Note on approximation by nonlinear optimization

Abstract

The purpose of this note is to discuss the use of nonlinear optimization techniques to solve approximation problems typical for example in signal identification. Different techniques based on classical and modern approaches to time series are available. The presented idea considers cases when signals are composed of a finite number of certain nonlinear functions distinct in their parameter sets, and realization of an additive random error. The focus is given to the sums of parameterized trigonometric functions. As the random error probability distribution is assumed unknown, the common LSQ criterion is replaced with its parameterized generalization. The obtained unconstrained non-smooth minimization problem can be solved either directly or after a smooth reformulation to the constrained problem. The initial values for computational procedures are estimated using heuristics and suitable statistical techniques, e.g., periodograms. The ideas are illustrated by simple explanatory examples accompanied by figures. Test results are shown for MS Excel Solver, MATLAB is used for visualization.