Fitting nature's basic functions. Part III: exponentials, sinusoids, and nonlinear least squares

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 produce exponentials, 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. Consider the problem of fitting an exponential function y(t) = C 0 e βt , (1) with unknown parameters C 0 and β, to a set of measured points {(t i ,y i), i = 1, 2, …, m} with additive random errors i in the y i. Because y(t) depends nonlinearly on β, making the fit by linear least squares is impossible. Linearization is possible by taking natural logarithms, ln[y(t)] = ln(C 0) + βt ≡ L 0 + βt, (2) but a linear fit of this model does not give the same result as a nonlinear fit of the original. As an example, we consider the record of fossil-fuel CO 2 emissions compiled 1 plots the annual total global emissions, expressed in megatons of carbon, as discrete points. Figure 2 plots the natural logarithms of these totals ; the dashed line represents a linear least squares fit of ln[y(t – t 0)] = L 0 + β(t – t 0), (3) with t 0 = 1856.0 chosen for consistency with our previous global temperature fits. The parameter estimates were (4) The fit confirms the growth's basically exponential character , despite the systematic variations around the straight line. We will address those variations later. For now, we note that the back transformed function , (5) plotted as a dashed curve in Figure 1, does not track the data nearly so well as the solid curve, which we obtained by a nonlinear fit of , (6) which gave the parameter estimates (7) Although the mathematical models in Equations 3 and 6 are equivalent, the statistical models (8) (9) are not. ln ln , , , ... y C e i m i t t i i …