ũ, κ i ∈ γwhere ẽ = y − X ˆτ − Zũ, ˆτ and ũ are solutions to (2.11). In this form the AImatrix is relatively straightforward to calculate.The combination of the AI algorithm with sparse matrix methods, in which onlynon-zero values are stored, gives an efficient algorithm in terms of both computingtime and workspace.Estimation/prediction of the fixed and random effectsTo estimate τ and predict u the objective functionlog f Y (y | u ; τ , R) + log f U (u ; G)is used. The is the log-joint distribution of (Y , u).Differentiating with respect to τ and u leads to the mixed model equations(Robinson, 1991) which are given by[ ] [ ] [ ]X ′ R −1 X X ′ R −1 Z ˆτ X ′ R −1 yZ ′ R −1 X Z ′ R −1 Z + G −1 =ũ Z ′ R −1 . (2.11)yThese can be written asC ˜β = W R −1 ywhere C = W ′ R −1 W + G ∗ , β = [τ ′ u ′ ] ′ and[ ]G ∗ 0 0=0 G −1 .The solution of (2.11) requires values for γ and φ. In practice we replace γ andφ by their REML estimates ˆγ and ˆφ.Note that ˆτ is the best linear unbiased estimator (BLUE) of τ , while ũ is the bestlinear unbiased predictor (BLUP) of u for known γ and φ. We also note that[ ] ([ ] )ˆτ − τ 0 ˜β − β = ∼ N , C −1 .ũ − u 0