Process Control with Highly Left Censored Data Javier Orlando Neira Rueda a, MSC. a

Transcription

1 Process Control wth Hghly Left Censored Data Javer Orlando Nera Rueda a, MSC. a Unversdad UNIMINUTO. Departamento de Ingenera Dagonal 81C Nº 72 B 81, Bogotá, Colomba javer.nera@unmnuto.edu Andrés Carrón García b Departamento de Estadístca e Investgacón Operatva Aplcadas y Caldad. Unversdad Poltécnca de Valenca, España. PhD. Camno de Vera, s/n, edfco 7A, 46022, Valenca, Span acarron@eo.upv.es 1 Abstract.- The need to control ndustral processes, detectng changes n process parameters n order to promptly correct problems that may arse, generates a partcular area of nterest. Ths s partcularly crtcal and complex when the measured value falls below the senstvty lmts of the measurng system or below detecton lmts, causng much of ther observatons are ncomplete. Such observatons are called ncomplete observatons or left censored data. Wth a hgh level of censorshp, for example greater than 70%, the applcaton of tradtonal methods for montorng processes s not approprate. It s requred to use approprate data analyss statstcal technques, to assess the actual state of the process at any tme. Ths paper proposes a way to estmate process parameters n such cases and presents the correspondng control chart, from an algorthm that s also presented. 1. INTRODUCTION Industral processes demands each tme better measurement performance, and n some cases ths requres measurng n the lmts of equpment senstvty. When the measured quantty s very small and ts true value falls below a certan lmt of detecton s sad that ths value s n the category of left censored data [1]. Wth ths non detectable values, the person n charge of control may be confused as to how to treat these observatons usng tradtonal statstcal methods such as Shewhart Control Chart [2]. Assumng that the detecton lmt s equal to a constant C and the engneer n charge of montorng the process knows that the measured quantty X s smaller than C, but wthout knowng ts exact value, four are the alternatves usually adopted: 1. The set of values below C are taken as zero: X = 0 2. Values below C are fxed n the md of the nterval [0, C]: X = C/2. 3. The set of values below C are taken as equal to the detecton threshold: X = C 4. Values under C are gnored, and substtuted by other readngs over C

2 2 If we analyze censored data usng the frst method wll tend to "underestmate" the true value of the mean from the sample taken. Thrd and fourth methods wll tend to "overestmate" the true value of the mean. If we analyze the data usng the second method, we see that t s an attempt to take the mddle poston between methods 1 and 3. The fourth method, a part of overestmatng the mean value, smply gnores undetectable, and the result can be serous. [2]. As an example of ths stuaton, we can cte the case of montorng polluton parameters, now very commonly controlled due to envronmental protecton regulaton, and nvolvng measurement of some parameters whose legal lmt s very low, close to what standard measurement equpment can capture [3]. Ths requres ncreasng the use of statstcal technques to relably measure or estmate n such stuatons [4]. 2. STATISTICAL DATA FOR CONTROL CEV CENSORED BY THE LEFT As already mentoned, there are many processes where the control outputs are censored n a large percentage and parameter estmates are sgnfcantly based. Even n relatvely smple stuatons one has to rely heavly on statstcal methods for large samples and asymptotc propertes. In ths secton, a control chart s obtaned to determne the mean and standard devaton n a process contanng censored data. Is assumed to develop: That the measured quantty T s normally dstrbuted wth mean μ and standard devaton σ and, respectvely. The observatons are censored by the left (the formula s smlar for rght censorng a [5]). Wth left-censored data, the target graph CEV Control (Condtonal Expected Value) s to detect ncreases n the mean and / or ncreases n the standard devaton of the process. In other words, the two control charts have a sngle control lmt as dscussed later. Moreover wth left censored data s very dffcult to detect decreases n the process mean that such changes ncrease the proporton of censorshp. Smlarly f the proporton of censorshp s greater than 50%, a decrease n the dsperson process also leads to more censored observatons. Subgroups wth all censored observatons provde lttle nformaton about changes n process parameters [5] and may addtonally generate a further based estmate. In ths case, stuatons where left censored observatons, ncreases n the average of the process and ncreases the dsperson of the data obtaned are of nterest. If T s the qualty characterstc that we wll control for changes n varablty. And T can be modeled as a normal random varable wth mean μ and standard devaton σ (T~N (µ, σ)). Then T wll have a probablty densty functon σ -1 [(t-μ)/σ], survvor functon Φ[(tμ)/σ] and a typfed value Z= [(t-μ)/σ] where t s the observed value. [6] The probablty of censure for a random varable T normally dstrbuted wth mean μ and standard devaton σ censored by the left of C s descrbed as:

3 3 Eq. 1 Pc = P T C C T C P Z Where ZC s typfed pont value censorshp C, and Φ(ZC), t s the functon of Normal Dstrbuton Model Typfed at that pont [7]: Eq. 2 Thus, we can wrte: C Z c Eq. 3 Pc = P 2 Z Z c 1 c 2 T C Z e dz c 2 For example, for data normally dstrbuted N (0,1) wth a fxed level by the left censorshp C = 1 s obtaned: Eq. 4 Where, Pc s the censorshp rato. 1 0 Pc (1) 0, WEIGHTS CEV FOR CENSORED DATA TO THE LEFT The control chart proposed n ths paper, s based on replacng each censored observaton by a condtonal expected value denoted as Wc, whch we wll call "Weghts CEV". These weghts are based on CEV, the sample mean and standard devaton s plotted subgroup smlarly to tradtonal graphcs X y S. Ths condtonal expected value or weght Wc for left censored observatons s obtaned as: Eq. 5 Zc Wc E T T C Zc

4 4 Where the term ( (Z C )/Φ(Z C )) can be denoted as the role of chance V(Z C ), defned as the rato of the densty functon and the dstrbuton functon [6]: Eq. 6 Eq. 7 Eq. 8 V C Zc Zc Zc C Snce (ZC) Densty Functon Standard Normal at the pont of censorshp C: C 1 2 Zc e 2 CϵR; µϵr; σ>0 Therefore, new data used to buld the defned control chart as CEV: t, s t C w w, s t C C 2 The Control Chart CEV s to montor the average and standard devaton of the subgroups wth weghts CEV (w ). It wll be called Control Chart CEV X for averages and Control Chart CEV S for standard devaton. [5]. As already seen, the calculaton of the weghts for the censored observatons depends on the parameters μ and σ under control. The procedure for estmatng the parameters μ and σ of a process under control, you wll see later n the ntal mplementaton phase for process montorng wth left censored observatons. The dea of usng weghts CEV s based on the lkelhood functon gven by Stener & Mackay, who n turn are based on the book Lawless, J.F., 1982.

5 5 4. COMPUTING CONTROL LIMITS Calculatng control lmts requres recourse to smulaton. In Fgures 1 and 2 are provded for constructng graphs of the control lmts of the graph CEV X and S, obtaned by smulatng 1000 estmates for each level of censorshp and for each model. We used a rsk of false alarm of (type I error). [8] [9] Control lmts shown on these graphs are standardzed, so they gve the control lmt for subgroups wth sample szes (n= 3, 5, 10, 20) and Pc proporton of censorshp, assumng the process s under control wth mean zero and standard devaton equal to one. An nterpolaton between the dfferent curves to locate a boundary of a subgroup sze n dfferent dce may be used. The horzontal axes for both graphs are presented n logarthmc scale. FIGURE 1. STANDARDIZED UPPER CONTROL LIMIT (UCL X ) FOR THE GRAPH CEV X MODEL CEV IZQ Once estmated process parameters μ and σ under control are placed the control lmts UCL X and LCL S, whch are standardzed control lmts. 4.1 Control Lmts. CEV X y S Charts. The approprate control lmts for any ssue rased can be obtaned usng the followng formulas: Eq. 9. Upper control lmt for the chart CEV UCL Eq. 10. Upper control lmt for the chart CEV UCL X S X s

6 6 Where μ and σ are process parameters controlled. Table 1 shows the coeffcents for the most common case n=5. FIGURE 2 STANDARDIZED UPPER CONTROL LIMIT (UCL S ) FOR DE GRAPH CEV S MODEL CEV IZQ. TABLE 1. COEFFICIENTS FOR CALCULATING THE CONTROL LIMITS (N=5) n=5 %c coefcente coefcente LCS X LCS S 0,02 1,61 2,36 0,03 1,47 2,15 0,04 1,42 2,09 0,07 1,42 2,08 0,10 1,42 2,07 0,16 1,42 2,09 0,24 1,43 2,09 0,31 1,42 2,09 0,50 1,43 2,09 0,69 1,42 2,09 0,84 1,43 2,09 0,98 1,42 2,08 Note: Values UCL X y UCL S for dfferent sample szes and censorng proportons are detaled n Appendx B.

7 7 4.2 ntal mplementaton Step commonly called ntal mplementaton phase nvolves collectng a set of samples when the process s under control. When workng wth uncensored data suggest workng wth 100 observatons or more for the ntal mplementaton of graphcs CEV X y S. Ths restrcton ensures that the sample sze estmates of the ntal parameters of the process are accurate and reasonably good. The followng steps are applcable to the model CEV Left; maxmum lkelhood procedures for ths model are detaled n Appendx A. The ntal mplementaton procedure for establshng the control chart CEV X y S for a fxed confdence level C s: 1. Takng K subgroups, each of sze n. 2. Estmate the mean and standard devaton under control µ y σ, usng the method presented n APPENDIX A. 3. Determne the weght Wc CEV for censored observatons wth the equaton gven n paragraph CEV Pesos for the Left censored observatons, based on the estmaton of μ and σ under control, and replace all censored observatons Wc value. 4. Calculate and create control lmts usng the desgn of the s gven for graphcs CEV (X y S), plottng the averages and devatons of the subgroups. 5. Search any sgn out of control n the graph (ponts outsde the control lmts). Browse process condtons, f any subgroup runaway was collected over tme, repeat the procedure from step 2 f some subset out of control was removed from the sample. The mprecson of the estmaton algorthm when censorshp s hgh can lead to bas n the process parameters. Remember that n the estmaton procedure process varablty full sample nstead of only the dsperson wthn the subgroup as typcally done for tradtonal control charts used. As the publcaton [5], maxmum lkelhood estmates work well for large samples. The maxmum lkelhood method s teratve, generatng a computatonal effort vares consdered estmaton model used. At the end of that all observatons are censored case; the maxmum lkelhood estmate s not possble. 5. EXAMPLE In trals of characterzaton of geotextles, specfcally n testng n-plane flow capacty for the so called dran geocompostes, we fnd a case n whch the above stuaton s present. Ths test conssts n applyng a confnng pressure over the geotextle and evaluate the amount of water (n lters) flowng (or dranng) durng certan tme at certan water level gradents [10], as shown n Fg. 3.

8 8 FIGURE 3. GEOCOMPOSITE TEXTILE DRAINAGE TEST: AN EXAMPLE OF ESTIMATE WITH LEFT CENSORING. The problem appears for certan combnatons of the test desgn parameters (ambent condtons, geotextle thckness and tme requred for testng). For geotextle of less than 2 mm thckness and wth certan water pressure gradent, the testng equpment has a lmt of detecton of water flow n 50ml/Hour. Therefore, when one wants to montor the performance of a geotextle whose average n-plane flow capacty s less than ths lmt, test wll generate left censored observatons. Consder for the process under control of a data matrx wth K=100 subgroups of sze n = 5 taken to estmate the mean and standard devaton under control wth censorshp C= 50ml/h. Table 2 shows the frst 25 samples of sze 5, wth means and standard devatons. The mean and standard devaton of the process was estmated gvng the followng results: Intal Mean Censored data: μ 0 = 50,0846 Intal Standard Devaton Censored: σ = 0,2720 Applyng the proposed method, estmatons of mean and standard devatons are: Estmated Mean Under Control: μ = 49,0279 Estmated Standard Devaton Under Control: σ = 0,9915 The CEV weght for censored data n the montorng process parameters under control CEV and estmated s: Error! No se encuentra el orgen de la referenca ,03 0,99 Wc 48, ,03 0,99 The proporton of theoretcal censorshp calculated as:

9 9 Eq ,03 Pc 0,843 0,99 Drawng control charts based on standardzed control lmts for the chart CEV X and S 3 and 4; these are 1.42 and 2.07, respectvely. TABLE 2. DATA EXAMPLE X S 1 50,0 50,0 50,0 50,0 50,0 50,0 0,0 2 50,0 50,0 50,0 50,0 50,0 50,0 0,0 3 50,0 50,0 50,0 50,0 50,0 50,0 0,0 4 50,3 50,0 50,0 50,0 50,0 50,1 0,2 5 50,0 50,2 50,0 50,7 50,0 50,2 0,3 6 50,4 50,0 50,0 50,0 50,0 50,1 0,2 7 50,0 50,3 50,8 50,0 50,0 50,2 0,3 8 50,6 50,0 50,0 50,0 51,2 50,4 0,5 9 50,0 50,5 50,9 50,8 50,6 50,5 0, ,0 50,0 50,0 50,0 50,7 50,1 0, ,0 50,4 50,0 50,0 50,0 50,1 0, ,0 50,0 50,0 50,0 50,0 50,0 0, ,0 50,0 50,0 50,0 50,0 50,0 0, ,0 50,0 50,0 50,0 50,0 50,0 0, ,0 50,0 50,0 50,0 50,0 50,0 0, ,0 50,0 50,0 50,0 50,0 50,0 0, ,9 50,0 50,0 50,0 50,0 50,2 0, ,0 50,0 50,0 50,0 50,0 50,0 0, ,0 50,0 50,0 50,0 50,0 50,0 0, ,0 50,0 50,0 50,0 50,0 50,0 0, ,3 50,0 50,0 50,0 50,0 50,1 0, ,0 50,0 50,0 50,0 50,5 50,1 0, ,0 50,0 50,0 50,2 50,0 50,0 0, ,0 50,0 50,0 51,0 50,0 50,2 0, ,0 50,0 50,0 50,0 50,0 50,0 0,0 Calculatng the control lmts for the control chart of the mean and standard devaton accordng to the formulas gven n paragraph 4 s obtaned: Eq. 9 y Eq. 10 Upper control lmt for the chart CEV x 1,42*0, , ,43583 Upper control lmt for the chart CEV S 0,9915*2,07 2,0524 In Fgure 4 and 5 lsts the results of the ntal deployment, where ponts are not removed. In ths case the ponts are wthn specfcatons. One can say that the data come from a process under control. As a result, they may contnue the montorng process usng the control lmts gven for the CEV model. The lower control lmt s unnecessary because no average subgroups of observatons wll be below Wc for graphc CEV X and 0 for graphc CEV S. Thus only ncreases were detected n the mddle of the process, whch n practce s what processes usually more concerned wth ths type of censorshp.

10 10 FIGURE 4. CONTROL CHART CEV X FOR THE MODEL CEV FIGURE 5. CONTROL CHART CEV S FOR THE MODEL CEV Both n Fgure 4 and 5, t s seen that there s no pont outsde the calculated control lmts, so t can be sad that the process s fully controlled. 6. CONCLUSIONS The stuatons n whch the measurng equpment are lmted n ther senstvty are not at all desrable, but nevertheless present and ts treatment requres a number of precautons to avod makng blunders. Ths paper has developed control charts statstcal problem solved wth a smple operaton, n lne wth Shewhart graphcs, but for stuatons censorshp left only makes sense to use upper lmts. The problem of estmaton of censored data s solved wth maxmum lkelhood estmators and an teratve calculaton process. Mssng values censorshp s replaced by these estmates. Ths provdes a more accurate montorng of assessment of the controlled varable wth other alternatve s acheved. Usng smulatons allows for the control lmts for the graph of the mean and range. Research s now orented towards assessng the effectveness of the graph and compare ther performance wth other alternatve charts.

11 11 APPENDIX A Lke the process of maxmum lkelhood for censored by the rght of Stener & Mackay processes, the process of maxmum lkelhood estmaton for censored by the left process s teratve and nvolves replacng each censored observaton wth a condtonal expected value calculated equatons 1 and 4 are set out n paragraph 3; on the bass of these weghts, mean and standard devaton of the process are re-estmated, ncludng: The estmated mean and standard devaton are obtaned by: Eq. 11 Eq. 12 ˆ n 1 w n ˆ n 1 ( w ˆ ) r ( n r ) ( Z ) 2 C (Ap1) Where r equals the number of uncensored observatons, n the total number of data, equals the number of teraton, for whch: Eq. 13 C Z C Zc Z c Zc Zc z Z It s always between 0 and 1. When t s close to 1 the percentage of censorshp s small and close to 0 when the proporton of censorshp s great. To calculate estmated mean and standard devaton s proposed for the CEV Model Left. The followng expressons: Eq. 14 Eq. 15 ˆ n 1 w n ˆ n ( 2 w ˆ ) 1 1 r ( n r ) ( Zc ) (Ap2)

12 12 Where r equals the number of uncensored observatons, n the total number of data and: Eq. 16 λ C ˆ C ˆ 1 1 ˆ ˆ 1 1 Zc z C ˆ C ˆ 1 1 ˆ 1 ˆ 1 Where s the number of teraton performed, Z C It s always between 0 and 1. Close to 1 when the percentage of censorshp s small and close to 0 when the proporton of censorshp s great. FIGURE 6. ESTIMATION PROCESS To fnd the maxmum lkelhood estmate s teratvely appled to the formula Ap2 data untl estmates converge. Fgure 6 show the estmaton process for the proposed model.

13 13 APPENDIX B Below we present several tables 3, 4, 5, and 6, wth the values of the coeffcents for calculatng the control lmts wth dfferent sample szes, and a probablty of error type I (α) TABLE 3. COEFFICIENTS FOR CALCULATING THE CONTROL LIMITS (N=20) n=20 %c coefcente coefcente LCS X LCS S 0,02 0,69 1,50 0,03 0,68 1,49 0,04 0,68 1,49 0,07 0,68 1,49 0,10 0,68 1,49 0,16 0,68 1,49 0,24 0,68 1,49 0,31 0,68 1,49 0,50 0,68 1,49 0,69 0,68 1,49 0,84 0,68 1,49 0,98 0,68 1,49 TABLE 4. COEFFICIENTS FOR CALCULATING THE CONTROL LIMITS (N=10) n=10 %c coefcente coefcente LCS X LCS S 0,02 1,02 1,80 0,03 0,97 1,71 0,04 0,97 1,71 0,07 0,97 1,71 0,10 0,97 1,71 0,16 0,98 1,72 0,24 0,97 1,72 0,31 0,97 1,72 0,50 0,97 1,71 0,69 0,97 1,72 0,84 0,97 1,71 0,98 0,97 1,71

14 14 TABLE 5. COEFFICIENTS FOR CALCULATING THE CONTROL LIMITS (N=5) n=5 %c coefcente coefcente LCS X LCS S 0,02 1,61 2,36 0,03 1,47 2,15 0,04 1,42 2,09 0,07 1,42 2,08 0,10 1,42 2,07 0,16 1,42 2,09 0,24 1,43 2,09 0,31 1,42 2,09 0,50 1,43 2,09 0,69 1,42 2,09 0,84 1,43 2,09 0,98 1,42 2,08 TABLE 6. COEFFICIENTS FOR CALCULATING THE CONTROL LIMITS (N=3) n=3 %c coefcente coefcente LCS X LCS S 0,02 2,46 3,23 0,03 2,11 2,78 0,04 1,94 2,54 0,07 1,92 2,53 0,10 1,94 2,55 0,16 1,95 2,56 0,24 1,95 2,56 0,31 1,95 2,56 0,50 1,95 2,56 0,69 1,95 2,56 0,84 1,95 2,57 0,98 1,95 2,56 REFERENCES [1] J. P. Klen, Survval Analss, Technques for Censored and Truncated Data, vol. Vol. 2, Sprnger, [2] R. L. Mason and J. P. Keatng, ""Under the Lmt: Statstcal methods to treat and analze nondetectble data"," Qualty Progress, vol. October, pp. pp , [3] R. H. Shumway, R. S. Azar and M. Kayhanan, Envronmental Scence and Tecnology, vol. vol. 36, no. 15, pp. pp , [4] M. F. Ulín, ""Análss de Datos Censurados para Ingenería y Cencas Bológcas".," Revsta de Matemátca, vol. Vol. 24, no. 2, pp. PP , 2007.

15 15 [5] S. H. Stener and R. J. Mackay, ""Montorng Processes Wth Hghly Censored Data"," Journal of Qualty Technology, vol. vol.32, no. No.3, pp. pp , [6] Lawless, J.F., Statstcal Models and Methods for Lfetme Data, New York: Jhon Wley & Sons,, [7] A. Neves Martnez, Análss de regresón con datos censurados. Aplcacacon al estudo de factores pronostco en la supervvenca al cancer de mama., Tess Doctoral, España: Departamento de Estadístca e Investgacón Operatva, Unversdad Poltécnca de Valenca., [8] D. Cox. D.R. y Oakes, Analss of survval data, Chapman and Hall., [9] D. C. Mongomery, Introducton to Statstcal Qualty Control, 5th ed., Jhon Wley & Sons, U.S.A., [10] A. Bamforth, "Interpretaton of In - Plane Flow Capacty of Geocomposte Dranage by Test to ISO wth Soft Foam ASTM D4716 Varous Natural Backfll Materals," n GIGSA GeoAfrca, Cape, 2009.