tailieunhanh - Ebook Data reduction and error analysis for the physical sciences (3rd edition): Part 2

(BQ) Part 2 book "Data reduction and error analysis for the physical sciences" has contents: Least-Squares fit to an arbitrary function, fitting composite curves, direct application of the maximum likelihood method, testing the fit. | CHAPTER 8 LEAST-SQUARES FIT TO AN ARBITRARY FUNCTION NONLINEAR FITTING The methods of least squares and multiple regression developed in the previous chapters are restricted to fitting functions that are linear in the parameters as in Equation m y x a7T x 8-1 j i This limitation is imposed by the fact that in general minimizing X2 can yield a set of coupled equations that are linear in the m unknown parameters only if the fitting functions y jr are themselves linear in the parameters. We shall distinguish between the two types of problems by referring to linear fitting for problems that involve equations that are linear in the parameters such as those discussed in Chapters 6 and 7 and nonlinear fitting for those problems that are nonlinear in the parameters. Example . In a popular undergraduate physics laboratory experiment a real silver quarter is irradiated with thermal neutrons to create two short-lived isotopes of silver 47Ag108 and 47Ag that subsequently decay by beta emission. Students count the emitted beta particles in 15-s intervals for about 4 min to obtain a decay curve. Data collected from such an experiment are listed in Table and plotted on a semi-logarithmic graph in Figure . The data are reported at the end of each 15-s interval just as they were recorded by a scaler. The data points do not fall on a straight 142 Least-Squares Fit to an Arbitrary Function 143 Time s FIGURE Number of counts detected from the decay of two excited states of silver as a function of time Example . Time is reported at the end of each interval. Statistical uncertainties are assumed. The curve was obtained by a nonlinear least-squares fit of Equation to the data. line because the probability function that describes the process is the sum of two exponential functions plus a constant background. We can represent the decay by the fitting function y x a J a2e t ữi a e a where the parameter dị corresponds to the background radiation and a2 and 3