Efficient nonlinear parameter estimation via interactive switching between function minimization algorithms
✍ Scribed by Jane B. Hazelrig; Eugene Ackerman; Malcolm E. Turner Jr.
- Publisher
- Elsevier Science
- Year
- 1979
- Tongue
- English
- Weight
- 910 KB
- Volume
- 12
- Category
- Article
- ISSN
- 0010-4809
No coin nor oath required. For personal study only.
✦ Synopsis
Nonlinear parameter estimates must be obtained by Iterative processes that can consume large amounts of computer time, so it is important that the function minimization or optimization process be efficient. Standard algorithms are based on varying hypotheses about the hypersurface of the objective function (the function to be minimized). The characteristics of this surface vary with the functional form of the objective function, the experimental data, and the current position of the search pattern in the parameter space. No one algorithm is best for all problems or even for all phases of the same problem. Strategies were developed for dynamically selecting the algorithm whose underlying hypotheses most closely match the local geometry of the parameter space. The results presented here indicate that interactive switching between algorithms can increase the robustness and efficiency of the optimization process.
Nonlinear parameter estimates are obtained by locating a point in ap-dimensional parameter space, denoted by the p-vector B = &,&,..., /?,J', at which an objective function, F(fl, assumes its optimal (minimum or maximum) value. Suppose, for example, that an experiment is performed involving n measurements of a response, y, indexed as Yi i=l,2,...,n. Suppose further that the ith observation is made under m experimental conditions denoted by the m-vector xi = (Xi], xi*, . . .) Xi,>'. A theoretical model is proposed that relates the observations to the experimental conditions through the unknown parameters 8. In other words, a function is defined, ,u(@, xi), for predicting the ith response given a set of experimental conditions, xi, and parametric values, /3. The problem is to determine the "best" estimates for 8. The standard objective function for nonlinear regression analysis is FW = i: [Yi -Pu(BT xJl2* i=l The #I that minimizes F(B) is the nonlinear least squares estimate.