By Allen Mclntosh (auth.)
The expanding energy and lowering cost of smalI pcs, especialIy "personal" desktops, has made them more and more well known in statistical research. The day will not be too remote whilst each statistician has on his or her computer computing strength on a par with the big mainframe desktops of 15 or two decades in the past. those similar elements make it quite effortless to procure and manage huge amounts of information, and statisticians can count on a corresponding raise within the dimension of the datasets that they need to examine. regrettably, as a result of constraints imposed by means of structure, measurement or fee, those smalI pcs don't own the most reminiscence in their huge cousins. hence, there's a becoming want for algorithms which are sufficiently low-cost of area to allow statistical research on smalI desktops. One region of study the place there's a desire for algorithms which are low-priced of house is within the becoming of linear models.
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Extra info for Fitting Linear Models: An Application of Conjugate Gradient Algorithms
Proof: For any i, 1~ i ~ s, multiplication of assumption (I) on the left by T; gives T; = T; TI + ... + T; T, By assumption (2), all transformations on the right hand side vanish except possibly T; T;= T;2 and so we have T; = T;2 as required. To obtain the direct sum result, let v be any vector in V. By assumption (1), we have v = Iv = T1v+ ... 2) so that v is written as a sum of vectors in the subspaces T; V. To show that this is unique, suppose that v = VI+ ... +v s where viE T; V i=l ... 3) and suppose that vj '¢: Tjv for some j.
X/l has no component in (4). J Thus, the decomposition corresponds to the common description of the estimation space. 1. We emphasize that this discussion applies to a balanced experiment only. The problems that occur in the unbalanced case are discussed below. 1. 1 This description suggests that we might write the four vector spaces listed above using orthogonal projection operators: 1. R(1)R" 2. [I-R(I))R(F])R" 3. [I-R(t))R(F2)R" 4. ) This can be generalized to an arbitrary model, and results in a direct sum decomposition, as we now show.
Suppose that we have some symmetric, positive definite matrix A so that the product A-Iv is easy to compute, and A is a "reasonable" approximation to G. Let B be any non-singular matrix such that B'B = A and put 9 = BfJ It can be shown (see for instance Buckley, 1978) that the key formulae in the Hestenes-Stiefel algo- rithm become ayl = _ p(kl'g(kl p(kl'Gp(kl where g(kl still represents the gradient of", with respect to fJ: In similar fashion, one can show that the formulae of Beale (I972) become k=I,2, ...