Fitting nature's basic functions part I: polynomials and linear least squares
Computing in Science and Engineering
Fitting Nature's Basic Functions Part II: Estimating Uncertainties and Testing Hypotheses
Computing in Science and Engineering
Fitting Nature's Basic Functions Part Iv: The Variable Projection Algorithm
Computing in Science and Engineering
Variable projections neural network training
Mathematics and Computers in Simulation - Special issue: Applied and computational mathematics - selected papers of the fifth PanAmerican workshop - June 21-25, 2004, Tegucigalpa, Honduras
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In Part I and Part II of this series, we used linear least squares to fit polynomials of various degree to the annual global temperature anomalies for 1856 to 1999. Polynomials are much beloved by mathematicians but are of limited value for modeling measured data. Natural processes often display linear trends, and occasionally a constant acceleration process exhibits quadratic variation. However, higher-order polynomial behavior is rare in nature, which is more likely to produceexponentials, sinusoids, logistics, Gaussians, or other special functions. Modeling such behaviors with high-order polynomials usually gives spurious wiggles between the data points, and low-order polynomial fits give nonrandom residuals. We saw an example of this syndrome in Figure 4 of Part I, where we attempted to model a quasicyclic variation with a fifth-degree polynomial. That example also illustrated that polynomial fits usually give unrealistic extrapolations of the data.