# Choosing and correcting σ m 29 852 when youre using

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Choosing and correcting σ m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.5.2 When you’re using equation 8.9. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.5.3 Think about your results! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 8.5.4 When your measurements are correlated. . . . . . . . . . . . . . . . . . . . . . 30 9 CHI-SQUARE FITTING AND WEIGHTED FITTING: DISCUSSION INCLUD- ING COVARIANCE 31 9.1 Phenomenological description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 9.2 Calculating the uncertainties of a single parameter—gedankenexperiment . . . . . . 34

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– 3 – 9.3 Calculating the uncertainties of two parameters—gedankenexperiment . . . . . . . . 34 9.4 Calculating the uncertainties of three parameters—gedankenexperiment . . . . . . . 35 9.5 Doing these calculations the non-gedanken easy way . . . . . . . . . . . . . . . . . . 35 9.6 Important comments about uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 37 10 BRUTE FORCE CHI-SQUARE AND THE CURVATURE MATRIX 37 10.1 Parameter Uncertainties in Brute Force chi-square Fitting . . . . . . . . . . . . . . . 37 11 USING SINGULAR VALUE DECOMPOSITION (SVD) 38 11.1 Phenomenological description of SVD . . . . . . . . . . . . . . . . . . . . . . . . . . 39 11.2 Using SVD for Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 11.3 Important Conclusion for Least Squares!!! . . . . . . . . . . . . . . . . . . . . . . . . 42 11.4 How Small is “Small”? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 11.4.1 Strictly Speaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 11.4.2 Practically Speaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 11.5 Doing SVD in IDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 11.5.1 IDL’s SVD routine la svd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 11.5.2 My routine lsfit svd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 12 REJECTING BAD DATAPOINTS II: STETSON’S METHOD PLUS CHAU- VENET’S CRITERION 43 12.1 Stetson’s sliding weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 12.2 Implementation of the weight in our matrix equations . . . . . . . . . . . . . . . . . 45 13 MEDIAN/MARS, INSTEAD OF LEAST-SQUARES, FITTING 46 13.1 The Median versus the MARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 13.1.1 For the Standard Median—it’s the MARS . . . . . . . . . . . . . . . . . . . 47 13.1.2 For an arbitrary function, e.g. the slope—it’s a weighted MARS . . . . . . . 47 13.2 The General Technique for Weighted MARS Fitting . . . . . . . . . . . . . . . . . . 49 13.3 Implementation, a Caution, and When To Stop Iterating . . . . . . . . . . . . . . . . 50
– 4 – 13.4 Errors in the Derived Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 13.5 Pedantic Comment: The MARS and the Double-sided Exponential pdf . . . . . . . 50 13.6 IDL’s related resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 14 FITTING WHEN MORE THAN ONE MEASURED PARAMETERS HAVE UNCERTAINTIES 52 14.1 A preliminary: Why the slope is systematically small . . . . . . . . . . . . . . . . . . 52 14.2 Jefferys’ Method: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 14.3 The Data Matrix and Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 14.4 The Data Covariance Matrix and Defining Chi-Square . . . . . . . . . . . . . . . . . 57 14.5 Formulation of the Problem and its Solution with Lagrange Multipliers . . . . . . . 58 14.6 The Derivative Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 14.7 The Specific Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 14.8 The Solution to the Lagrangian: Two Matrix Equations . . . . . . . . . . . . . . . . 61 14.9 Solving Equations 14.18a and 14.18b Iteratively . . . . . . . . . . . . . . . . . . . . 62 14.10Taking all those derivatives! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 14.11The Initial Guess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 14.12The Covariance Matrix (and errors) of the Derived Parameters . . . . . . . . . . . . 64 15 NOTATION COMPARISON WITH NUMERICAL RECIPES 64 0. LEAST-SQUAREs FITTING FOR TWO PARAMETERS, AS WITH A STRAIGHT LINE. 0.1. The closed-form expressions for a straight-line fit First consider the least-squares fit to a straight line. Let y m be the m th measurement of the observed quantity (in this example, y m is zenith distance; t m be the time of the m th measurement; M = the total number of observations, i.e. m runs from 0 to M 1. Remember that in the least- squares technique, quantities such as t m are regarded to be known with high accuracy while the quantity y m has uncertainties in its measurement.

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