An ad-hoc modified Likelihood Function Applied to Optimization of Data Analysis in Atomic Spectroscopy

A n “ad - hoc” modified Likelihood Function Applied to Optimization of Data Analysis in Atomic Spectroscopy. *Leonardo Bennu n * Applied Physics Laboratory, Physics Department, University of Concepci on; Casilla 160 - C, Concepcion, Ch ile. E-ma il: lbennun@u dec.cl (L. Bennu n) Runnin g title: “ A d — hoc ” Likelih ood Function for opti mization of spe ctroscopic resul ts. Keywords: “ Ad — hoc ” Likelihood Function ; Optimi zation of Statistical Results; Improving accuracy and un certainty; M etrological traceabilit y; Least Sq uares M ethods; Spectral Anal ysis ; Evaluati on of Repeatabili ty. Abstract: In this paper we p ropose an “ ad — hoc ” construction of the Lik elihood Functi on, in order to d evelop a data analysi s p rocedure, to be appl ied in atomic and n uclear spectral anal ysis. The cl assical Li kelihood Function was modi fied taking into account the u nderlying statistics of the ph enomena studi ed , by the inspection of th e residues of th e fitting, which should behave wit h spe cific statisti cal propertie s. Thi s n ew f ormulati on was anal yti cally developed, but the sough t parameter should be evaluated numeri cally, since it cannot be obtained as a function of ea ch one o f the independent vari ables. For th is simpl e numerical evaluation, along wi th the acquired data, we also should process many sets of external data, with specific properties — This new data should be u ncorrel ated with the acquired sign al. Th e devel oped statistical method was e valuated usi ng computer simulated spectra. The num erical estimations of the calcul ated parameter appl ying this method, indicate an improvement o ver accuracy and precision, bei ng one order of magnitude better than those produ ced by l east squares approaches . W e still have to evaluate the improvement produced by thi s met hod over Detection and Quan tit ation Limit s, in TXRF spectral analysi s. INTRODUCTION In atomi c and nucl ear spectroscopi es, al most al l of th e methods for spe ctral an alysis are based on t he least squares algori thms. These data p rocessing an d in terpretatio n techniqu es ar e usual ly appli ed i ndistin ctly to l inear or non — li near physical systems. The great succ ess of the application o f this method i s based ess entially on a deep agreement between the unde rlying physi cal phenomena studi ed and th e ap plied math emati cal theory. In m ost of the atomi c and nucl ear spect roscopi c techni ques, i n order to obtain the sample/sy stem characte rization, we fo u nd a comm on sequ ence of e vents , wh ich can be detail ed as fol lows : a ) a generation of ener getic p articles requi red as an exci tation source, b ) the de-exci tation process of the samp le , c ) the detecti on (u sually trough a solid stat e det ector) and d ) an acqui siti on process (typi cally th e el ectronic chai ns are composed by a preamplifier, amplifi er, and a mu ltichannel analyzer). Th e intri nsic fluctuati ons of the e x citati on an d d e-excitati on of the sampl e, being discrete event s, are ruled by the Po i sson´s Distri bution like all of the atomic or nuclear interactions . Moreover, i n the cl assical texts of Probabili ty, t he radi oactive decay an d nuclear d ecay reactions are used a s iconic examples of the Poisson´s Stati stics. Thi s probability also rules the cha racteristi c backgrounds proper of the matrix of th e sample. At this stage, we should menti on that the Poisson´s Di stribu tion, when it is applied to a large number of events (usually, n >30) is very w ell descr i bed by a Norma l D i stribution. So , t he joint probability th at describes al l kind or sequ ence of atomi c or nuclear events i n the sampl e, is a Normal Distri bution. The Monte Carlo meth ods app ly th is property in order t o i nfer the ave rage b ehavior of the parti cles in th e physi cal system from the average behavi or of simulat ed particles.[ 1 ] At each one of the rest of the steps that follow the el ectrical si gnal leaving the detector until it is processed in the multi channel an alyzer , it is affected by characteri stics fluctuati ons, li ke temperature an d gai n voltage va riati ons , e l ectrical environmental noise, small systematics errors, etc. Again , by applyin g the Central Limit theorem, we can assu re that the Di stribution that rul e the acqui red atomi c or nuclea r si gnal, at each channel in th e multichann el analyzer, w ould be a Normal Distrib ution. The Central Limit theorem is i mplicitly applied in the data p rocessin g of a large number of compl ex s yst ems, whi ch are stud ied wi th the l east squares approach and many of th eir multiple variati ons.[ 2 ] These methodologi es are applied with many strategies in computational ma thematics , i n order to optimi ze the resul ts in scientific and engineering applications. Some of the a reas of appl ications are: image and video processing , medical treatments, etc.[ 3 ] In the specific case of spectral analysi s, the application of the Maximum Likelihood formulati on i s almost perfectl y suited , si nce it is constructed consi dering a Normal Distribution at each channel . But, a que stion remains : How go od i s the qual ity of the results obtai ned from a Maxi mum Li kelihood est ima tion for a g iven paramete r? What th e Likelihood Fun ction i s computi ng i s h ow li kely the measured data i s to have com e from the distribu tion assuming a particular val ue for the hidd en parameter; the mo re likely this i s, the closer one woul d think t hat th is particul ar choice for hi dden p arameter is to the t rue value. So , i n atomic spectroscopi es, the r esults obtai ned from least squares algori th ms should be the best, and no method of improvement of the results could be proposed. M oreover, the pr operties of t he obt ained resul ts were larg ely stu died. The evaluated p arameters from th e least squ ares formul ation are unbiase d ( in the li mit of infinite measurements) and have mini mum variance among all unbiased linear estimators. This means that the estimates “get us as close t o t he t rue un known parameter values as w e can get ” . For these reasons, a n improvement on the qual ity o f the results over t hose obtai ned by the l east squares meth od seems t o b e unreali stic. Moreover, these resul ts are c onsidered as the limi t of hi ghest quality, to whi ch tend the results, for instance, obtai ned from the Neural Netw ork approach [ 4 , 5 ] when they are applied to data an alysi s in atomic or nuclear spe ctroscopies, an d relat ed systems. However , in another w ork [ 6 ] we devised a n ew smoothin g method wh ich was applied to simulated spectroscopic data , produ cing result s with better accuracy th an those obtain ed from the least squares approach es. In thi s paper, we propos e a modi fication in the constructi on of the Li kelihood function, which leads t o a remarkable i mprovement on the qu ality of the result s provided by the least squares ap proach es . Thi s modi ficati on was made takin g i nto account the underlying statisti cs of the ph enomena studi ed , by the i nspection of th e resi dues of th e fitting, which shoul d b ehave wi th specific stati stical properties. Thi s new formu lation was analyti cally developed, but the cal cul ated p arameter shou ld be evaluated n umericall y, since i t cannot be obtained as a fun ction of each o n e of th e independen t variabl es. For the requi red nu merical evaluat ion, al ong wi th the a cquired spectrum, we should process many sets of external d ata wi th specifi c properties. Thi s arbitrary t erm is a random sequence of data wh ich i s uncorrelated wi th the acquired sig nal. It should be ruled by a Gaussian di stribution, havi ng mean val ue zero and standard deviati on ∆ =1. This stati stical method was evalu ated u sin g computer simul ated spectra . The numerical estimations of th e calcul ated parameter appl ying this method, in dicate an improvement over accuracy an d p recision , one order of magni tude better th an those produced by the l east squares appr oaches. We still have to evaluate the imp rovement produced by thi s method over Detection and Quan tit ation Limit s, in TXRF spectral analysi s. THEORETICAL A Maximum Li kelihood esti mate for some hid den paramete r γ (o r parameters, pl ural) of some pr obability distribution i s a numb er   computed from an Independent and Identically Distributed sample (IID) M 1 , ..., M n from the given di stributi on that maxi miz es somethin g called the “ Likelihood F u nction”. Let´s sup pose that th is distri bution i s governed by a probabili ty density fun ction ( pdf ) G ( X ; γ 1 , ..., γ k ) , where th e γ i ’s are all hidden parameters. The Likelihood Functi on associated to th is sample is:  󰇛         󰇜    󰇛           󰇜     󰇛󰇜 Note that in al l cases t he estimated val ues are represented by English letters while parameter val ues are represented b y Greek lette rs. If the distribu tion is N( µ ,σ 2 ) , the Likeli hood Functi on is:                    󰇛 󰇜  󰇭        󰇛      󰇜               󰇛      󰇜     󰇮  󰇛 󰇜 where the symb ol “ ^ ” over the va riabl es µ i and σ i 2 indi cates that they are estimators o f their real valu es. In atomi c spectros copies (like TXRF, µSR-XRF , PIXE, e tc.) the    values co u ld be linearly related with an specifi c known function, that i s,      . The fun ction F can be understood as a parti cular per fe ctly defined signal , d irectly linked to the presence of a given element i n the sample.[ 7 , 8 ] The Fi sequence take speci fic values at each channel i ; and the α valu e is l inearly rel ated mai nly wi th the ab undance of th is el ement, and others fundamental parameters .[ 9 ] In thi s case, the Maximu m Likelih ood yields [ 10 ]:     󰇛       󰇜           󰇧     󰇨     󰇛󰇜 At Eq. (3) t he m i values represent t he discret e data acqu ired wi th t he multi channel analyzer; and th e σ i values ar e thei r standard devi ation, at each channel i . In this case, t he total number of chan nels analyzed is n . The mo st p robable   value wi th its c onfidence interval can be calcul ated from Eq. (3) as [ 10 ]:                                       󰇛󰇜 The same resul t is obtained applying least squares mini mization to this heteroscedasti c system. As was discuss ed in t he Introduction, i n at omic and n uclear spectrosc opi es, i t is advisable to mention that the results at Eq. (4 ) are obtained from an adequate representation of th e studied physical system. Each one of t he underlying processes are indeed well described by the probabili ties appli ed (Normal Distri butions) in the devel oped model. In th is sense, t he val ues obtai ned at Eq. (4) shoul d be th e best, and no method of improvement o f the r esults could be propos ed. If we ask: C ould the quality (accura cy and uncertainty) of a measurem ent be imp roved beyond that the Maximum Likeli hood Criterion establ ishes in atomi c spectroscopy ? Being these sp ectroscopi es wel l described by the proposed stati stical model , the answer shoul d be negative. But, let´s in spect the residues of the adjustmen t, which also shoul d behave with specific statistical pro pe rties. On ce the α param eter is evaluated, the d ifference         , shoul d on the a verage: i ) have equal nu mber/quanti ty of p ositi ve and negative values , ii ) si nce it i s ruled by a Poi sson stati stics, i ts uncertaint y at each channel i should be:       . Taking into account these elements we can propose an i mproved versi on of t he li kelihood functi on Ŧ [ 11 , 12 , 13 , 14 ] whi ch i ncludes the stat isti cal properties of the studied system , as:                            󰇧     󰇨     󰇛󰇜 At E q . (5) t he t erm   i s an arbitrary random sequence of n dat um , un correlat ed with the acqui red sign al. This data i s ruled by a Gaussian distribu tion, h aving mean valu e zero and stan dard d eviati on ∆ =1 . Thi s external dat a is processed simultaneousl y wi th the acquired sign al. Now we derivate the Eq. (5) with respect to α , in order to obtain the most probable α value, which maxi mize L .                                                                            󰇛󰇜 The α parameter canno t be analytically evaluated at Eq. (6), since it cannot be obtained as a function of each one of the i ndependent va riables. For the required/obli gatory nu merical eval uation, i ) an i nitial α value must b e c alculat ed, whi ch can be obt ained fr om Eq. (4) . Later, i t shoul d be refi ned nu merically ; an d ii ) an arbitrary sequence of data    should be incl uded, wit h the properties requi red at Eq. (5). In order to obtai n the numeri cal evaluation of the parameter α we apply to Eq. (6) a prop erty o f th e spectroscopic measur ements, where the standard d eviation at each channel i s   󰇛  󰇜     . Then, we can wri te:                                                        󰇛󰇜 The standard error pr opagati on form ula is required in order t o evaluate the uncertainty of the parameter α . Thi s formula i s expressed as: [ Ŧ ] The re are ma ny deno minations about the “ad hoc ” modifications ov er the likel ihood function. It is not clear in which ca tegory this proposition should b e included.    󰇛   󰇜   󰇧  󰇛   󰇜   󰇨     󰇛   󰇜  󰇛󰇜 Being 󰇛  󰇜 the uncertainty of the fun ction  󰇛   󰇜 , and    the uncertainty of every one of the k in dependent i nput variables. The Eq. (8 ) is implicitly appli ed over Eq. (6) . Taking int o account that          , th e uncertainty of th e parameter α , is:                                                                                                                     󰇛󰇜 The α parameter previously obtained at Eq. (7) is required f or th e eval uati on of Eq. (9 ). We appli ed this method over c omputer simul ated spectra. The p roposed method produces resul ts wi th a noto rious enhancem ent on th e quali ty of th e accuracy an d precision of the results, compa red with those provided by the l east square s algorith ms. APPLICATION EX AMPLES This method was appl ied to simul ated data in order to evaluate its ch aracterist i cs and the improvement obtained ove r the quali ty of the results. We appli ed t his method over a n a rbitrary sinusoidal function  󰇛  󰇜   󰇛 󰇜   . I n this particul ar case, the functi on is defined as:  󰇛  󰇜    󰇛     󰇜    󰇛  󰇜 At each channel i , i n a simul ated spectros copic acqu isi tion the “measu red” mi signal i s obtain ed from the Fi functi on affected by fluctu ations rul ed by t he P oisson´s statistics, as:              󰇛  󰇜 At Eq.(11) a p articul ar Gaussian random seq uence ( bgi ) 1 is required, h aving mean value zero an d standard deviati on ∆=1 . The Fi function and one of its possible “measured” mi sig nals are shown in Fig. (1 ). 0 20 40 60 80 100 10 20 30 40 50 60 Counts Channel (arbitrary units, usu ally energy) Fi mi Figure 1. A sinusoidal function (  , black points) and one of its simulated spectroscopic results ( mi , red poin ts) obtained by an atomic sp ectroscopic te chnique . This simulated measurement is affected by fluctu ations ruled by t he Pois son´s statisti cs. The random s equence ( bgi ) 1 u sed i n order to o btain th e mi data i s shown i n the Fig. (2), wh ere al so a second sequence ( bgi ) 2 with the sam e prop erties is included. Bo t h sets of data coul d be used i n order to evalu ate the Eq.(7). 0 20 40 60 80 100 -3 -2 -1 0 1 2 3 background fluctuations Channel (arbitrary units, usually energy) (bgi)1 (bgi)2 Figure 2. Two sequ ences of arbitrary bg data. The first set (( bg ) 1 , red points) was used to construct the “measured” sequence mi , shown in the Fig.(1 ). This sequence can be applied in Eq.(7). Th e second set (( bg ) 2 , blue points) or any oth er Gaussian random sequence (with mean val ue zero and standard deviat ion ∆=1 ) also can be used in the Eq.(7) in order to evaluate the α parameter. In the Fi g. (1) the mi data i s obtai ned from th e F function wh en i t is affected by a particular realiz ation of the statisti cal fluctuations ( bgi ) 1 , so in this case the α paramet er is strictly 1. In order t o check th e prop er devel opment of the Eq.(7), w e cal culated t he α parameter from the m i data shown in th e Fig.(1). I n this particular case the ( bgi ) 1 set of data i s used tw i ce, first in order to c onstruct the mi data from the F function, and second as the bgi s et of data requi red in Eq.(7). An i nitial guess of th e alph a value can be obtained ap plying the Eq. (4) to th is m i data. In thi s case w e get : α       . L ater we refine this value d etermining the most appropriate number which makes zero th e second membe r of the Eq.(7 ). The num erical eval uation of the pa ramet er α i s shown in Fig. (3). 0,98 0,99 1,00 1,01 1,02 -80 -60 -40 -20 0 20 40 60 0,990 0,995 1,000 1,005 1,010 -30 -20 -10 0 10 20 30 Second Member Eq.(7)  parameter Numerical evaluation of the most appropiate  parameter Figure 3. Nu merical eval uation of the m ost appropriate α val ue, wh ich makes zero th e second memb er of Eq. (7 ). In this id eal ca se, a) th e exact α value is strictly 1 and b) the bg set of dat a is used t wice, first t o construct the mi data, and second as t e rm required in Eq.(7 ). A zoom of th e region of i nterest is in cluded, wh ere is observed that the alpha value is r ecovered with high accuracy . As can be seen from Fi g.(3) , the al pha value is recovered wi th h igh accuracy from the dev eloped formulati on . The very sm all di fference obtained (~0.01%) can be attribut ed to numerical errors (r ound-off e rror ). This is a cha racteristi c result , wi th independency of the i ntensity of the α param eter , if the bg set of data is simult aneously used in order to p roduce the mi data and as a t erm in th e Eq.(7). But i n a real measurem ent w e d on´t know which is the random noise th at aff ects the pur e si gnal. In this case, w e shoul d propose an other set of bg d ata i n order to evaluate Eq.(7) , as is shown i n the Fig.(2 ) (bl ue point s). For the cas e bein g anal yzed the al pha parameter obtai ned wit h this new bg dat a is α =0.995. Evaluation of the Acc uracy and Precisi on For th e evalu ation of the accurac y and precisi on obtai ned with this meth od, i n the Tabl e (1) we show ten calcul ations of the α param eter f rom the mi data shown in Fig.(1). In each case, w e require a di fferent bg s et of data i n order to eval uate the Eq.(7) . From the o b tained α values we can estimate the quality o f the result s o btai ned with this method. Real Value a) Eq.(4) b) Eq.(7) #Evaluati on α ∆ α α ∆ α α 1 1 0 0.985 0.02 0.995 2 2 0.987 1 3 0.9992 4 1.0157 5 1.0058 6 1.0215 7 1.0105 8 0.9895 9 1.0017 10 1.0096 Table 1. T en evaluatio ns of the α p arameter from the same mi dat a. Case a) Evaluation from a p reviously develop ed least squares method , Eq.(4) . b) Results obtained from th e prop osed method , Eq.(7 ); in each ca se a dif ferent realization of random noise is required as the bg term. The resul ts at Table 1 are also sh own at Fig.(3), where is observed that usually this method produc es better resul ts than the least squares appr oach. 1 2 3 4 5 6 7 8 9 10 0,985 0,990 0,995 1,000 1,005 1,010 1,015 1,020 1,025 Real Value Least Squares Evaluation Calculated  parameter Evaluation Number Calculations made with this method Figure 3. Evaluations o f the α parameter f rom the same mi data. a) Real value of the α param eter (red line); b) Evaluation from a least squ ares approach (blue line), and c) Ten results o b tained from th e pr oposed meth od (bl ack lines) , f or each case a di fferent realization of random noise is r equired as the bg term, in ord er to evaluate the Eq.(7). From the same mi data we can obtai n a l arge n umber of ev aluati ons. For each evaluation, a new seque nce of bg data is requi red. In this case the a veraged ten results in Fig.(3 ) produce the val ue ,        . We observe that the qu ality of the accuracy and precisi on in the results obtai ned with this method, are i mproved one order of magnitude, compared with those produced by t he l east squares algori thms. Sin ce more eval uations can be simply obtained, the quality of the resul ts can be imp roved easil y (in thi s ideal system). CONCLUSIONS In data spectral anal ysis, th e applicati on of the Max imu m Li kelihood formulati on i s almost perfectl y suited, si nce it i s constructed b ased on a d eep agreem ent between the underlying ph ysical phenomena studi ed and the appli ed mathematical th eory. In thi s sense, th e valu es obtai ned from the classi cal l east squ ares approach shou ld be the best, and no m ethod of i mprovement of th e results could be proposed. Moreover, the properties of the obtain ed resul ts wer e largel y stud ied : i ) The results from th e l east squares formul ation are unbi ased ( in the limi t of i nfinite measur ements) an d ii ) they have mi nimum vari ance among all u nbiased li near estimat ors. Thi s means that the estimates “get us as close t o the true u nknown p arameter values as we can get”. For these r easons, a n imp rovement on the qualit y of the results over those o btained by least squares approaches seems to be unreal istic. However, in another work w e developed a new s mo othin g method [6] wh ich was applied to si mulated spectroscopi c data, produ cing resul ts with better accuracy than those obtained from th e least squ ares approach . In thi s pap er, w e prop ose a modi fication in the c onstruction of t he Likelihood function, which leads to a rema rkable i mprovement on t he qual ity of the result s. This modifi cation was mad e taking i nto account the underlying statisti cs of the p henomena studied , by the i nspection of the residues o f the f itting, wh ich shoul d behave with specific statistical properti es. Thi s new formulati on was anal ytically developed, but the cal culated parameter should be e v aluat ed numerically, since it cannot be obtained as a function of each one of the in dependent vari ables. For the requi red nu merical evaluati on, we require: i ) the sough t signal F , to b e evaluat ed from the acqui red spectrum, should be accurately known, ii ) along with the acqui red sp ectrum, we sh ould process many sets o f external d ata with specific properti es. Thi s arbi trary term is a random sequence of d ata which is uncorrelated wi th the acqu ired signal . It should be ruled b y a Gaussian distributi on, having mean value zero and standa rd deviati on ∆= 1. As was sho wn, the results obtained fr om Ma ximum Li kelihood can be improved, but a greater n umber o f data sh ould be handled . I n t his method, if we made n evaluations, the total numb er of data process ed is th e n ti mes greater t han the number of the a cquired sig nal – this external data is artificially in sert ed i n the nu meric proc edure . But, i f we only process th e acqui red data , the quali ty of t he results from least squares approach cannot b e enhanced. The developed stati stical method was evaluated usi ng computer si mulated spectra . The nu merical esti mations of the calcul ated parameter appl ying this method, indicate an imp rovement over accura cy and precisi on, one order of mag nitude better than th ose produced by th e common least squ ares appr oaches (see Table 1). Si nce m ore evaluations can be simp ly obtained, the qual ity of the r esults can be i mprov ed e a sil y (in this ideal system). I n spectral analysi s, w e still have to eval uate th e improvem ent produc ed by this method i n conjuncti on wi th a n ew smoothing p rocedure [6 ] , ov er Detection and Quantit ation Limits, when they are applied over real experimental results. These kind of evaluations are formally required i n al l spectroscopic techniqu es, i n ord er to establ ish their metr ological traceability, definin g carefull y their operational procedures. 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