Nonlinear Least Squares Curve Fitting

Summary There is no general solution for the minimization of chi-squared to find the best estimates of the parameters when the model is nonlinear. A number of methods have been developed that can often find the best estimates and their uncertainties with arbitrary precision. Iterative methods are usually used to minimize a nonlinear chi-squared. Another…

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Generalized Linear Least Squares Curve Fitting

Summary Least squares curve fitting can be generalized to work with any linear model to determine unique parameter estimates and uncertainties in terms of matrix equations. Linear models follow a general form. All other models are classified as nonlinear. While nonlinear models do not have a general analytic solution, some nonlinear models can be transformed…

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Linear Least Squares Curve Fitting of Straight Line Models

Summary Curve fitting is a useful procedure for testing theoretical models and estimating model parameters and their uncertainties. Curve fitting is carried out by choosing the set of estimated parameters that minimize an error function. The conventional error function for analytic results with calculus is the sum of the squares of residuals. The residuals are…

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Propagation of Error with Single and Multiple Independent Variables

Summary Often it is necessary to calculate the uncertainty of derived quantities. This procedure and convention is called propagation of error. Calculating the propagation of error formula requires knowing how to take derivatives/ We give the propagation of error formulas for a single variable and multiple independent variable functions. We list some common error propagation…

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Reporting Measurements and Uncertainties, Significant Digits, and Rounding

Summary In this lesson, we talk about the number of significant digits used to report measurements and uncertainties. There are conventional rules for the number of significant digits in a number. Experts find that the actual number of significant digits in a measurement is a function of how many significant digits in the uncertainty are…

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