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systems of equations with matrices; data fitting 185the reader to [B&R 02, Pres 94, M&W 65, Thom 92]. However, we will emphasizethree points:1. If the data being fit contain errors, then the “best fit” in a statistical senseshould not pass through all the data points.2. If the theory is not an appropriate one for the data (e.g., the parabola inFigure 8.3), then its best fit to the data may not be a good fit at all. This isgood, for it indicates that this is not the right theory.3. Only for the simplest case of a linear least-squares fit can we write down aclosed-form solution to evaluate and obtain the fit. More realistic problems areusually solved by trial-and-error search procedures, sometimes using sophisticatedsubroutine libraries. However, in §8.7.6 we show how to conduct sucha nonlinear search using familiar tools.Imagine that you have measured N D data values of the independent variable y asa function of the dependent variable x:(x i ,y i ± σ i ), i=1,N D , (8.50)where ±σ i is the uncertainty in the ith value of y. (For simplicity we assume thatall the errors σ i occur in the dependent variable, although this is hardly ever true[Thom 92]). For our problem, y is the number of decays as a function of time, and x iare the times. Our goal is to determine how well a mathematical function y = g(x)(also called a theory or a model) can describe these data. Alternatively, if the theorycontains some parameters or constants, our goal can be viewed as determining thebest values for these parameters. We assume that the model function g(x) contains,in addition to the functional dependence on x, an additional dependence upon M Pparameters {a 1 ,a 2 ,...,a MP }. Notice that the parameters {a m } are not variables,in the sense of numbers read from a meter, but rather are parts of the theoreticalmodel, such as the size of a box, the mass of a particle, or the depth of a potentialwell. For the exponential decay function (8.49), the parameters are the lifetime τand the initial decay rate dN(0)/dt. We indicate this asg(x)=g(x; {a 1 ,a 2 ,...,a MP })=g(x; {a m }). (8.51)We use the chi-square (χ 2 ) measure as a gauge of how well a theoretical function greproduces data:χ 2def=∑N D( ) 2 yi − g(x i ; {a m }), (8.52)σ ii=1where the sum is over the N D experimental points (x i ,y i ± σ i ). The definition(8.52) is such that smaller values of χ 2 are better fits, with χ 2 =0occurring if the−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 185