But always look at the normalized covariance matrix

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But always look at the normalized covariance matrix. Suppose one pair of off-diagonal elements departs significantly from zero. Then their corresponding functions are far from being orthogonal and the variances of the derived coefficients will suffer as a result. You might be able to eliminate one of the parameters to make the fit more robust. For example, suppose one function is t cos( t ) and the other is sin( t ) cos( t ). If the range of t is small, these two functions are indistinguishable and have a large covariance; you should eliminate one from the fit. If the range of t is large, there is no problem. For further discussion of covariance, see § 9. Also, you might also want to try out another example in Taylor’s § 8.5.
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– 18 – 6. REJECTING BAD DATAPOINTS I.: CHAUVENET’S CRITERION Least-squares fitting is derived from the maximum likelihood argument assuming the datapoint residuals δy m have a Gaussian pdf. This means that the errors are distributed as p ( δy ; σ ) = 1 2 πσ e - parenleftbigg δy 2 2 σ 2 parenrightbigg , (6.1) where σ 2 is the true variance of the datapoints, i.e. s 2 in equation 3.1 (to be precise, s 2 needs to be averaged over many experiments). More importantly, the probability of finding datapoints inside the limits ± Δ y is P ( | δy | < Δ y ) = integraldisplay y - Δ y p ( δy ; σ ) d ( δy ) = erf parenleftbigg Δ y 2 σ parenrightbigg , (6.2) where we use the commonly-defined error function erf( X ) = 1 π integraltext + X - X e - x 2 dx . A particularly im- portant value is for Δ y = σ , for which P ( | δy | ) = 0 . 683 . (6.3) If we have an experiment with M datapoints, then the number of datapoints we expect to lie outside the interval ± Δ y is M ( outside Δ y ) = M bracketleftbigg 1 erf parenleftbigg Δ y 2 σ parenrightbiggbracketrightbigg . (6.4) Chauvenet’s criterion simply says: 1. Find Δ y such that M ( outside Δ y ) = 0 . 5. This is given by Δ y σ = 2 erf - 1 parenleftbigg 1 1 2 M parenrightbigg . (6.5) This criterion leads to the numbers in the associated table, which is a moderately interesting set of numbers. Many astronomers adopt 3 σ , which is clearly inappropriate for large N ! 2. Discard all datapoints outside this range. We offer the following important Comments :
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– 19 – Chauvenet’s criterion versus M M Δ y σ 100 2.81 1000 3.48 10 4 4.06 10 5 4.56 This assumes data are Gaussian-distributed. In real life this doesn’t often happen because of “glitches”. Examples of glitches can be interference in radio astronomy, meteors in optical astronomy, and cosmic rays on CCD chips. These glitches produce bad points that depart from Gaussian statistics. They are often called outliers . It is very important to get rid of the outliers because the least-squares process minimizes the squares of the residuals. Outliers, being the points with the largest residuals, have a disproportionately evil effect on the result.
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