For example, if we are fitting to two parameters, param1 and param2, the chi-squared as a function of the values of the parameters might have two or more local minima. If we imagine a plot of the value of the sum of the squares or the chi-squared as a function of the parameters to which we are fitting, in general for a nonlinear fit there may be many local minima instead of one big one, as is the case for linear fitting. This again will be analytic for a linear fit.ДЇor a nonlinear fit, no such analytic solutions are possible, so iteration is required to find the minimum in the sum of the squares or the chi-squared. Similarly, when there are explicit errors in the data, we form the chi-squared,, and we solve the corresponding equations.

This, in general, can be done analytically, provided the model to which we are fitting is linear in the parameters.

, a, the answer is found by solving a set of simultaneous equations. Recall that if sos is the sum of the squares of the residuals, then we are seeking the minimum in its value. In this chapter we discuss nonlinear models and the EDA program FindFit that can often find a reasonable fit to them. To briefly review, the terms refer to the way in which the parameters to which we are fitting enter into the model. The previous chapter, 'Fitting Data to Linear Models by Least-Squares Techniques,' introduced the distinction between linear and nonlinear models. The primary differences are that FindFit: (1) recognizes EDA's data format, including errors in both coordinates (2) estimates errors in fit parameters (3) by default displays graphical information about the fit and (4) uses an algorithm that has been optimized for speed and stability for the types of nonlinear fits commonly performed in the physical sciences and engineering. The main EDA program introduced here is FindFit, which accomplishes fits to arbitrary models.ДЇindFit is similar to the NonlinearFit function in the Statistics`NonlinearFit` package which is standard with Mathematica. However, since sometimes nature really is nonlinear, such fits are often unavoidable, and the principles and some tools for nonlinear fitting are the topics of this chapter. In addition, computational methods of nonlinear fitting are still a current research topic in computer science. These difficulties are one of the reasons that, as we shall see, the whole topic of spectral line shapes is still a very active subject of research spanning the fields of chemistry, physics, astronomy, and more. Often such fits require large computational resources and great skill, patience, and intuition on the part of the analyst. One of the most difficult topics in all of data analysis in the physical sciences is fitting data to nonlinear models.

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Mathematica 7 fit nonlinear packages