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International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossman and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 18.1, p. 373
Section 18.1.8.4. Singularity in refinement
a
San Diego Supercomputer Center 0505, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0505, USA, and bStructural, Analytical and Medicinal Chemistry, Pharmacia & Upjohn, Inc., Kalamazoo, MI 49001-0119, USA |
Unless there are more linearly independent observations than there are parameters to fit them, the system of normal equations has no solution. The inverse of the matrix does not exist. Second-order methods fail in these circumstances by doing the matrix equivalent of dividing by zero. However, the objective function is still defined and has a defined gradient at all points. First-order methods will find a point at which the gradient is close to zero, and zero-order methods will still find a minimum value for the objective function. The difficulty is that the points so found are not unique. If one computes the eigenvalues and eigenvectors of the matrix of normal equations, one will find in this case that there are some eigenvalues that are very small or zero. The eigenvectors corresponding to these eigenvalues define sets of directions in which the parameters can be moved without affecting the value of the objective function. This region of the parameter space simply cannot be determined by the available data. The only recourses are to modify the model so that it has fewer parameters, add additional restraints to the problem, or collect more data. The real hazard with this situation is that the commonly used refinement methods do not detect the problem. Careful use of cross validation and keeping careful count of the parameters are the only remedy.
