COPYRIGHT 2008, PRINCETON UNIVERSITY PRESS

systems of equations with matrices; data fitting 1874001.6y(x)300200100y(x)1.20.800 400 800 1200 1600 2000x0.41 1.2 1.4 1.6 1.8 2xFigure 8.5 Left: A linear least-squares best fit of data to a straight line. Here the deviation oftheory from experiment is greater than would be expected from statistics, or in other words, astraight line is not a good theory for these data. Right: A linear least-squares best fit of differentdata to a parabola. Here we see that the fit misses approximately one-third of the points, asexpected from the statistics for a good fit.In this case (also known as linear regression and shown on the left in Figure 8.5)there are M P =2parameters, the slope a 2 , and the y intercept a 1 . Notice that whilethere are only two parameters to determine, there still may be an arbitrary numberN D of data points to fit. Remember, a unique solution is not possible unless thenumber of data points is equal to or greater than the number of parameters. Forthis linear case, there are just two derivatives,∂g(x i )∂a 1=1,∂g(x i )∂a 2= x i , (8.55)and after substitution, the χ 2 minimization equations (8.53) can be solved [Pres 94]:a 1 = S xxS y − S x S xy∆∑N D1∑N DS = , S x =σ 2 i=1 i∑N DS xx =x 2 iσ 2 i=1 i, a 2 = SS xy − S x S y, (8.56)∆x iσ 2 i=1 i∑N D, S xy =i=1∑N D, S y =y iσ 2 i=1 i, (8.57)x i y iσi2 , ∆=SS xx − Sx. 2 (8.58)Statistics also gives you an expression for the variance or uncertainty in the deducedparameters:σ 2 a 1= S xx∆ , σ2 a 2= S ∆ . (8.59)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 187