RE: Re[2]: Covariance: Matrix=S or Matrix=R

From: mark.e.sale@gsk.com Subject: RE: Re[2]: [NMusers] Covariance: Matrix=S or Matrix=R Date: Wed, 14 Sep 2005 20:03:50 -0400 Steve et al. To continue beating this horse ... A saddle point is to be distinguished from a local minima. All local search algorithms have the potential for local mimima - period. A saddle point is different. A saddle point is a point at which the first derivative of the OBJ wrt parameters is (close to) zero - i.e., the function is flat, locally, but one dimension if curving up and one dimension is curing down. Minima and saddle points can be distinguished by using the second derivative (is the surface curving up or down). If the second derivative (Hessian) is poorly defined, you can't be certain that the flatness isn't due to being at the top of a peak (curving down - a maxima) in one dimension vs being a the bottom (curving up - a minima) in another. My understanding (for what it is) is that modern non-linear regression algorithms are pretty robust to not getting stuck in saddle points - of course depending on how well defined the surface is. If the surface is flat as far as the algorithm can see, it has a hard time telling if this is maxima or a minima. But, again, this is a known problem for non-linear regression and great effort has be applied to getting modern algorithms (which NONMEM actually uses) to address it robustly. There are non-linear regression-like algorithms that are (more) robust to local minima. They are complex, inefficient and rarely used. Other algorithms that are robust to local minima include the convexity stuff from the USC group, and I suppose MCMC could be included as well, seems to me it should not have a problem with local minima, but I'm not sure. I think the text is unclear, the R and S matrix tell you nothing about whether this is a local or global minima, only if it is a minima (or either kind) or a saddle point. I think the work global should be ignored. Mark Sale M.D. Global Director, Research Modeling and Simulation GlaxoSmithKline 919-483-1808 Mobile 919-522-6668