Least Squares Fitting (LSQF) with a complicated function Theexampleswehavelookedatsofarhavebeenlinearintheparameters

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1 Leas Squares Fg LSQF wh a complcaed fuco Theeampleswehavelookedasofarhavebeelearheparameers ha we have bee rg o deerme e.g. slope, ercep. For he case where he fuco s lear he parameers we ca fd a aalc soluo usg e.g. mar verso for a sragh le he mar s. However, we ofe ecouer he case where he fuco s o-lear some of he parameers, e.g. he fuco lab 5: / A, B, C,,, Ae B C Ths fuco s lear A, B,C, bu o-lear σ ad μ. Oher eamples of fucos wh a o-lear parameer are: / A / A Ae For mos o-lear fucos we cao wre dow a soluo for he parameers a closed form usg, for eample, he echques of lear algebra.e. marces. Usuall o-lear problems are solved umercall o a compuer. Somemes b a chage of varables we ca ur a o-lear problem o a lear oe. We wll do hs Lab 6 whe he lfeme of Ba37 s measured. R. Kass/Sp5 P3700/Lecure 8

2 Deermao of a Lfeme Lab 6 The amou of a radoacve subsace as a fuco of me s gve b: 0 e / = he amou of he subsace prese a me. 0 = amou of he subsace a he begg of he eperme = 0. = he lfeme of he subsace. We ca ur hs o-lear problem o a lear oe b a chage of varables Take he aural log of boh sdes of he above epoeal equao: =l =l 0 -/=C+D ow a lear problem he parameers C ad D! I fac s jus a sragh le! = -/D To measure he lfeme Lab 6 we frs f for D he slope & he rasform D o =-/D. R. Kass/Sp5 P3700/Lecure 8

3 Lfeme Daa = l D / D secods The ercep s gve b: C = 4.77 = l 0 or 0 = 7.9 = R= l[ Y ] me, R. Kass/Sp5 P3700/Lecure 8 3

4 Fd he values 0 ad akg o accou he uceraes he daa pos. Weghed formulas are Talor P98 ad Prob. 8.9 w =/ The ucera he umber of radoacve decas s govered b Posso sascs. The umber of cous a b s assumed o be he average of a Posso dsrbuo: = = Varace The varace of = l ca be calculaed usg propagao of errors: The slope & ercep from a sragh le f ha cludes uceraes he daa pos: / l / / / l l =4 75 ad = l l If all he 's are he same =4.75 ad = = -/ = -/ = 0.7 sec To calculae he error o he lfeme τ, we frs mus calculae he error o he slope, β: : he he epressos are decal o he uweghed case / / R. Kass/Sp5 P3700/Lecure 8 4 The epermeall deermed lfeme s: = 0.7 ±.3 sec / / 3

5 Leas Squares Fg-Epoeal Eample We ca calculae he o see how good he daa fs a epoeal deca dsrbuo: For hs problem: l 0 = =.73 ad = 0.7 sec / 0e 0 / 0 0 / 0e e 5.6 Posso appromao Mahemaca Calculao: Do[{csq=csq+c[]-a*Ep[-[]/au]^/a*Ep[-[]/au]},{,,0}] Pr["The ch sq per dof s ",csq/8] v=-cdf[chsquaredsrbuo[8],csq]; Pr["The ch sq prob. s ",00*v,"%"] 8 DOF sce we calculaed 0 & from he 0 daa pos The ch sq per dof s.96 The ch sq prob. s 4.9 % Ths s o such a good f sce he probabl s ol ~4.9%. R. Kass/Sp5 P3700/Lecure 8 5

6 The Error o he Mea Error o he mea revew from Lecure 4 Queso: If we have a se of measuremes of he same qua: Q f f m m f m q Wha's he bes wa o combe hese measuremes? How o calculae he varace oce we combe he measuremes? Assumg Gaussa sascs, he Mamum Lkelhood Mehod sas combe he... measuremes as: / / weghed average If all he varaces = = 3.. are he same: uweghed average The varace of he weghed average ca be calculaed usg propagao of errors: / / Ad he dervaves are: 4 / / / / R. Kass/Sp5 P3700/Lecure 8 6 =error he weghed mea

7 If all he varaces are he same: / The error he mea ges smaller as he umber of measuremes creases. Do' cofuse he error he mea wh he sadard devao of he dsrbuo! If we make more measuremes: The sadard devao of he gaussa dsrbuo remas he same, BUT he error he mea decreases Lecure 4 Eample: Two epermes measure he mass of he proo: Eperme measures m p=950±5 MeV Eperme measures m p =936±5 MeV Usg jus he average of he wo epermes we fd: m p = /=943MeV p Usg he weghed average of he wo epermes we fd: m p =950/5 +936/5 //5 +/5 =937 MeV ad he varace: =//5 +/5 =4MeV =4.9 =5 MeV Sce eperme was more precse ha eperme we would epec he fal resul o be closer o eperme s value. m p =937±5 MeV R. Kass/Sp5 P3700/Lecure 8 7