Linear Calibration Is It so Simple?

Transcription

1 ecto: Chemstr Lear Calbrato Is It so mple? Daa Arsova, ofa Babaova, Petko Madjukov Departmet of Chemstr, outh-west Uverst Neoft lsk, 66 Iva Mhajlov tr., 700 Blagoevgrad, Bulgara Abstract. Calbrato procedure s a mportat part of strumetal aalss. Usuall t s ot the major ucertat source whole aaltcal procedure. However, mproper calbrato mght cause a sgfcat bas of the aaltcal results from the real (certfed) value. tadard Gaussa lear regresso s the most frequetl used mathematcal approach for estmato of calbrato fucto parameters. I the preset artcle are dscussed some ot qute popular, but hghl recommeded certa cases methods for parameter estmato, such as: weghted regresso, orthogoal regresso, robust regresso, bracketg calbrato etc. ome useful appromatos are also preseted. pecal atteto s pad to the statstcal crtera whch to be used for selecto of proper calbrato model. tadard UV-VI spectrometrc procedure for determato of phosphates water was used as a practcal eample. everal dfferet approaches for estmato of the cotrbuto of calbrato to the geeral ucertat of the aaltcal result are preseted ad compared.. INTODUCTION: The aaltcal methods ca be classfed two geeral groups: absolute ad relatve methods[-3]. Most of the classcal aaltcal methods (e.g. varous gravmetrc ad volumetrc methods) are absolute. The are based o smple measuremet of quatt - mass of the sample or reaget volume ad subsequet calculatos based o fudametal relatos. Most of the strumetal methods for aalss are relatve. I such case the relato betwee aalte cotet ad drectl measured aaltcal sgal s ether complcate, or dfferet from case to case, or depedet o factors whch are mpossble to cotrol. uch methods requred calbrato. Accordg to the offcal defto calbrato s: et of operatos that establsh uder specfed codtos the relatoshp betwee values of quattes dcated b the measurg strumet or measurg sstem, or values represeted b a materal measure or referece materals, ad the correspodg values realzed b stadards. [4]. More smpl sad the calbrato s a comparso betwee two quattes aalte cotet ad the aaltcal sgal. The am of ths comparso s to evaluate parameters of emprcal mathematcal fucto, allowg estmato of the aalte cotet ukow sample. 35

2 Facult of Mathematcs& Natural cece FMN 009 Accordg to the temporar metrologcal requremets, all sources of ucertat should be take to accout whe the ucertat of the fal aaltcal result s beg estmated. Tpcal ucertat sources for a relatve method are preseted graphcall Fg.. ubject of the preset work s the cotrbuto of calbrato procedure to the geeral ucertat. Due to the complet of the problem, ol the smplest lear calbrato model wll be dscussed. Fg. : Fsh boe dagram presetg tpcal sources of ucertat for relatve aaltcal method.. UE AND ABUE OF LINE EGEION The most frequetl used approach calbrato procedure s the well kow lear (Gaussa) regresso [5,6] usg calbrato fucto: () a 0 + a I most of the cases the respose for cocetrato 0 s epected to be 0. Thus, the commol used form of lear calbrato fucto s: () a where: s depedet varable (aalte cotet), s fucto of (aaltcal sgal). Equato should be used as a calbrato model after provg the statstcal sgfcace of a 0. egresso parameters a 0 ad a are evaluated usg equatos: (3) a 36

3 ecto: Chemstr 37 (4) a a 0 Most popular ad dsputable characterstc of the qualt of lear regresso s the correlato coeffcet, calculated accordg to equato: (5) However, ths calbrato approach mples fulfllg of umber of requremets, whch are ot wdel kow ad usuall ot tested (Fg. )[7-0]. uch detaled statstcal aalss of the calbrato fucto has to be doe ol durg aaltcal method developmet ad valdato procedures. Oce proved applcablt of Gaussa regresso, t ca be appled route aalss wthout further tests. Alteratve methods for lear calbrato parameters calculato wll be dscussed further. Fg. : Flow chart dagram presetg the requremets for proper applcato of stadard lear regresso procedure for calbrato. Of course, all tests have to be doe after provg leart of the calbrato fucto.

4 Facult of Mathematcs& Natural cece FMN UNCETAINTY ETIMATION Other usolved problem remas the estmato of the ucertat, troduced the fal result b the calbrato procedure. It should be oted that besdes the regresso procedure tself, the calbrato stadards preparato ad measuremets are also cotrbutors to the ucertat of calbrato. There are three dfferet appromatos most frequetl to estmate calbrato cotrbuto to the ucertat of the results. 4. APPOXIMATION - GAUIAN TATITIC. Ths approach s based o suggesto that the ol source of ucertat s the spread of the epermetal pots aroud the calbrato le. tadard devato of the regresso ( ) s a basc characterstc of the ucertat troduced b the regresso procedure. It mght be estmated accordg to the equato: (6) Where ( ˆ ) are the measured aaltcal sgals for stadard solutos; ŷ are the values calculated accordg to the calbrato fucto for correspodg cocetratos. I case of good agreemet betwee epermetal pots ad calbrato (regresso) fucto the value of s close to 0. The value s essetal for further steps ucertat estmato. Ucertat of the calbrato procedure preseted as stadard devato mght be calculated accordg to equato: (7) Where: + + a m a. ( avg ) ( avg ) s stadard devato of calculated cocetrato correspodg to aaltcal sgal ; a s the slope of the calbrato le, ad avg avg are the average values of aaltcal sgals ad cocetratos of all calbrato stadards; s the cocetrato of the stadard ; s 38

5 ecto: Chemstr umber of stadards ad m s umber of parallel measuremets of the sample resultg the aaltcal sgal. The cofdece terval ( ) of the obtaed sample cocetrato, takg to accout ol ca be calculated usg t-test of tudet wth probablt α ; ad degrees of freedom ν c ( umber of stadards; c- umber of coeffcets the calbrato equato): (8) s t( α;ν ) The specfed above cofdece terval has a specfc hperbolc shape (Fg. 3, le ) depedg o the value of ad umber of stadards. It should be oted that the best, terms of precso, s the mddle part of workg rage. 5. APPOXIMATION - IMPLIFIED UNCETAINTY POPAGATION APPOACH The stadard devato of the coeffcet a ( ) ca be calculated accordg to: a (9) a / The ucertat of sample cocetrato, preseted form of stadard devato, s calculated accordg to the ucertat propagato law [-3] usg equato: (0) a + a Ths equato correspods to oe coeffcet calbrato model (Eq. ). 6. APPOXIMATION 3 - BACKETING CALIBATION APPOXIMATION Bracketg tself s a smplfed calbrato method. The cocetrato of the sample s calculated accordg to Eq. wth slope ad tercept estmated usg ol two closes eghbor stadards. Ths method s especall 39

6 Facult of Mathematcs& Natural cece FMN 009 useful whe the aalte cotets ma samples var a arrow cocetrato rage. It s also applcable as appromato case of comple olear relato betwee cocetrato ad aaltcal sgal. Calbrato usg two stadards makes easer applcato of the ucertat propagato law ad thus, to take to accout the ucertates of cocetratos ad aaltcal sgals for stadards. Aalte cocetrato the sample s calculated accordg to the equato: 40 ( ) ( ) hgh low () low + low ( ) Where: low, low ad hgh, hgh are the correspodg cocetratos, aaltcal sgals for the lower ad hgher stadards. 4. Comparso betwee ucertat estmato models The comparso s based o stadard UV-VI method for determato of phosphates water []. 7. ANALYTICAL POCEDUE The method s based o a reacto of orthophosphate os wth acdfed soluto cotag molbdate ad atmo os, formg a atmo phosphomolbdate comple. After reacto wth ascorbc acd a blue colored molbdeum comple s formed. The quatfcato s carred out usg spectrophotometer at wavelegth 880 m. stadard solutos are used for calbrato coverg cocetrato rage from 0.05 to.50 mg/l calculated as phosphorus cotet. The ucertat of cocetrato was calculated usg ucertat propagato law ad precso data for the prmar stadard soluto ad all volumetrc devces used. Ucertates of aaltcal sgals were estmated from s depedet parallel measuremets of each stadard. esults are preseted Table. I order to compare the dfferet approaches, the calbrato ucertat was estmated for vrtual samples coverg the workg rage from absorbace 0.0 to 0. wth cremet 0.0. Ucertat of absorbace was suggested as equal to (the average of correspodg values for measured stadards). esults are preseted Fgure 3. The Appromato shows qute eve ucertat dstrbuto alog the workg rage. Most probabl the calculated values are uderestmated sce ucertat of aaltcal sgals ad cocetratos of the stadards are ot take to accout. O the other had, ths s the ol approach demostratg the tpcal for regresso hperbolc dstrbuto of the ucertat. Ths approach seems to be the best for cases wth ver good leart hgh low

7 ecto: Chemstr (low value of r ) ad low values of cocetrato ad aaltcal sgal ucertates for all stadards. Table : Cocetratos, absorbace ad correspodg ucertates for 6 stadard solutos used for calbrato. Cocetrato / mg/l P ( ) Ucertat ( ) Absorbace ) ( Ucertat ( ) tadard 0,05 0,0 0,0057 0,0004 tadard 0, 0,0 0,056 0,0003 tadard 3 0,5 0,04 0,0537 0,0005 tadard 4 0,50 0,06 0,050 0,000 tadard ,08 0,8 0,0008 tadard 6,50 0,0 0,456 0,0005 Fg. 3: Ucertat of calbrato, preseted as a half wdth of the cofdece terval, estmated usg three dfferet approaches: - Appromato accordg to Equato 7. - Appromato accordg to Equato Appromato 3 step b step ucertat propagato approach for the ucertat of calculated b Equato. 4

8 Facult of Mathematcs& Natural cece FMN 009 I case of Appromatos ad 3 there s a clear uderestmato of the ucertat the lower part of the workg rage. The tred for creasg the wdth of the cofdece terval wt creasg of cocetrato s also ver clear. Appromato 3 shows adequatel hgh ucertat values the hgher cocetrato/sgal rage of the calbrato graphcs. Ths s also the approach whch s most sestve to a fluctuatos the stadards because ol two of the stadards are used calculatos. I order to acheve most precse ucertat estmato s ecessar to modf the Appromato order to mplemet the model the ucertates of cocetratos ad aaltcal sgals of calbrato stadards. Ths wll be a subject of further works. 8. EFEENCE []. Keller, J.-M. Mermet, M. Otto, H. Wdmar (00), Aaltcal Chemstr, WILEY-VCH, Vehem. []. abovch, (005), Measuremet Errors ad Ucertates. Theor ad Practce, prger cece ad Meda, Ic., New York. [3]. alcoe, (007), Measuremet Ucertat. Approach va the Mathematcal Theor of Evdecs, prger cece ad Busess Meda, LLC, New York. [4] Iteratoal Vocabular of Basc ad Geeral Terms Metrolog (VIM) Iteratoal Orgazato for tadardzato, 003. [5] Pollard, (98), Hadbook of umercal methods of statstcs. Faces ad tatstcs, Moscow. ( ussa) [6] D. Paulso, (007), Hadbook of egresso ad Modellg, Capma & Hall/CC, Boca ato.k. [7] Dazer, L. Curre, (998) Calbrato. Part, Pure & Appled Chemstr, 70(4), [8] K. Dazer, M. Otto, L. Curre, (004) Calbrato. Part, Pure & Appled Chemstr, 76(6), 5-5. [9] Measuremet cece Chemstr ummer school 008, 3-4 August, 008, Celje, lovea. [0] Measuremet cece Chemstr ummer school 009, - Jul, 009, Blagoevgrad, Bulgara. [] Iteratoal stadard IO