RE: Slow PRED in tables?

Doug and Bob, This is point that I have long been interested in. For each individual, the original WRES in NONMEM is based on a Schur decomposition of the estimated covariance matrix (I have verified this numerically using MATLAB) of the residual vector. Thus if the individual covariance matrix of the residuals is C, and the residual vector is r, WRES is define as C^(-1/2)*r, where C= U*diag(lambda)*U' , and the columns of U are the eigenvectors of C, and lambda is the diagonal matrix of eigenvalues. The main point of the computation is that the covariance of C^(-1/2)*r is a unit matrix, where C^(-1/2) = U*lambda(^-1.2)*U' has the rough interpretation of being a square root of C^(-1). Further diagnostic analysis of the components of WRES treats them as if they were independent N(0,1) random variables since indeed are are uncorrelated and normalized to have unit variance (although the normality assumption is doubtful for hgihly nonlinear models). Thus the computation of WRES in NM7.2 requires the computation of all eigenvalues and eigenvectors of the C matrix, which for large matrices can start to be expensive (although the best LAPACK implementation of a symmetric positive definite eigendecomposition uses Jim Demmel's O(N^1/3) routine and maybe considerably faster than what you currently seeing in NM7.2 - I have no way of knowing what NM7.2 actually uses.) But in any event, all the WRES computation is doing is de-correlating and normalizing the residuals. Any matrix W such that W*r has a covariance matrix equal to the unit matrix will do. In particular, choosing W = lower triangular Cholesky decompostion of C^(-1), works just fine and is much faster to compute than to compute than a Schur eigen-decompostion. Based on the name, I assume that WRESCHOL is just WRES implemented with such a Cholesky decomposition. WHich is better - WRES or WRESCHOL ? This is difficult question without further information. It does point out that the details of the WRES results, no matter how done, should be take with a large grain of salt (WRESCHOL numerically will look totally different than WRES, yet they are both equally valid de-correlation/normalizations) Electrical engineers have long been aware of this de-correlation/normalization ambiguity. Basically if C(^-1/2) is any matrix that decorrelates and normalizes the residuals, so is U*C^(-1/2), where U is any orthogonal matrix (U'*U=I) Thus there are an infinite number of choices to use for computing a reasonable WRES for further diagnostic analysis. In the electrical engineering community, any matrix of the form U*C^(-1/2) is called a whitening matrix, since it transforms correlated data into 'whitened', decorrelated data with unit variances. In order to decide which decorrelation is best, one must impose an optimization criterion to pick the particular best whitening matrix, This is not so easy to do,but has given rise to a large literature under the general name of 'projection pursuit' methods. I have long suspected there may be something here worth looking into with respect to population PK/PD model evaluation techniques, but his is still an open question.