Transcription of Nonlinear Least Squares Data Fitting
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Nonlinear Least Squares Data Fitting
Appendix DNonlinear Least SquaresData IntroductionA Nonlinear Least Squares problem is an unconstrained minimization problem of theformminimizexf(x)=m i=1fi(x)2,where the objective function is defined in terms of auxiliary functions{fi}.Itis called Least Squares because we areminimizingthe sum ofsquaresof thesefunctions. Looked at in this way, it is just another example of unconstrained min-imization, leading one to ask why it should be studied as a separate topic. Thereare several the context of data Fitting , the auxiliary functions{fi}are not arbitrarynonlinear functions. They correspond to the residuals in a data Fitting problem (seeChapter 1). For example, suppose that we had collected data{(ti,yi)}mi=1consist-ing of the size of a population of antelope at various times. Hereticorresponds tothe time at which the populationyiwas counted. Suppose we had the datati:12458yi:3461120where the times are measured in years and the populations are measured in hun-dreds.
Example D.2 Gauss-Newton Method. We apply the Gauss-Newton method to an exponential model of the form y i ≈ x1e x2ti with data t =(12458)T y =(3.2939 4.2699 7.1749 9.3008 20.259)T. For this example, the vector y was chosen so that the model would be a good fit to the data, and hence we would expect the Gauss-Newton method to perform much ...
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