# Rather it is given by χ 2 δ y t χ v 1 δ y χ 811a

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Rather, it is given by χ 2 = δ Y T χ · V - 1 · δ Y χ . (8.11a) Of course, this leads to a different expression for a , which replaces equation 8.8f, 4 To understand this comment about Δ χ 2 = 1, see § 9.

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– 31 – a = ( X T χ · V - 1 · X χ ) - 1 · X T χ · V - 1 · Y χ , (8.11b) and also to a different equation for the covariance matrix, [ α χ ] - 1 = ( X T χ · V - 1 · X χ ) - 1 . (8.11c) Correlated datapoints can occur when the measured y m are affected by systematic errors or instrumental effects. Cowan § 7.6 discusses this case. For example, suppose you take an image with a known point-spread function (psf) and want to fit an analytic function to this image. Example: a background intensity that changes linearly across the field plus a star. Here the independent variables in the function are the ( x, y ) pixel positions and the data are the intensities in each pixel. You’d take the intensity in each individual pixel and fit the assumed model. But here your data values are correlated because of the psf. Because you know the psf, you know the correlation between the various pixels. Such a formulation is required for CBR measurements because of the sidelobes of the radio telescope (which is just another way of saying “psf”). Another case of correlated measurements occurs when your assumed model is incorrect. This is the very definition of correlation, because the residual δy m is correlated with the data value y m . But how do you calculate V ? If you could do a large number J of experiments, each with M datapoints producing measured values y m,j , each measured at different values of x m , then each element of the covariance matrix would be V mn = j ( y m,j y m )( y n,j y n ). You don’t normally have this opportunity. Much better is to look at your residuals; if the model doesn’t fit, use another one! Normally, and in particular we assume everywhere in this tutorial, the measurements are uncorrelated, so one takes V = I (the unitary matrix). 9. CHI-SQUARE FITTING AND WEIGHTED FITTING: DISCUSSION INCLUDING COVARIANCE 9.1. Phenomenological description Consider the first two coefficients in our example of § 5.2. In this example, the fit gives y = A 0 + A 1 t + A 2 t 2 , where the numerical values are given in vector form by equation 4.12. The coefficient A 0 is the y -intercept and A 1 is the slope. They have derived values A 0 = 96 ± 34 and A 1 = 4 ± 9. Remember what these uncertainties really mean: in an infinity of similar experiments, you’ll obtain an infinity of values of ( A 0 , A 1 ) that are normally distributed with dispersions (34,9). Loosely
– 32 – speaking, this means that A 0 lies between (96 34 = 62) and (96 + 34 = 130) and A 1 lies between 5 and 13. Suppose you are interested in knowing about A 0 without regard to A 1 . By this we mean that as A 0 is varied from its optimum value of 96, χ 2 increases from its minimum value. As we vary A 0 , if we allow A 1 to take on whatever value it needs to for the purpose of minimizing χ 2 , then this is what we mean by “knowing about A 0 without regard to A 1 ”. For this case, the uncertainty of A 0 is indeed 34. Ditto for A 1 . In other words, equations 3.7, and 8.9 apply.

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