Around the best values a a 2 pick values for δa δa

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) around the best values ( a 0 , a 2 ). Pick values for ( δa 0 , δa 2 ) and redo the least-squares solution for a 1 . This gives a new value for χ 2 which is, of course, larger than the minimum value that was obtained with ( δa 0 , δa 2 ) = 0. Call this difference Δ χ 2 ( δa 0 ,δa 2 ) . As above, this follows a chi-square distribution, but now with ν = 2. Determine the dependence of Δ χ 2 ( δa 0 ,δa 2 ) upon ( δa 0 , δa 2 ) and find the set of values of ( δa 0 , δa 2 ) such that Δ χ 2 ( δa 0 ,δa 2 ) = 2 . 3. This is the desired result, namely the ellipse within which the actual values ( δa 0 , δa 2 ) lie with a probability of 68 . 3%, without regard to the value of a 1 .
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– 35 – These values can be defined in terms of the curvature matrix [ α χ ], as we discuss below. Consider now what you’ve done in this process. For each least-squares fit you used trial values of ( δa 0 , δa 2 ). In specifying them you had exactly two degrees of freedom because you are fixing two parameters. This distribution follows a chi-square distribution with two degree of freedom ( ν = 2). So the uncertainty σ a 0 is that value for which Δ χ 2 δa 0 = 2 . 3, which follows from the integrated probability for the chi-square distribution for ν = 2. (The chi-square fit for the third parameter a 1 has M 1 degrees of freedom, but again this is irrelevant.) One can expand this discussion in the obvious way. Consider finally. . . 9.4. Calculating the uncertainties of three parameters—gedankenexperiment Suppose we want to know the values of all three parameters (or, generally, all N parameters). Then we pick trial values for all three. There is no least-squares fit for the remaining parameters, because there are none. For each combination of the three (or N ) parameters we obtain Δ χ a , which defines a 3- (or N -) dimensional ellipsoid. This follows a chi-square distribution with ν = 3 (or N ). We find the (hyper)surface such that Δ χ a is that value within which the integrated probability is 68 . 3%. This defines the (hyper)surface of σ a . 9.5. Doing these calculations the non-gedanken easy way The obvious way to do the gedanken calculations described above is to set up a grid of values in the parameters of interest ( δa n ); perform the chi-square fit on the remaining variables, keeping track of the resulting grid of χ 2 ; and plot the results in terms of a contour plot (for two parameters of interest) or higher dimensions. There’s an easier way which is applicable unless you are doing a nonlinear fit and the parameter errors are large 5 . The curvature matrix [ α χ ] of equation 8.8d contains the matrix of the second derivatives of χ 2 with respect to all pairwise combinations of δa n , evaluated at the minimum χ 2 ; it’s known as the curvature matrix for this reason. Clearly, as long as the Taylor expansion is good we can write Δ χ 2 a = δ a T · [ α χ ] · δ a . (9.1) Knowing the curvature matrix, we don’t have to redo the fits as we described above. Rather, we can use the already-known matrix elements.
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