Point Estimation Method of Electromagnetic Flowmeters Life Based on Randomly Censored Failure Data

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1 Sesos & Tasduces, Vol. 76, Issue 8, August 4, pp. 7- Sesos & Tasduces 4 y IFSA Pulshg, S. L. Pot Estmato Method of Electomagetc Flowmetes Lfe Based o Radomly Cesoed Falue Data Zhe Zhou, Wezh Jag, ogchao Gua, B Nu, Dezhog Ma School of Measuemet Techology ad Commucato Egeeg, Ha Uvesty of Scece ad Techology, Ha, Cha Tel.: , fax: E-mal: zhzh49@6.com Receved: 8 Jue 4 /Accepted: 3 July 4 /Pulshed: 3 August 4 Astact: Ths pape aalyzes the chaactestcs of the etepse afte-sale sevce ecods fo feld falue data, ad summazes the types of feld data. Maxmum lkelhood estmato ad the least squaes ae peseted fo the complexty ad dffculty of feld falue data pocessg, ad Mote Calo smulato s poposed. Mote Calo smulato, the elatvely smple calculato, s a effectve, whose esult s closed to that of the othe two s. Though the afte-sale sevce ecods aalyss of a specfc electomagetc flowmete etepses, ths pape llustates the effectveess of feld falue data pocessg s. Copyght 4 IFSA Pulshg, S. L. Keywods: Falue data, Relalty, Mote Calo smulato, Estmato, Poduct lfe.. Itoducto Relalty data s the ass of elalty aalyss ad evaluato. The poducts elalty data fom suvey ad collecto could seve as poduct elalty evaluato ad falue aalyss data samples. The feedg ack the esults to the desg, maufactug ad maagemet, ad we ll ultmately acheve the pupose of mpovg poduct elalty, ad povde mpotat fomato fo futue ew poduct desg [-]. The etepse poducts sevce ecod cotas a lage ume of poduct feld falue data. These data help to get the elalty level of the wokg poducts ude feld codtos, ad help to aalyze the falue phase, ad the occuece ste. These data ae mpotat fomato fo falue mode aalyss ad elalty assessmet. Some feld data ae fom the sevce cete, ad some ae fom the uses sample suvey. I ode to eflect the actual stuato moe accuately, the data should e tue ad elale, ut the pofessoalsm of the ste pesoel ecodg, jo esposlty, the alty to detfy falue modes, the desg of sample tale, omatve dusty temology, o the complete ecoded fomato, would affect the authetcty of the data feld, ad theefoe pose dffculty ad complexty to the feld data pocessg [3-4]. But to eally udestad the elalty level of wokg poducts ude feld codtos, we must fst aalyze the elalty of feld data [5-6].. Chaactestcs of Feld Falue Data Now we mostly apply laoatoy o feld data to the assessmet of elalty dcatos. The statstcal aalyss of feld data s dscussed the atcle, sce the applcato of feld data ca e fully le wth actual codtos of use, save a lot of mateals, eegy, lao ad othe costs ad show sgfcat ecoomc eefts [7]. Feld data shows the followg chaactestcs. 7

2 Sesos & Tasduces, Vol. 76, Issue 8, August 4, pp. 7- The souces of fomato ae vey complex. The same type of poduct ca e used dffeet felds, due to dffeet opeatos, dffeet evomets, ad dffeet occuece tme, easos, o postos, poducg accuate o complete ecods. The statg tme s dffeet fom each othe. Due to the dffeet tmes uts puchasg poducts, the statg momet of use dffes, ad theefoe they would ot wok smultaeously as smulato ude laoatoy codtos, eve fo the same poduct. 3 Some poducts, though wth o falue, lose fomato mdway fo vaous easos. Ude the actual codtos of use, the poduct fomato wll e lost mdway. 4 Whe the statstcal eds, some poducts ae faled, ut some ae ot. 3. Seveal Types of Feld Falue Data Poduct feld data ae egulaly cesoed complete lfe test, cludg the followg asc codtos. Fxed ume cesog data. Let poducts whch ae depedet of the same type stat tmg at the same tme t, ad temate at the momet the th poduct fals. I ths way, the poduct lfe data of uts ahead ae avalale: T( T(... T, ( ad that s cesoed data. Fxed tme cesog data. Smla to cesoed, the ume of falue at a pedetemed tme t s a adom vaale. If uts of poduct ae faled whe osevato s temated, the we could get the data T( T(... T( t, that s tme cesoed stuato. 3 Radomly cesoed data [4]. Poduct elalty data o-ste ascally elogs to the adomly cesoed data, whch s composed of teupted osevatoal data ad fault data adomly. Suppose the lfe of feld poduct s TT,,..., T, whch s depedet ad detcally dstuted o the dstuto fucto Ftθ (; o ts desty fucto f (; t θ of ts oveall T, θ s ts ukow paamete. Ad suppose the cesoed tme { L } of coespodg poduct s depedet, ad thee s the dstuto fucto { Gt (,,,..., } o desty fucto { g (, t,,..., }, assumg that the dstuto of the cesoed tme s uelated to the ukow paametes. Also suppose { T } ad { L } ae depedet. If toducg m{ T, L }(,,..., ( to show the stuato that the poduct ca e oseved, ad the etoducg, ( T L δ (,,...,, ( ( T > L we ca get "adom vaales" (o two-dmesoal adom vaale, that s (, δ (,,...,, (3 dcatg whethe the poduct s cesoed. Thus, poduct cesoed tme s a adom vaale, that s, adom cesog. 4 Douly cesoed data. Suppose the poduct lfe data o-ste TT,,..., T ae depedet ad detcally dstuted, t ( t (... t ( ae the osevatos, assumg samples avalale as pat of the tecepto data osevatoal, whch meas teceptg elated fomato the mddle peod of wokg tme, wth oth stat tme ad dow tme ukow, whch cludes the left ad ght cesoed, doule cesoed. 4. Aalyss of Feld Falue Data Pot ad teval estmato s ca e used paamete estmato of dstuto model elalty egeeg. By ths meas, the ukow sample s gve closed to the tue value of the paamete y oseved samples. Thee ae may s of pot estmates, such as the poalty plots estmate [8], least squaes [9-], ad maxmum lkelhood estmato [-]. Poalty plots estmate s smple ad easy, oe ca estmate all the paametes eeded the poalty pape, coveet fo the majoty of egeeg ad techcal pesoel to gasp, whch s wdely used. But thee ae too may huma factos dawg, the least squaes, ad othe quattatve estmato s elmate ths dawack. The pecso of maxmum lkelhood estmato s hgh, ut wth a lage amout of calculato ad complexty. The est lea uased estmato ad est lea vaat estmato must ely o specal foms, whch ae dedcated ad lage, so they ca t e used wthout the tale [3]. Besdes, least squaes estmato ft all kds of codtos, oth complete - samples ad complete samples, oth cesoed ad tme cesoed stuato. Cuetly, the Mote Calo smulato s also a popula paamete estmato [4-6]. Ths atcle gves thee s of paamete estmato: the maxmum lkelhood estmato, least squaes estmato, ad Mote Calo smulato. 4.. The Maxmum Lkelhood Estmato Method Suppose the jot desty fucto of the sample,,..., s f (; x θ, whee paamete θ s ukow, fo a gve x, we ame 8

3 Sesos & Tasduces, Vol. 76, Issue 8, August 4, pp. 7- L ( θ, f ( x; θ (4 fo the lkelhood fucto paametes. Accodgly, we call l L( θ, the Logathmc lkelhood fucto of paamete θ. If thee s statstcal quatty θ, whch ca make L( ˆ, θ max L( θ,, (5 θ we call θ as the maxmum lkelhood estmato (MLE of θ. Geeally, the commo to fd the MLE of paameteθ s to solve the lkelhood equato: l L( θ,,,,..., k, (6 θ whee k s the ume of ukow paamete θ. 4.. Least Squaes Estmato Method Assumg a lea equato as: + ts esdual sum of squaes s If we ota ths x, (7, ( Q (, (8 θ δ (9 θ (, ( ( whee s the ume of falue data, ad, s the value coespodg to scatte. Take expoetal dstuto as a example, as ( t γ F ( t e, ( afte the coveso, we ca get If l Ft ( l t μ F ( t (, x t, ad keepg a lea elatoshp, we could ota that, γ. Thus the aveage lfe expectacy of the poduct s: m + γ ( Mote Calo Smulato Method If the poduct lfe T sujects to two-paamete expoetal dstuto, ts dstuto fucto s ( t γ F T ( t P{ T t} e, (4 whee, γ ae fo the scale paamete ad posto paametes espectvely, esdes, >, γ>, t>. Takg T o tasfomato t, the dstuto fucto of the tasfomato o s F ( ( P{ x} e + <, (5 whee, γ. Ad the the lea tasfomato s: + (6 So, the dstuto fucto fo covetg s F ( y P{( + y < y} e, (7 Tucatg test (, wth fxed ume, the sample of T was T,,,...,,, theey otag the ode statstcs T ( T (... T (. Calculate the aveage ad covaace of the aove equato, E( ( cov( E( (, ( j ( + cov( (, ( j (8 Mote Calo stochastc smulato gves the calculated value. Accodg to fomula (7, assgg U F y ( y e (9 we could get to kow that U oeys ufom dstuto o the teval U(,. 9

4 Sesos & Tasduces, Vol. 76, Issue 8, August 4, pp. 7- The at-solvg l accodg to F ( t fomula (9. I ths case s gve y the expesso of U. It ca fst e compute-geeated ufomly dstuted adom sample U, sotg of U (, o the teval (,, esdes, gvg the coespodg, ad (, fally we ca fd the coespodg paamete. Mote Calo smulato algothm s as follows: Fo the gve,, ad smulato tmes N, N sets of samples may e geeated. Repeatg the followg steps a ~ would help to geeate N sets of aalog sample values { y,,,...,,,,...,... } ( j j N : a Geeatg adom vaale samples U sujected to ufom dstuto U (, ; Accodg to y l, calculate a set of U samples of adom vaales y, y,..., y ; c By sotg, ode statstcs would e otaed y(, y(,..., y(,..., y ( ; d Fxed data cesog, ota the samples { y, y,..., y }. ( j ( j ( j 5. The Example Aalyss Takg vald samples selected a ceta type of electomagetc flowmetes afte-sale sevce ecods a etepse fo aalyss, the sample data ae show Tale. No. Sales volume /sets Tale. The sample data. Sales date Rewok ume /sets Rewok Date Lfe tme /days Packet No A Example fo Maxmum Lkelhood Estmato Usg Weull++7. elalty aalyss softwae, ad maxmum lkelhood estmato fo the aove sample data aalyss, we could ota that two-paamete expoetal dstuto s the est dstuto. The dstuto of expoetal poalty s show Fg.. Fg.. Poalty gaph of expoetal dstuto. The poalty desty fucto fo the twopaamete expoetal dstuto s: f( t e.4e ( t γ.4( t 43 Accodg ths, the aveage lfe expectacy of ths type of electomagetc flowmete could e estmated as: MTBF + γ 543 (days 5.. A Example fo Least Squaes Estmato Method Accodg to fomula (8, the esdual sum of squaes s: Q (,.339 Accodg to fomula (9, the value of ad ae:.566, So, 4.5, γ 39 (days Ad the aveage lfe expectacy of the poduct s: MTBF + γ 58 (days 5.3. A Example fo Mote Calo Smulato Method Electomagetc flowmete mateace data could e vewed as esultg fom the ume fxed

5 Sesos & Tasduces, Vol. 76, Issue 8, August 4, pp. 7- cesoed pogam (, (5,, ad the samples ae T: 43, 83, 89, 95,, 3, 4, 63, 86, 4, (ut: day. Assumg that electomagetc flowmete sujects to two-paamete expoetal dstuto, wth Tale showg the esults of the thee s, we ca see that the Mote Calo smulato (N, s also a effectve, ad the esults of t s close to the othe two s of esults. Tale. Compaso of thee s esults. The maxmum lkelhood estmato Least squaes estmato Mote Calo smulato 6. Coclusos γ MTBF (days Ths pape pesets seveal effectve elalty aalyss s fo the complexty ad dffculty of feld falue data pocessg, whch ae vefed y a electomagetc flowmete sale data. The esults show that the maxmum lkelhood estmato ad least squaes estmato esults of the assessmet ae closed. Ad although the Mote Calo smulato has ts ow advatages, the esults have a lttle as, dcatg that ths assessmet stll eeds to e mpoved. Though the study of these thee assessmet s, we tally acheve that the lfetme of such electomagetc flowmete poducts sujects to two-paamete expoetal dstuto. Ths cocluso ca futhe povde the ass fo elalty lfe test ad elalty gowth test. Ackowledgemets The wok was suppoted y Natual Scece Foudato of Helogjag Povce of Cha (F35 ad Natoal Hgh-tech R&D Pogam of Cha (863 Pogam (8AA47. Refeeces []. Guofag He, Collecto ad aalyss of elalty data, Natoal Defese Idusty Pess, 995. []. u Zhao, Aalyss of elalty data, Natoal Defese Idusty Pess,. [3]. Dezhog Ma, Zhe Zhou, aoyag u, Aalyss of lfe dstuto of electomagetc flowmetes ased o falue data, Joual of Scetfc Istumet, Vol. 3, Issue 6, 9, pp [4]. Jadg Che, Maxmum Lkelhood Estmato Cosstecy of adomly cesoed Weull dstuto paametes, Chese Joual of Appled Poalty ad Statstcs, Vol. 5, Issue 3, 989, pp [5]. Chagju We, ug Zhog, Relalty aalyss of feld falue data, Hue Isttute of Techology, Vol. 8, Issue 5, 3, pp [6]. aopg Sh, Jahua Da, Baq Mu, Statstcal aalyss of elalty data ased o mateace ecods, Joual of Cha Uvesty of Scece ad Techology, Vol. 38, Issue 9, 8, pp [7]. Chagju We, ug Zhog, Relalty aalyss ad applcatos of automotve epa feld data, Hue Isttute of Techology, Vol., 5. [8]. ualog u, Repg u, Lfetme Dstuto Aalyss ad Relalty Estmato of Poducto, Developmet ad Iovato of Machey ad Electcal Poducts, Vol. 6, 7, pp [9]. aoyog Ja, Chuasheg u, Ba, The veto ad way of thkg o least squaes, Joual of Nothwest Uvesty (Natual Scece Edto, Vol. 3, 6, pp []. Che upg, Gog Qgwu, Fu Fega, Wu Su, A accuate fault locato ad ts data pocessg y least squae, Poceedgs of the Iteatoal Cofeece o 'Powe System Techology',, pp. 8-. []. Jadg Che, Pocessg of adomly cesoed data, Mathematcal Statstcs ad Maagemet, Vol. 4, 987, pp. 6-. []. Lawless J. F., Statstcal models ad s fo lfetme data, Joh Wley ad Sos,. [3]. Wemg Gu, Guagx Na, Jgxa Ma, A mpovemet of the est lea uased estmato fo chaactestc le of Weull dstuto, Joual of Mechacal Stegth, Vol., 994, pp [4]. g J, aj Hog, Mote Calo system egeeg, Natoal Defese Idusty Pess, 3. [5]. D A., Mote Calo s system elalty, 'a Jaotog Uvesty Pess, 7. [6]. a Jag, Ruyg L, Ru Kag, Ng Huag, The of etwok elalty ad avalalty smulato ased o Mote Calo, Poceedgs of the Iteatoal Cofeece o 'Qualty, Relalty, Rsk, Mateace, ad Safety Egeeg (ICQRMSE, Chegdu, Schua, 5-8 Jue, pp Copyght, Iteatoal Fequecy Seso Assocato (IFSA Pulshg, S. L. All ghts eseved. (