[R] LME: internal workings of QR factorization
Izmirlian, Grant (NIH/NCI) [E]
izmirlig at mail.nih.gov
Thu Apr 12 22:44:08 CEST 2007
Hi:
I've been reading "Computational Methods for Multilevel Modeling" by Pinheiro and Bates, the idea of embedding the technique in my own c-level code. The basic idea is to rewrite the joint density in a form to mimic a single least squares problem conditional upon the variance parameters. The paper is fairly clear except that some important level of detail is missing. For instance, when we first meet Q_(i):
/ \ / \
| Z_i X_i y_i | | R_11(i) R_10(i) c_1(i) |
| | = Q_(i) | |
| Delta 0 0 | | 0 R_00(i) c_0(i) |
\ / \ /
the text indicates that the Q-R factorization is limited to the first q columns of the augmented matrix on the left. If one plunks the first
q columns of the augmented matrix on the left into a qr factorization, one obtains an orthogonal matrix Q that is (n_i + q) x q and a nonsingular upper triangular matrix R that is q x q. While the text describes R as a nonsingular upper triangular q x q, the matrix Q_(i) is described as a square (n_i + q) x (n_i + q) orthogonal matrix. The remaining columns in the matrix to the right are defined by applying transpose(Q_(i)) to both sides. The question is how to augment my Q which is orthogonal (n_i + q) x q with the missing (n_i + q) x n_i portion producing the orthogonal square matrix mentioned in the text? I tried appending the n_i x n_i identity matrix to the block diagonal, but this doesn't work as the resulting likelihood is insensitive to the variance parameters.
Grant Izmirlian
NCI
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