Transcription of Nonlinear Least Squares Data Fitting
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Example: tourism industry
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. It is common to model populations using exponential models, and so wemight hope thatyi x1ex2tifor appropriate choices of the parametersx1andx2.
746 Appendix D. Nonlinear Least Squares Data Fitting This can be rewritten as ∇f(x1,x2)= e x2 t1 e 2 2 ex2 3 ex2t4 e 2t5 x1t1ex2t1 x1t2ex2 t2 x1t3ex2t3 x1t4ex2t4 x1t5ex2 5 x1ex2t1 −y1 x1ex2t2 −y2 x1ex2t3 −y3 x1ex2t4 −y4 x1ex2t5 −y5 sothat ∇f(x1,x2)=∇F(x)F(x).TheHessianmatrixis∇2f(x)=∇F(x)∇F(x)T+ m i=1 f i(x)∇ 2f i(x)= ex2 t1 e x2 2 …
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