# We want to combine these m j measurements to derive n

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We want to combine these ( M × J ) measurements to derive N coefficients. We denote the matrix of measured parameters by X , which is ( M × J ) [using conventional matrix notation, not IDL notation, in which in the vertical dimension (number of rows) is M and the horizontal dimension (number of columns) is J ]. We will want to concatenate this matrix to a one-dimensional vector x of length ( MJ ). Specifically, the M × J datapoint matrix is

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– 55 – Conventional Fit -3 -2 -1 0 1 2 3 t -3 -2 -1 0 1 2 3 4 y Proper Fit -3 -2 -1 0 1 2 3 t -3 -2 -1 0 1 2 3 4 y Fig. 14.1.— Comparing a conventional fit for to y = a 0 + a 1 t (left panel) to a proper one when both measured variables have errors. On the right, the ellipses denote the correlated errors in ( t, y ); these are the generalization of the errorbars on the left. The right-hand slope is a bit steeper than the left-hand one. X = x 0 , 0 x 0 , 1 x 0 , 2 . . . x 0 ,J - 1 x 1 , 0 x 1 , 1 x 1 , 2 . . . x 1 ,J - 1 x 2 , 0 x 2 , 1 x 2 , 2 . . . x 2 ,J - 1 . . . . . . . . . . . . . . . . . . . . . x M - 1 , 0 x M - 1 , 1 x M - 1 , 2 . . . x M - 1 ,J - 1 (14.4a) We don’t use this big matrix in the solution. Instead, we turn it into a vector in which the first J elements are the data for m = 0, the second J elements are for m = 1, etc. So the vector has dimensions ( MJ ) × 1, like this:
– 56 – x = x 0 , 0 x 0 , 1 x 0 , 2 . . . x 0 ,J - 1 x 1 , 0 x 1 , 1 x 1 , 2 . . . x 1 ,J - 1 x 2 , 0 x 2 , 1 x 2 , 2 . . . x 2 ,J - 1 . . . . . . . . . x M - 1 , 0 x M - 1 , 1 x M - 1 , 2 . . . x M - 1 ,J - 1 (14.4b) One important reason for writing the whole set of data as a vector instead of a matrix is to make it possible to write the covariance matrix for all measured data in the conventional form, as we now discuss.

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– 57 – 14.4. The Data Covariance Matrix and Defining Chi-Square The whole object of fitting is to minimize the chi-square. When all measured parameters have uncertainties, their uncertainties can be correlated. We have to generalize the definition of chi-square accordingly. First, suppose that the observational errors in the datapoints are uncorrelated. Then the intrinsic variance of each datapoint is described by a single number. In our example, for uncorrelated observational errors we’d have the variances in the y values be ( σ 2 y 0 , σ 2 y 1 , . . . ), and similarly for the t values; this would give χ 2 = summationdisplay m δy 2 m σ 2 ym + δt 2 m σ 2 tm (14.5) However, it is common that errors are correlated. For example, if we were fitting y to a polynomial in t , then the errors in the various powers of t would certainly be correlated. More generally, then, the covariances among the different measured values are nonzero. These covariances are the off-diagonal terms in the covariance matrix. Thus, if we denote the covariance matrix for
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