2.1.3 Eigenvalues Analysis of the Approximate Hessian Matrix



next up previous contents index
Next: 2.1.4 Parameters Constraints Up: 2.1 Nonlinear Least Squares Previous: 2.1.2 Calculation of the

2.1.3 Eigenvalues Analysis of the Approximate Hessian Matrix

    Let C denote the approximate Hessian matrix:

C is a symmetric, positive semidefinite matrix that can be decomposed into a product of three matrices:

where L is a diagonal matrix of non-negative eigenvalues and A is a matrix of eigenvectors columns. Given the decomposition of C, its inverse which is needed for solving the linear system in (2.26) can be calculated as:

where is a diagonal matrix whose elements are the inverse of the eigenvalues.

An analysis of the eigenvalues spectrum of C highlights important information about the modeling problem characteristics. The ratio between the highest and lowest eigenvalues defines the condition number of the extraction. A large condition number denotes an ill-conditionned problem typical of models with correlated parameters or with a weak response to certain parameters.

A typical remedy in such cases is to fix the values of parameters associated with small eigenvalues to a constant. In some cases, this solution is not adequate and a multi-stage update of the model parameters to resolve the uncertainty is more appropriate. The model parameters are divided into groups that contain uncorrelated parameters with eigenvalues of similar magnitude. During every iteration of the Levenberg-Marquardt algorithm, the parameters in the same subgroup are updated while keeping the parameters in the other subgroups constant. This divides the parameter spaces into orthogonal subspaces where the modeling problem is well conditioned. Correlated parameters, and parameters with varying sensitivity level can then be extracted separately. The current implementation provides the capability of dividing model parameters into distinct subgroups. In addition, a granular control of the optimization allows a great flexibility in devising the appropriate extraction strategy gif.


Martin Stiftinger
Tue Aug 1 19:07:20 MET DST 1995