On a New Definition of a Stochastic-based Accuracy Concept of Data Reconciliation-Based Estimators

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1 On a New Defnton of a Stochastc-based Accuacy Concept of Data Reconclaton-Based Estmatos M. Bagajewcz Unesty of Olahoma 100 E. Boyd St., Noman OK 73019, USA Abstact Tadtonally, accuacy of an nstument s defned as the sum of the pecson and the bas. Recently, ths noton was genealzed to estmatos. Howee, the concept used a maxmum undetected bas, as well as gnoed the fequency of falues. In ths pape the defnton of accuacy s modfed to nclude expected undetected bases and the fequency. Keywods: Instumentaton Netwo Desgn, Data Reconclaton, Plant Montong. 1. Intoducton Tadtonally, accuacy of an nstument s defned as the sum of the pecson and the bas (Mlle, In a ecent pape (Bagajewcz, 2004 ths noton was genealzed to estmatos agung that the accuacy of an estmato s the sum of the pecson and the maxmum nduced bas. Ths maxmum nduced s the maxmum alue of the bas of the estmato used, that s, a esult of a cetan specfc numbe of bases n the netwo whch hae not been detected. Ths lead to a defnton of accuacy that s dependent on the numbe of bases chosen. Asde fom many othe shotcomngs of the defnton, two stand out as the most mpotant: The defnton has no tme hozon assocated to t, no states anythng about the fequency at whch each senso wll fal, o the tme t wll tae to epa t. In addton, the defnton could be moe ealstc f expected bas, nstead of maxmum bas s used. In ths pape, we eew the defntons and dscuss the esults of a Montecalo technque that can help detemne an expected alue of accuacy. 2. Bacgound Accuacy was defned fo nddual measuements as the sum of the absolute alue of the systematc eo plus the standad deaton of the mete (Mlle, Snce the bas s usually not nown, the defnton has lttle pactcal alue. Bagajewcz (2004 ntoduced a new defnton of accuacy of an estmato (o softwae accuacy defned as the sum of the maxmum undetected nduced bas plus the pecson of the estmato: â ˆ (1 = σ +

2 whee â and σˆ ae the accuacy, the maxmum undetected nduced bas and the pecson (squae oot of the estmato s aance Ŝ, espectely. In tun, the accuacy of the system can be defned n aous ways, fo example mang an aeage of all accuaces o tang the maxmum among them. Snce ths noles compang the accuacy of measuements of dffeent magntude, elate alues ae ecommended. ct T The maxmum undetected nduced bas s obtaned fom the assumpton that a patcula goss eo detecton test s used. In the case of the maxmum powe measuement test, and unde the assumpton of one goss eo beng pesent n the system ths alue s gen by: ˆ ( p, 1 ( p [( I SW s ] = Z ct Max (2 s Wss ( p whee Z s the ctcal alue fo the test at confdence leel p, S s the aance- T coaance matx of the measuements and W = A ( ASA A (A s the ncdence matx. When a lage numbe of goss eos ae pesent n the system, an optmsaton model s needed. Thus, fo each set T we obtan the maxmum nduced and undetected bas by solng the followng poblem: ˆ ( p ( T = Max ct, ( SW s ct, s s T s T s. t (3 ( p W Z W s ct, s ct s T Theefoe, consdeng all possble combnatons of bas locatons, we wte ˆ ( p, T ( p = Max ( T (4 As t was mentoned aboe, ths defnton states what the accuacy of the system s, when and f a cetan numbe of goss eos ae expected to tae place. In othe wods, t epesents the wost case scenao and does not dscuss the fequency of such scenao. We now dscuss a new defnton and how to obtan an expected alue next 3. Stochastc Based Accuacy We defne the stochastc based maxmum nduced based as the sum oe all possble bases of the expected facton of tme ( Γ n whch these bases ae pesent. ˆ = ˆ E (5 ( p ( p, [ Γ ] The fomula assumes that When eos n a cetan numbe of sensos occu they eplace othe exstng set of undetected eos.

3 Sensos wth detected eos ae epaed nstantaneously. Sensos hae the own falue fequency, whch s ndependent of what happens wth othe sensos. Fo example, the pobablty of one senso falng at tme t, when all sensos whee functonng coectly between tme zeo and tme t s Φ 1 = f ( t 1 f ( t, whee f (t s the sece elablty functon of senso f [ s ] s sensos ae not epaed. When sensos ae epaed, one can use aalablty and wte f (t = /( + µ, whee s the epa ate and µ s the falue ate. The second ssue, the epa tme, s moe poblematc because t also affects the alue of σˆ, whch becomes the esdual pecson dung that peod of tme. So, E[ Γ ] can only be estmated by dentfyng the pobablty of the state wth the fequency of the state n the case of neglgble epa tme. Howee, when epa tme s sgnfcant [ ] dffcult to estmate and thee ae no expessons aalable. E Γ s moe In addton, multple goss eos do not ase fom a smultaneous eent, but athe fom a goss eo occung and addng to an exstng set of undetected goss eos. In addton, poblem (3 assumes the wost case n whch all wll flag at fst, but t does not say what wll happen f some ae elmnated. We now defne the stochastc-based expected nduced based as the sum oe all possble bases of the expected facton of tme ( Γ n whch these bases ae pesent. ~ ( p ~ ( p, E = E E Γ (6 [ ] [ ] [ ] To undestand how the stochastc-based nduced bas (and by extenson, the stochastcbased accuacy can be calculated. Assume that a system s bas fee n the peod [0, t 1 ] and that senso fals at tme t 1. Thus, f the bas s not detected, then thee s an expected nduced bas that one can calculate as follows: ~ ( p, 1,ct E ( = I SW θ h( θ ;, ρ dθ (7 [ ] [ ],ct whee h( θ ;, ρ s the pdf of the bas θ wth mean alue and aance ρ. Note that we ntegate oe all alues of θ, but we only count absolute alues, as the accuacy defnton eques. Thus, n between t 1 and the tme of the next falue of ( p, 1 some senso t 2, the system has an accuacy gen by σˆ + E[,un det ( ]. In tun, f the bas s detected, the senso s taen out of lne fo a duaton of the epa tme R. Dung ths tme (and assumng no new falue taes place, the system has no nduced bas, but t has a lowe pecson, smply because the measuement s no longe used to pefom data econclaton. Thus, dung epa tme, the expectaton of the accuacy due to detected bases s gen by the esdual pecson σ ˆ R (. Afte a

4 peod of tme R the accuacy etuns to the nomal alue when no bases ae pesent σˆ. Thus, n the nteal [0,t 2, the accuacy s gen by [ σˆ t 1 + σˆ R( R + σˆ *(t 2- ( p, 1 t 1- R ]/t 2 when bas s detected and [ σˆ t 1 + E[,un det ( ] (t 2- t 1 ]/t 2 when bas s undetected. The expectaton s then gen by multplyng the undetected poton by the coespondng pobablty p un det (,ct =,ct h( θ ;, ρ dθ and the detected by ts complement [ ( ] 1. p un det Assume now that the bas n senso s undetected at t 1 and anothe bas n some othe senso occus at t 2, whch can be n tun detected o not detected. If t s undetected, then the expected nduced bas s gen by: ~ ( p, E[ (, ],ct 2,ct = [ I SW ] θ + [ I SW ] θ,ct,ct (9 h( θ ;, ρ h( θ ;, ρ dθ dθ whee, fo smplcty of pesentaton we hae assumed that, ct and, ct can be used as ntegaton lmts. (n ealty, the ntegaton egon s not a ectangle. We leae ths detal fo futue wo. In tun, f the eo n senso s detected, then we assume that the nduced bas emans. Qute clealy, the scenao shown s one of many, and whle one s able to obtan the expected nduced eos n each case, the poblem of calculatng the expected facton of tme n each state pessts. Thus, we esot to Montecalo smulatons to assess ths. (8 3.1 Montecalo smulatons Consde a scenao s, composed of a set of n s alues of tme (t 1, t 2,, t ns wthn the tme hozo h. Fo each tme t, one consdes a sample of one senso falng wth one of two condtons: t s bas s detected o undetected. Sensos that hae been based between t -1 and t and whee undetected at t, contnue undetected. Thus, when bas n senso s detected, fo the tme between t and t +R we wte R ~ ( p,m E [ a ] = σˆ [ ( + E ( l1,,l2,,..., lm, ] (10 whee the second tem s the expected bas due to the pesence of m -1 undetected eos. n ~ ( [ p,m,ct,ct E ( l1,,l2,,...,lm, ] =... z( [ I SW ] θ 1,,ct = 1 (11 h( θ ;, ρ dθ Fo the nteal (t +R,t +1, we wte E[ a ] ˆ E[ ˆ = σ +,un det ( l1,,l2,,...,lm, ] (12 In tun, f the eo was not detected, then we wte t +1, we wte

5 ~ ( p,m [ a ] σˆ ( + E[ ( l,l,..., l ] E (13 = 1, 2, m, The aboe fomula s ald fo l, 1, = 1,... m. Othewse, the same fomula s ~ ( p,m used, but s emoed fom ( l1,,l2,,..., lm,. To obtan an aeage accuacy of the system n the hozo h and fo the scenao s, the accuacy n each nteal o sub-nteal s multpled by the duaton of such nteal and dded by the tme hozo h. Fnally all the alues ae added to obtan the expectaton fo that scenao. The fnal accuacy s obtaned usng the aeage of all scenaos. Fnally, scenaos ae sampled the followng way. Fo each senso a set of falue tmes s obtaned by samplng the elablty functon epeatedly and assumng that sensos ae as good as new afte epa (AGAN mantenance. Of these, undetectablty s sampled usng a pdf gen by p un det ( and ts complement. 4. Example Consde the example of fgue 1. Assume flowmetes wth σ =1, 2 and 3, espectely. We also assume that the bases hae zeo mean and standad deaton ρ =2,4 and 6 espectely, falue ate of 0.025, 0.015, (1/day and epa tme of 0.5, 2 and 1 day espectely. The system s baely edundant (Only one goss eo can be detemned, and when t s flagged by the measuement test, hadwae nspecton s needed to obtan ts exact locaton; Ths s due to goss eo equalency (equalency theoy: Bagajewcz and Jang, The poblem was un wth scenaos contanng 20 eent samples. A poton S 2 S 1 of one such sample s fo example depcted able 1. Conegence s S 3 acheed ey qucly (see fgue 2 to Fgue 1. Example a alue of accuacy of (The sold lne s the aeage alue. Compaately the accuacy defned fo maxmum bas of one bas pesent s Ths hghlghts the fact that usng a maxmum expected undetected bas s too conseate 5. Dscusson and Conclusons The poblems wth an exstng defnton of accuacy hae been hghlghted and a new defnton, whch ges a moe ealstc alue has been pesented. In addton a Montecalo samplng technque was suggested to detemne the alue of the accuacy. Some shotcomngs stll eman: The expected alue of exstng undetected bases s detemned usng ectangula ntegaton egons, when t s nown these egons hae othe moe complex foms. Ths can be addessed analytcally somehow, but one can also esot to sample the bas szes as well. All ths s pat of ongong wo.

6 Table 1. Example of one scenao (Poton Tme Bas n senso Bas detected 15.6 S 1 No 43.6 S 1 No 62 S 2 Yes 90 S 2 Yes 100 S 2 Yes 115 S 1 Yes 150 S 3 Yes 160 S 1 Yes 170 S 2 No 185 S 2 No 189 S 1 Yes 193 S 1 No 208 S 2 Yes Fgue 2. Montecalo Iteatons conegence. Refeences Bagajewcz, M., 2004, On the Defnton of Softwae Accuacy n Redundant Measuement Systems. To appea. AIChE J., (aalable at Bagajewcz M. and Q. Jang. Goss Eo Modelng and Detecton n Plant Lnea Dynamc Reconclaton. Computes and Chemcal Engneeng, 22, 12, (1998. Mlle R. W. Flow Measuement Engneeng Handboo. McGaw Hll, (1996 Acnowledgements Fundng fom the US-Natonal Scence Foundaton. Gant CTSXXXXXX s acnowledged.