Fall Analyss o Experental Measureents B Esensten/rev S Errede General Least Squares wth General Constrants: Suppose we have easureents y( x ( y( x, y( x,, y( x wth a syetrc covarance atrx o the y( x easureents y( x Suppose the theory predcton ( ; λ ( ( ; λ, ( ; λ,, ( ; λ nvolves M (< paraeters λ ( λ, λ,, λm y x y x y x y x n soe general (e not necessarly lnear anner Addtonally, suppose there are unctons ( λ ( λ, ( λ,, ( λ that relate (e constran the M λ -paraeters n soe general (but not necessarly lnear anner va use o Lagrange Multplers α ( α α α he χ ( ; λα s dened as:,,, y( x χ λα ; χ λ + α λ y x y x; λ y x y x; λ + α λ where y( x s the easureents, and the syetrc nverse o the covarance atrx o the y ( x colun vector {nb In the lnear constrant case ( λ Bλ b ( λ ay be non-lnear unctons o the M λ -paraeters} We nze the χ ( λα ; by takng dervatves wrt ( ; λ contans the constrant equatons However, n general the constrant equatons λ α We (agan use the teraton technque here too Suppose that ater ν teratons, we have obtaned a set o approxate values o the M λ -paraeters and Lagrange Multplers α : ν λ ν ν λ λ ν λ M and: ν α ν ν α α ν α We then expand (e lnearze χ ( λα ; n a aylor seres around these ponts ( ν ν ; ν ν ν ν ν ν ν ν then solve or Δλ ( Δλ, Δλ,, ΔλM, Δα ( Δα, Δα,, Δα and terate urther λ α, slar to the dscusson n 598AEM Lect Notes (p 5-9 For addtonal detals, see eg ndvdual progra wrte-ups or eg advanced texts on ths subject * * Let us assue that we have deterned the best values ( ; the Lagrange Multpler constraned LSQ t ethod λ α o these paraeters usng We can obtan a better estate, we wsh, o the easured rando varables y x hs procedure goes by the nae Adjustent o Observatons : 598AEM Lecture Notes
Fall Analyss o Experental Measureents B Esensten/rev S Errede We dene a colun vector o easured values o the rando varables (nb these ay not necessarly be ndependent, wth correspondng syetrc covarance atrx o the easureents We want to know the true values (e expectaton values o the easureents: E ˆ ˆ ˆ ˆ [ ],,, We wll estate the usng a LSQ ttng ethod, and call the estates the tted values o the easureents We obtan the tted values o the easureents by adjustng the easureents so that: Each easureent s allowed to ove by an aount deterned ro the sze o the uncertanty on the easureent, σ he resultng tted values o the easureents satsy one or ore constrants We dene a colun vector: o tted values o, e the estates o ˆ Let there be constrants whch can be expressed n the or: ( (,,,,,,,,, ( or, denng a nb In general, these wll be non-lnear equatons colun vector: ( (,,, (,,, Reeberng the teratve χ nzaton ethod(s, we choose to work wth lnearzed correctons : (,,, c c c or, denng a colun vector: c In ters o χ nzaton, snce the s are just constants, nzng c s equvalent to nzng χ wth respect to χ wth respect to 598AEM Lecture Notes
Fall Analyss o Experental Measureents B Esensten/rev S Errede What should we actually nze? I we use χ ( ( c c χ ( c the soluton s (obvously, e the best estate o ˆ s, tsel In order to do better, we ust add n soe new noraton n ths case, the requreent that the constrants be satsed by the s hus, we nstead nze: ( c χ ; α χ + α where: c c+ α akng dervatves o χ ( c; α α α α α, we obtan: s a colun vector o Lagrange Multplers ( χ α ( c; α ( ( c; χ ( α χ α c ; (e the constrants wll be satsed ( c+ α Note that s a atrx ( B( wth jk th k eleent: B jk, j where j,,, ranges over the tted varables and k,,, ranges over the constrants hus, the equatons that we need to solve n order to accoplsh ths χ ( c; ( and: c + B α α nzaton are: For the general non-lnear case, we ust resort to approxaton ethods We aylor seres expand (e lnearze the constrant equatons around, an ntal estate o the tted values o the easureents hen we requre that: As usual, we assue that ( ( + + s sall enough so that we can saely neglect/gnore the and the hgher- ters n the aylor seres expanson nvolvng hgher powers o ( order dervatves o ( (hs step s known as lnearzng the constrants 598AEM Lecture Notes 3
Fall Analyss o Experental Measureents B Esensten/rev S Errede hen: ( + + B A neat trck exsts or solvng ths convenently We wrte: hen: ( ( ( ( c c where: c ( We rewrte ths as: + B ( cc and: c ( + c c + B cc where t s plctly understood that the dervatves and the constrants are evaluated at the an ntal estate he other equaton we ust solve s: yelds: c Bα hus: ( r B c B B α Hα where: H B B hen: B c B c r c+ Bα, whch, ultplyng on the LHS by By constructon, H B B s a square, syetrc (and real atrx, and thereore, t has a square, syetrc (and real nverse H ( B B hus, ultplyng r B c ( B B α Hα on the LHS by H ( B B Lagrange Multplers: α H r and Fnally, the result o ths step s: + c c B BH r gves the α + gves the correcton We explctly need to check/very whether or not ths new satses the constrants: ( I t does, then we re done I not, then we use ths as a new and repeat (e terate the above procedure untl ( s satsed I ( s satsed, then ( also 598AEM Lecture Notes 4
Fall Analyss o Experental Measureents B Esensten/rev S Errede Now let us calculate χ ( c; I α ro the quanttes that we have obtaned s satsed, recallng that and H B B are syetrc atrces, then: hus, ( c; χ α r α χ c; α c c+ α c c ( BH r ( BH r r H B ( BH r r H ( B B H r H r H H H r H r α r α hs s the value o r χ ater the step to + c Next, we deterne the covarance atrx o the tted values usng error propagaton: Now t s just algebra But: r B c B ( ( BH r + c c r + + + BH, thus: ( r B B B and thus: ( BH B hen: BH B BH B ( ( BH B BH B Multplyng ths out on the RHS and agan usng H ths sples to: B B B H B 598AEM Lecture Notes 5
Fall Analyss o Experental Measureents B Esensten/rev S Errede As beore, snce atrx s syetrc, t has postve dagonal eleents Lkewse, the syetrc H B B H B B also has postve dagonal eleents, and so does hereore, ro ( B H ( B ( B( B B ( B, we see that the dagonal eleents o are saller than the dagonal eleents o hus, the -standard devaton uncertantes assocated the adjusted (e tted easureents are less than the -standard devaton uncertantes on the orgnal easureents ull Quanttes: ull quanttes are dstrbutons o noralzed/ractonal derences between the tted easured quanttes whch can be very helpul n veryng the valdty o the LSQ ttng procedure We dene the th c pull quantty as the noralzed correcton: p c where the brackets are synonyous wth the expectaton value, e: c E[ c ] E[ ] Note that there s no bas, then: c I everythng s nce e the nput easureents are Gaussan/norally-dstrbuted and ther uncertantes, as contaned n the ndvdual eleents o the covarance atrx o the easureents have all been correctly / properly assgned and the varous approxatons and assuptons are all vald, then the N, p should be dstrbuted as By explctly lookng at the dstrbutons (eg hstogras o the p or any ndependent easureents o each o the, we can turn ths around and check the ngredents lsted above, especally whether the uncertantes on the ndvdual have ndeed been correctly assgned or not, by seeng whether the pull dstrbuton p or each N, or not s ndeed dstrbuted as Let us suppose that we have perored the Adjustent o Observatons, startng wth our ntal easureents and arrvng at nal adjusted/tted values It s not trval to evaluate the c he colun vector correcton c We also have the covarance atrx o the easureents and that o the adjusted/tted easureents E[( ( ] ( ( Forally: ˆ ˆ ˆ ˆ j j j j j and: ( E ˆ ˆ ˆ ˆ [] j j j j j 598AEM Lecture Notes 6
Fall Analyss o Experental Measureents B Esensten/rev S Errede I the easureents are truly unbased, then: ˆ ˆ, e E [ ] E[ ] hus: c ( ˆ ( ˆ For convenence, we dene the colun vectors: ( ˆ hen: ( ˆ ( ˆ and: δ and: ( ˆ c δ δ or: δ δ c E[ δδ ] δδ δ c δ c δ δ + c c cδ or: + c c cδ or: c c + cδ c δ hs s what we need, snce the dagonal eleents o the covarance atrx are the c But we need to evaluate cδ n order to nsh the job c c c Let us evaluate c δ or the case where δ Dδ Note that ths s a lnear relatonshp, wth D beng a square atrx hen: δ Dδ ( ˆ D( ˆ Or: ( ˆ ( ( D Dˆ D Dˆ ˆ D D ˆ Now: ( ˆ ( ˆ c δ δ cδ δ δ δ δ δ δδ δδ But ro δ Dδ we get: D ( δ, and ro: D ( D ˆ ( δ we get: D cδ δδ δδ D δδ D D Or: c ( D δ, snce s a syetrc atrx hus: δ c D 598AEM Lecture Notes 7
Fall Analyss o Experental Measureents B Esensten/rev S Errede r But we earler derved: + BH ( BH B and: ( B H ( B cδ ( B H B BH B ( B H ( B ( BH B + hus or the lnear case where δ Dδ : c c + cδ c c p c ( p c ( c ( or: p σ σ Navely, one ght expect between and Snce ( B H ( B c σ σ + σ, but ths gnores/neglects the correlaton, then σ calculatng the p pulls > σ and thus we won t get nto trouble n Exaples o LSQ t pulls are shown n the gures below or a oy Monte Carlo progra that carres out LSQ ts to branchng ratos o neutral and charged chared D esons, ro a paper by Werner M Sun, Sultaneous least-squares treatent o statstcal and systeatc uncertantes, Nucl Inst Meth hys Res A 556 35-33 (6 598AEM Lecture Notes 8