ADV ASTRONOMICAL DATA ANALYSIS
ADV ASTRONOMICAL DATA ANALYSIS AST 5765
University of Central Florida
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This 4 page Class Notes was uploaded by Virgil Bartell on Thursday October 22, 2015. The Class Notes belongs to AST 5765 at University of Central Florida taught by Staff in Fall. Since its upload, it has received 42 views. For similar materials see /class/227569/ast-5765-university-of-central-florida in Astronomy at University of Central Florida.
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Date Created: 10/22/15
y x MN 5 UCF Physics AST 57654762 Advanced Astronomical Data Analysis Fall 2009 Lecture Notes 10 Error Propagation Fitting Check In 1030 1035 5 min 0 Questions before we start Go over HW3 1035 1055 20 min Error in Addition 1055 1100 5 min 0 This is important enough to memorize spending extra class time on it o For addition f u 1 and all the partial derivatives are 1 so a 2 0395 03203v 0 Add in quadrature o What is it for subtraction Same since the l in the partial derivative squares to l o What if it s just f u a with constant 1 Same as if U 0 0f 2 a Uf2au Error in Multiplication 1100 1105 5 min 0 For multiplication f M a 2 11203 uzag2uvaiv since f M L 2 Li 2livm f2 u2 112 m o How about multiplication by a constant f au Then 1 a and U 0 a 2 1205 of 2 aau o How about division f auv Then 11 171 and there s a l in front of squared to l in the straight term but not in the cross term a 2a Wu Z2 1 2 3 4 5 6 7 It gets 8 5 G Error in Mean and Wrapup 1105 1110 5 min 0 How about the error in the mean of many measurements 1 z N Ex 9 That s a sum and a multiplication If Um Um then 1 2 02 2 2 m 7 2 7 N 7 10 a lt N a N a 039 2 i ll 0 The error in the mean improves with N 0 NOTE This is not the scatter in the measurements it is the error in the mean estimated from the measurements 0 Remember errors add in quadrature sometimes with a weighting 0 It s easiest to work with variances rather than uncertainties 0 Recall that in the Poisson distribution the variance is the value 0 Chapters 3 and 4 show how to average according to weights calculated from uncertainties 0 They also discuss pragmatic reasons for doing this vs discarding old data and the limitations to taking more data to improve 039 through large N Computational Error Analysis Monte Carlo 1110 1115 5 min 0 MODERN LIFE IS EASIER 0 Program the formula into a computer 0 Generate samples of fake measurements distributed according their uncertainties 0 Evaluate the function for each draw in the sample 0 Calculate the uncertainty 0 Note This gives an answer but no information on which parameters it s sensitive to 0 To do that vary each 0 and do the above many times 0 Look at how of changes with changes in each 0 7 Fitting Basics 1115 1125 10 min Have a set of theoretical equations or a model Some parameters are known some are not Unknowns are called free parameters Want to nd values of the free parameters so the model ts the data as well as possible MUST have more data points than free parameters Can estimate both parameters and their uncertainties if enough data Generally use maximum likelihood to decide which model is best least squares is a special case Sometimes can be done with minimization calculus other times must be done numerically see Bevington Ch 8 There are Python routines for line ts and for generalized ts Often problems can be reduced to or approximated by line ts Always check goodness of t not just errors Linear Fitting 1125 1150 25 min Model y a bx measurements xi Assume uncertainty 01 mainly in y or ip Want to minimize Ay y 7 y 7 a 7 bx Assume Gaussiandistributed measurements Then the probability of getting any single mea surement is 1 1 if 7b i 2 12 iexp 7 y a 9 H 2710 2 71 Probability for the whole dataset is 1 1 if 7b i 2 Pab HP H 7 exp 77 w 13 M27103 2 71 We want to maximize P 17 b 17 b appear in the exponent which is negated so minimize it 2 X2 2 least squares l4 l 9y 7 a 7 bzi o Rearrange o Solve to get 32 X i 2 619 OR 6 2 2 6 2 1 i zquot 51 Ufa a 96 0 1 9yi 7 a 7 bm 1 i zquot 51 0112 a 96 1 9yiiaibm 0 M E ii i 1 xi 023ng 27 0207ng aDbE aEbG q W 18 Ewe 7 EF 1 iDFiEO A DGiEZ 15 16 17 18 19 20 21 22 23 24 25 26
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