Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. with respect to λ. 1. χ λ χ λ ( ) λ, and thus:
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1 More on χ nd errors : uppose tht we re fttng for sngle -prmeter, mnmzng: If we epnd The vlue χ ( ( ( ; ( wth respect to. χ n Tlor seres n the vcnt of ts mnmum vlue χ ( mn χ χ χ χ mn mnmzes χ, nd thus: Then: χ ( s the mnmum vlue of χ 0. : χ, whch s eplctl desgnted s such, s: mn χ. Thus, ner the mnmum of χ,.e. n the vcnt of, then: χ χ χmn ( + χ χ χ mn + ( χ + ( + Note tht the ove epresson s nlogous to one we developed when we were dscussng the M.L.M. for severl prmeters (ee 598AEM Lect. Notes 3, p. 3-4: M M M ( + ( k k + ( k k ( r r + k k k r k r For onl sngle prmeter, ths epresson reduces to: When then: ( 0, nd ner the mmum of (,.e. n the vcnt of ( ( ( + : 598AEM Lecture Notes 7
2 The nlog cn e mde even stronger recllng tht for ndependent Gussn/normll-,,, of the rndom vrle (, wth..f. dstruted mesurements ( then: f ( ( ; e π L ( ; ( ( ( ; ( ; ln ( χ π ( ln ln π Thus, we see tht: χ ( ; whch phscll mens tht the (negtve curvture of ( of the (postve curvture of χ ( ; t ts mnmum., t ts mmum s equl to Recll tht when we were llowed to truncte the Tlor seres epnson for ( t the qudrtc term, when the numer of mesurements ws ver lrge, we found tht: ( ˆ V, j ( E j ws the -j th element of the nverse of the covrnce mtr Vˆ of the ftted prmeters: ( V E ( ( j j ˆ [ ] For the stuton here, wth just sngle prmeter, we hve: mn, j χ ( ; E E + E + χ χ χ χ ( + ( mn + where we hve ssumed tht we cn replce ( : ( χ ( ; E E + E + 598AEM Lecture Notes 7
3 Ths epresson now descres how χ : χ vres n the neghorhood of mn ( mn + χ χ In prtculr, when the -prmeter vres n,,3,... stndrd devtons: ( ± n ( ± mn + mn + n χ χ χ or: χ ( χ ( χ ( Δ ± ± mn n Thus, we see tht chnge of χ ( χ ( χ mn ( mnmum vlue mn Δ ± ± from ts χ corresponds to stndrd devton chnge n the -prmeter. Lter on/shortl, we wll generlze ths to the cse of mn -prmeters Lest-qures Ft to trght Lne: Now we wll eplore n some detl the use of the LQ prncple n the cse where the ; depends lnerl on the -prmeters. theor We wll egn wth the χ ( ft of strght lne to mesured dt ponts (,,,. We mke ndependent mesurements ( ± t the correspondng ponts (n.. whch re ssumed to e known precsel,.e. 0 here. We mke the nstz,.e. hpothess tht: ( +, whch s the theor tht we use to predct the ( ; ( ;,,.e. ( ;, +. Best Ft Lne n.. The est ft lne s sometmes clled the regresson lne non-phscs tpes... We defne the χ (, χ for the LQ ft to strght lne ( + s: ( ( ;, ( ( ( + ( ( 598AEM Lecture Notes 7 3
4 The est estmtes of the slope nd the ntercept nmel I. II. (, ( ( ( χ 0 (, ( ( ( χ 0 We rewrte these equtons s: nd re otned from: 0 0 I. II. + + It s conventonl to defne: nd: Then the ove equtons cn e wrtten compctl s: I. II. + + Two equtons & two unknowns solve these two equtons smultneousl nd: where: Net, we work on determnng the elements of the covrnce mtr of the ftted prmeters: Vˆ ( cov, cov (, We cn use generl methods to clculte the mtr elements, ut here we wll revert to more prmtve technque for purposes of llustrton: ropgton of Errors. We lred know tht the covrnce mtr of the mesurements, whch we re lws ssumng re ndependent, s of the dgonl form: V ( AEM Lecture Notes 7 4
5 We lso hve the two functons: ( (, (,, ( nd: ( (, (,, ( Then, usng error propgton to determne e.g. the vrnce ssocted wth : And: Where: But: And: Thus: ( Then: ( + + And now, summng over ll terms, we otn: Thus:. Usng error propgton on., we otn: cov,, whch s < 0, re negtvel/nt-correlted wth ech other. Wth t more effort, we lso otn the covrnce: ( hence we see tht nd Thus, the covrnce mtr of ftted prmeters ssocted wth LQ-ft to strght lne + s: Vˆ cov, cov (, 598AEM Lecture Notes 7 5
6 In prctce, we must NEVER forget tht nd re (nt-correlted. Ther (nt- correlton s mportnt when, for emple, we tr to nterpolte or etrpolte the ftted lne. The result of the LQ ft s the est-ft lne : + Usng error propgton, the vrnce of ( + s: ( ( ( ( cov (, Wth lttle lger, nd rememerng tht we cn wrte ths s: cov (, ( + + If we hd gnored the fct tht nd re (nt- correlted, we would hve nsted otned: ( + + whch s qute dfferent (nd ncorrect. However, compre these two results when Let s work through numercl strght-lne ft emple whch ws generted to e n del cse: ˆ We egn wth the theor : ˆ Net, we choose the 0,,,3, 4,5 ( 6 nd t ech -pont, we specf vlue for. Fnll, fnd 4 + nd then dd to t the product of nd rndom numer R, chosen from the Gussn/norml dstruton N ( 0, ths smultes Gussn error, or kck to ech ndvdul mesurement. Then cll the fnl numer ( ( 4 ( R Thus, the set of Monte Crlo mesurements + +. s gurnteed to e set of Gussn/ normll-dstruted rndom vrles wth gven/specfed nd followng known theor / 3/4 5/4 3/ R R AEM Lecture Notes 7 6
7 Let us now ft these mesurements to + : Gvng: 4.7 nd: wth: Compre these results to ther true vlues: ˆ 4.0 nd ˆ The est ft lne s: We lso determne the numercl vlues of the elements of the covrnce mtr of ftted prmeters: Vˆ cov, cov (, 598AEM Lecture Notes 7 7
8 Tkng the squre root of the dgonl elements of the covrnce mtr of ftted prmeters nd gnorng the (nt- correlton etween nd, we would s: Ftted slope: ± 4.3 ± 0. Ftted ntercept: ± 0.88 ± 0.45 whch re certnl consstent wth ˆ 4.0 nd ˆ.0. The correlton coeffcent s: ρ Is ths n greement wth our ntuton? As The mnmum vlue of χ occurs t: χ ( mn ( cov, 0.67 ncreses, does 6 ( χ,.078 decrese, nd vce vers? Yes! nce there re M 6 4 degrees of freedom, from the Vlue CL upper vs. χ grph on p. 4 of 598AEM Lect. Notes 6, we would epect to fnd χ >.078 to occur out ~ 70% of the tme, upon repetng ths eperment gzllon tmes. The χ per degree of freedom s: χ N of ( <. The grphcl LQ ft result (see ove looks good, ee nd s good, quntttvel, from the χ LQ ft result. Now we nvestgte t more nltcll ths specfc cse tht of LQ ft wth two lner prmeters nd. We hve: χ (, or: ( χ, nce nd must e postve, nd χ, s 3-dmensonl surfce whose cross-sectons ( slces or contours of constnt χ n the horzontl - plne re ellpses, s shown n the fgure elow: s lso postve, we cn consder ( ( ( ( χ (, χ + + ρ χ + n mn mn ρ ( ellpse lng n, plne t heght n ove χmn 598AEM Lecture Notes 7 8
9 χ (, (, n.. the mjor/mnor es of the ellpse ren t prllel to the nd es unless 0, n whch cse: cov (, 0. For ths emple: χ, AEM Lecture Notes 7 9
10 If we collpse the contours of constnt χ χ mn +, χ χ mn +, χ χ mn + m onto the - plne, we get feelng for how the mnmum looks, usng onl -dmensonl plot: "est ft" (, (0.88, contours of constnt χ + m mn "true" (, (.0, Below, we plot the sngle contour χ ( the error ellpse would not e tlted, nd, χ +. If nd were uncorrelted, nd would e the sem-mjor/mnor es. mn 598AEM Lecture Notes 7 0
11 For fed/constnt vlue of χ, the ellpses contours of constnt χ n the - plne re: ( ( ( ( + ρ n ρ The ellpse contour(s mke n ngle ϕ wth respect to the horzontl -s of: ρ ( cov, ϕ tn tn Alterntvel, we cn slce the 3- fgure through plne of constnt or χ, s functon of, or vce vers, s shown n the two fgures elow: nd plot Ths nterpretton s clerl WRONG f correltons est etween nd! 598AEM Lecture Notes 7
12 Etrpolton Fnd outsde the rnge of mesured : Wht s the vlue of ( t 0? From the theor we know tht Wht does the LQ ft predct? From the LQ ft result: ( + we get: And, from p. 6 of these Lecture Notes (ove: We eplctl see tht the Thus, here: cov (, ( ( ( term s the domnnt contrutor to for lrge, whch s ssocted wth the - uncertnt on the slope of the ftted lne: The - uncertnt on ( 0 Thus, we quote: s: ±.83. ( ( 598AEM Lecture Notes 7 ( n.. Hd we not ncluded the (nt- correlton etween the LQ-ftted slope nd the ntercept, we would hve nsted otned: ( > Interpolton Fnd nsde the rnge of mesured : Wht s the vlue of ( t 0.5? From the theor we know tht From the LQ ft results we get: ( nd ( ( ( Thus, we eplctl see tht for ner zero, the domnnt contruton s from the - uncertnt on the ntercept of the ftted lne: The - uncertnt on ( 0.5 Thus, we quote: nd s: ( 0.5 ( ± Hd we gnored the (nt- correlton etween, we would hve nsted otned: > Comment on the correlton term: In ths nce emple, the omsson of the cov (, would not hve chnged the estmtes of ( term ver much However, n the rel world, the error mde gnorng the correlton term s sometmes ver lrge. Morl: never gnore t!
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