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188 chapter 8This is a measure of the uncertainties in the values of the fitted parameters arisingfrom the uncertainties σ i in the measured y i values. A measure of the dependenceof the parameters on each other is given by the correlation coefficient:ρ(a 1 ,a 2 )= cov(a 1,a 2 )σ a1 σ a2, cov(a 1 ,a 2 )= −S x∆ . (8.60)Here cov(a 1 ,a 2 ) is the covariance of a 1 and a 2 and vanishes if a 1 and a 2 are independent.The correlation coefficient ρ(a 1 ,a 2 ) lies in the range −1 ≤ ρ ≤ 1, with apositive ρ indicating that the errors in a 1 and a 2 are likely to have the same sign,and a negative ρ indicating opposite signs.The preceding analytic solutions for the parameters are of the form found instatistics books but are not optimal for numerical calculations because subtractivecancellation can make the answers unstable. As discussed in Chapter 2, “Errors& Uncertainties in Computations,” a rearrangement of the equations can decreasethis type of error. For example, [Thom 92] gives improved expressions that measurethe data relative to their averages:a 1 = y − a 2 x,a 2 = S xyS xx,x = 1 ∑N dx i ,Ni=1y = 1 N∑N dy ii=1∑N dS xy =i=1(x i − x)(y i − y)σ 2 i∑N d(x i − x) 2, S xx =. (8.61)i=1σ 2 iC DIn JamaFit.java in Listing 8.2 and on the CD, we give a program that fits aparabola to some data. You can use it as a model for fitting a line to data, althoughyou can use our closed-form expressions for a straight-line fit. In Fit.java on theinstructor’s CD we give a program for fitting to the decay data.8.7.2 Exponential Decay Fit AssessmentFit the exponential decay law (8.49) to the data in Figure 8.4. This means findingvalues for τ and ∆N(0)/∆t that provide a best fit to the data and then judging howgood the fit is.1. Construct a table (∆N/∆t i , t i ), for i =1,N D from Figure 8.4. Because timewas measured in bins, t i should correspond to the middle of a bin.2. Add an estimate of the error σ i to obtain a table of the form (∆N/∆t i ±σ i ,t i ). You can estimate the errors by eye, say, by estimating how much thehistogram values appear to fluctuate about a smooth curve, or you can takeσ i ≃ √ events. (This last approximation is reasonable for large numbers, whichthis is not.)−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 188