Notes on Statistics for Physicists, Revised

Transcription

1 Lortor for ucler Studes Cornell Unverst Ithc, Y 4853 CLS 8/5 Jul 8, 98 otes on Sttstcs for Phscsts, Revsed J Orer Jul 8, 98 Lortor for ucler Studes Cornell Unverst Ithc, Y 4853 Prnted for the U. S. Atomc Energ Commsson

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3 CLS 8/5 OTES O STATISTICS FOR PHYSICISTS, REVISED Contents Prefce -. Drect Prolt -. Inverse Prolt 3 -. Lkelhood Rtos Mmum-Lkelhood Method Gussn Dstrutons Mmum-Lkelhood Error, One Prmeter Mmum-Lkelhood Errors, M Prmeters, Correlted Errors 8 -. Progton of Errors, the Error Mtr Sstemtc Errors 8 -. Unqueness of Mmum-Lkelhood Soluton 8 -. Confdence Intervls nd ther Artrrness -. Bnoml Dstruton 3 -. Posson Dstruton 4 -. Generlzed Mmum-Lkelhood Method Lest-Squres Method Goodness of Ft, the χ -Dstruton 33 Aend I : Predcton of Lkelhood Rtos 36 Aend II: Dstruton of the Lest Squres Sum 37 Aend III: Lest Squres wth Errors n Both Vrles 39 Aend IV: Mmum Lkelhood nd Lest Squres Solutons umercl Methods 4 Aend V: Cumultve Gussn nd Ch-Squred Dstrutons

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5 CLS 8/5 otes on Sttstcs for Phscsts, Revsed J Orer Prefce These notes re sed on seres of lectures gven t the Rdton Lortor n the summer of 958. I wsh to mke cler m lck of fmlrt wth the mthemtcl lterture nd the corresondng lck of mthemtcl rgor n ths resentton. The rmr source for the sc mterl nd roch resented here ws Enrco Ferm. M frst ntroducton to much of the mterl here ws n seres of dscussons wth Enrco Ferm, Frnk Solmtz nd George Bckus t the Unverst of Chcgo n the utumn of 953. I m grteful to Dr. Frnk Solmtz for mn helful dscussons nd I hve drwn hevl from hs reort "otes on the Lest Squres nd Mmum Lkelhood Methods" []. The generl resentton wll e to stud the Gussn dstruton, noml dstruton, Posson dstruton, nd lestsqures method n tht order s lctons of the mmum-lkelhood method. August 3, 958 Prefce to Revsed Edton Lwrence Rdton Lortor hs grnred ermsson to reroduce the orgnl UCRL Ths revsed verson conssts of the orgnl verson wth correctons nd clrfctons ncludng some new tocs. Three comletel new endces hve een dded. Jul 8, 98 [] Frnk Solmtz, "otes on the Lest Squres nd Mmum Lkelhood Methods" Insttute for ucler Studes Reort, Unverst of Chcgo. - -

6 CLS 8/5 - Drect Prolt Books hve een wrtten on the "defnton" of rolt. We shll merel note two roertes: ( sttstcl ndeendence (events must e comletel unrelted, nd ( the lw of lrge numers. Ths ss tht f s the rolt of gettng n event of Clss nd we oserve tht out of events re n Clss, then we hve: lm A common emle of drect rolt n hscs s tht n whch one hs ect knowledge of fnl-stte wve functon (or rolt denst. One such cse s tht n whch we know n dvnce the ngulr dstruton f(, where cosθ, of certn sctterng eerment. In ths emle one cn redct wth certnt tht the numer of rtcles tht leve t n ngle n n ntervl s f(, where, the totl numer of scttered rtcles, s ver lrge numer. ote tht the functon f( s normlzed to unt: f ( d As hscsts, we cll such functon "dstruton functon". Mthemtcns cll t "rolt denst functon". ote tht n element of rolt, d, s: d f( d - Inverse Prolt The more common rolem fcng hscst s tht he wshes to determne the fnl-stte wve functon from eermentl mesurements. For emle, consder the dec of sn rtcle, the muon, whch does not conserve rt. Becuse of ngulr momentum conservton, we hve the ror knowledge tht: f( + However, the numercl vlue of s some unversl hscl constnt et to e determned. We shll lws use the suscrt zero to denote the true hscl vlue of the rmeter under queston. It s the o of the hscst to determne. Usull the hscst does n eerment nd quotes result ±. The mor orton of ths reort s devoted to the questons "Wht do we men nd?" nd "Wht s the est w to clculte nd?". These re questons of etreme mortnce to ll hscsts. - -

7 CLS 8/5 Crudel sekng, s the stndrd devton [], nd wht the hscst usull mens s tht the "rolt" of fndng: ( - < < ( + s 68.3 % (the re under Gussn curve out to one stndrd devton. The use of the word "rolt" n the revous sentence would shock mthemtcn. He would s the rolt of hvng: ( - < < ( + s ether or The knd of rolt the hscst s tlkng out here s clled nverse rolt, n contrst to the drect rolt used the mthemtcn. Most hscsts use the sme word, rolt, for the two comletel dfferent concets: drect rolt nd nverse rolt. In the remnder of ths reort we wll conform to ths slo hscst-usge of the word "rolt". 3 - Lkelhood Rtos Suose t s known tht ether Hothess A or Hothess B must e true. And t s lso known tht f A s true the eermentl dstruton of the vrle must e f A (, nd f B s true the dstruton s f B (. For emle, f Hothess A s tht the K meson hs sn zero, nd Hothess B tht t hs sn, then t s "known" tht f A ( nd f B (, where s the knetc energ of the dec π - dvded ts mmum vlue for the dec + + mode K π + π. If A s true, then the ont rolt for gettng rtculr result of events of vlues ; ;...; s: d A f ( A d The lkelhood rto R s: [] In 958 t ws common to use role error rther thn stndrd devton. Also some hscsts delertel multl ther estmted stndrd devtons "sfet" fctor (such s π. Such rctces re confusng to other hscsts who n the course of ther work must comne, comre, nterret, or mnulte eermentl results. B 98 most of these msledng rctces hd een dscontnued

8 CLS 8/5 R f ( A fb ( f ( A B f ( ( Ths s the rolt tht the rtculr eermentl result of events turns out the w t dd, ssumng A s true, dvded the rolt tht the eerment turns out the w t dd, ssumng B s true. The foregong length sentence s correct sttement usng drect rolt. Phscsts hve shorter w of sng t usng nverse rolt. The s Eq. ( s the ettng odds of A gnst B. The formlsm of nverse rolt ssgns nverse roltes whose rto s the lkelhood rto n the cse n whch there est no ror roltes fvorng A or B [3]. All the remnng mterl n ths reort s sed on ths sc rncle lone. The modfctons led when ror knowledge ests re dscussed n Secton. An mortnt o of hscst lnnng new eerments s to estmte eforehnd how + + mn events he wll need to "rove" hothess. Suose tht for the K π + π one wshes to estlsh ettng odds of 4 to gnst sn. How mn events wll e needed for ths? Ths rolem nd the generl rocedure nvolved re dscussed n Aend I: Predcton of Lkelhood Rtos. 4 - Mmum-Lkelhood Method The recedng secton ws devoted to the cse n whch one hd dscrete set of hotheses mong whch to choose. It s more common n hscs to hve n nfnte set of hotheses;. e., rmeter tht s contnuous vrle. For emle, n the µ-e dec dstruton: f(; + the ossle vlues for elong to contnuous rther thn dscrete set. In ths cse, s efore, we nvoke the sme sc rncle whch ss the reltve rolt of n two dfferent vlues of s the rto of the roltes of gettng our rtculr eermentl results,, ssumng frst one nd then the other vlue of s true. Ths rolt functon of s clled the lkelhood functon L(. L( F( ; ( [3] An equvlent sttement s tht n the nverse rolt roch (lso clled Besn roch one s mlctl ssumng tht the ror roltes re equl

9 CLS 8/5 The lkelhood functon, L(, s the ont rolt denst of gettng rtculr eermentl result, ; ;...; ssumng f(; s the true normlzed dstruton functon: f ( ; d The reltve roltes of cn e dsled s lot of L( versus. The most role vlue of s clled the mmum-lkelhood soluton. The RMS (root-men-squre sred of out s conventonl mesure of the ccurc of the determnton. We shll cll ths. ( Ld Ld (3 In generl the lkelhood functon wll e close to Gussn (t cn e shown to roch Gussn dstruton s --> nd wll look smlr to Fg. (. L( Fg. ( L( Fg. ( Good Sttstcs Poor Sttstcs Fg.. Two emles of lkelhood functons L( Fg. ( reresents wht s clled cse of oor sttstcs. In such cse, t s etter to resent lot of L( rther thn merel quotng nd. Strghtforwrd rocedures for otnng re resented n Sectons 6 nd 7. A confrmton of ths nverse-rolt roch s the Mmum-Lkelhood Theorem, whch s roved n Crmer [4] use of drect rolt. The theorem sttes tht n the lmt of lrge, --> ; nd furthermore, there s no other method of estmton tht s more ccurte. In the generl cse n whch there re M rmeters, ; ;...; M to e determned, the rocedure for otnng the mmum-lkelhood soluton s to solve the M smultneous equtons: - 5 -

10 CLS 8/5-6 - w where w log L( ; ;...; M (4 5 - Gussn Dstrutons As frst lcton of the mmum-lkelhood method, we consder the emle of the mesurement of hscl rmeter, where s the result of rtculr te of mesurement tht s known to hve mesurng error. Then f s Gussn-dstruted [4], the dstruton functon s: f( ; π e ( For set of mesurements, ech wth ts own mesurement error, the lkelhood functon s: L( ( e π then:. ( const w + w w, ( (5 [4] A dervton of the Gussn dstruton nd ts relton to the noml nd Posson dstrutons s gven n Chter II of R. B. Lnds, "Phscl Sttstcs", Wle, ew York, 94, or E. Segrè, "ucle nd Prtcles" + const w. (

11 CLS 8/5 (6 s the mmum-lkelhood soluton. ote tht the mesurements must e weghted ccordng to the nverse squres of ther errors.. When ll the mesurng errors re the sme we hve: whch s the conventonl determnton of the men vlue. et we consder the ccurc of ths determnton. 6 - Mmum-Lkelhood Error, One Prmeter It cn e shown tht for lrge L( roches Gussn dstruton. To ths romton (ctull the ove emle s lws Gussn n, we hve: L( e h ( where s the RMS sred of out : h h w - ( + const. w - h ( - w - h Snce s defned n Eq. (3 s, we hve: h w Mmum-lkelhood Error (7-7 -

12 CLS 8/5 It s lso roven n Crmer [5] Tht no method of estmton cn gve n error smller thn tht of Eq. (7 (or ts lternte form Eq. (8. Eq. (7 s ndeed ver owerful nd mortnt. Let us now l ths formul to determne the error ssocted wth (Eq. (6. We dfferentte Eq. (5 wth resect to. The nswer s: w Usng ths n Eq. (7 gves: Lw of Comnton of Errors Ths formul s commonl known s the "Lw of Comnton of Errors" nd refers to reeted mesurements of the sme quntt whch re Gussn-dstruted wth "errors". In mn ctul rolems, nether nor m e found nltcll. In such cses the curve L( cn e found numercll trng severl vlues of nd usng Eq.( to get the corresondng vlues of L(. The comlete functon s then otned usng smooth w curve through the onts. If L( s Gussnlke, s the sme everwhere. If not, t s est to use the verge: w w ( Ld Ld A luslt rgument for usng the ove verge goes s follows; f the tls of L( dro w off more slowl thn Gussn tls, s smller thn: w Thus, use of the verge second dervtve gves the requred lrger errors. ote tht use of Eq. (7 for deends on hvng rtculr eermentl result efore the error cn e determned. However, t s often mortnt n the desgn of eerments to e [5] H. Crmer, Mthemtcl Methods of Sttstcs, Prnceton Unverst Press,

13 CLS 8/5 le to estmte n dvnce how mn dt wll e needed n order to otn gven ccurc. We shll now develo n lternte formul for the mmum-lkelhood error, whch deends onl on knowledge of f(;. Under these crcumstnces we wsh to w determne verged over mn reeted eerments consstng of events ech. For one event we hve: w log( f fd for events: w log( f fd Ths cn e ut n the form of frst dervtve s follows: log( f f - f f f f + f log( f f fd - f f d + d The lst ntegrl vnshes f one ntegrtes efore the dfferentton ecuse f ( ; d. Thus: w f - f d nd Eq. (7 leds to [6] : + [6] Emle : Assume n the µ-e dec dstruton functon, f(;, tht. How mn µ-e decs re needed to estlsh to % ccurc (. e., 3? f f ; f d ( + d 3 + log - 9 -

14 CLS 8/5 f f d Mmum-lkelhood Error (8 See Emle. 7 - Mmum-Lkelhood Errors, M-Prmeters, Correlted Errors When M rmeters re to e determned from sngle eerment contnng events, the error formuls of the recedng secton re lcle onl n the rre cse n whch the errors re uncorrelted. Errors re uncorrelted onl for: ( for ll cses wth. ( For the generl cse we Tlor-end w( out : w( w( + M w M M β H β β +... where β - nd: H - w Covrnce Mtr (9 The second term of the enson vnshes ecuse w re the equtons for : log L( w( M M H β β log ote tht ( lm > 3 3 For - ; 3.8 For ths rolem ;.5 5 events 3 - -

15 CLS 8/5 eglectng the hgher-order terms, we hve: M M L( C e H β β (n M-dmensonl Gussn surfce. As efore, our error formuls deend on the romton tht L( s Gussnlke n the regon. As mentoned n Secton 4, f the sttstcs re so oor tht ths s oor romton, then one should merel resent lot of L(. Accordng to Eq. (9, H s smmetrc mtr. Let U e the untr mtr tht dgonlzes H: U H U - h h h M h where T U U - ( Let β (β ;β ;...;β M nd γ β U -. The element of rolt n the β-sce s: d M C e ( γ T U H( γ U d M β Snce U s the Jcon reltng the volume elements d M β nd d M γ, we hve: d M C e h γ d M γ ow tht the generl M-dmensonl Gussn surfce hs een ut n the form of the roduct of ndeendent one-dmensonl Gussns we hve [7] : [7] Proof: We must evlute the followng ntegrl: γ γ M γ γ γ M C e h d γ whch cn e trnslted nto: γ γ C e h γ dγ γ e hγ dγ γ, e hγ dγ - -

16 CLS 8/5 γ γ δ h Then: β β M M γ γ U U M U h U (( U h U Accordng to Eq. (, H U - h U, so tht the fnl result s: ( ( (H - where H - Mmum-lkelhood Errors, M rmeters I w Averged over reeted eerments: f f H f d ( Snce e hγ dγ for,, nd hγ dγ the result s γ γ for. If then: γ e for,, γ γ C e hγ dγ γ, e hγ dγ And snce γ e hγ dγ, γ γ h h Therefore γ γ δ h - -

17 CLS 8/5-3 - An emle of use of the ove quoted formuls s gven n [8]. [8] Emle : Assume tht the rnges of monoenergetc rtcles re Gussn-dstruted wth men rnge nd strgglng coeffcent (the stndrd devton. rtcles hvng rnges ;...; ; re oserved. Fnd, nd ther errors. Then: ( L e, ( π ( log( log( π w ( w ( w 3 The mmum-lkelhood soluton s otned settng the ove two equtons equl to zero: ( The reder m rememer stndrd-devton formul n whch s relced ( : ( Ths s ecuse n ths cse the most role vlue,, nd the men, do not occur t the sme lce. Men vlues of such qunttes re studed n Secton 6. The mtr H s otned evlutng the followng qunttes t nd w ; ( 4 3 w + + ; ( 3 w

18 CLS 8/5 A rule for clcultng the nverse mtr H s: H - (- + th mnor of H determnn t of H If we use the lternte notton V for the error mtr must e relced wth V H ;.e., the lkelhood functon s:, then whenever H ers, t T L( e β V β ( We note tht the error of the men s error on the determnton of s where s the stndrd deevton. The Correlted Errors The mtr V ( s defned s the Error Mtr (lso clled the ( Covrnce Mtr of. In Eq. ( we hve shown tht V H where: when, H nd H - Accordng to Eq. (, the errors on nd re the squre roots of the dgonl elements of the error mtr H - : nd where the lst s sometmes clled the "error of the error"

19 CLS 8/5 w H - The dgonl elements of V re the vrnces of the s. If ll the off-dgonl elements re zero, the errors n re uncorrelted s n Emle. In ths cse contours of constnt w lotted n (, sce would e ellses s shown n Fg.. The errors n nd would e the sem-mor es of the contour ellse where w hs droed unt from ts mmum-lkelhood vlue. Onl n the cse of uncorrelted errors s the RMS error H nd then there s no need to erform mtr nverson. ( ( w - / ((H - / w - / w w (H -/ Fg.. Contours of constnt w s functon of nd. Mmum lkelhood soluton s t w w. Errors n nd re otned from ellse where w ( w. ( Uncorrelted errors V H V H. ote tht ( Correlted errors. In ether cse ( nd ( t would e serous mstke to use the ellse hlfwdth rther thn the etremum for. In the more common stuton there wll e one or more off-dgonl elements to H nd the errors re correlted (V hs off-dgonl elements. In ths cse (Fg. the contour ellses re nclned to the, es. The RMS sred s whch cn e shown to V e the etreme lmt of the ellse roected on the s (the ellse hlfwdth s s H whch s smller. In cses where Eq. ( cnnot e evluted nltcll, the s cn e found numercll nd the errors n cn e found lottng the ellsod where w s unt less thn w. The etremums of ths ellsod re the RMS errors n the s. One should llow ll the to chnge freel nd serch for the mmum chnge n whch mkes w w. Ths mmum chnge n s the error n nd s V. 8 - Progton of Errors: the Error Mtr - 5 -

20 CLS 8/5-6 - Consder the cse n whch sngle hscl quntt,, s some functon of the 's: (,..., M. The "est" vlue for s then (. For emle could e the th rdus of n electron crclng n unform mgnetc feldwhere the mesured qunttes re τ, the erod of revoluton, nd v, the electron veloct. Our gol s to fnd the error n gven the error n. To frst order n ( - we hve: - ( M ( M M ( ( ( rms M M H ( ( A well known secl cse of Eq. (, whch holds onl when the vrles re comletel uncorrelted, s: ( rms ( M In the emle of ort rdus n terms of τ nd v ths ecomes: ( ( ( ( 4 4 v v v v R R R + + π τ τ π τ τ n the cse of uncorrelted errors. However, f v τ s non-zero s one mght eect, then Eq. ( gves: ( ( v v v v R + + τ π τ π π τ τ π 4 4 It s common rolem to e nterested n M hscl rmeters,..., M, whch re known functons of the. If the error mtr H -, of the s known, then we hve: ( ( M M H ( (3

21 CLS 8/5 In some such cses the otnle [9]. Then: cnnot e otned drectl, ut the re esl (J -, where J 9 - Sstemtc Errors "Sstemtc effects" s generl ctegor whch ncludes effects such s ckground, selecton s, scnnng effcenc, energ resoluton, ngle resoluton, vrton of counter effcenc wth em oston nd energ, ded tme, etc. The uncertnt n the estmton of such sstemtc effect s clled "sstemtc error". Often such sstemtc effects nd ther errors re estmted serte eerments desgned for tht secfc urose. In generl the mmum-lkelhood method cn e used n such n eerment to determne the sstemtc effect nd ts error. Then the sstemtc effect nd ts error re folded nto the [9] Emle 3: Suose one wshes to use rdus nd ccelerton to secf the crculr ort of n electron n unform mgnetc feld;. e., r nd. Suose the orgnl mesured qunttes re τ ( ± µs nd v ( ± km/s.also snce the veloct mesurement deended on the tme mesurement, there ws correlted error τ v.5-3 m. fnd r, r,,. vτ Snce r.59 m nd πv 6.8 m/s we hve: π τ nd π. Then,, π π π π π,. The mesurement errors secf the error mtr s: s V 3.5 m Eq. (3 gves: m 3 m s ( V + V + V π π π π v vτ τ 4 V + V + V 3.39 m 4π π 4π Thus r (.59 ±.84 m For, Eq. (3 gves: π π π 9 ( V + V + V.9 m 4 Thus (6.8 ±.54 m/s. π s - 7 -

22 CLS 8/5 dstruton functon of the mn eerment. Idell, the two eerments cn e treted s one ont eerment wth n dded rmeter M+ to ccount for the sstemtc effect. In some cses sstemtc effect cnnot e estmted rt from the mn eerment. Emle cn e mde nto such cse. Let us ssume tht mong the em of monoenergetc rtcles there s n unknown ckground of rtcles unforml dstruted n rnge. In ths cse the dstruton functon would e: f(,, 3 ; C π ( e + 3 where m C(,, 3 fd mn The soluton 3 s sml relted to the ercentge of ckground. The sstemtc error s otned usng Eq. (. - Unqueness of Mmum-Lkelhood Soluton Usull t s mtter of tste wht hscl quntt s chosen s.. For emle, n lfetme eerment some workers would solve for the lfetme, τ, whle others would solve for λ, where λ /τ. Some workers refer to use momentum, nd others energ, etc. Consder the cse of two relted hscl rmeters λ nd. The w mmum-lkelhood soluton for s otned from. The mmum-lkelhood soluton for λ s otned from w. But the we hve: λ w, nd λ w Thus the condton for the mmum-lkelhood soluton s unque nd ndeendent of the rtrrness nvolved n choce of hscl rmeter. A lfetme result τ would e relted to the soluton λ τ λ. The sc shortcomng of the mmum-lkelhood method s wht to do out the ror rolt of. If the ror rolt of s G( nd the lkelhood functon otned for the eerment lone s H(, then the ont lkelhood functon s: L( G( H( - 8 -

23 CLS 8/5 w ln G + ln H w ln + G ln H ( - ln H ln G( gve the mmum lkelhood soluton. In the sence of n ror knowledge the term on the rght hnd-sde s zero. In other words, the stndrd rocedure n the sence of n ror nformton s to use n ror dstruton n whch ll vlues of re equll role. Strctl sekng, t s mossle to know "true" G(, ecuse t n turn must deend on ts own ror rolt. However, the ove equton s useful when G( s the comned lkelhood functon of ll revous eerments nd H( s the lkelhood functon of the eerment under consderton. There s clss of rolems n whch one wshes to determne n unknown dstuton n, G(, rther thn sngle vlue.for emle, one m whsh to determne the momentum dstruton of cosmc r muons. Here one oserves: L(G G ( H ( ; d where H(; s known from the nture of the eerment nd G( s the functon to e determned. Ths te of rolem s dscussed n Reference []. - Confdence Intervls nd Ther Artrrness So fr we hve worked onl n terms of reltve roltes nd RMS vlues to gve n de of he ccurc of the determnton. One cn lso sk the queston, "Wht s the rolt tht les etween two certn vlues such s ' nd ''?". Ths s clled confdence ntervl. P(' < < '' '' ' Ld Ld [] M. Anns, W. Cheston, H. Prmkoff, "On Sttstcl Estmton n Phscs", Revs. Modern Phs. 5 (953,

24 CLS 8/5 Unfortuntel such rolt deends on the rtrr choce of wht quntt s chosen for. To show ths consder the re under the tl of L( n the Fg. 3. L( Fg. 3 Fg. 3. Shded re s P( >, sometmes clled the confdence lmt of. P( > ' ' Ld Ld If λ λ( hd een chosen s the hscl rmeter nsted, the sme confdence ntervl s: P(λ > λ' λ' Ldλ Ldλ ' λ L d P( > ' Ldλ Thus, n generl, the numerc vlue of confdence ntervl deends on the choce of the hscl rmeter. Ths s lso true to some etent n evlutng. Onl the mmumlkelhood soluton nd the reltve roltes re unffected the choce of. For Gussn dstrutons, confdence ntervls cn e evluted usng tles of the rolt ntegrl. Tles of cumultve noml dstrutons nd cumultve Posson dstrutons re lso vlle. Aend V contns lot of the cumultve Gussn dstruton. - Bnoml Dstruton Here we re concerned wth the cse n whch n event must e one of two clsses, such s u or down, forwrd or ck, ostve or negtve, etc. Let e the rolt for n event of Clss. Then ( - P s the rolt for Clss, nd the ont rolt for oservng events n Clss out of totl events s: - -

25 CLS 8/5 P(,!!(! ( The Bnoml Dstruton (4 ote tht (, [ + ( ] P. The fctorls correct for the fct tht we re not nterested n the order n whch the events occurred. For gven eermentl result of out of events n Clss, the lkelhood functon L( s then: L/(!!(! ( w ln( + ( - ln( - + const. w - (5 w - - ( (6 From Eq. (5 we hve: (7 From (6 nd (7: ( + ( ( (8 The results, Eqs. (7 nd (8, lso hen to e the sme s those usng drect rolt []. Then: [] Emle 4: In Emle on the µ - e dec ngulr dstruton we found tht 3 s the error on the smmetr rmeter. Suose tht the ndvdul cosne of ech event s not known. In ths rolem ll we know s the numer u vs. the numer down. Wht s then? Let e the rolt of dec n the u emshere; then we hve: - -

26 CLS 8/5 nd ( ( Posson Dstruton A common te of rolem whch flls nto ths ctegor s the determnton of cross secton or men free th. For men free th λ, the rolt of gettng n event n n ntervl d s d. Let P(, e the rolt of gettng no events n length. Then we λ hve: dp(, - P(, λ d ln P(, - λ + const. P(, λ e (t, P(,, the sme s for the rdoctve dec lw (9 Let P(, e the rolt of fndng events n length. An element of ths rolt s the ont rolt of events t d,..., d tmes the rolt of no events n the remnng length: d ( 4 4 B Eq. (8, rememerng tht, 4 4 ( For smll ths s s comred to when the full nformton s used. - -

27 CLS 8/5 d P(, d λ λ e ( The entre rolt s otned ntegrtng over the -dmensonl sce. ote tht the ntegrl: λ λ d does the o ecet tht the rtculr rolt element n Eq. ( s swet through! tmes. Dvdng! gves: P(, ( λ! λ e The Posson dstruton ( As check, note: P (, e λ λ λ e! λ e λ! e λ ( λ Lkewse t cn e shown tht (. Equton ( s often eressed n terms of : P(,! e the Posson dstruton ( Ths form s useful n nlzng countng eerments. Then the "true" countng rte s. We now consder the cse n whch, n certn eerment, events were oserved. The rolem s to determne the mmum-lkelhood soluton for nd ts error: - 3 -

28 CLS 8/5 L(! e w ln - - ln! w - w - Thus we hve: nd Eq. (7 In cross secton determnton, we hve ρ, where ρ s the numer of trget nucle er cm 3 nd s the totl th length. Then: ρ nd In concluson we note tht L( d L( d + e e d d ( +! +! 4 - Generlzed Mmum-Lkelhood Method So fr we hve lws worked wth the stndrd mmum-lkelhood formlsm, where the dstruton functons re lws normlzed to unt. Ferm hs onted out tht the normlzton requrement s not necessr so long s the sc rncle s oserved: nmel, tht f one correctl wrtes down the rolt of gettng hs eermentl result, then ths lkelhood functon gves the reltve roltes of the rmeters n queston. The onl requrement s tht the rolt of gettng rtculr result e correctl wrtten. We shll now consder the generl cse n whch the rolt of gettng n event n d s F( d, nd: m mn Fd ( - 4 -

29 CLS 8/5 s the verge numer of events one would get f the sme eerment were reeted mn tmes. Accordng to Eq. (9, the rolt of gettng no events n smll fnte ntervl s: + P(, e Fd The rolt of gettng no events n the entre ntervl mn < < m s the roduct of such eonentls or: P(,( m - mn e m Fd mn e The element of rolt for rtculr eermentl result of events occurrng t,..., s then: d e F( d Thus we hve: L( e ( F ( ; nd: w( ln F( ; m mn F( ; d The solutons re stll gven the M smultneous equtons: w The errors re stll gven : ( ( (H - where: - 5 -

30 CLS 8/5 H - w The onl chnge s tht no longer ers elctl n the formul: w F F d F A dervton smlr to tht used for Eq. (8 shows tht s lred tken cre of n the ntegrton over F(. In rvte communcton George Bckus hs roven, usng drect rolt, tht the Mmum-Lkelhood Theorem lso holds for ths generlzed mmum-lkelhood method nd tht n the lmt of lrge there s no method of estmton tht s more ccurte. Also see Sect. 9.8 of []. In the sence of the generlzed mmum-lkelhood method our rocedure would hve een to normlze F(; to unt usng: f(; F ( ; Fd For emle, consder smle contnng ust two rdoctve seces, of lfetmes nd. Let 3 nd 4 e the two ntl dec rtes. Then we hve: F( ; 3 e + 4 e where s the tme. The stndrd method would then e to use: f(; e e wth whch s normlzed to one. ote tht the four orgnl rmeters hve een reduced to three usng 5 4. Then 3 nd 4 would e found usng the ulr equton: 3 [] A. G. Frodesen, O. Skeggestd, H. Tofte, Prolt nd Sttstcs n Prtcle Phscs (Colum Unverst Press, 979 ISB The ttle s msledng, ths s n ecellent ook for hscsts n ll felds who whsh to ursue the suect more deel thn s done n these notes

31 CLS 8/5 Fd the totl numer of counts. In ths stndrd rocedure the equton: ( must lws hold. However, n the generlzed mmum-lkelhood method these two qunttes re not necessrl equl. Thus the generlzed mmum-lkelhood method wll gve dfferent soluton for the, whch should, n rncle, e etter. Another emle s tht the est vlue for cross secton s not otned the usul rocedure of settng ρl (the numer of events n th length L. The fct tht one hs ddtonl ror nformton such s the she of the ngulr dstruton enles one to do somewht etter o of clcultng the cross secton. 5 - The Lest-Squres Method Untl now we hve een dscussng the stuton n whch the eermentl result s events gvng recse vlues,..., where the m or m not, s the cse m e, e ll dfferent. From now on we shll confne our ttenton to the cse of mesurements (not events t the onts,...,. The eermentl results re ( ±,..., ( ±. One such te of mesurement s where ech mesurement conssts of events. Then nd s Posson-dstruted wth L ( e! (. In ths cse the lkelhood functon s: nd w ln( ( ( + const. We use the notton ( ; for the curve tht s to e ftted to the eermentl onts. The est-ft curve corresonds to. In ths cse of Posson-dstruted onts, the solutons re otned from M smultneous equtons: ( ( ( - 7 -

32 CLS 8/5 Fg. 4 ( Fg. 4. ( s the functon of known she to e ftted to the 7 eermentl onts. If ll the >>, then t s good romton to ssume ech s Gussn-dstruted wth stndrd devton (t s etter to use rther thn for where cn e otned ntegrtng ( over the -th ntervl. Then one cn use the fmous lest squres method. The remnder of ths secton s devoted to the cse n whch the re Gussn-dstruted wth stndrd devton (see Fg. 4. We shll now see tht the lest-squres method s mthemtcll equvlent to the mmum-lkelhood method. In ths Gussn cse the lkelhood functon s: L ( e π ( (3 w( S( ln( π where S( ( ( (4 The solutons re gven mnmzng S( (mmzng w : S ( (5-8 -

33 CLS 8/5-9 - Ths mnmum vlue of S s clled S, the lest-squres sum.the vlues of whch mnmze re clled the lest-squres solutons.thus the mmum-lkelhood nd lest-squres solutons re dentcl. Accordng to Eq. (, the lest squres errors re: ( ( (H, where H S Let us consder the secl cse n whch ; ( s lner n the : M f ( ; ( (Do not confuse ths f( wth the f( on ge Then: ( ( M f f S (6 Dfferenttng wth resect to gves: f f H ( ( (7 Defne: f U ( (8 Then: M H U S In mtr notton the M smultneous equtons gvng the lest-squres soluton re: u - H (9 u H -

34 CLS 8/5-3 - s the soluton for the s. The errors n re otned usng Eq. (. To summrze: Equton (3 s the comlete rocedure for clcultng the lest-squres solutons nd ther errors. ote tht even though ths rocedure s clled "curve-fttng" t s never necessr to lot n curves. Qute often the comlete eerment m e comnton of severl eerments n whch severl dfferent curves (ll functons of the m ontl e ftted. Then the S-vlue s the sum over ll the onts on ll the curves. ote tht snce w( decreses unt when one of the hs the vlue ( +, the S-vlue must ncrese one unt. Tht s: (,...,,..., + + S S M See Emles 5 [3], 6 [4], 7 [5]. [3] Emle 5: ( s known to e of the form ( +. There re eermentl mesurements ( ±. Usng Eq. (3 we hve f, f, H, H - ( ( ( These re the lner regresson formuls whch re rogrmmed nto mn ocket clcultors. The should not e used n those cses where the re not ll the sme. If the re ll equl, the errors re ( ( ( ( H H or ( ( If M f ( ; ( ( M H f ( ( ( H where: H f f ( ( (3

35 CLS 8/5 [4] Emle 6: The curve to e ftted s known to e rol. There re four eermentl onts t -.6, -.,.,.6. The eermentl results re 5 ±, 3 ±, 5 ±, nd 8 ±. Fnd the est-ft curve. ( ; f f, f, 3 H H 4 4 ; H ; H ; H3 H ; H H.6.6 (the error mtr.6.68 u [ ] H V ;.85 ; 3.7 ;.96 ; ; ; 3 ( (3.685 ±.85 + (3.7 ±.96 + (7.88 ± 4.94 s the est-ft curve. Ths s shown wth the eermentl onts n Fg

36 CLS 8/5 Lest squres when the re not ndeendent Let: V ( ( Y X Fg. 5. Ths rol s the lest squres ft to the 4 eermentl onts n Emle 6. [5] Emle 7: In Emle 6 wht s the est estmte of t? Wht s the error of ths estmte? Soluton: uttng nto the ove equton gves s otned usng Eq. (. f + f f V V + f V + f3 V33 + f f V + f f 3V Settng gves: 5.37 So t, ±

37 CLS 8/5 e the error of the mesurements. ow wwe shll tret the more generl cse where the off dgonl elements need not e zero;.e., the qunttes re not ndeendent. We see mmedtel from Eq. ( tht the log-lkelhood functon s: w T ( V ( + const. The mmum-lkelhood soluton s found mnmzng: S ( V ( T. where: V ( ( Generlzed Lest Squres Sum 6 - Goodness of Ft, the c Dstruton The numercl vlue of the lkelhood functon t L( cn, n rncle, e used s check whether one s usng the correct te of functon for f(;. If one s usng the wrong functon, the lkelhood functon wll e lower n heght nd of greter wdth. In rncle one cn clculte, usng drect rolt, the dstruton of L( ssumng rtculr true f( ;. Then the rolt of gettng n L( smller thn the vlue oserved would e useful ndcton of whether the wrong te of functon f(; hd een used. If for rtculr eerment one got the nswer tht there ws one chnce n 4 of gettng such low vlue of L(, one would serousl queston ether the eerment or the functon f(; tht ws used. In rctce, the determnton of the dstruton of L( s usull n mossl dffcult numercl ntegrton n -dmensonl sce. However, n the secl cse of the lestsqures rolem, the ntegrton lmts turn out to e the rdus vector n - dmensonl sce. In ths cse we use the dstruton of S( rther thn of L(. We shll frst consder the dstruton of S(. Accordng to Eqs. (3 nd (4 the rolt element s: d S P e d ote tht S ρ, where ρ s the mgntude of the rdus vector n -dmensonl sce. The volume of - dmensonl shere s U ρ. The volume element n ths sce s then: d ρ d S S Thus: ds

38 CLS 8/5 dp( S S e S ds The normlzton s otned ntegrtng from S to S. S dp( S S e ds Γ( (3 where S S(. Ths dstruton s the well-known χ dstruton wth degrees of freedom. χ tles of: P(χ S S dp( S S for severl degrees of freedom re n the "Hndook of Chemstr nd Phscs" nd other common mthemtcl tles. From the defnton of S (Eq. (4 t s ovous tht S. One cn show, usng Eq. (9, tht ( S S. Hence, one should e suscous f hs eermentl result gves n S-vlue much greter thn: ( + Usull s not known. In such cse one s nterested n the dstruton of S S( Fortuntel, the dstruton s lso qute smle. It s merel the χ dstruton of ( M degrees of freedom, where s the numer of eermentl onts, nd M s the numer of rmeters solved for. Thus we hve: dp(s χ dstruton for ( M degrees of freedom S ( M nd S ( M (3 Snce the dervton of Eq. (3 s somewht length, t s gven n Aend II [6][7]. [6] Emle 8: Determne the χ rolt of the soluton to Emle

39 CLS 8/5 S 5 ( (. + 5 (. + 8 (.6 S.674 comred to S 4-3 Accordng to the χ tle for one degree of freedom the rolt of gettng S >.674 s.4. Thus the eermentl dt re qute consstent wth the ssumed theoretcl she of: ( [7] Emle 9 Comnng Eerments: Two dfferent lortores hve mesured the lfetme of the K rtcle to e (. ±. - sec nd (.4 ±. - sec resectvel. Are these results rell nconsstent? Accordng to Eq. (6 the weghted men s.8 - sec.(ths s lso the lest squres soluton for τ. Thus: K S S - Accordng to the χ tle for one degree of freedom the rolt of gettng S > 3. s.74. Therefore, ccordng to sttstcs, two mesurements of the sme quntt should e t lest ths fr rt 7.4 % of the tme

40 CLS 8/5 Aend I: Predcton of Lkelhood Rtos An mortnt o for hscst who lns new eerments s to estmte eforehnd ust how mn events wll e needed to "rove" certn hothess. The usul rocedure s to clculte the verge logrthm of the lkelhood rto. The verge logrthm s etter ehved mthemtcll thn the verge of the rto tself. We hve: f A log R f A( d ssumng A s true, (3 f B or f A log R f B ( d ssumng B s true. f B Consder the emle (gven n Secton 3 of the K + meson. We eleve sn zero s true, nd we wsh to estlsh ettng odds of 4 to gnst sn. How mn events wll e needed for ths? In ths cse Eq. (3 gves: log log( d - log( d; 3 Thus out 3 events would e needed on the verge. However, f one s luck, one mght not need so mn events. Consder the etreme cse of ust one event wth ; R would then e nfnte nd ths one sngle event would e comlete roof n tself tht the K + s sn zero. The fluctuton (RMS sred of log(l for gven s: ( log R log R f log f d A f A B log f f A B f Ad

41 CLS 8/5 Aend II: Dstruton of the Lest-Squres Sum We shll defne: the vector ote tht: Z nd the mtr F s f ( H F T F Eq. (7, Z F H Eq. (8 nd (9 (33 Then Z F H - (34 S M [( ( ] Z F + F where the unstrred s used for. S M f ( + S S + T T T ( Z F F ( + ( F F ( T T T T T T ( Z F F F ( + ( Z F H H H H ( H F Z H H usng Eq. (34. The second term on the rght s zero ecuse of Eq. (33. S S T T T T T ( Z F F F H H H ( F Z F F ( Q S Z Z Z Z where T T F Z nd Q F H F (35 T ote tht: Q T T T ( F H F ( F H F ( F H F Q If q s n egenvlue of Q, t must equl q, n egenvlue of Q. Thus q or. The trce of Q s: TrQ T FH c F HcH c TrI c,, c, c M Snce the trce of mtr s nvrnt under untr trnsformton, the trce lws equls the sum of the egenvlues of the mtr. Therefore M of the egenvlues of Q re one, nd

42 CLS 8/5 ( - M re zero. Let U e the untr mtr whch dgonlzes Q (nd lso ( - Q. Accordng to Eq. (35, S η U ( - Q U - T η, where η (Z - Z U - S m η where m re the egenvlues of ( - Q. S Thus: M η snce the M nonzero egenvlues of Q cncel out M of the egenvlues of. dp( S S e d ( M η where S s the squre of the rdus vector n ( - M-dmensonl sce. B defnton (see Secton 6 ths s the χ dstruton wth ( - M degrees of freedom

43 CLS 8/5 Aend III: Lest-Squres wth Errors n Both Vrles Eerments n hscs desgned to determne rmeters n the functonl reltonshetween qunttes nd nvolve seres of mesurements of nd the corresondng. n mn csess not onl re there mesurement errors δ for ech, ut lso mesurement errors δ for ech. Most hscsts tret the rolem s f ll the δ usng the stndrd lest squres method. Such rocedure loses ccurc n the determnton of the unknown rmeters contned n the functon f ( nd t gves estmtes of errorswhch re smller thn the true errors. The stndrd lest squres method of Secton 5 should e used onl when ll the δ << δ. Otherwse one must relce the weghtng fctors n Eq. (4 wth ( δ where: ( δ ( δ f δ + (36 Eq. (4 then ecomes S n f ( δ A roof s gven n Reference [8]. We see tht the stndrd lest squres comuter rogrms m stll e used. In the cse where + one m use wht re clled lner regresson rogrms, nd where s olnoml n one m use multle olnoml regresson rogrms. f The usul rocedure s to guess strtng vlues for nd then solve for the rmeters f usng Eq. (3 wth relced δ. The new cn e evluted nd the rocedure reeted. Usull onl two tertons re necessr. The effectve vrnce f method s ect n the lmt tht s constnt over the regon δ. Ths mens t s lws ect for lner regressons [9]. [8] J. Orer, Lest Squres when Both Vrles Hve Uncertntes, Amer. Jour. Phs., Oct. 98. [9] Some sttstcs ooks wrtten secfcll for hscsts re: H. D. Young, Sttstcl Tretment of Eermentl Dt, McGrw-Hll Book Co., ew York,

44 CLS 8/5 Aend IV: umercl Methods for Mmum Lkelhood nd Lest-Squres Solutons In mn cses the lkelhood functon s not nltcl or else, f nltcl, the rocedure for fndng the nd ther errors s too cumersome nd tme consumng comred to numercl methods usng modern comuters. For resons of clrt we shll frst dscuss n neffcent, cumersome method clled the grd method. After such n ntroducton we shll e equed to go on to more effcent nd rctcl method clled the method of steeest descent. The grd method (w 3/ (w (w / w (w / - w Fg. 6. Contours of fed w enclosng the mmum lkelhood soluton w. Fg. 7. A oor sttstcs cse of Fg. 6. If there re M rmeters,..., M to e determned one could n rncle m out fne grd n M-dmensonl sce evlutng w ( (or S ( t ech ont. The mmum vlue otned for s the mmum lkelhood soluton w. One could then m out contour surfces of w w, w ( w, etc. Ths s llustrted for M n Fg. 6. In the cse of good sttstcs the contours would e smll ellsods. Fg. 7 llustrtes cse of oor sttstcs. P. R. Bevngton, Dt Reducton nd Error Anlss for the Phscl Scences, McGrw- Hll Book Co., ew York, 969 W. T. Ede, D. Drrd, F. E. Jmes, M. Roos, B. Sdoulet, Sttstcl Methods n Eermentl Phscs, orth Hollnd Pulshng Co., Amsterdm-London, 97 S. Brndt, Sttstcl nd Comuttonl Methods n Dt Anlss, secon edton, Elsever orth-hollnd Inc., ew York, 976 S. L. Meer, Dt Anlss for Scentsts nd Engneers, John Wle nd Sons, ew York,

45 CLS 8/5 Here t s etter to resent the S + surfce thn to + tr to quote errors on. If one s to quote errors t should e n the form < where w contour surfce (or the ( < + nd re the etreme ecursons the surfce mkes n (see Fg. 7. It could e serous mstke to quote + or s the errors n. w In the cse of good sttstcs the second dervtves H could e found numercll n the regon ner w. The errors n the s re then found nvertng the H-mtr to otn the error mtr for ;. e.: ( ( (H - The second dervtves cn e found numercll usng: [ w( +, + + w(, w( +, w( + ] w, In the cse of lest squres use H S So fr we hve for the ske of smlct tlked n terms of evlutng w ( over fne grd n M-dmensonl sce. In most cses ths would e much too tme consumng. A rther etensve methodolog hs een develoed for fndng mm or mnm numercll. In ths end we shll outlne ust one such roch clled the method of steeest descent. We S. (Ths s the sme s fndng shll show how to fnd the lest squres mnmum of ( mmum n w (. Method of Steeest Descent At frst thought one mght e temted to vr (keeng the other s fed untl mnmum s found. Then vr (keeng the others fed untl new mnmum s found, nd so on. Ths s llustrted n Fg. 8 where M nd the errors re strongl correlted. But n Fg. 8 mn trls re needed. Ths stewse rocedure does converge, ut n the cse of Fg. 8, much too slowl. In the method of steees descent one moves gnst the grdent n -sce: S S S ˆ ˆ So we chnge ll the s smultneousl n the rto: - 4 -

46 CLS 8/5 S S : S : 3 :... Strtng ont ( ( Strtng ont ( (4 (3 ( (3 (S + (S + S S Fg. 8. Contours of constnt S vs. nd. Stewse serch for the mnmum. Fg. 9. Sme s Fg. 8, ut usng the method of steeest descent. In order to fnd the mnmum llong ths lne n -sce one should use n effcent ste sze. S s vres qudrtcll from the mnmum oston s An effectve method s to ssume ( where s s the dstnce long ths lne. Then the ste sze to the mnmum s: s s 3S 4S + S s + S S + S 3 3 where S, S, nd 3 S s long s wth ste sze S strtng from s,. e., s s + s, s3 s + s. One or two tertons usng the ove formul wll rech the mnmum long s shown s ont ( n Fg. 9. The net reetton of the ove rocedure tkes us to ont (3 n Fg. 9. It s cler comrng Fg. 9 wth Fg. 8 tht the method of steeest descent requres much S thn does the one vrle t tme method. fewer comuter evlutons of ( Lest Squres wth Constrnts S re equll sced evlutons of ( In some rolems the ossle vlues of the re restrcted susdr constrnt reltons. For emle, consder n elstc sctterng event n ule chmer where the mesurements re trck coordntes nd the re trck drectons nd moment. However, the comntons of tht re hscll ossle re restrcted energmomentum conservton. The most common w of hndlng ths stuton s to use the 4 S. Then S s mnmzed wth resect to constrnt equtons to elmnte 4 of the s n ( the remnng s. In ths emle there would e (9 4 5 ndeendent s: two for orentton of the sctterng lne, one for drecton of ncomng trck n ths lne, nd one for sctterng - 4 -

47 CLS 8/5 ngle. There could lso e constrnt reltons mong the mesurle qunttes. In ether cse, f the method of susttuton s too cumersome, one cn use the method of Lgrnge multlers. In some cses the constrnng reltons re nequltes rther thn equtons. For emle, suose t s known tht must e ostve quntt. Then one could defne new set of s where (,, etc. ow f S ( s mnmzed no non-hscl vlues wll e used n the serch of the mnmum

48 CLS 8/5 Aend V: Cumultve Gussn nd Ch-Squred Dstrutons The e.: χ confdence lmt s the rolt of Ch-Squred eceedng the oserved vlue;. P χ CL ( χ dχ Where P for degrees of freedom s gven Eq. (3 [].. n D Confdence Level C.L... For n D CL π > 3, e d. wth χ n D c Fg (. χ confdence level vs. χ for n D degrees of freedom ( χ >. Gussn Confdence Lmts Let χ. Then for n D, [] Fg. s rernted from: Rev. Mod. Phs. 5, o., Prt, Arl 98 (ge

49 CLS 8/5 dp d ( e e Γ π Thus CL for n s twce the re under sngle Gussn tl. D For emle the rolt of gettng n D curve for d χ 4 hs vlue of CL.46. Ths mens tht the s 4.6% for Gussn dstruton n D Confdence Level C.L c Fg (. χ confdence level vs. χ for n D degrees of freedom ( χ <