# M reflects that single datapoint and this colors all

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m reflects that single datapoint and this colors all the derived errors. This isn’t fixed by using the reduced chi-square hatwider χ 2 , which will be nowhere near unity. Replacing the sum of residuals squared by the ARS seems reasonable until you realize that the MARS coefficient values are completely independent of the residuals—yet, you’d expect the errors to be smaller for better data! A similar concern holds for replacing the ARS by its weighted counterpart. My current—but untested—recommendation is this. Suppose you have M datapoints. Con- ventionally, we define the +1 σ dispersion by the boundary where 34 . 2% of the points lie outside the limit, and ditto for the 1 σ boundary. These boundaries are defined only by the number of daapoints outside the boundaries, not how big the residuals are. We can do the same here. Define the sample variance not by the sum-of-squares of residuals as in equation 3.1, but rather by how far away from the MARS fitted line you need to go before 34 . 2% of the points lie outside the plus-and-minus boundaries. Then use this fake variance in equation 3.7. 13.5. Pedantic Comment: The MARS and the Double-sided Exponential pdf In fact, there is a specific pdf for which the MARS is the theoretically correct solution: the double-sided exponential. Here the pdf of the measured datapoints is p y m ) = e -| Δ y m | m 2 σ m . (13.7)

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– 51 – where, again, Δ y m = ( y m a · f ( x m )). For this, the logarithm of the likelihood function is (we exclude the term involving log Π M - 1 m =0 1 σ m for simplicity) L y m ) = log( L y m ) ) = M - 1 summationdisplay m =0 bracketleftbigg | y m a · f ( x m )) | σ m bracketrightbigg . (13.8a) The absolute value signs are horrible to deal with, so we rewrite this as L y m ) ) = summationdisplay Δ y m > 0 y m a · f ( x m ) σ m summationdisplay Δ y m < 0 y m a · f ( x m ) σ m . (13.8b) Now we take the derivative of L with respect to a n and set it equal to zero to find the maximum. This gives dL da n = summationdisplay Δ y> 0 f n ( x m ) σ m summationdisplay Δ y< 0 f n ( x m ) σ m = 0 , (13.9) which is the MARS fit. 13.6. IDL’s related resources IDL provides the median function, which uses sorting and is much faster than our general weighted-MARS technique—but of course cannot deal with functional forms. IDL also provides ladfit (“least absolute deviation fit”), which does a weighted-MARS fit for a straight line. My routine, polyfit_median , does a weighted-MARS fit for an arbitrary polynomial; it is slightly less accurate as ladfit for an odd number of points but is slightly better for an even number.
– 52 – 14. FITTING WHEN MORE THAN ONE MEASURED PARAMETERS HAVE UNCERTAINTIES We’ve mentioned that one of the essential assumptions of least squares is that the independent variables are known with high precision and the errors occur only in the measured data. Suppose you’re fitting two variables, t and y , as in equation 0.1. This essential assumption means that t is known with high precision and all the uncertainty is in y , and you are minimizing the squares of the residuals in the y -direction only. If

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