ASReml-S reference manual - VSN International

2.2 Estimation 12where H ij = ∂ 2 H/∂κ i ∂κ j .− ∂2 l R∂κ i ∂κ j= 1 2 tr (P H ij) − 1 2 tr (P H iP H j )+ y ′ P H i P H j P y − 1 2 y′ P H ij P y (2.7)The elements of the expected information matrix are( )E − ∂2 l R= 1 ∂κ i ∂κ j 2 tr (P H iP H j ) . (2.8)Given an initial estimate κ (0) , an update of κ, κ (1) using the Fisher-scoring (FS) algorithmisκ (1) = κ (0) + I(κ (0) , κ (0) ) −1 U(κ (0) ) (2.9)where U(κ (0) ) is the score vector (2.6) and I(κ (0) , κ (0) ) is the expected informationmatrix (2.8) of κ evaluated at κ (0) .For large models or large data sets, the evaluation of the trace terms in either (2.7) or(2.8) is either not feasible or is very computer intensive. To overcome this problem theAI algorithm [Gilmour et al., 1995] is used. The matrix denoted by I A is obtained byaveraging (2.7) and (2.8) and approximating y ′ P H ij P y by its expectation, tr (P H ij )in those cases when H ij ≠ 0. For variance components models (that is, those linear withrespect to variances in H), the terms in I A are exact averages of those in (2.7) and (2.8).The basic idea is to use I A (κ i , κ j ) in place of the expected information matrix in (2.9)to update κ.The elements of I A areI A (κ i , κ j ) = 1 2 y′ P H i P H j P y. (2.10)The I A matrix is the (scaled) residual sums of squares and products matrix ofy = [y 0 , y 1 , . . . , y k ]where y i , i > 0 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) and y 0 = y, the data vector.In this form the AI matrix is relatively straightforward to calculate.The combination of the AI algorithm with sparse matrix methods, in which only nonzerovalues are stored, gives an efficient algorithm in terms of both computing time andworkspace.One process involves estimation of τ and prediction of u (although the latter may notalways be of interest) for given θ, φ and γ. The other process involves estimation of thesevariance parameters.