MTH
lsfit_2008.pdf

# We expect the zenith distance y m to change linearly

• Lesson Plan
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We expect the zenith distance y m to change linearly with time as follows:

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– 5 – A + Bt m = y m . (0.1) Given this, one does the maximum likelihood (ML) estimate assuming Gaussian statistics. When all measurements have the same intrinsic uncertainty, this is the same as looking for the solution that minimizes the sum of the squares of the residuals (which we will define later). This leads to the pair of equations (Taylor 8.8, 8.9), called the normal equations AN + B summationdisplay t m = summationdisplay y m (0.2a) A summationdisplay t m + B summationdisplay t 2 m = summationdisplay t m y m . (0.2b) Two equations and two unknowns—easy to solve! The closed-form equations for ( A, B ) are Taylor’s equations 8.10 to 8.12. 0.2. Better is the following generalized notation. We want a way to generalize this approach to include any functional dependence on t and even other variables, and to have an arbitrarily large number of unknown coefficients instead of just the two ( A, B ). This is very easy using matrix math. We will ease into this matrix technique gently, by first carrying through an intermediate stage of notation. First generalize the straight-line fit slightly by having two functional dependences instead of one. We have something other than the time t m ; call it s m . For example, we could have s m = cos( t m ) or s m = t 2 m ; or we could have s m = x m , where x m is the position from which the observation was taken. To correspond to equation 0.1, s m = 1. Then we rewrite equation 0.1 to include this extra dependence As m + Bt m = y m . (0.3) There are still only two unknown parameters, so this is an almost trivial generalization; later we’ll generalize to more parameters. We have M equations like equation 0.3, one for each measurement. They are known as the equations of condition because they are the equations that specify the theoretical model to which we are fitting the data. There are M equations of condition and only two unknowns ( A and B ). This is too many equations! We have to end up with a system in which the number of equations is equal to the number of unknowns.
– 6 – To accomplish this, from equation 0.3 we form the normal equations . The number of normal equations is equal to the number of unknowns, so in this case we will have two. We could carry through the same ML derivation to derive equations equivalent to equation 0.2; the result is A summationdisplay s 2 m + B summationdisplay s m t m = summationdisplay s m y m (0.4a) A summationdisplay s m t m + B summationdisplay t 2 m = summationdisplay t m y m . (0.4b) We can rewrite these equations using the notation [ st ] = s m t m , etc.: A [ s 2 ] + B [ st ] = [ sy ] (0.5a) A [ st ] + B [ t 2 ] = [ ty ] . (0.5b) This is, of course, precisely analogous to equation 0.2. And now it’s clear how to generalize to more parameters! 1. LEAST-SQUARES FITTING FOR MANY PARAMETERS, AS WITH A CUBIC With this notation it’s easy to generalize to more ( N ) unknowns: the method is obvious because in each equation of condition (like equation 0.3) we simply add equivalent additional terms such as Cu m , Dv m , etc; and in the normal equations (equation 0.5) we have more products and also more normal equations.

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